- Research article
- Open Access
A multivariate prediction model for microarray cross-hybridization
BMC Bioinformatics volume 7, Article number: 101 (2006)
Expression microarray analysis is one of the most popular molecular diagnostic techniques in the post-genomic era. However, this technique faces the fundamental problem of potential cross-hybridization. This is a pervasive problem for both oligonucleotide and cDNA microarrays; it is considered particularly problematic for the latter. No comprehensive multivariate predictive modeling has been performed to understand how multiple variables contribute to (cross-) hybridization.
We propose a systematic search strategy using multiple multivariate models [multiple linear regressions, regression trees, and artificial neural network analyses (ANNs)] to select an effective set of predictors for hybridization. We validate this approach on a set of DNA microarrays with cytochrome p450 family genes. The performance of our multiple multivariate models is compared with that of a recently proposed third-order polynomial regression method that uses percent identity as the sole predictor. All multivariate models agree that the 'most contiguous base pairs between probe and target sequences,' rather than percent identity, is the best univariate predictor. The predictive power is improved by inclusion of additional nonlinear effects, in particular target GC content, when regression trees or ANNs are used.
A systematic multivariate approach is provided to assess the importance of multiple sequence features for hybridization and of relationships among these features. This approach can easily be applied to larger datasets. This will allow future developments of generalized hybridization models that will be able to correct for false-positive cross-hybridization signals in expression experiments.
Expression microarrays are powerful tools for disease diagnosis, prognosis and treatment , offering unparalleled insight into the function of the entire genome and the dynamic interactions among genes. Two common platforms are oligonucleotide and cDNA microarrays. Oligonucleotide microarrays are generated by either robotic deposition of pre-synthesized oligos or in situ synthesis of ~25-mer oligo probes ontosolid slides [2, 3], while cDNA microarrays are created by spotting long strands of amplified cDNA sequences, such as expressed sequence tags (ESTs) .
Specific hybridization is the desired type of hybridization between a probe and the target sequence that comes from the same transcript. By contrast, cross-hybridization may occur between parts of the probe and target sequences that do not come from the same transcript as the probe. Cross-hybridization can be a significant contributor to false-positive noise in array data and is known to happen in both oligo and cDNA microarray platforms. Duplex stabilities and re-association kinetics for nucleic acid hybridization is complex, and many factors are involved. Experimental conditions such as hybridization temperature, salt concentration, viscosity of the solvents, pH value are important. Concentration, complexity, lengths, and GC contents, as well as the secondary structures of nucleic acids are also critical. A comprehensive review can be found in .
Hybridization in solvents is different from that on solid surfaces, and different surfaces and platforms have different properties. Several studies have been conducted to model the expression intensities using binding kinetics based on physical properties or oligo composition in the popular oligonucleotide microarrays made by Affymetrix [6–8]. Cross-hybridization is an especially severe problem for cDNA microarrays because of the lengths of the probes . Because predictions of binding free energy cannot yet be achieved for longer sequences, the models developed for oligo arrays cannot be generalized to cDNA microarrays. Several univariate studies have attempted to correlate the hybridization intensities and sequence characteristics between the probe-target pair for cDNA or DNA microarrays using genomic sequences [10–13]. Most of these studies [10–12] reached the same (and non-surprising) conclusion that sequences sharing a high degree of identity have a higher chance to cross-hybridize. Another approach to studying cross-hybridization is to investigate the relationships between contiguous pairing segments and hybridization intensity . All these studies acknowledged some exceptions that could not be accommodated by their univariate analyses. To the authors' knowledge, no systematic multivariate predictive modeling has been attempted for cDNA microarray hybridizations.
A field relevant to the microarray cross-hybridization issue is the design of short interfering RNA (siRNA) sequences (10 ~25 nucleotides) leading to RNA interference (RNAi). In particular, the selection of effective siRNA sequences that minimize off-target silencing effects is a topic of great interest in computational and functional genomics [14–16]. As in the field of cDNA array analysis, these reports point to the fact that more studies focus on the hybridization between short sequences (such as oligo arrays or siRNA design) rather than on cross-hybridization between long sequences.
Specific signal quantification is crucial for correct interpretation of microarray experiments, and probe selection has been the major task for array design in the past decade [17–29] to avoid cross-hybridization. However, the number of probes spotted on both oligo and cDNA arrays increases dramatically as the technology advances, and cross-hybridization almost becomes inevitable. A computational method validated by proper experiments to quantify platform-specific cross-hybridization is needed to derive correct quantification of sequence-specific signals. The challenge is that cross-hybridization is the result of complex interactions between multiple target and probe sequences on the arrays (see Figure 1a in Additional file 1). It seems very difficult to attack this problem in generality at this point. Therefore, as a first step toward understanding this complex phenomenon of a many-to-many relationship, we propose to investigate a simplified system with hybridization between one target and multiple probes spotted on the arrays; that is, to quantify the hybridization of one target to many probes (see Figure 1b in Additional file 1).
