Gene expression data are part of the Grand Challenges in Global Health Project: Grant Number 37772, “Biomarkers of protective immunity against Tuberculosis in the context of HIV/AIDS in Africa” (funded by the Bill & Melinda Gates Foundation through the Grand Challenges in Global Health Initiative; http://www.biomarkers-for-tb.net/. PBMC from 40 TB cases and from 40 healthy household contact controls were extracted and analyzed by flow cytometry for proportions of CD3⁺ T-lymphocytes and CD14⁺mononuclear phagocytes as described before 1. All donors gave informed consent. This study was approved by local ethics committees in Stellenbosch (South Africa) (N05/11/187) and Berlin (EA 10 1/176/07, Germany).

Signals of gene expression in whole blood as well as in CD3⁺ cells, for the Human Whole Genome Oligo 44K Agilent arrays (GE2_44k_1005) were measured according to manufacturer's protocols. The microarray design was an independent swop design as recommended by Landgrebe et al. 13: 50% of each group ("TB", "TST+", "TST-") were labelled with Cy3, the other half using Cy5. Pairs for hybridization on an array were chosen to match regarding age and gender. For validation of the deconfounding algorithm we used CD3⁺ proportions. CD3⁺ cells sorted by fluorescence-activated cell sorting (FACS) were subjected to RNA extraction and microarray measurements of gene expression following the same procedure as for the whole blood samples. More details about the observational field study as well as the gene expression dataset will be published separately (see http://www.biomarkers-for-tb.net/publications).

Gene expression data were normalized using R-package limma 14: background correction using the method normexp 15, lowess normalization was applied for each array (within array normalisation), quantile normalisation on the set of all arrays (between array normalisation) as recommended 16. As proposed by 17, gene expression intensities for both groups were obtained as in Equations 4 and 5 from re-parameterizing the normalized log-ratios (M) and mean log-intensities (A) resulting from the limma analysis.

Summarizing, for each of the 40 TB cases and the 40 healthy household contact controls we were able to analyze gene expression data of whole blood as well as for the sorted CD3⁺ cells of the same samples together with their FACS-measured cell type proportions for the CD3⁺ cell population.

The basis of our deconfounding algorithm was implemented as proposed by Venet et al. (2001) 10 and Lahdesmaki et al. (2005) 4 using R 18:

   input X and n

  normalize columns of X (either centre, or by quantile normalization)

   generate start values for S and C

   apply constraints to S and C (see below)

(*) fix S, calculate C using lsqnonneg-algorithm

   apply constraints for S

   fix C, calculate S using lsqnonneg-algorithm

   apply constraints for C

   if | X - SC | < a or number iterations > b then EXIT and report S and C

   else continue at (*)

where X is the gene expression matrix measured from heterogeneous tissue (rows: genes, columns: samples), S and C as in equation 2, iteration exit criteria were set a = 0.1 and b = 100. The Least squares non-negative matrix factorization algorithm is implemented as in the MATLAB function lsqnonneg 19. The constraints are:

1. S non-negative and normalized (either centered, or by quantile normalization 16)

2. 0 ≤ c_ij ≤ 1 for all elements of C (cell type i, sample j)

3. ∑_i c_ij = 1 for all samples j (i.e. cell type proportions sum to 100%)

Our implementation is available as an R-package and has additional options for using quantile normalization instead of global normalization proposed previously 10. Moreover, it is possible to run the deconfounding on log-values of the normalized intensities or on non-log data. Finally, our implementation does not apply the de-correlation proposed by Venet et al. 10.

To assign the right cell type for each of the estimated profiles, our implementation relies on a majority count decision involving the estimated gene expression profiles from n_marker = 9 markers. Five of these markers are considered to be expressed exclusively for a specific cell type (positive marker genes) and the remaining four exclusively not in this cell type (negative marker genes). Marker genes were chosen according to a priori molecular immunological knowledge. For our experimental dataset we used CD3D, CD3E, CD3G, CD2 and CD7 as positive markers, and CD19, FCGR1A, CD14 and MARCO as negative markers for the CD3⁺ T cells.

