Washing experiments
Aliquots of the RNA sample solution were hybridized to three Affymetrix GeneChip HGU133plus2 arrays (A to C) and equilibrated for 16 hours in the hybridization oven. For one of the three arrays (C) washing and staining was performed according to the manufacturer's instructions. Briefly, the standard protocol includes a low stringent wash (900 mM Na^{+}) at 30°C followed by 6 stringent wash cycles at 50°C with decreased salt concentration (100 mM Na^{+}). After these washes the array is stained with streptavidin phycoerythrin (SAPE) in two rounds which are intermitted by a round of antiSAPE antibody staining (staining step) and nonstringent washes.
For another array (A) the first scan was done immediately after low stringent wash and staining without subsequent stringent washing. Then the array was stringently washed and scanned in alternating order three more times where each washing step consists of a definite number of washing cycles (see Figure 1). The third array (B) was low stringently washed followed by two stringent washing cycles and staining before the first scan. Subsequently it was analogously processed as array A. Each measurement (scan) is characterized by the array (A, B or C), the scan number and the total number of washing cycles performed after hybridization. The first and second scans of array A (A1 and A2) resemble the design of the previous washing experiment [4].
All three chips are repeatedly processed in a second series of alternating wash/scancycles which was performed using the same protocol for each chip as in the first series as described above. As in the first series the arrays were also stained a second time to compensate for any loss of bleached fluorescent dye. The set of raw intensity data (celfiles) is available from the Gene Expression Omnibus (GEO) repository under accession number GSE18161.
Selected results of a preexperiment performed to test the fluidic script of the washingscanning cycles are given in the additional file 1 (supplementary text) to support the results of the main experiment described in the remainder of the paper. The results of this preexperiment also served as basis for the design of the main experiment as discussed in the supplement.
Hybridization and washing of microarrays: Theory and basic equations
Microarray experiments include several steps: (i) RNAextraction, purification and preparation which includes amplification, in vitro transcription and biotinlabelling; (ii) hybridization, i.e. the addition of the RNAsample onto the microarray and equilibration during which the added RNAfragments are intended to bind to the oligonucleotide probes attached to the chip surface; (iii) staining, i.e. the addition of fluorescent marker (streptavidinphycoerythrin; SAPE) which bind to the biotinlabels covalently attached to a certain fraction of cytosines of the hybridized RNAfragments. Primary SAPE association is further amplified in a second round of SAPEtoSAPE binding via antiSAPE antibody staining; (iv) washing, i.e. rinsing of the chip with buffer. Essentially two washing regimes are applied, namely a mild lowstringent one intended to remove predominantly unhybridized markers and RNAtargets; and a more severe high stringent regime to wash off weakly bound nonspecific RNAfragments; (v) scanning and subsequent image analysis to quantify the probe intensities.
The hybridization step (ii) can be described by two coupled reversible secondorder reactions of specific (S) and nonspecific (N) target binding to the microarray probes,
The superscript "f" indicates free species and PS and PN are the probes duplexed with specific and nonspecific transcripts, respectively. K^{P,S} and K^{P,N} are the equilibrium constants of specific and nonspecific transcript binding. The superscript 'P' accounts for the fact that the constants depend on the particular sequence of each probe and thus they vary in a probespecific fashion. In particular we will use P = PM,MM below to differentiate between the properties of perfect match (PM) and mismatched (MM) probes used on GeneChip microarrays.
The reactions (1) provide the hyperbolic adsorption law for the probe/target duplex formation under equilibrium conditions,
where Θ^{P,h}(0) is the fraction of probes occupied by species 'h' immediately after the hybridization step. The square brackets denote the concentrations of the respective species. The socalled binding strengths are defined as
[h] = [N], [S] are the total concentrations of the respective transcripts.
Upon washing the microarray is rinsed with RNAfree buffer solution which causes the partial unbinding of specific and nonspecific transcripts according to [6, 8, 9]
where the k^{P,h} denote the dissociation rate constants upon washing. Eq. (4) assumes that the supernatant solution acts as a concentration sink which removes the unbound transcripts from the system. This assumption seems reliable because in practice washing is performed in discrete cycles in each of which the exhausted buffer is replaced by new one (see also the detailed discussion of the process given in [6]). This first order reaction kinetics gives rise to the exponentially decaying washing function [3, 12],
which provides the fraction of probe/target duplexes surviving after washing time t. If each washing cycle is performed using the same protocol (amount of buffer and dwelltime of the buffer in the cell) then the argument of the washing function can be substituted by the number of washing cycles. In this case w^{P,h}(t) with t = 1, 2,... defines the reduction of the probe occupancy after t washing cycles.
