The annotation of the PM probe sequences was obtained with the alignment to the genome sequence of S. cerevisiae strain S288c (SGD of August 7, 2005) as provided in the package davidTiling of Bioconductor. Available data correspond to 3 replicates of poly(A), 2 replicates of total RNA and 3 replicates of genomic DNA. The CEL files were read and the normalized signals (poly(A) and total RNA) were obtained using Equation 4. The analysis steps (denoising, segmentation and detection of TARs) were performed on the poly(A) signal as it showed an improved hybridization quality 16 . Once the signal is constructed from CEL and annotation files we used tilingArray package functions to obtain equally-spaced samples. Other resampling methods can be applied without loss of generality.

The annotation files for S. aureus microarray are provided in the ArrayExpress database (A-AFFY-165). The microarrays of the experiment correspond to three replicates of genomic DNA, three replicates of RNA of the 15981 wild-type strain, and three replicates of the sigB deletion. All the preprocessing steps were performed as previosly described for S. cerevisiae dataset.

The denoising was evaluated using the signal to noise ratio (SNR), a quantitative measure of its performance. In order to compare the results obtained with those from Huber et al. 15 based on a variance stabilization and normalization transformation, the same definition of SNR was used. We looked at a set of control regions, two positive control regions (pos) within the ORFs of RPN2 and SER33 at coordinates 217860−220697 and 221078−222487and two negative control regions (neg) in the background at coordinates 216800−217700 and 222800−227000 of S. cerevisiae (see Figure 2). We assumed (as in 15 ) that the differences between positive and negative controls give an estimation of the signal level, whereas variations from the mean intensity within each region are due to noise.

The SNR was computed as

SNR = Δμ σ = 1 σ ∑ r ∈ pos μ r | pos | − ∑ r ∈ neg μ r | neg | ,

with the noise standard deviation σcalculated as the average of the differences between 0.975 and 0.025 quantiles of the data within each of the control regions. Namely,

σ = ∑ r ∈ pos , neg ( Q r 0 . 975 − Q r 0 . 025 ) ( Q N 0 . 975 − Q N 0 . 025 ) ( | pos | + | neg | )

where the symbol r counts over the different regions and Q _N refers to the standard normal distribution N ( 0 , 1 ) . Table 1 shows the SNR of the normalized signal and the wavelet-based denoised signal using Donoho’s method 29 and the SUREShrink approach 30 in relation to the best SNR obtained in 15 . We observe that the use of wavelets for denoising results in a large increase in the SNR (18.94% with Donoho’s method and 30.63%with SUREShrink approach), especially when the SUREShrink denoising is applied. This could be due to the elimination of most part of the non-Gaussian noise component and the consequent reduction in the estimated variance. In the rest of the paper, the SUREShrink is the method applied for denoising.

Estimated SNR values of the tiling signal. The normalized and the wavelet denoised signal using Donoho’s and SUREShrink on which the calculation was performed are shown in Figure 3.

A descriptive example of the denoising and segmentation for S. cerevisiae is shown in Figure 3. The analysis corresponds to a 140 Kb segment of chromosome 1 from position 20000 to position 160000. The results are given for the three algorithms compared (SCM, PMSW, ZCL). The CWT computation of ZCL used as mother wavelet the second derivative of a Gaussian with 100 scales. Zero-crossing lines were calculated and only those with a length greater than a pre-defined threshold were considered to correspond to signal transitions.

The TAR start and end positions were defined as the transition locations for which the difference between the mean intensity of neighboring segments is greater than 10% of the dynamic range of the tiling signal. Moreover, the inspection of the intensity histogram of chromosome 1 forward strand was used to set the minimum normalized transcription level value to −2. The same parameters were adopted to process the other strands of the organism. The R function segmentZCL (provided as Supplementary Material) implements the whole segmentation procedure.

Another representative example of segmentation results is given in Figure 4. In this case, the S. aureus signal of 15981 and sigB mutant were segmented using ZCL and fixing only the number of scales to 100. No denoising was applied prior to the segmentation. The figure represent a fragment of the signals from position 2.1 Mb to 2.3 Mb. In Additional file 4: Figure S1 and Additional file 5: Figure S2, the results for the PMSW algorithm are shown. In Additional file 6: Figure S3 and Additional file 7: Figure S4, the results for the SCM algorithm are presented.