Cell type specific gene expression profiles (columns of the signature matrix S) were simulated according to a gamma distribution such that expectation value and variance were those of the experimental data (shape a = 12.5 and scale b = 0.65):

As by Venet et al. 10, biological variance was modeled as multiplicative error term ϵ, technical variance as additive error term ϵ. For our experimental data, variation was found to increase with mean signal intensities. Therefore, we decided to model a constant coefficient of variation instead of standard deviation:

where η = 0.17 and ϵ ~ N(0, χ · I_gene), using χ = 0.1 as estimated from our experimental data.

Gene expression values for negative marker genes had expression X_{marker, neg} = 6.0, positive marker genes had X_{marker, pos} = 12.0 in the expressing cell type - as observed for the marker genes in our experimental study. Cell type proportions, C_sim, were drawn from the uniform distribution between cell type specific maximum and minimum values as in our experimental flow cytometry data. The simulated gene expression matrix, X_sim, was calculated from simulated cell type-specific gene expression profiles, S_sim, and simulated cell type proportions, C_sim, corresponding to equation 2:

To investigate the algorithm's capabilities regarding detection of differential expression of single features and for classification, two groups of gene expression profiles were simulated, e.g. corresponding to TB patients and TST+ controls in our experimental data. We simulated n_sample = 100 individuals in each group. For each gene expression profile n_genes ∈ {1000, 10000} genes were considered, with n_markers = 10 and n_diff ∈ {20, 600} differentially expressed biomarkers.

Differential expression was simulated by adding Δ_diff ∈ 1, 2, 5} to the expression values of the biomarker genes in the first cell type. Figure 3 illustrates the generation of simulated profiles.

We simulated a gene expression experiment with samples mixed out of two cell types (CD3 and other) for 10,000 genes, where 600 genes were differentially expressed. For the differentially expressed genes we simulated all eight possible combinations of NEUTRAL, UP and DOWN. Sample sizes of the two groups under comparison (alike TB and TST⁺ healthy control) varied from n_samples ∈ {4, 10, 20, 40, 80, 120}. Simulated gene expression data were analyzed as the experimental data. As for the latter we were able to analyze a simulated whole blood sample (mixture of the two cell types) as well as the two cell type-specific gene expression profiles after deconfounding. Simulated whole blood gene expression data were analyzed using the t-test, ranking candidates for differential expression using p-values and - to enable a direct comparison - considering the 100 top candidates as positive candidates for differential expression. The cell type-specific gene expression profiles (columns of the signature matrix ) estimated from deconfounding were ranked using absolute log-fold-change values. Also here the 100 top candidates were chosen.

The worst-case scenario for biomarker detection in heterogeneous tissues arises when cell types involved express differentially regulated biomarkers in opposite directions. In this case, in the tissue RNA isolate, signals for differential expression likely cancel each other and hamper detection of respective biomarkers markedly. To identify a possible exit strategy, we conducted a simulation study for this worst-case scenario, again considering noise values estimated from the experimental data in this study.

To exemplify the worst-case classification task, we simulated differential gene expression as above, but also subtracted the same value from the expression values of the second cell type. This way, for all cells in the mixture averaged over all samples, no differential expression is expected, while for the single cell types it is more or less evident. Gene expression profiles for new samples, for validation of the trained classifiers in the classification scenario, were generated using the identical signature matrices, S_sim, as for the training step, but with new values for the concentration matrices as well as for the noise term realizations.

R-package deconf implementing the deconfounding algorithm and options, R-scripts for data simulation, data analysis and an anonymized part of the experimental dataset is available as additional file 1 (Windows R-package) and additional file 2 (tar-gz archive).

The deconfounding algorithm was applied to the whole blood gene expression matrices for both groups of individuals (TB and TST⁺) both using the quantile normalization as well as global mean normalization approach for log- and non-log-intensities. Numbers of cell types was set to n_CT = 2. Deconfounding results - estimated cell type-specific gene expression profiles as well as cell type proportions - could be compared to the actual experimental data (see figure 4, figure 5 and table 1):

Figure 4 displays mean values of the measured CD3⁺ expression profile in TB patients against both estimated columns of the signature matrix .