We assume in agreement with previous studies [8, 9] that the dissociation rate is related to the stability of the respective duplex in terms its free energy of probe/target hybridization, which in turn depends on the equilibrium binding constant introduced in Eq. (1), ΔG^{P,h} = RT·lnK^{P,h} (R is the gas constant), i.e.
where const = K_{0}^{γ} and γ are sequenceindependent scaling constants which apply to all probes on the microarray. K_{0}, for example, depends on the rate constant of probe/target formation and the washing conditions (salt concentrations, temperature, see [6] for a detailed argumentation and references given therein).
The probe intensities measured in the scanning step (v) are directly related to the total probe occupancy due to specific and nonspecific binding surviving after t washing cycles, [9, 13–16] (see Eqs. (2) and (5)),
where M(t) is the maximum intensity upon complete saturation of the probes and O(t) its minimum value due to the optical background. The maximum intensity rescales the dimensionless occupancy into intensity units. It depends on the amount and quality of optical labelling of the targets, on the sensitivity of the scanner and on the imaging software which transforms the intensity spot of each probe into one intensity value. Saturation of optical detection is typically characterized by the appearance of a constant upper limit of intensity values which was not observed in most of our data. We assume therefore linear and timeindependent calibration of the scanner. Exceptions observed in the 2^{nd} series will be discussed below. More critically, repeated scanning will potentially bleach the fluorescent labels [17, 18]. Such bleaching gives rise to a timedependence of M(t)≈ M·b(t), where b(t) is the bleaching factor decaying with time, i.e. b(0) = 1 and 1 ≥ b(t > 0) ≥ 0. The optical background depends, besides other factors, on the amount of residual fluorescent markers and thus is also basically a function of washing and scanning cycles due to washing and bleaching as well. Throughout the paper we will consider net intensities which have been corrected for the optical background before further analysis, I^{p} (t) = I^{p} * (t) O(t), using, the zonealgorithm provided by Affymetrix for estimating O(t) for each chip measurement [19].
Washing efficiency is related to intensity
Figure 2 compares the intensities of the PM and MMprobe intensities of four selected probe sets before (t = 0) and after t = 17 stringent washing cycles taken from scans A1 and A4 (see Figure 1). The mean intensity level clearly decreases after washing. The mean decrement of the MM slightly exceeds that of the PM probes (see the horizontal dashed lines in Figure 2).
Each probe set is intended to interrogate one transcript. The target concentration is therefore assumed to be a constant for all probes of each set in a first order approximation which neglects effects such as the 3'/5'amplification bias of RNAfragments. The observed variability of the intensities of the individual probes about their setaverage results mainly from the sequence dependence of the binding constant for probe/target association. In Figure 2 two probes of relatively high (see labels a and b) and two probes of relatively low (labels c and d) intensity are indicated. Washing leaves the intensities of the former ones nearly unchanged whereas the intensities of the latter probes strongly decrease. Both PM and MM behave similarly. The scattering width of the probes about their set average clearly increases after washing.
To generalize these trends we calculate the intensity distributions of all PM and MMprobes of the respective chip measured in the first and the last scan after t = 0 and t = 17 washing cycles (see Figure 3 and also Figure 1 for experimental protocol). Basically, washing broadens the intensity distribution and shifts their center to the left. The highintensity limit of the right flank of the distribution remains essentially unaffected whereas the low intensity flank considerably shifts towards smaller intensities (panel a of Figure 3). Washing obviously affects the probes in an intensitydependent manner. The decrease of the effect of washing with increasing intensity can be explained by the fact that higher intensities are associated with stronger probe/target interactions which in turn are relatively stable against washing.
The probe intensity decomposes additively into contributions due to nonspecific and specific hybridization (see Eq. (7)). To study the effect of washing on both hybridization modes we separately calculate the intensity distributions for probes which are predominantly hybridized with nonspecific (panel b of Figure 3) and with specific transcripts (panel c). The respective ensembles of probes are obtained using the classification criteria provided by the hook method as described below.
Washing has an almost identical effect on the nonspecifically hybridized PM and MM probes, their respective distributions being almost identical both before and after washing. This observation can be explained by the fact that the discrimination between the probetypes PM versus MM is relevant only with respect to the specific transcripts not to nonspecific transcripts.