A dataset of CYP450 PCR products spotted on microarrays following the experimental design proposed in  was used for our model development. The genes in the cytochrome P450 family are known to have varying degrees of sequence similarities, thus making them good candidates for studying cross-hybridization phenomena on microarrays [11, 30]. Because hybridization is influenced by sequence characteristics as well as many experimental factors, the experimental/hybridization conditions, such as target/probe concentration, salt concentration, and hybridization temperature, were kept consistent throughout this study.
The immediate goal of our current research is to identify efficacious sets of sequence features for predicting hybridization between probe-target pairs in a multivariate fashion and to determine how different factors synergistically influence hybridization. Our ultimate goal, which reaches beyond the scope of this paper, is to estimate specific hybridization features after correcting for false-positive cross-hybridization.
A dataset of CYP450 PCR products spotted on microarrays  was used to validate the proposed multivariate approach. Thirty-one different cDNAs from the CYP450 family (with lengths ranging from 500 to 1200 bp) were hybridized individually with each of 31 arrays. Triplicates were generated, for a total of 93 arrays. The target/probe concentrations and other experimental conditions (such as temperature and salt concentration) were constant across arrays. Details of the experiments and array manufacturing processes are described in Methods and .
Triplicate data were used to estimate the parameters λ and α in the generalized log transformation of the hybridization intensities [Equation (1) in Methods]. The estimated parameters were = 1.39*10-20 and = 1.79*10-12. Hybridization experiments were highly reproducible among replicates (0.94 < Spearman correlation coefficient < 0.97; see Table 1 in Additional file 1). Hybridization intensities of target 17 in all three replicates were consistently lower than others (see Figure 2a in Additional file 1). These low intensities, including specific (self-self) hybridization, indicate that systematic errors were introduced in this target sample. Therefore, the data of target 17 were excluded, and the remaining data were used to re-estimate α and λ. The re-estimated parameters were = 4.71*10-22 and = 2.78*10-13 (see Figure 2b in Additional file 1). A total of 69 data points outside the dynamic range were excluded from further analyses (see Result 2.1 and Figure 2c in Additional file 1). To avoid over-fitting, only one of the three replicates, Replicate 1, was used for model development. Replicate 1 was chosen (907 data points) because it had the highest similarity to the other replicates (see Table 1 in Additional file 1); i.e., it was closest to the centroid of the replicate set.
Twelve potential predictors were included in the model (see Methods, Table 1). The pairwise correlations between all pairs of variables and hybridization intensities (X1 to X12 and TY) were summarized in Result 2.2 in Additional file 1 (see Figure 3 in Additional file 1). As expected, some of the variables were correlated. The probe-target pairs with more most-contiguous-base-pairs (X11) and long overlaps (X8) often had higher intensities (TY > 6.5) than others (Figure 1).
Three multivariate methods, multiple linear regression (MLR), regression tree (RT) analysis, and feed-forward artificial neural network (ANN) analysis were performed to predict hybridization (for details see Methods). The results from these analyses were compared with that of the third-order polynomial regression, using percent identity (X7) as the sole predictor, as proposed by Xu and collaborators  [Equation (3) in Methods]. Five-fold cross-validation (CV) was performed to estimate the generalized errors  for all types of models so that the estimated errors were directly comparable. Models with all possible combinations of 12 potential predictors (4,095 combinations) were fitted and evaluated in each CV fold, and the model with the minimum sum of square errors was selected when p variables were included in the model (p = 1, 2, ..., 12). In the case of a closed-form solution for the model identification procedure (as in MLR), one-step CV was performed. Otherwise, two-step CV was performed: first-step CV to make decisions on the most appropriate internal model complexity and second-step CV to estimate the generalized errors of the final model (such as RTs and ANNs; for details see Methods).
Third-order polynomial regression (PR)
The third-order polynomial model using percent identity (X 7 ) as the single predictor  was significant (R2 = 0.31, p < 10-4). The polynomial terms were statistically significant, and the point estimates were = -53.28, = 253.21, = -365.11, = 173.35. The estimated CV error was 0.9981 (± 0.0889) [Equations (4) and (5) in Methods]. The residuals were examined with respect to the predictor, and no obvious pattern was detected to suggest any model violation.
Multiple linear regression (MLR)
A total of 20,475 (= 4,095 × 5) multiple linear regression models [Equation (6) in Methods] were computed, and the model with the minimum sum of square errors at a given subset size p was selected (see Figure 4a in Additional file 1). The CV errors of all subset sizes were estimated (Figure 2a). The multiple linear regression with minimum CV errors (0.9123) contained two variables (Figure 2a). The most parsimonious model within one standard error of the minimum CV errors, the model with p = 1, was chosen . Its only variable was the most contiguous base pairs (X11) (Figure 3a). The regression coefficients were estimated using the full dataset after the model subset size was decided. The regression model was significant (R2 = 0.35, p < 10-4). The transformed hybridization increased 0.029 units as the most contiguous base pair increased by one unit. The residuals were examined, and no obvious pattern was detected to suggest model violation.
