The general assumption is that the (natural) logarithm of the background intensity can be approximated by a function linear in some fitting parameters ω_α. Once these parameters are set to their optimal values _α an estimate of the background intensity for the i-th (i = 1, 2, ..., N_dim) PM probe can be obtained as

where N_f is the number of fitting parameters. Ω_iα is a sequence- and position-dependent element of the N_dim × N_f -dimensional matrix Ω, which will be defined below.

The optimized values for the model parameters _α are obtained by minimizing the difference (of the logarithms) of the observed and estimated intensities

i.e. solving a linear least square problem. The sum extends over the training set which is a subset (with K elements) of the intensities of all annotated features. The choice of the elements of will be discussed later (see Data Set - Parameter Optimization). The minimum is found by imposing vanishing partial derivatives of S w.r.t. ω_α. This yields the following N_f linear equations

which can be rewritten as

Here, we have introduced the matrix

and the vector Γ_α

(Note that M = Ω^TΩ is symmetric and dim(M) = N_f × N_f).

The solution of Eq. (5) is given by the vector

of the optimal parameter values. If the matrix M is singular, M^-1 has to be replaced by its pseudoinverse M⁺which can be obtained by means of Singular value decomposition (SVD). In this work a standard SVD algorithm based on Golub and Reinsch is used (see e.g. 1112).

Due to the symmetry of M only half of the off-diagonal elements need to be generated, hence reducing the computational effort. For the chips tested in this paper with dimensions up to 1164 × 1164 features, the computational time on a standard PC (×86_64 Intel Core 2 Duo with 3 GHz, 3 GB RAM) required to estimate the background intensities is 8 to 10 seconds for the larger chips, and faster for the smaller ones.

This makes our algorithm an order of magnitude faster than our previous version 10, 3 to 5 times faster than GC-RMA, PDNN and MAS5, and about twice as slow as RMA and DFCM (Bioconductor packages were used for the testing). Note that for our algorithm, the time estimate includes both reading in the CEL-file and the background calculation, as it is done in one step.

This computation involves the generation of the matrix M and vector Γ (from Eqs. (6) and (7)), the SVD of M to solve Eq. (5) and the estimation of the background intensity for all PM probes through Eq. (2). Differently from other approaches in which the cost function is minimized by means of Monte Carlo methods 13 or other dynamical algorithms 10, the solution of SVD provides the exact minimum of the cost function Eq. (3). Hence, there is no risk in getting stuck in local minima different from the global one.

As mentioned above, probes in Affymetrix form PM/MM pairs. Consider now a target sequence at a concentration c in solution. The analysis of Affymetrix spike-in data (see e.g. 14) shows that not only the PM signal increases with increased target concentration c but also the MM intensity. This is an indication that a single MM nucleotide only partially prevents probe-target hybridization. Therefore the intensity of MM probes can also be decomposed in a non-specific and specific part as in Eq. (1). Supported by Affymetrix spike-in data analysis, our assumption is that the non-specific part of the hybridization is about equal for PM and MM probes: . The specific part of the signal is different in those two cases; equilibrium thermodynamics suggests a constant ratio , independent of the target concentration, as observed in experiments 3.

These insights are useful for the selection of probes for the optimization set in Eq. (3): includes all MM probes whose intensities are below a certain threshold I₀ and whose corresponding PM intensities also fulfill <I₀ (a similar selection criterion was recently used by Chen et al. 15). The threshold I₀ is chosen on the basis of the total distribution of the intensities. contains a significant fraction of the mismatch probes: typically 35%. Since the specific signal of MM intensities is lower than that of their corresponding PM's, they provide more reliable information on the background. The coordinates and sequences of the probes in are then fitted to the intensities of these probes yielding the parameters ω. With those newly acquired parameters ω the background signal of all MM probes is estimated based upon the assumption .

The choice of the matrix elements of Ω is dictated by input from physical chemistry as well as by the architecture of the microarray. Different schemes involving different choices for Ω with a varying number of parameters N_f were tested. Given a choice of Ω and in particular the number N_f of fitting parameters, the accuracy of the background estimation is reflected by the value of S from the minimization of Eq. (3). While the addition of fitting parameters always yields lower values of S, a too large set of fitting parameters runs the risk of "overfitting". The final choice of Ω is a compromise between a minimization of S and the use of the smallest possible set of parameters.

In the present model the number of parameters is N_f = 50. Similarly to the previous work 10 these parameters can be split into two groups: a first group describes the correlation of the background intensities with features which are physical neighbors on the chip; the second group are nearest-neighbor parameters which describe affinities for non-specific hybridization to the chip.

The parameters ₂ to ₉ and ₁₁ to ₁₈ describe the coupling of the background intensities to the physically neighboring features on the chip. As already mentioned, our estimate of the PM background is based on the non-specific intensity of the MM sequence. An Affymetrix chip is designed such that MM and PM are found in rows at equal y-coordinates. In addition, given a PM at (x, y), the corresponding MM feature is at (x, y + 1).