On the other hand, PM and MM probes respond differently under specific hybridization: the distributions of the MM probes before and after washing are shifted towards smaller intensities compared with the respective PMdistributions. The smaller onaverage intensity of the MM signal reflects the weaker binding of specific transcripts to these probes, owing to the mismatched pairing at the middle position of their probe sequence. Note that the highintensity flank of the specifically hybridized probes remains essentially fixed after washing whereas the left flank shifts downwards considerably. The steep decay of the right, washingindependent flank of the PMdensity distribution can be attributed to the maximum intensity value referring to saturated probe spots with strongly bound specific transcripts (see the vertical dotted line in Figure 3).
Note that bleaching is independent of the hybridization mode and thus it is expected to affect all probe intensities equally. Its effect on the time dependence of the intensity will therefore not exceed the weakest time course observed in repeated scanning. The virtual invariance of the specifically hybridized probes of largest intensity (see Figure 3c and probes a and b in Figure 2) gives consequently strong indication that bleaching adds, if at all, only a tiny contribution to the time dependence of the intensities provided that the scanner works below the saturation level of its calibration curve [20]. Saturation of the scanner is observed for a small fraction of probes in the 2^{nd} series but not in the 1^{st} one (see below). We will therefore neglect bleaching in the remainder of the paper to a good approximation.
Part d of Figure 3 shows the distributions of the logaveraged intensities for each probe set probing the same transcript. Trivially, averaging considerably reduces the variability of the intensity values giving rise to the narrowing of the distributions. Moreover, averaging over the probe set partly removes the sequence dependence of the intensities and this way stresses the effect of transcript abundance on the intensities. The respective distributions consequently reflect the effect of the varying amount of specific transcripts on the washing efficiency: Their right flank is governed essentially by specific hybridization (compare with panel c) whereas their peak and the left flank are obviously dominated by nonspecific binding (compare with panel b). Note also that setaveraging reduces the apparent value of the saturation intensity because nonsaturated spots also contribute to the average value.
Probelevel kinetics of washing
Figure 4 (panel a) illustrates the effect of washing on the intensities of single P = PM and MM probes taken from three representative probe sets of high, medium and small average intensities. The obtained probe intensities decrease with increasing number of washing cycles, t, as expected. Moreover, the effect of washing decreases with intensity in agreement with the general trends discussed in the previous section. The decay law is obviously not singleexponential (Eq. (5)). Instead, it seems to follow a multiphase decay. This behavior can be rationalized in terms of heterogeneous desorption of different transcripts with different binding free energies. Let us approximate this behaviour using an empirical simple twocomponent decay function which considers a fast shorttime and an asymptotic longtime component,
Accordingly, the decay is characterized by two parameters: the exponential decay time τ^{P} (in units of the number of cycles after which the probe intensity is expected to decay to 1/e of its initial value) and the asymptotic intensity level, w^{P}∞. The latter term neglects the kinetics of the slow component and subsumes its effect in terms of the fraction of probes which survived after extensive washing in the timewindow of the experiment. Below we will apply a complementary approach to analyze the kinetics of the slow component more in detail.
Part b of Figure 4 shows examples of fits of Eq. (8) to the averaged PM and MM decays taken from part a of the figure (note the logarithmic scale). One finds that the limiting intensity values, log(w∞^{P}), correlate with the initial intensities, logI(0), i.e. large intensity levels give rise to relatively large limiting values and vice versa. This correlation also becomes evident in the right column of Figure 4a which shows the logdifference of the PM and MMintensities of the considered probe pairs: The smaller MMintensities of specifically hybridized probes are associated with faster decay rates compared with the respective PMdata, causing an increasing course of the logdifference in most cases.
To generalize these results we estimate the long and shorttime parameters for all PM and MM probes of the array A by means of
The obtained limiting values of the individual probes (see grey dots in panel a of Figure 5) are smoothed using a moving average over 10^{3} probes to filter out the average relation between w∞^{P} and the initial intensity value, I^{P}(0) (thick lines). It turns out that extensive washing reduces the intensities to about 10% of their initial values in a wide range of relatively small intensities, logI^{P}(0) < 3.5. For intensities larger than a certain threshold, logI^{P}(0) > 3.5, the limiting washing level increases with intensity up to w∞^{P} > 0.9 (i.e. log w∞^{P} < 0.05 in Figure 5). In other words, up to 90% of the initial intensity value of probes of high intensity survives, whereas weak intensity probes are dimmed to less than 10% after extensive washing. Importantly, there is virtually no difference in the washing efficiency between the PM and MM probes indicating that both probetypes behave identically at the same intensity level. This result is consistent with the model of ref. [8].