Regression tree (RT)
A total of 4,095 large trees was grown for each of the five CV training sets (for details see Methods and Methods 1.1 in Additional file 1). Each large tree was then pruned. The first-step CV was performed to compute the cost for each subtree. The smallest tree within one standard error of the minimum-cost subtree was selected . The model with the minimum sum of square errors at a given subset size p was selected (see Figure 4b in Additional file 1). The generalized errors were estimated in the second-step CV (Figure 2b). The model with minimum CV errors was the model of subset size 2, and it was also the most parsimonious model within one standard error (Figures 2b). The models of subset size 2 were not all the same across the five CV training sets (Figure 3b), and the majority (four of the five) contained X11 (most contiguous base pairs) and X4 (target GC content). We therefore fitted the model using the entire dataset with X11 and X4 to derive the optimal regression tree. This subtree partitioned the feature space into five decision regions (Figure 4). Node 1 at the root is the most contiguous base pairs (X11 > 19.5), which can separate strong hybridizations from others. When there are more than 20 contiguous base pairs, the transformed hybridization intensities were stronger than 8.68 (Figure 4). The space became dichotomized three times (Nodes 2 to 4) after the first node, by the target GC content (X4). That is, target GC content influenced the hybridization levels in a nonlinear fashion. The residuals were examined, and no obvious pattern was detected.
Artificial Neural Network (ANN)
The first-step CV for early stopping was performed to select the appropriate number of hidden nodes to avoid overfitting for the 4,095 models in each training sets (see Figure 4c in Additional file 1). The model with the minimum sum of square errors at a given subset size p was selected (see Figure 4c in Additional file 1). The generalized errors were calculated in the second-step five-fold CV to decide the appropriate number of variables to retain in the models (Figure 2c). The model with minimum CV errors was of subset size 5 (CV error = 0.7487). The most parsimonious model within one standard error (0.067) was the model with four predictors (Figure 2c). The majority contained variables X3 (target length), X4 (target GC content), and X10 (target di-nucleotide distance), and X11 (most contiguous base pairs) (Figure 3c). Two exceptions were the models having X11replaced by X5 (Smith-Waterman score). This variable substitution is not surprising because X5 and X11 are linearly highly correlated (r = 0.98, p < 10-165). Interestingly, the rank correlation is much lower than the linear correlation (r = 0.14, p < 3.88*10-5). The target GC content and lengths influenced the hybridization intensities in a nonlinear fashion in addition to the effects of the most contiguous base pair. The residuals were examined with respect to the predictor, and no obvious pattern was detected.
Comparison of CV errors among models showed that the multivariate models were superior to the univariate third-order polynomial model proposed earlier , and indicated that more than one variable was important for hybridization prediction (Figure 5). Regression trees and artificial neural networks improved the prediction by including additional nonlinear effects (see Table 2 in Additional file 1). The CV correlation provides a summary measure of prediction quality [Equation (7) in Methods]. The selected regression tree using the most contiguous base pairs (X11) and target GC content (X4) outperformed all other chosen models (R-k(i)= 0.75, p < <10-4; see Table 2 in Additional file 1).
DNA microarrays are widely used for transcriptomic profiling, where the expression of thousands of genes is monitored simultaneously. The correct interpretation of all such microarray experiments depends on reliable and specific signal quantification.
We combined a systematic variable selection scheme with multiple competing multivariate models to improve current predictability of hybridization models for cDNA microarrays. Variable selection progression using five-fold cross-validation clearly showed that neither the sequence percent identity (X7), the variable identified in previous univariate studies [10–12], nor the E-value (X6), the variable heuristically used to measure hybridization potentials for arrays [26, 33], was the most predictive independent variable. Instead, we found the most contiguous base pairs (X11) to be most predictive when only a single variable was selected (Figure 3). Prior to our final analysis using all 12 potential predictors, X1 to X12, we had performed a preliminary analysis using the first 10 potential variables, X1 to X10, for all three multivariate models with the same systematic search scheme. The results were fairly consistent with what we found using all 12 variables, with the noticeable exception that the most contiguous base pair, X11, was replaced by the Smith-Waterman alignment score, X5, for all three models, MLRs, RTs, and ANNs (see Figures 5 and 6 in Additional file 1). This variable substitution is to be expected because X5 and X11 are linearly highly correlated (r = 0.98, p < 10-165). The performance of the most parsimonious models for all methods of our final analyses, which included variable X11, was slightly improved over the preliminary analyses, which used variable X5 (see Tables 2 and 3 in Additional file 1). Although both ANNs and RTs do not have closed-form solutions, the consistent results yielded by the models using 10 or 12 variables showed the robustness of this method we used.