The results from the ZCL segmentation were compared to those obtained with PSMW 6 and SCM 15 . The robust PMSW method is based on the calculation of a pseudo-median within a sliding window. The local expression level is computed with the Hodges-Lehmann estimator 31 on the RNA normalized signal. To be able to do this, the Tilescope pipeline 7 was implemented. Once the candidate transcript regions were determined, the TARs were assembled by the combination of a normalization intensity threshold and a max-gap and min-run criteria. The former is defined as the maximum distance below which two adjacent transcribed probes are included in the same TAR. The later as the minimum length of a feature to be classified as a transcribed region.

Huber’s method is based on the structural change model (SCM). The SCM model 15 16 is used in econometrics for the modeling of sharp transitions in financial time series. It has been applied to the segmentation of comparative genomics hybridization (CGH) data 32 . The signal is modeled as a piecewise constant function of chromosomal coordinates described using the segment boundaries, the maximum number of segments and the mean signal value for each segment. The method is applied independently to each chromosome and, if the signal is strand-specific, to each of its two strands. A dynamic programming algorithm part of the tilingArray package of Bioconductor computes a globally optimal set of parameters for segmentations of increasing number of segments.

Due to the lack of a biologically validated ground truth to evaluate the outputs, we compared the methods in terms of two metrics, sensitivity and positive predictive value (PPV) at probe-level. We define sensitivity as the number of probes in the detected TARs that overlap with annotated regions (true positives, TP) divided by the total number of probes in the annotated regions (sum of true positives and false negatives, TP + FN): Sensitivity=TP/(TP + FN). The PPV is defined as the number of probes in the detected TARs that overlap with annotated regions (TP) divided by the total number of probes in the detected TARs (sum of true and false positives, TP + FP): PPV=TP/(TP + FP). A sensitivity of 100% is not expected since in any given tissue or cell line at any given experimental condition, not all known genes will be expressed. Also, a PPV of 100% is not expected since an accurate and complete gene annotation is not available 5 33 .

The PMSW and SCM methods were applied to the S. cerevisiae using the parameters previously reported in the literature 6 15 . In particular, the PMSW method used a bandwidth size (BW) of 3, a normalized intensity threshold equal to −2, a separation between the probes in a TAR (maxgap) of 10 and a minimum acceptable TAR size (minrun) of 90. The maximum number of segments for the SCM method was fixed to 1500. In the case of ZCL we selected the following parameters: 100 wavelet scales, minimum TAR size of 10 and minimum value of normalized intensity transcription equal to −2. The zero-crossing line length threshold was computed based on the histogram of line lengths.

The graphical representation of the results obtained after processing the S. cerevisiae tiling signal are shown in Figure 5. For each chromosome we calculated the number of detected TARs, PPV, sensitivity and computation time for the forward and reverse strands. Table 2 presents the performance metrics mean value. The segmentation with PMSW includes a larger number of detected TARs. The highest PPV values are obtained with the ZCL method (with or without denoising) at the cost of a slightly reduced sensitivity. PMSW and ZCL outperform SCM in term of computation time. The best sensitivity value corresponds to PMSW.

Mean number of detected TARs, probe-level PPV, probe-level sensitivity and computational time for PMSW, SCM and ZCL methods (all chomosome and strands of S. cerevisiae).

In-depth analysis of chromosome 1 gives interesting insights into concerning the relationship between methods. In the forward strand, the number of probes annotated as genes is 12796 representing 19.35% of the total number of probes. 65.16% of probes are correctly classified by the three algorithms (12.21% of gene probes and 52.95% of non-gene probes). From the annotated probes, 63.12% are detected by all methods, while only 11.85% of the probes are not detected by either of them. This means that 88.15% of the annotated probes are detected by at least one of the methods. The reverse strand contains 11866 annotated probes (17.90% of probes located in this strand), from which 19.74% are considered part of a TAR by all the methods and 28.35% are true negative probes. In this strand, 51.41% of the annotated probes are included in a TAR by any method while only 14.35% are never detected. In other words, 85.65% of the probes in the strand are detected by at least one algorithm. In light of this outcome, we considered it worthwhile to evaluate if the combination of results computed with the different methods would improve the performance of the segmentation.