The same result was obtained if one separately studies probes which predominantly hybridize with nonspecific or specific transcripts (see thin lines in Figure 5) despite the fact that the respective probes accumulate in the low and high intensity range, respectively. These results show that the limiting washing level is governed by the probe intensity, independent of the probe type (PM or MM) and of the hybridization mode (specific or nonspecific).
The decay constant shows a similar mean trend with increasing intensity as the limiting washing level despite the wider scattering of the individual probe data (panel b, Figure 5): Larger intensities are obviously associated with larger decay times τ^{P} indicating the slowing down of washing efficiency. Also the behaviour of this parameter is mainly driven by the intensity independently of probe type and hybridization mode (data not shown).
In the Methodssection we present a simple theoretical approach to express the two washing parameters studied as a function of the probe intensity. The theoretical curves obtained reproduce well the experimental data, and particularly the gradual increase of w∞ and τ at intensities above a certain threshold (see dashed curves in Figure 5). The theory assumes that the intensity is directly related to the probe/targetbinding constant which in turn determines the washing rate in terms of a power law in agreement with Eq. (6). It gives rise to a relatively sharp intensity threshold above which both washing constants start to increase, in agreement with the experimental data.
Selective washing of PM and MM probes and apparent concentration dependence
In panel a of Figure 6 we plot the mean intensities (logscale) and the washing parameters as a function of the logmean of the PM and MM intensities averaged over each probe set (Σ, see Eq. (19)). The abscissa is governed by the concentration of specific transcripts interrogated by the respective probe set, i.e. Σ ~ [S] [10] to a good approximation. The virtually identical mean intensities of the PM and MMprobes at small Σvalues are characteristic indicators for the predominance of nonspecific hybridization of these probes because of the absence of specific transcripts. At a certain threshold of Σ (see the vertical dotted line in Figure 6) the PM and MM curves split into two branches due to the onset of specific binding. The observed intensity difference between both probe types can be simply explained by the larger binding constant of specific binding of the PM probes compared with that of the mismatched ones, i.e. K^{PM,S} > K^{MM,S}.
The analogous plot of the asymptotic washing rate and of the decay constant (parts b and c of Figure 6) shows a similar split of the respective PM and MMcharacteristics. This result again illustrates the direct relation between the probe intensities and the washing parameters discussed in the previous section. The washing step removes about 90% of the initially bound nonspecific transcripts (w^{N}∞ < 0.1) from both the PM and MM probes. In contrast, only 10% of the specific transcripts associated with the PMprobes (w^{PM,S}∞ > 0.9) but about 50% bound to the MM probes (w^{MM,S}∞~0.50.6) are washed off. The higher limiting washing rate of the MM probes is obviously the consequence of their smaller specific binding constant, K^{PM,S} > K^{MM,S} (see above).
The washing step consequently improves the performance of the chip experiment because it removes nonspecific transcripts much more strongly than specific ones [3, 21]. Washing thus selectively reduces the relative contribution of the nonspecific signal. Also the relative signal of the MMprobes is reduced upon washing. The vertical difference between the PM and MMbranches at larger abscissavalues in panel a of Figure 6 indicates that washing decreases the saturation level of the MMprobes to a larger extent than that of the PM. The different saturation intensities of PM and MM probes have been observed previously and attributed to different washing rates of both probe types [9].
Note also that the limiting survival fraction w∞ (and also the characteristic washing time) monotonically increases with Σ, which suggests less effective washing at larger transcript concentrations. The observed intensity represents however the superposition of contributions due to the more stable specific transcripts and less stable nonspecific transcripts (Eq. (7)). The observed trend therefore reflects the increasing contribution of specific hybridization and not the change of the individual washing rate as a function of transcript concentration. At large abscissa values the discussed washing parameters level off to their asymptotic values referring to the average values for specific transcripts (see Figure 6).
Hybridization regimes
The different hybridization regimes can be more clearly identified and characterized by direct comparison of the respective PM and MM values. The GeneChip technology uses these probe pairs of perfect matched and mismatched probes where the latter are intended to serve as an intrinsic reference for the former ones. Specific differences between the characteristics of both probe types can be extracted using a special version of the MA(differenceversussum) plot which relates the logarithmic difference of the PM and MM intensities, Δ, to their logmean, Σ (Figure 7, see also Methods, Eq. (19) and [10, 11]).