Our result showed that the most contiguous base pair (X11) and target GC content (X4) were the most predictive predictors in the selected regression tree (Figure 4), and it resonates with the finding by Wren et al. , but with significant improvements. Wren et al. only used one predictor, the most contiguous hydrogen bonds, in their model while we examined the relationships between all possible combinations of potential predictors and hybridization. They found that signals above background levels begin at ~45 hydrogen bonds (HBs) and become prominent after ~60 HBs . As expected, the most contiguous hydrogen bond is highly correlated with the most contiguous base pair (X11) in our study (r = 0.9988, p ≈ 0). The selected regression tree in our study showed that hybridizations were strong when more than 20 contiguous base pairs were found between probe and target pairs (Node 1 in Figure 4). Using the same hydrogen bond conversion (GC having 3 hydrogen bonds and AT having 2 hydrogen bonds), the hydrogen bond numbers for 20 base pairs segment are between 40 and 60. After separation at Node 1 (X11) in the regression tree, target GC content (X4) was found to influence hybridizations in a nonlinear fashion by further dichotomizing the decision space three times (Nodes 2 to 4 in Figure 4). Node 2 separated the second highest intensities with the remaining according to whether GC content exceeds 60%, supporting the intuition that targets having higher GC content have higher hybridization strength with probe sequences. The remaining two nodes divide the remaining space into three regions. The need of nonlinearities in hybridization model is not surprising because there is no straight forward prediction algorithm for prediction of secondary structure or folding energy for long sequences, such as the target sequences in our study. However, folding energy of sequences is generally correlated to GC content as illustrated by the high correlation found in probe GC content and estimated probe folding energy (see Figure 3 in Additional file 1). The nonlinear relationship between target GC content and hybridization may reflect the complex effects and interactions between secondary structure of target sequences and the hybridization between probe and target sequences for microarrays.
Predictability of the model could be improved in the future, for instance, by accounting for thermodynamic features, as it is sometimes done for oligonucleotide arrays [6–8]. Efforts are also under development to improve the computation speed for large dataset  and accommodate the constraints of unequal lengths between probe/target sequences and for long sequences in real world data .
Recently, after "jaw dropping" discordant results  among array platforms were reported , reproducibility across-platform has become a research topic of intense interest [38–42]. One of the contributing factors to the inconsistencies across platforms is thought to be due to the intrinsic differences of each array platforms . The systematic multivariate approach proposed here can easily be applied to understand platform-specific hybridization processes, and this can potentially improve the comparability across platforms.
The major limitation of our model development is the use of a small dataset. At this point, the analysis of a relatively simple and small system seems to be the only way forward. Our proposed method should thus be seen as the first step toward understanding more fully the complexities surrounding cross-hybridization in other, larger systems. The hope behind our work is that scientists will begin to generate larger and more generalized datasets with hybridization between many targets and probes (see Figure 1a in Additional file 1), so that better and more widely applicable models may be developed in the near future.
We proposed and validated a systematic strategy using multiple competing multivariate models to select critical sequence characteristics and quantify their relationship with hybridization on microarrays. The multivariate models outperformed the currently used univariate model in all cases. The most contiguous base pairs and the target GC content were found to be significant predictors of hybridization. Our systematic approach offers a quantitative method to correct for cross-hybridization signals on microarrays and shows the benefit of modeling nonlinear interdependencies between predictors and hybridization intensities.
A dataset of CYP450 PCR products spotted on microarrays  was used in this study. Thirty-one different DNAs from the CYP450 family (with lengths ranging from 500 to 1,200 bp) were hybridized individually with each of 31 arrays. Triplicates were generated, for a total of 93 arrays. Each array had 31 probes spotted at 1 μM. The probes were ~150 mer (ranging from 129–170 bp) PCR products, which corresponded to the 31 transcripts. The array manufacture details were described in . Target/probe concentrations within a dynamic range were kept constant . Other hybridization conditions (such as consistent buffer composition, salt concentration, 42°C in 50% forrnamide-based hybridization condition) in this study were consistent across experiments . The hybridization intensities in our study can be viewed as the "conditional binding affinities" (i.e., binding affinities conditioned on a constant probe/target concentration, experimental temperature, etc.).
Triplicate data were used to estimate the parameters λ and α in the generalized log transformation of the hybridization intensities [Equation (1)]. This transformation with slightly different parameterizations, was developed independently by two research groups [43, 44]:
where z = y - α, and .
This transformation is based on the expression model
, where y represents the measured raw hybridization intensity, α is the background noise, μ is the true hybridization level, ε and η are normally distributed error terms with mean 0 and variances and , respectively, and n denotes sample size. The transformation not only agrees with the widely used log transformation , but also stabilizes the variance, satisfying the equal-variance assumption for linear models . Maximum likelihood estimation implemented in the software package R  was used to estimate the parameters α and λ. The hybridization intensities used in the analyses were transformed according to the estimated form of Equation (1) and are denoted by TY. Even though triplicates were used for the estimation of α and λ, only one of the triplicates was used for model fitting and cross-validation so that estimates would not be overly optimistic.
Twelve potential predictors, reported to be important for predicting hybridization, were included in our study (Table 1). Probe/target sequence lengths and GC contents, variables X1 to X4, are important for hybridization [5, 13, 30]. Sequence alignment features are always thought to be important. For instance, sequence percent identity, X7, is considered the best predictor for cross-hybridization on cDNA microarrays, based on several univariate models [10–12]. Other alignment features were Smith-Waterman alignment score (X5), E-value (X6), and overlap length (X8). They were indicated as potential good predictors in univariate studies or used empirically for predicting hybridization [10, 12, 26, 33]. The program ssearch34, [49, 50], a rigorous and efficient implementation of the Smith-Waterman algorithm , was used to calculate these alignment features.