For the displayed example in figure 4, non-log data were quantile normalized: Experimental data show considerable variation when compared to the estimates after deconfounding. As expected, cell type 1 is evidently better correlated with the experimental CD3⁺ profile than cell type 2. The correlation is best for large expression values.

Also, referring to figure 5, even though there is lower correlation between experimental and estimated cell type proportions, the indicated regression lines in the scatter plots for experimental and estimated proportions show the correct tendencies for the respective cell types.

Table 1 (first row) depicts correlations of mean measured profiles with the estimates from deconfounding results for the comparison between the four methodological algorithmic alternatives.

To investigate the influence of sample size on the quality of deconfounding results, we had to rely on simulation studies which were aimed at mirroring experimental data distribution and noise as realistically as possible. Figure 6 (middle panel) shows the simulation results for n_sample = 20, which approximates the sample size for the GC6 experimental data (cf. figure 4) - and also a typical value for such type of clinical study involving high-throughput analyses. The effect of sample size is clearly distinguishable for simulation results using n_sample = 4 (figure 6, left) and n_sample = 120 (figure 6, right) respectively.

The deconfounding algorithm itself does not assign a cell type to the estimated cell type specific expression profiles (columns of ). Therefore, to find out in which of the two possible orders the two estimated cell type profiles (CD3⁺ and others) reside, one has to rely on expression signals of cell type specific markers. Regarding the analysis of the experimental data, such markers were chosen based on a priori immunological knowledge. In our simulation studies, we simulated 5-10 positive CD3⁺ marker genes, which were expressed at high levels (simulated level for X_{marker, CD3} = 12), whereas these marker genes showed a low mean expression in the alternative cell type (simulated level for X_{marker, other} = 6). These expression levels were used as observed for the experimental data. Another group of marker genes was simulated in the reverse manner. Figure 7 shows the distributions of estimated marker gene expression levels from simulated data after deconfounding employing global mean (A) or quantile normalization (B). Here, use of the robust quantile normalization was rewarding for this critical step: Lack of a possibility to assign the right cell types thwarts the analysis as a whole. It is also evident, that marker gene expression levels were estimated mostly correctly regarding relative values in both cell types, whereas absolute gene expression levels were scaled down in the estimates. However, to be able to use these cell type specific estimated marker gene expression levels to assign the right cell types it is only necessary that positive markers have top expression levels in the cell type exclusively expressing them.

Because we want to study the use of deconfounding for biomarker discovery, in our power-study we compared the t-test and our deconfounding approach regarding their power to detect differential gene expression (candidate biomarkers). Figure 8 shows the central results: t-test and deconfounding approach show comparable results for higher sample sizes (40 ≤ n_sample ≤ 120) and cases A and B, for which differential gene expression is either in the same direction in both cell-types or differential in one cell type only. However, for small sample sizes in all cases, and especially also for large sample sizes in figure 8C, the deconfounding ranking approach detects more of the true differentially expressed genes than the t-test. As it is for this worst-case scenario (figure 8C), where differentially expressed signals of the cell types involved cancel each other, we aimed at assessing application of the deconfounding ranking approach for the classification objective for this case.

As power for detection of differential expression ("DGE power") we define the proportion of truly differentially expressed genes in the 100 top-ranked 100 candidates. Table 1 (second row) depicts this power for detection of differential expression for four algorithmic alternatives. Choosing quantile normalization for intensity values and using non-log values gives optimal results.

As an important objective is to find biomarkers from the estimated cell type specific gene expression signatures resulting from the deconfounding, we have to show how such biomarkers could be applied to a new patient's whole blood expression dataset. The deconfounding algorithm results in estimates for the signature matrix and the concentration matrix for a given group of samples. In our case, the procedure uses simulated gene expression profiles of 40 individuals (per study group) to estimate two cell type-specific gene expression profiles (CD3+ and NotCD3+). It is, however, not possible to use a single individual's profile for deconfounding, as for a single case there is no information available about how a change in cell type proportions influences measured gene expression signals. To enable the use of the deconfounding results for classification of a new individual, we have to either measure or estimate a single individual's cell type proportions. To estimate cell type proportions from a single whole blood expression profile we employed a random forest machine to learn to predict cell type proportions from simulated whole blood gene expression data using the training dataset and the deconfounding estimates of . For a new individual, this trained random forest was then used to estimate cell type proportions.