This socalled hook plot reveals details of the probehybridization more clearly than the logintensity plots shown in Figure 6 (part a): The curve obtained can be divided (from the left to the right) into the nonspecific (N), mixed (mix), specific (S), saturation (sat) and asymptotic (as) ranges (see also [10] for a detailed description). In the Nrange the probes hybridize predominantly nonspecifically. Specific hybridization comes progressively into play in the mixrange which causes the marked increase of the Δvalues and the steep increasing slope of the hook curve. The hookcurve reaches its maximum in the subsequent Srange with dominating specific hybridization of the probes. The probes progressively saturate in the subsequent decaying satrange of the curve. Finally, it reaches the asymptotic regime referring to the maximum possible PM and MMintensity values.
The experimental hook curve of the unwashed chip (t = 0) is accurately described using the Langmuir binding model (see [10] for details, Methodssection and the respective dashed curve in Figure 7). We selected subsets of probes from the Nrange (i.e. to the left from the break point) and from the S and satranges to calculate the intensity distributions shown in panel b and c of Figure 3 above.
The effect of washing on the position and dimensions of the hookcurve will be discussed in the next section. Here we calculate analogous PM/MMdifference characteristics for the washing parameters, Δ_{w} = (log w∞^{PM}  log w∞^{MM}) (Eq. (37)) and Δ_{τ} = (τ^{PM}  τ^{MM}), and compare them with the logintensity difference, Δ(t = 0) = log I^{PM}(0)  log I^{MM}(0) (Eq. (19)) in Figure 7. The hookcurves obtained for the washing parameters partly resemble the course of the 'intensity' hook indicating the close relation between washing parameters and the intensity in agreement with the results discussed above. Note however two distinct differences: Firstly, the maximum of the 'washing' hooks is shifted to the left from the S towards the mixrange which results in an asymmetric shape of the curve. Secondly, the endpoint of the curves decay to less than 50% of the maximum value in contrast to the logintensity hook which decays only to about 90% (see the curves in Figure 7).
The asymmetrical shape of the washinghook can be rationalized by the fact that the PMprobes hybridize to a larger degree with specific transcripts in the mixrange than the MMprobes, the specific binding affinity of which is reduced by the mismatched pairing in the middle of their sequence. In particular, the asymptotic washing level (w∞) of the PM and MM in the mixrange is governed by the competition between specifically and nonspecifically bound transcripts. The latter Ntranscripts dominate the washing of the MM probes whereas the former Stranscripts dominate the washing of the PM probes. This difference results in markedly smaller values of the asymptotic washing level of the MM (and of their decay constant) and thus in relatively large Δ_{w} and Δ_{τ} values compared with Δ(t = 0). In the Srange also the hybridization of the MM becomes dominated by specific binding with larger asymptotic intensity levels. As a consequence the 'washing' hooks start to decrease in the Srange at smaller Σvalues than the respective intensityhook which decays due to the onset of saturation only in the satrange. This 'delayed' decrease of the logintensity difference also explains its larger final level observed in the experiments.
Upon complete saturation of the probes one expects a vanishing PM/MMlog intensity difference Δ(t = 0)→0. The experimental data however indicate that the probes are not yet fully saturated. In the Methodssection we propose a simple fit equation for the w∞hook which is based on the washing kinetics established above and on the Langmuir binding isotherm. It accurately approximates the experimental data and, moreover, allows extrapolation of the asymptotic Δ_{w}value referring to complete saturation (see the dashed curve in Figure 7). This asymptotic Δ_{w}value is inversely related to the specific binding constant of the MMprobes (Eq. (39)).
'Hook' characterization of washing
The hook curve provides simple overview characteristics of each hybridized chip in terms of its position (start coordinates, Σ_{start}(t), Δ_{start}(t); see also Eqs. (22) below) and dimensions (height and width, α(t) and β(t), respectively; see Eq. (23)). Panel a of Figure 8 shows the hook curves of the studied chips (A and B) scanned after each washing step according to the experimental protocol as illustrated in Figure 1. It turns out that washing essentially increases the vertical and horizontal dimensions of the curve. Particularly, (i) the left, increasing branch of the curve shifts markedly to the left towards smaller Σvalues whereas the right, decreasing branch and the Σcoordinate of the endpoint remains nearly invariant; (ii) the Δcoordinates of the maximum and of the endpoint distinctly increase whereas the Δcoordinate of the startpoint remains virtually unchanged.