Secondary structures of sequences are important for hybridization interference, and the free energy for the 31 probe DNA sequences, X9, was estimated using Mfold . The target sequences were long (many over 800 bp) so that the existing algorithm had no reasonable prediction performance for their folding energy or hybridization potential (cf. ). Thus, no prediction of the folding energy of the target transcripts was included in the model. One important feature to determine the hybridization potential between oligo sequences is the magnitude of pairwise base stacking of hybridization free energy by summing up all pairs of the free energy between neighboring two-base pairs, called the nearest-neighbor model [5, 53, 54]. There is no simple way to generalize this model for long and unequal-length sequences. Therefore, the standardized Euclidean distance between target-probe pairs, X10, using the alignment-free method with di-nucleotide word frequency  was used as a variable to mimic the empirical effect of nearest-neighborhood model for oligo sequences. Short segments of strong hybridization have been believed to be critical for predicting hybridization potentials [13, 56]. As suggested by an anonymous reviewer, we included two more variables as potential predictors, X11 and X12, in our final analyses. The 'most contiguous base pairs between probe and target pairs' (or the length of identical substring) was included as variable X11. The most contiguous hydrogen bonds, considering GC having three hydrogen bonds and AT having two hydrogen bonds, would be an interesting variable to include. However, this variable is highly correlated to the most contiguous base pairs (r = 0.9988, p <10-10), and therefore, we included a more independent variable, the GC content of the most contiguous segment, as X12.
The pairwise linear and rank (Spearman) correlations between all 12 variables and the transformed hybridization intensities (X1 to X12 and TY) were examined. The correlations among triplicates were also examined to confirm the reproducibility and quality of the dataset.
Three multivariate methods, multiple linear regression (MLR), regression tree (RT) analysis, and feed-forward artificial neural network (ANN) analysis were performed to model hybridization. The comparative use of these three methods was to cover a range of possibilities that stretches from the computationally straightforward use and interpretation of multiple linear regression to computationally intensive and algorithmically intricate machine learning methods using artificial neural networks with early stopping and topology optimization . Between these extremes, we also considered decision trees which iteratively dichotomize the complex domain based on different combinations of variables and correspondingly produce models that are easy to interpret . The results from these analyses were compared with that of the third-order polynomial regression proposed by Xu and collaborators using percent identity as sole predictor  :
All multivariate model identification strategies proceeded as follows. Five-fold cross-validation (CV) was performed to estimate the generalized errors of the model . The data were split into k (= 5) roughly equal-size parts. For the kth part, the model was fitted to the other k-1 (= 4) parts of the data. Models with all possible combinations of variables (4,095 combinations) were fitted and evaluated, and the model with the minimum sum of square errors was selected when p variables were included in the model (p = 1, 2, ..., 12). The prediction error of the selected model was then calculated for the kth part. A total of 20,475 models (5 × 4095) were trained over five folds. This approach may be viewed as five "CV training sets" and five "CV testing sets". The procedure was carried out for k = 1, 2,..., 5, and then the CV estimate of the prediction error (CV errors) was computed as
where k(i) is the part containing observation i, and is the fitted value for observation i, computed with the k(i)th part of the data removed. The estimate of the standard error of the CV error  is
In the case of a closed-form solution (i.e., MLR), one-step CV was performed to estimate the generalized errors. For models that use CV to make decisions on the most appropriate internal model complexity in the first step, a second-step CV was used to estimate generalized errors. This step for regression trees and ANNs ensured that the generalized errors were estimated from data outside those training data that were used to fit (train/validate) the model in the first-step CV. The resulting CV errors were compared with those estimated by one step CV errors of MLR. The most parsimonious model within one standard error of the model with the minimum CV error was chosen  for each of the three multivariate methods. CV residuals of the selected models were examined with respect to the predictors to assess model assumptions.
Multiple linear regression (MLR)
The simplest relationship between the predictors and the hybridization intensities is linear:
Y = β0 + β1X1 + β2X2 + ... + β p X p + ε. (6)
The appropriate number of variables, p, was determined using CV errors [Equation (4)] and its estimated standard errors [Equation (5)].
Regression tree (RT)
A regression tree, also known as CART (classification and regression tree), represents a multistage decision process, where a binary decision is made at each stage  to partition the d-dimensional space into smaller and smaller regions . Three standard steps for regression tree modeling are growing a large tree, pruning, and finally selection of a subtree tree based on CV [32, 58]. More details are summarized in Method 1.1 in Additional file 1. We performed a fourth step, namely second-step CV, to estimate generalized errors, using the data external to the training-testing data set used for the first-step CV in earlier steps. This second-step CV yielded an estimate of generalized errors, and determined the appropriate number of variables to retain in the models.
Artificial Neural Network (ANN)
An ANN is a two-stage nonlinear regression or classification method . It identifies arbitrary multiparametric functions directly from experimental data as universal approximators . The first-stage nonlinear regression is between the input predictors and the hidden layer, and the second-stage regression is between the hidden layer and the response variable.