These were multiplied to the group-specific signature matrices estimated by deconfounding from the two groups in the training data. The resulting group-specific gene expression matrices - based on cell type proportions as in the new individual - were used in a majority votes comparison approach and the individual classified accordingly.

We show that this deconfounding ranking approach significantly improves classification results regarding prediction error rates, if the differential expression of a biomarker panel relies on genes that are regulated in the opposite direction in the cell types involved. Figure 9 shows distributions of classification errors in 100 validation runs. Clearly, the t-test-LDA approach is not better than mere guessing, whereas -dependent on noise and numbers of differentially expressed genes - the deconfounding ranking approach correctly classifies most of the simulated cases.

We also regressed cell type proportions on marker gene expression (CD3G and MARCO) in the experimental whole blood dataset and achieved a correlation of 34% between leave-one-out samples and their estimated proportions of CD3+ cells. Figure 10 shows a scatterplot of the leave-one-out samples and their estimated proportions, as well as the distribution of correlations with 200 permutated values for the cell type proportions. Prediction is significant, and its precision comparable to what the deconfounding is able to reproduce in the simulated data (compare figure 5).

For the purpose of biomarker detection, homogeneous cell populations are not generally a prerequisite as there may be markers so clear that their signal can be read in spite of the considerable variation introduced by tissue heterogeneity. This is mostly a desired result. However, especially in experiments where biomarkers are sought for cases which are not easily separable otherwise (e.g. prospective studies), they might be detected better after taking tissue heterogeneity into account - with our work and manuscript we want to propose an approach for such cases.

Others have implemented and studied principles of in-silico deconfounding 48910111222, but our study for the first time combines the following results:

- validates in-silico deconfounding results using experimental data of a molecular field study;

- implements a realistic simulation study with noise parameters estimated from the experimental dataset;

- systematically investigates the influence of sample size on quality of estimated cell type specific gene expression profiles;

- compares the power to detect differential expression (i.e. univariate biomarker candidates) with a classical t-test approach;

- optimizes the deconfounding algorithm employing a quantile normalization step as well as marker-assisted cell type profile recognition under realistic noise conditions;

- proposes a classification approach using the results of a deconfounding ranking analysis and compares these results with a classical t-test-LDA approach for the worst-case scenario of biomarkers regulated in opposite directions.

Our results show that, even under noisy, realistic conditions of a molecular field study - involving field-collected whole blood samples and considerable individual variations between enrolled individuals -the deconfounding ranking approach using non-log, quantile-normalized gene expression data from whole-blood RNA can facilitate identification of valid differential gene expression signals. These biomarker candidates can then be used in a classification approach which - for the case where biomarkers are regulated in opposite directions in different cell-types - is far more powerful than canonical discriminant analysis. In the applied clinical situation, our approach will of course be not more than an initial step for the identification of candidate biomarkers for classification - which then would be entered into further validation studies before applicable for cost efficient clinical routine diagnostics.

A critical prerequisite of our deconfounding approach is that, in principle, we assume independence of a cell type-specific gene expression profile and the proportion of the respective cell type within the heterogeneous tissue. Figure 2C, illustrates the unfavorable case for which gene expression on the single-cell level is regulated as a function of the expressing cell-type's proportion in the tissue. It is conceivable that such a regulation is indeed real for some genes - and this would not only blur estimates of cell type-specific gene expression profiles, but also produce false estimates for such specific genes. As shown in our validation study, however, in general the independence assumption does not lead to false results for the estimated profile as a whole. Thus, biosignature detection will still be enhanced by use of deconfounding ranking even if the independence assumption for single-cell gene expression and cell type proportion does not hold in every respect.