Comparative segmentation analysis using ZCL and PMSW and SCM algorithms was applied for the hybridization data obtained with a custom designed Affymetrix tiling array of S. aureus. Segmentation results for S. aureus are summarized in Table 4. In this case, the performance measures are almost identical for all methods. These results suggest that the performance of the methods depends on the quality of the signals, decreasing for PMSW and SCM algorithms as the SNR of the signal get worse. In spite of this, other advantages such as computation time, automatic selection of parameters and the possibility of parallel computation makes ZCL our preferred option to segment tiling signals.

Mean number of detected TARs, probe-level PPV, probe-level sensitivity and computational time for PMSW, SCM and ZCL methods.

The most frequent transcriptional analysis is the detection of genes that have changed their expression in the conditions under study (differential expression analysis). As sigma B affects the expression of more than one hundred genes, we decided to test whether it is possible to use the intensity of all the probes included in each detected TAR with the ZCL segmentation procedure to calculate the expression level of the transcript in a particular environmental condition. In order to carry out this analysis using tiling microarrays we need to compress the intensity of all the probes included in each detected TAR into one value. Standard methods for microarray normalization can be applied, for example RMA (Robust Multichip Average) algorithm in the case of Affymetrix microarrays 34 . This processing can be performed using the packages affxparser, affy and limma of Bioconductor for CDF (chip definition file) generation, normalization and differential expression analysis.

We introduced a simple analytical tool to be used independently of the microarray platform to measure the gene expression level based on the median value of the TAR probe intensities. We calculated this value for each wild-type and sigmaB mutant sample. We applied a statistical analysis (t-test) to obtained the p-value associated with the expression change taking into account the biological variability of the samples. Considering well-defined TARs in the S. aureus annotation, we found previously described alterations in several genes 21 . In Figure 6, we show the boxplots that represents these expression level changes. We confirmed the down-regulation of sigB and other σ ^B -regulated genes, as the alkaline shock protein 23 (asp23) 22 23 and lysine-specific permease (lysP) 21 , although the latter is not statistically significant (p > 0.05). We also found genes up-regulated in the sigmaB mutant, as the staphylococcal nuclease (nuc) 23 24 , the zinc metalloprotease aureolysin (aur) 24 25 and the α-hemolysin (hla) 24 26 , the latter without a statistically significant p-value.

The CWT of a continuous signal s(x)is defined as 40

CWT ( a , b ) = 1 a ∫ − ∞ + ∞ s ( x ) ψ ∗ b − x a dx

where a ∈ R + − { 0 } is the scale, b ∈ R is the translation, ψ(x) is the mother wavelet, ψ ^∗((b−x)/a) is the complex conjugated, scaled and translated wavelet and CWT is the 2D matrix of wavelet coefficients. The continuous input signal s(x) interpolates the discrete input samples s k,k=1,…,nwhere n is the length of the signal.

The CWT can be interpreted as the correlation of the input signal with a position reversed version of ψ rescaled by a factor a. For an 1D input signal, the result is a 2D description of the signal with respect to the position b and scale a and shifted by b. The scale a is inversely proportional to the central frequency of the dilated wavelet ψ _a =ψ(x/a), which is typically a bandpass function; b represents the position location at which we analyze the signal. The larger the scale a, the wider the analyzing function ψ _a , and hence the smaller the corresponding analyzed frequency. The output value is maximized when the frequency of the signal matches that of the corresponding dilated wavelet. The CWT computation for arbitrary scales can be easily adapted to a parallel implementation with a linear computational complexity 41 .

Mallat’s fast wavelet algorithm 42 uses the multiresolution properties of the wavelet to compute the CWT at dyadic scales a=2 ⁱ and time shifts b=2 ⁱ k, k ∈ , Z , resulting in what is known as DWT. For additional information about the wavelet transform and its properties the reader is referred to 17 .

The analysis starts with background correction and quantile normalization as describe by the RMA algorithm 34 . Next, we calculate the geometric mean of the RNA intensities and the geometric mean of the DNA replicates to get a signal score s k at position k proportional to the transcription level in the reference genome 16

s [ k ] = ∑ j = 1 n log RNA j [ k ] ∑ j = 1 m log DNA j [ k ] ,

where n is the number of RNA samples and m is the number of DNA samples.