We applied a series of measures to avoid potential pitfalls associated with ANN, such as model overfitting, input scaling problems, arbitrary numbers of hidden nodes, and multiple minima . In our study, one hidden layer was used because it has been shown to be sufficient for approximating all functional forms . All predictors were scaled between 0 and 1 in the feed-forward ANN to eliminate scaling effects. First-step CV for early stopping was performed to select an appropriate number of hidden nodes to avoid model overfitting . In a similar manner to our treatment of regression trees, a second-step CV was performed with the ANN to decide the appropriate number of variables to retain. More details on ANN model fitting and topology optimization can be found in a comprehensive review .
The estimates of CV errors [Equation (4)] were compared among three multivariate models and the third-order polynomial model. Furthermore, the CV correlation, calculated as
provides a summary measure of prediction quality.
Steinmetz LM, Davis RW: Maximizing the potential of functional genomic. Nature Reviews Genetics 2004, 5: 190 -1201. 10.1038/nrg1293
Lipshutz RJ, Morris D, Chee M, Hubbell E, Kozal MJ, Shah N, Shen N, Yang R, Fodor SP: Using oligonucleotide probe arrays to access genetic diversity. Biotechniques 1995, 19(3):442–447.
Okamoto T, Suzuki T, Yamamoto N: Microarray fabrication with covalent attachment of DNA using Bubble Jet technology. Nat Biotech 2000, 18(4):438. 10.1038/74507
Schena M, Shalon D, Davis RW, Brown PO: Quantitative monitoring of gene expression patterns with a complementary DNA microarray. Science 1995, 270(5235):467–467.
Ptijssen P: Overview of principles of hybridization and the strategy of nucleic acid probe assays. In Laboratory Techniques in Biochemistry and molecular biology: hybridization with nucleic acid probes Part I: theory and nucleic acid preparation. Volume 24. Amsterdam, The Netherlands, Elsevier Science Publishers BV; 1993:19–78.
Hekstra D, Taussig AR, Magnasco M, Naef F: Absolute mRNA concentrations from sequence-specific calibration of oligonucleotide arrays. Nucl Acids Res 2003, 31(7):1962–1968. 10.1093/nar/gkg283
Held GA, Grinstein G, Tu Y: Modeling of DNA microarray data by using physical properties of hybridization. PNAS 2003, 100(13):7575–7580. 10.1073/pnas.0832500100
Zhang L, Miles FM, Aldape KD: A model of molecular interactions on short oligonucleotide microarrays. Nature Biotechnology 2003, 21(7):818–821. 10.1038/nbt836
Kothapalli R, Yoder S, Mane S, Loughran T: Microarray results: how accurate are they? BMC Bioinformatics 2002, 3(1):22. 10.1186/1471-2105-3-22
Miller NA, Gong Q, Bryan R, Ruvolo M, Turner LA, LaBrie ST: Cross-hybridization of closely related genes on high-density macroarrays. Biotechniques 2002, 32(3):620–625.
Xu W, Bak S, Decker A, Paquette SM, Feyereisen R, Galbraith DW: Microarray-based analysis of gene expression in very large gene families: the cytochrome P450 gene superfamily of Arabidopsis thaliana. Gene 2001, 272(1–2):61–74. 10.1016/S0378-1119(01)00516-9
Evertsz EM, Au-Young J, Ruvolo MV, Lim AC, Reynolds MA: Hybridization cross-reactivity within homologous gene families on glass cDNA microarrays. Biotechniques 2001, 31(5):1182–1192.
Wren JD, Kulkarni A, Joslin J, Butow RA, Garner HR: Cross-hybridization on PCR-spotted microarrays. IEEE Eng Med Biol Mag 2002, 21(2):71–75. 10.1109/MEMB.2002.1046118
Santoyo J, Vaquerizas JM, Dopazo J: Highly specific and accurate selection of siRNAs for high-throughput functional assays. Bioinformatics 2005, 21(8):1376–1382. 10.1093/bioinformatics/bti196
Yamada T, Morishita S: Accelerated off-target search algorithm for siRNA. Bioinformatics 2005, 21(8):1316–1324. 10.1093/bioinformatics/bti155
Huesken D, Lange J, Mickanin C, Weiler J, Asselbergs F, Warner J, Meloon B, Engel S, Rosenberg A, Cohen D, Labow M, Reinhardt M, Natt F, Hall J: Design of a genome-wide siRNA library using an artificial neural network. Nature Biotechnology 2005, 23(8):995 -1001. 10.1038/nbt1118
Nielsen HB, Knudsen S: Avoiding cross hybridization by choosing nonredundant targets on cDNA arrays. Bioinformatics 2002, 18(2):321–322. 10.1093/bioinformatics/18.2.321
Rouillard JM, Zuker M, Gulari E: OligoArray 2.0: design of oligonucleotide probes for DNA microarrays using a thermodynamic approach. Nucl Acids Res 2003, 31(12):3057–3062. 10.1093/nar/gkg426
Tolstrup N, Nielsen PS, Kolberg JG, Frankel AM, Vissing H, Kauppinen S: OligoDesign: optimal design of LNA (locked nucleic acid) oligonucleotide capture probes for gene expression profiling. Nucl Acids Res 2003, 31(13):3758–3762. 10.1093/nar/gkg580