Figure 9 shows ordinate (Δ, panel a) and abscissa (Σ, panel b) coordinates of the first and last measurements at t = 0 and t = 17, and their difference to illustrate the washing effect observed in the experiment. As discussed in the previous sections, the increment of the PM/MMlog difference, δΔ = Δ(17) Δ(0)≈ δα, virtually disappears in the Nrange because PM and MM probes are equally affected by washing on the average (δα^{N} ≈ 0, see also Eq. (28) below). The maximum in the mixrange simply reflects the larger amount of specific hybridization of the PM whereas the "final" level at large Σvalues is caused by the less effective washing of the PM probes due to the more strongly bound specific transcripts. This difference gives rise to different saturation levels of the PM and MMprobes which is characterized by the mean logintensity ratio, δα^{S} ≈ 0.2.
In contrast, the increment of the logmean intensity, δΣ = Σ(17) Σ(0) ≈ δβ, reflects the mean washing rate of the PM and MMprobes in the respective hybridization range (see above). The washing rate is maximum in the Nrange due to the relatively weak binding of the nonspecific transcripts (δβ^{N} ≈ 0.95, see also Eq. (28) in the Methodssection). It progressively decreases by about one order of magnitude with increasing amount of specific binding (δβ^{S} ≈ 0.13).
In summary the hybridization regimes can be associated with specific washing rates which reflect the binding characteristics of the PM and MM with the respective targets.
Washing kinetics of hook parameters
In the previous sections we analyzed the effect of washing on the intensity of the PM and MM probes in the different hybridization ranges after the first and the last washing cycle to discover the basic changes after the washes. In this section we address in more detail the washing time dependence using all measured time points.
The hook analysis provides a straightforward method to summarize the hybridization characteristics of each chip in terms of a small number of selected parameters which are obtained by fitting a theoretical hook curve to the experimental one. The fitted equation is derived by combining the twospecies Langmuir adsorption isotherm with posthybridization washing kinetics of the PM and MM probes (see Methodssection, Eq. (20)). The proposed function fits well to the experimental data obtained from the washing experiments (see panel b of Figure 8). For automatic fits in standard analysis we used the available hook program which, to a good approximation, applies also to the washing experiment (see below, Eqs. (29)  (32)). In particular, we estimated and plotted the following hook parameters as a function of the number of washing cycles t (see Figure 10):

the mean intensity levels of saturation Σ(∞,t) and of nonspecific hybridization Σ(0,t);

the width β(t) and the height α(t) of the hook curve, defined by Eq. (23). These parameters serve as measures of the binding strength of nonspecific hybridization and of the apparent PM/MMgain, respectively;

the mean S/Nratio (specificto/nonspecific hybridization strength) of the signal R(t) defined by Eq. (21). It represents a sort of mean 'signaltonoise' ratio of the measurement;

the mean level of specific hybridization φ(t) defined by Eq. (24). It estimates the mean expression level of all detected genes.

In addition we also take into account the washingtime dependence of the optical background O(t) which has been subtracted from the raw probe intensities before the hook analysis as described in the methods section. The optical background is partly attributed to a residual amount of free labels which are expected to be progressively removed upon washing.
Figure 10 shows the obtained hook parameters as a function of the number of washing cycles for the studied chips A (four scans, first scan before washing), B (four scans, first scan after washing) and C (one scan only after 6 standard washing cycles). The observed kinetics can be approximated in the loglog plots for t >1 (i.e. considering all time points except the first one) by linear functions of the form
(see lines in Figure 10). Note that both types of measures Ψ and F scale logarithmically with the intensity and/or binding strength: Ψ~ logF~ logI~ logX (see, e.g. Eqs. (19) and (22)). The slopeparameter η scales the washing time exponentially, i.e., (Ψ(t)Ψ(1)) = log t^{η}. The obtained slopes agree well between the independent washing experiments using chips A  C.
Note that the parameters which are related to the probe intensities (the start and end points, Σ(0,t) and Σ(∞,t), respectively; and the optical background, O(t)) decrease with washing time (η < 0) reflecting the progressive washingoff of bound targets. Contrarily, the dimensions of the hook (height and width, α(t) and β(t), respectively) and the mean S/Nratio (<R(t)>) increase with washing (η > 0). One sees from the interpretation of the hook dimensions (see Eq. (23)) that the positive signs of the slopes reflect a stronger removal of (i) nonspecific transcripts compared with specific ones (β(t)) and (ii) MMbound specific transcripts compared with the removal of PMbound specific transcripts (α(t)). The selective washing of nonspecific transcripts gives rise to the progressive increase of the mean S/Nratio <R(t)>. Hence, the washes effectively improve the specificity of the expression measurement. The average signaltonoise ratio increases roughly by the factor of 2  2.5 per 10 stringent washing cycles (η_{R}~ 0.33  0.43). On the other hand, the mean expression level (φ(t), see Eq. (24)) remains virtually constant after washing (η_{φ}≈ 0). We will discuss this result below.