One of the most established methods of wavelet-based denoising was proposed by Donoho and Johnstone 29 and it is based on the thresholding of the DWT coefficients at scale a=2. This method is composed of three steps: (i) calculate the DWT of the tiling signal s k at scale a=2; (ii) threshold the wavelet coefficients; (iii) compute the inverse wavelet transform of the thresholded coefficients. A universal threshold, T, is proposed 29 to remove white noise which it is given by

T = σ 2 log ( n ) with σ = MAD / 0 . 6745 ,

where n is the length of s, σis the noise level and MAD is the estimated median absolute deviation in the first scale. An important issue is the selection of a suitable wavelet function. As the signal can be roughly approximated to a zero-order polynomial, a boxcar-like function such as the Haar wavelet gives a reasonable level of correlation (i.e., a good pattern matching) with the target signal.

Another well established method of wavelet shrinkage is SUREShrink 30 . This is based on Stein’s Unbiased Estimator for Risk (SURE). A subband adaptive threshold is applied. If the wavelet coefficients in the jth subband are {x _i :i=1,…,n}, we consider a soft thresholding procedure and apply Stein’s result. The quantity

SURE ( T ; x ) = n − 2 · ♯ { i : | x i | ≤ T } + ∑ i = 1 n ( x i ∧ T ) 2

is an unbiased estimate of risk, where T is the threshold and x _i ∧t=min(x _i T). This estimator can be used to select a threshold:

T SURE = argmi n 0 ≤ T ≤ 2 logn SURE ( T ; x )

For a large dimension n the law of large numbers will ensure that T _SURE will be almost the optimal threshold 30 .

An important issue in signal processing is to define an appropriate representation able to compress most of the signal information into few representative features. Sharp variations in amplitude (i.e., transitions and peaks) are among the most meaningful features of a signal. For that reason, many segmentation algorithms rely on their detection. Previous studies have detected the peaks in mass spectrometry data using either the ridge lines 43 or the zero-crossing lines 44 in a multi-scale decomposition of the signal. Zero-crossing lines seems a more consistent description as they belong to connected curves, are more robust to noise and easier to detect that ridge lines 44 .

It has been previously shown that the position of multiscale sharp transitions can be obtained from the zero-crossings of the signal convolved with the Laplacian of a Gaussian 45 . We define a wavelet at scale a as

ψ ( x ) = d 2 θ a ( x ) d x 2

where θ _a is a Gaussian function dilated by a factor a. Since the wavelet transform can be represented as

CWT ( a , b ) = ( s ∗ ψ a ) ( x ) | x = b

we derive that

CWT ( a , b ) = s ∗ a 2 d 2 θ a d x 2 ( x ) | x = b = a 2 d 2 d x 2 ( s ∗ θ a ) ( x ) | x = b

Hence, the wavelet transform of s(x) is proportional to the second derivative of s(x)smoothed by θ _a (x). The zero-crossings of CWT(a b) correspond to the inflection points of s∗θ _a . The identification of transcript start and end sites is achieved by computation of the redundant CWT over a wide scale range followed by zero-crossing line detection and length thresholding. The chosen mother wavelet is the second derivative of a Gaussian. The redundancy of the CWT yields enhanced information on the position-scale localization of the features of interest (in this case, the transitions) 46 .

An illustrative example is given in Figure 7. We generated a simulated transcriptional unit with a rectangular pulse signal of 721 samples with additive Gaussian noise (mean 0 and standard deviation 0.5) (see Figure 7(a)). The absolute values of the wavelet transform coefficients and the zero crossing lines are shown in Figure 7(b) and (c), respectively. Observe how the position of these lines corresponds to abrupt intensity transitions in the noisy signal and the longest connected curves identify the start and end points of the rectangular pulse. The R functions provided as supplementary material detect the zero-crossing lines and identify them as transcription start sites (TSS) and transcription end sites (TES) depending on the slope sign.

The candidates start and end sites detected as described in the previous section, are filtered to remove incorrect assignments. The purpose of this procedure is to filter those transitions that do not correspond to variations in signal intensity. For the generation of TARs we considered the signal transitions in which variation in intensity is at least 10% of the dynamic range of the analyzed signal. We also eliminate from the list of detected TAR all the start and end points that are not correctly paired off. We use the sign of the zero-crossing lines to separate start and end points and we match each start site with its corresponding end site. Finally, we define the minimum normalized intensity threshold required for the segments to be considered as transcriptional active regions. This value is calculated as the median of the signal intensity distribution, but this threshold can also be user-defined. In order to improve the definition of TARs, we cluster together consecutive segments for which the mean normalized intensity value is over the threshold.