Emrich SJ, Lowe M, Delcher AL: PROBEmer: a web-based software tool for selecting optimal DNA oligos. Nucl Acids Res 2003, 31(13):3746–3750. 10.1093/nar/gkg569
Li F, Stormo GD: Selection of optimal DNA oligos for gene expression arrays. Bioinformatics 2001, 17(11):1067–1076. 10.1093/bioinformatics/17.11.1067
Talla E, Tekaia F, Brino L, Dujon B: A novel design of whole-genome microarray probes for Saccharomyces cerevisiae which minimizes cross-hybridization. BMC Genomics 2003, 4(1):38. 10.1186/1471-2164-4-38
Wang X, Seed B: Selection of oligonucleotide probes for protein coding sequences. Bioinformatics 2003, 19(7):796–802. 10.1093/bioinformatics/btg086
Xu D, Li G, Wu L, Zhou J, Xu Y: PRIMEGENS: robust and efficient design of gene-specific probes for microarray analysis. Bioinformatics 2002, 18(11):1432–1437. 10.1093/bioinformatics/18.11.1432
Tomiuk S, Hofmann K: Microarray probe selection strategies. Briefings in bioinformatics 2001, 2(4):329–340. 10.1093/bib/2.4.329
Chen YA, Mckillen DJ, Wu S, Jenny MJ, Chapman R, Gross PS, Warr GW, Almeida JS: Optimal cDNA microarray design using expressed sequence tags for organisms with limited genomic information. BMC Bioinformatics 2004, 5(1):191. 10.1186/1471-2105-5-191
Nordberg EK: YODA: selecting signature oligonucleotides. Bioinformatics 2005, 21(8):1365–1370. 10.1093/bioinformatics/bti182
DasGupta B, Konwar KM, Mandoiu II, Shvartsman AA: DNA-BAR: distinguisher selection for DNA barcoding. Bioinformatics 2005, 21(16):3424–3426. 10.1093/bioinformatics/bti547
Chen DT, Chen JJ, Soong S: Probe rank approaches for gene selection in oligonucleotide arrays with a small number of replicates. Bioinformatics 2005, 21(12):2861–2866. 10.1093/bioinformatics/bti413
Chou CC, Chen CH, Lee TT, Peck K: Optimization of probe length and the number of probes per gene for optimal microarray analysis of gene expression. Nucl Acids Res 2004, 32(12):e99. 10.1093/nar/gnh099
Hastie T, Tibshirani R, Friedman J: The elements of Statistical learning: Data mining, inference, and prediction. New York, NY, Springer-Verlag; 2001:533.
Breiman L, Friedman JH, Olshen RA, Stone CJ: Classification and regression tree. New York, Wadsworth Inc; 1984.
Huang JC, Morris QD, Hughes TR, Frey BJ: GenXHC: a probabilistic generative model for cross-hybridization compensation in high-density genome-wide microarray data. Bioinformatics 2005, 21(suppl_1):i222–231. 10.1093/bioinformatics/bti1045
Leber M, Kaderali L, Schonhuth A, Schrader R: A fractional programming approach to efficient DNA melting temperature calculation. Bioinformatics 2005, 21(10):2375–2382. 10.1093/bioinformatics/bti379
Garel T, Orland H: Generalized Poland-Scheraga model for DNA hybridization. Biopolymers 2004, 75(6):453 -4467. 10.1002/bip.20140
Marshall E: Getting the noise out of gene arrays. Science 2004, 306(5696):630–631. 10.1126/science.306.5696.630
Tan PK, Downey TJ, Spitznagel ELJ, Xu P, Fu D, Dimitrov DS, Lempicki RA, Raaka BM, Cam MC: Evaluation of gene expression measurements from commercial microarray platforms. Nucl Acids Res 2003, 31(19):5676–5684. 10.1093/nar/gkg763
Shi L, Tong W, Fang H, Scherf U, Han J, Puri R, Frueh F, Goodsaid F, Guo L, Su Z, Han T, Fuscoe J, Xu ZA, Patterson T, Hong H, Xie Q, Perkins R, Chen J, Casciano D: Cross-platform comparability of microarray technology: Intra-platform consistency and appropriate data analysis procedures are essential. BMC Bioinformatics 2005, 6(Suppl 2):S12. 10.1186/1471-2105-6-S2-S12
Bammler T, Beyer RP, Bhattacharya S, Boorman GA: Standardizing global gene expression analysis between laboratories and across platforms. Nat Meth 2005, 2(5):351. 10.1038/nmeth754
Larkin JE, Frank BC, Gavras H, Sultana R, Quackenbush J: Independence and reproducibility across microarray platforms. Nat Meth 2005, 2(5):337. 10.1038/nmeth757
Irizarry RA, Warren D, Spencer F, Kim IF, Biswal S, Frank BC, Gabrielson E, Garcia JGN, Geoghegan J, Germino G, Griffin C, Hilmer SC, Hoffman E, Jedlicka AE, Kawasaki E, Martinez-Murillo F, Morsberger L, Lee H, Petersen D, Quackenbush J, Scott A, Wilson M, Yang Y, Ye SQ, Yu W: Multiple-laboratory comparison of microarray platforms. Nat Meth 2005, 2(5):345. 10.1038/nmeth756
Yauk CL, Berndt ML, Williams A, Douglas GR: Comprehensive comparison of six microarray technologies. Nucl Acids Res 2004, 32(15):e124. 10.1093/nar/gnh123
Durbin BP, Hardin JS, Hawkins DM, Rocke DM: A variance-stabilizing transformation for gene-expression microarray data. Bioinformatics 2002, 18(Suppl 1):S105-S110.