Some methodological details of our study remain an illustrative approach, and further investigations are thus called for. The normalization procedure has apparent influence on the quality of cell specific profile reconstruction as well as on the power of detection of differential expression. Our decision to use quantile normalization was based on the finding that using the original overall mean normalization by Venet et al. 10 led to poor recognition of cell-types using marker gene expression signals. Single outlier measurements could significantly shift the whole profile, thus thwarting cell-type identification. The quantile normalization approach resulted in a robust, more reliable marker-assisted cell type recognition. An improvement of the algorithm's capability to reconstruct cell type-specific gene expression profiles could be obtained if the starting profiles for the iterative optimization were already seeded with an approximate guess of what the specific cell type profile may look like. Such information could be provided by FACS analysis, or by expression profiles available in the literature (see for example the work of Watkins et al., 2009 23). Caution, however, is necessary to avoid inadequate influences on study group-specific differences. Also, averaging multiple deconfounding optimization runs could lead to a stabilizing effect for the resulting predicted cell-type profiles. Here as well, detailed studies are necessary.

Estimates of cell type-specific gene expression profiles were optimal given that deconfounding was run on log-intensities, whereas detection of differential expression was optimal using non-log input values. We may speculate about the reasons for this difference: Possibly, non-log inputs filter out or down-weight small expression values - which in turn often play a minor role in differential expression.

For the simulated worst-case scenarios, i.e. genes which are reciprocally regulated in the participating cell types, the deconfounding ranking approach produced promising results - both for achieving valid estimates of differential gene expression and for the classification task. However, the existing implementation could be improved by implementing a bootstrap test for differential expression, such that not only a ranking of candidates for differential expression, but also an estimate of the number of differentially expressed features becomes feasible. A first approach could be to draw bootstrap samples and compute 95% confidence intervals as quantiles from the bootstrap distribution of the resulting bootstrap estimates for (b denoting a bootstrap index). Such a bootstrap approach could also enable analysis of gene set enrichment with currently available methods (e.g. 2425).

The proposed deconfounding ranking approach to classification has to be considered as a first heuristic approach. Its performance sufficiently demonstrates superiority over approaches that do not take into account confounding with cell-type proportions (figure 9). However, a multivariate model of gene expression patterns (biosignatures) is still missing. It would be desirable to arrive at an analysis interface enabling the use of the plethora of available statistical learning methods. Also, the classification approach is dependent on either measurements or estimates of cell type proportions in the sample that is to be classified. If the field of application was gene expression signatures in blood, it is certainly conceivable that a cell type proportions profile is measured, as the necessary laboratory equipment is now available in labs all over the world. However, in our work we propose to try a regression approach based on the expression profiles of the marker genes which are also used to identify the cell type specific expression signatures after deconfounding. This approach worked well for our simulation study, figure 10 shows that it also delivers sufficient results for experimental data - comparable to what the deconfounding algorithm delivers (compare figure 5). However, there is certainly room for improvement - as apparently better estimates of cell type proportions based on single sample whole blood expression profiles would enable improved classification performance.

The presence of up- and down-regulated biomarkers suggest two further possible improvements. First, gene filtering with regard to absolut expression signals, i.e. focussing on medium to highly expressed genes may provide more robust signatures. Second, the identification of gene pairs as in the top scoring pair method 2627 may be an alternative to the ranking approach taken in our initial study here - and improve reliability in the presence of noisy field measurements.

There also exist alternative approaches to the non-negative matrix factorization approach taken by us and 410. For example, Ghosh proposes a mixture model approach 9, and there also exist Bayesian approaches for this task 2228. A comparison of existing methods for the application with biological data from heterogeneous tissues would certainly be an exciting and rewarding field of further work. Especially modern Bayesian methods promise to further improve the results, also regarding more than two cell types in the heterogeneous tissue to be resolved.

In our contribution, the deconfounding ranking approach is applied to gene expression profiles in peripheral blood samples. In principle, it is also applicable for other molecular profiles from heterogeneous tissues, e.g. metabolome or proteome profiles.