The considered hook parameters depend on the washing functions w^{P,h}(t) which characterize the removal of specific and nonspecific transcripts from the PM and MM probes (see Eqs. (21) (26)). The direct link between the kinetics of the hook parameters (Eq. (10)) and the washing functions can be established by assuming analogous powerlaw kinetics of the latter ones, i.e., log(w^{P,h}(t)) ~  η^{P,h}·log t (Eq. (33), see Methodssection for details). Accordingly, the three relevant exponents of the washing function can be expressed as linear combinations of the slopes of the hook parameters (Eq. (36)). With the values of the slopes given in Figure 10 one gets for the kinetic exponents of the washing functions for nonspecific transcripts η^{PM, N} = η^{MM, N} = 0.5 ± 0.1, and for the washing exponents of specific transcripts of the PM and MM probes, η^{PM,S} < 0.05 and η^{MM, S} = 0.15  0.2, respectively. These values reflect the fact that washing basically removes weakly bound nonspecific transcripts whereas specific transcripts, especially if bound to the PM probes, remain relatively stable against washing.
The analysis using a power law (Eq. (10)) complements our simple initial analysis in terms of a twocomponent decay function (Eq. (8)): In particular, the power law exponent obtained enables the effect of washing to be extrapolated to times exceeding the time range of the experiment. Note that the exponent estimates the decrease of the respective washing function upon increasing the number of washing cycles by one order of magnitude, i.e.  η = log(w^{P,h}(10^{1})/w^{P,h}(10^{0})). With this interpretation we can estimate the number of washing cycles after which the number of bound targets is expected to reduce on the average by one order of magnitude, t* = (0.1)^{1/η}. Particularly, one gets t*~ 10^{2} washing cycles for nonspecific targets, but about t*~ 10^{20} for specific targets bound to PM probes and still more than 10^{5} cycles for specific targets bound to MM probes. Hence, most of the specific transcripts are virtually unwashable from the PM probes, whereas the nonspecific transcripts, on average, can generally be removed by further washing.
The above powerlaw kinetics can be interpreted as the superposition of a large number exponential decay functions with a broad distribution of decay constants which are typically observed in heterogeneous systems with a large variety of different energetic states [22]. This description is equivalent to a timedependent washing rate leading to the progressive slowing down of washing with increasing number of washing cycles given below in Eq. (34). The application of this interpretation to the chosen summary characteristics of the hook analysis seems reasonable because the respective parameters are mean values averaged over a large number of probes with a broad distribution of individual binding constants.
Relabelling (second staining/washing series)
The experimental pipeline of consecutive washing cycles was repeated a second time for all three arrays A  C after completing the first washing series (see Figure 1). This second series starts with a second staining protocol before washing which consists of three alternating cycles of SAPE/antiSAPE labelling. The additional labelling of the bound targets markedly increases the intensity level of the scanned arrays of the second series as indicated by rightshift of the intensity distributions and of the hook curves compared with the respective characteristics of the first series (see Figure 11 and also Figure 8, panel a).
Generally, the kinetics of the hook parameters obtained from this second series follow the same linear trends as the first series according to Eq. (10) (compare large and small symbols in Figure 12). The systematically larger values of the optical background, O(t), and of the start and end points of the hook curve, Σ(0,t) and Σ(∞,t), respectively, reflect the increased intensity level of the second series as discussed above. The very similar slopes, especially for the shift of the maximum intensity Σ(∞,t) of both series indicates that nonlinearities of the calibration curve of the scanner near its saturation level [20] can be neglected to a good approximation because otherwise one would expect to observe a decreased slope of the larger intensity values of the 2^{nd} series.
Surprisingly, the width of the hook curve, β(t), is virtually identical in both series and changes similarly with washing time. Naively it might be expected that the final washing level of nonspecific hybridization reached in the first series would provide the starting level of the second series which then will be further reduced upon continuing the washes. However in contrast to this expectation that washing in the second series simply continues the first one, we observe similar widths of the hook curves in both series (see Figure 12 and also part b of Figure 11). In other words, the original level of the nonspecific background before the first washing series is actually reconstituted after relabelling of the targets in the second staining step.