Huber W, von Heydebreck A, Sultmann H, Poustka A, Vingron M: Variance stabilization applied to microarray data calibration and to the quantification of differential expression. Bioinformatics 2002, 18(Suppl 1):S96–104.
Rocke DM, Durbin B: A Model for Measurement Error for Gene Expression Arrays. Journal of Computational Biology 2001, 8(6):557–569. 10.1089/106652701753307485
Chen Y, Kamat V, Dougherty ER, Bittner ML, Meltzer PS, Trent JM: Ratio statistics of gene expression levels and applications to microarray data analysis. Bioinformatics 2002, 18(9):1207–1215. 10.1093/bioinformatics/18.9.1207
Kleinbaum DG, Kupper LL, Muller KE, Nizam A: Applied regression analysis and other multivariateable methods. 3rd edition. Pacific Grove, CA, Duxbury; 1998.
Durbin B, Rocke DM: Estimation of transformation parameters for microarray data. Bioinformatics 2003, 19(11):1360–1367. 10.1093/bioinformatics/btg178
Huang XQ, Hardison RC, Miller W: A space-efficient algorithm for local similarities. Comput Appl Biosci 1990, 6(4):373–381.
Pearson WR: Searching protein sequence libraries: comparison of the sensitivity and selectivity of the Smith-Waterman and FASTA algorithms. Genomics 1991, 11: 635–650. 10.1016/0888-7543(91)90071-L
Smith TF, Waterman MS: Identification of common molecular subsequences. Journal of Molecular Biology 1981, 147: 195–197. 10.1016/0022-2836(81)90087-5
Zuker M: Mfold web server for nucleic acid folding and hybridization prediction. Nucl Acids Res 2003, 31(13):3406–3415. 10.1093/nar/gkg595
SantaLucia JJ, Allawi HT, Seneviratne PA: Improved Nearest-Neighbor Parameters for Predicting DNA Duplex Stability. Biochemistry 1996, 35(11):3555 -33562. 10.1021/bi951907q
SantaLucia JJ: A unified view of polymer, dumbbell, and oligonucleotide DNA nearest-neighbor thermodynamics. PNAS 1998, 95(4):1460–1465. 10.1073/pnas.95.4.1460
Vinga S, Almeida J: Alignment-free sequence comparison--a review. Bioinformatics 2003, 19(4):513–523. 10.1093/bioinformatics/btg005
Flikka K, Yadetie F, Laegreid A, Jonassen I: XHM: A system for detection of potential cross hybridizations in DNA microarrays. BMC Bioinformatics 2004, 5: 117. 10.1186/1471-2105-5-117
Almeida JS: Predictive non-linear modeling of complex data by artificial neural networks. Curr Opin Biotechnol 2002, 13: 72–76. 10.1016/S0958-1669(02)00288-4
Martinez WL, Martinez AR: Computational statistics handbook with MATLAB. Boca Raton, Florida, Chapman&Hall/CRC; 2002.
Castro JL, Mantas CJ, Benitez JM: Neural networks with a continuous squashing function in the output are universal approximators. Neural Networks 2000, 13(6):561–563. 10.1016/S0893-6080(00)00031-9
We would like to thank Dr. David Rocke's suggestion and R-code for data transformation, and Dr. Trevor Hastie's suggestion on the cross-validation and variable selection procedure. We appreciate the comments and the biological insights from the researchers in the Marine Genomics Group in Charleston, SC. We would also like to thank the insightful suggestions from the anonymous reviewers. YAC was supported by the South Carolina Sea Grant (NA16RG2250, P. S. Gross, PI) and NHLBI proteomics grant (N01-HV-281-81-000, D. Knapp, PI).
JSA supervised the entire project and implemented ANNs. YAC performed the computational model development and implementation under the supervision of XL, EHS, EOV and JSA. YAC drafted the manuscript. CCC and KP provided the data and assistance with biological interpretations of the parametric sensitivities observed. WX proposed the third-order polynomial regression and provided biological insights. All authors contributed to the writing and revision of the manuscript and approved the final manuscript.
Electronic supplementary material
About this article
Cite this article
Chen, Y.A., Chou, CC., Lu, X. et al. A multivariate prediction model for microarray cross-hybridization. BMC Bioinformatics 7, 101 (2006). https://doi.org/10.1186/1471-2105-7-101
- Artificial Neural Network
- Topology Optimization
- Regression Tree
- cDNA Microarrays
- Subset Size