Figure 1 schematically represents the influence of the neighboring intensities on the background value. The strength of the correlation of the eight neighboring spots (i.e. the magnitude and sign of the corresponding _α) with the central MM feature is indicated by the color. The numbers in the figure identify the associated parameters _α. For example ₆ and ₁₅ are associated to the intensity at (x, y - 1) with respect to the reference MM intensity with coordinates (x, y). There does not seem to be any evidence that the middle-nucleotide classification in purines and pyrimidines reveals any insight concerning the background. Instead, our results show that the absolute values of two "corresponding" (purine-pyrimidine pair) _α 's are generally of the same order of magnitude. Across all species, we find typical average outputs of

Since ₆, ₁₅ respectively, reflects the correlation of the MM signal with its corresponding PM, as expected its magnitude is greatest among all parameter values. Next, the MM intensity shows strong correlations with its direct nearest neighbors positioned at (x ± 1, y), i.e. _2/11 and _3/12. Hence, the stronger the MM-neighboring intensities, the stronger their influence on the background signal of MM. However, in order to somehow compensate for strong MM-neighboring signals which might be caused by the presence of complementary target sequences, parameters _4/13 and _5/14 have negative sign. The remaining three parameters, i.e. the top three neighbors limit their influence to a basically negligible minimum which appears to indicate the weak sequence correlation in y-direction as previously found. The results indicate that the influence of neighboring intensities on the background noise is significant. In fact, our analysis shows that ≈ 30% of log are constituted by neighboring probes in terms of absolute intensities (see Table 3). It appears that for a few probes, the latter might even play a crucial role. It is unlikely that the correlations summarized in Fig 1 could be explained only as simple optical effects, as light from bright probes spilling into their neighborhood. Indeed, optical effects should produce an isotropic correlation pattern in the x and y directions which is not seen in our analysis. Another cause for this correlation might be that neighboring probes share common sequences, as is the case in Affymetrix chips 10.

The parameters _α (19 ≤ α ≤ 34) are the analogues of the nearest-neighbor free energy parameters. The nearest-neighbor model is commonly used to study the thermodynamics of hybridization of nucleic acids in solution (see e.g. 17). In this model it is assumed that the stability and thus the hybridization free energy ΔG of a dinucleotide depends on the orientation and identity of the neighboring base pairs. For RNA/DNA duplexes there are 16 hybridization free energy parameters which were measured in aqueous solution by Sugimoto et al. 20.

Recent experiments 21 focusing on specific hybridization show a good degree of correlation between the hybridization free energies in solution and those directly determined from microarray data. Concerning background data, we also expect a certain degree of correlation between the parameters _α (19 ≤ α ≤ 34) and their corresponding Sugimoto free energy parameters.

In order to test the relationship between the experimentally determined so-called Sugimoto parameters and our results, we calculate the correlation coefficient between these two sets. The results are reported in Table 2. In general, the correlation coefficients vary between 0.53 and 0.83 with a median value of 0.71. Figure 2 shows two typical results for C. Elegans and D. Melanogaster. Both plots indicate that the relationship between _α (19 ≤ α ≤ 34) and the nearest-neighbor free energy parameters of 20 is approximately linear.

Figure 3 shows a plot of position-dependent nearest-neighbor parameters ₃₅ to ₅₀ as obtained from the minimization of the cost function of Eq. (3) for four different sets of experiments. The data are plotted as function of the corresponding nearest neighbor free energy parameters of 20. We note that in all experiments shown there is a negative correlation between the two data sets. The parameters ₃₅ to ₅₀ reflect the difference in effective free energy between the ends and the middle of the probe sequence. Since weakly binding probes suffer more from end-effects (unzipping, etc.), ₃₅ to ₅₀ and their corresponding nearest-neighbor free energy parameters are negatively correlated. Hence, the negative correlation in Fig. 3 and those observed in all other cases (see correlation coefficients reported in Table 2) indicates that the dominant contribution to the background intensity comes from the middle nucleotides. This conclusion complies with other types of analysis which use position-dependent free energy parameters (see e.g. 716).

In this section we compare the estimated background signal (as given in Eq. (2)) with the intensities of given probe sets corresponding to non-expressed genes in the samples analyzed. We start with publicly available data taken from spike-in experiments on HGU95A chips http://www.affymetrix.com where genes have been spiked-in at known concentrations, ranging from 0 to 1024 pM (picoMolar). The data at 0 pM correspond to the absence of transcript in solution. Figure 4a compares the measured and predicted background for probeset 37777_at. Except for probes 2 and 15 for which the measured signal is higher than the predicted background, there is a nice agreement between our prediction and the experimental background intensity: the standard deviation of the absolute difference between the intensity of the PM and I^est is 28 in Affymetrix intensity units. Also shown are the background estimations as obtained with other algorithms (the data for PDNN is missing in Fig. 4a due to the unavailability of one of the supporting files from Bioconductor packages). Figure 4b presents the same information for probeset 209606_at. Here, the standard deviation is 7 in Affymetrix intensity units.

Next, we go beyond the spike-in experiments. To select non-expressed genes we considered probe sets with very low expression values as obtained from the RMA algorithm. In Figure 5 the PM intensities as well as the calculated background signal of two probesets from A. Thaliana and C. Elegans experiments are shown. The absolute intensities of both probesets are very low. As a consequence, we can safely assume these genes are not expressed and hence any measured signal can be attributed to the background. Figures 4 and 5 both show that the present model captures the essentials of and correctly predicts background intensities. Both Figures are representative for the CEL-files analyzed in this work.