A hint for a first, tentative explanation of this result is given by the small peak at the right end of the intensity distributions of the PM and MMintensities (see part a of Figure 11). We attribute this peak to optical saturation of the scanner giving rise to a certain fraction of probes with an essentially identical maximal intensity value, O^{max}. Optical saturation leads to underestimation of the level of chemical saturation for O^{max} < M, which in turn underestimates the width of the hook and finally will overestimate nonspecific binding. Optical saturation however affects only large intensity values I > O^{max}. Detailed comparison of the intensity distributions of both series indicates essentially identical shapes (part a of Figure 11). The discussed shift between both series applies obviously to all intensity values over the whole intensity range and not only to large intensities in the saturation range of the scanner.
We therefore suggest a second, alternative explanation which is schematically illustrated in Figure 13. In particular, we assume that the fluorescent markers bound to the targets drastically increase the washing yield of labelled transcripts compared with nonlabelled ones. Note that only a certain minor fraction of bound targets becomes fluorescently labelled with SAPE, whereas the remaining major fraction remains unlabelled and does not actually contribute to the measured probe intensity [23]. SAPE represents a bulky, watersoluble proteindye complex which covalently binds to the biotins attached to a small fraction of cytosines in the targetsequences. The molecular weight of SAPE of about ~300 kDa [24] by far exceeds that of unlabelled RNAtarget fragments (e.g., ~30 kDa for target lengths of 100 nucleotides). Hence, properties of the SAPE/RNA complexes such as molecular weight, size and hydrophilicity are expected to be dominated by the SAPE component. It is well established that the nature and position of attached fluorescent labels affect the signal intensity of surfacebound probe/target duplexes [25]. Our proposed explanation is that the SAPEmarkers drastically reduce the resistance to dissociation by washing of labelled targets compared to that of nonlabelled ones. As a consequence, washing is expected to remove essentially only the labelled targets whereas nonlabelled dark ones remain bound to the probes. In the second series, a certain fraction of these dark targets becomes labelled and thus visible after repeated staining of the chips.
For a simple estimation of the enrichment after two rounds of staining/washing we set the initial amount of labelled bright targets in each type of duplexes to 100% and assume duplexspecific washing yields in rough agreement with our analysis (see also legend of Figure 13). We also assume that the same amount of dark transcripts is transferred into bright ones in both staining rounds. This approximation applies for a large excess of dark transcripts. About 90% of the specific transcripts bound to the PM probes consequently survive the first washing round. Then, relabeling increases the total amount of bright targets again to 190% compared with the result of the first labelling round. The assumed washing yield of 90% reduces the final amount of bright targets to 190%·0.9 = 171% after the second washing round. The bright specific targets bound to the PM probes enrich consequently by a factor of about 171%/90% = 1.9 compared with one staining/washing round. Analogously one can estimate the enrichment factors for the different duplexes and their ratios. They decrease according to PMS > PMS/PN > PMS/MMS > PN (see the schematic plot in Figure 13). The nonspecific background only weakly enriches by a factor of 1.1, which is hard to observe experimentally. Indeed, we found similar nonspecific background levels β(t) in both series. Comparison of the hook parameters φ(t), <R(t)> and α(t) of both series provides experimental estimates of the enrichment factors (see Figure 12, right part; note the logarithmic scale). Importantly, the values obtained rank in the same order as that predicted by our simple model.
Sequence effects
In early papers the washing efficiency of surfacebound probe/target duplexes was used to estimate the stability of selected sequence motifs in terms of thermodynamic parameters such as the free energy of probe/target association [12]. Two high (labels a and b) and two lowintensity (c and d) probes are labelled in Figure 2 together with their sequences and the respective total numbers of nucleotides A, C, G and T per sequence. One of the highintensity probes (label a) contains eight cytosines, five of them are assembled in runs of two and three adjacent Cs. These motifs are associated with high binding affinities and thus weak washing potential (see also below). The overall base compositions of the remaining probe sequences (b  d) however look essentially similar although their intensities differ by up to three orders of magnitude.
To extract the relation between the probe sequence and washing efficiency in more detail we make use of the positional dependent sensitivity model which, in its simplest version, estimates the mean contribution of each nucleotide letter at each sequence position to the probe intensity [10, 26, 27] (see below, Eq. (41)). These socalled sensitivity profiles were separately calculated either for probes which are predominantly hybridized with nonspecific or with specific transcripts before and after washing (see panel a and c of Figure 14). In addition we calculated the respective differenceprofiles 'after washingminusbefore washing' to extract the effect of washing on the profiles (panel b and d of Figure 14).
The nonspecific profiles of nucleotides A and C show the typical parabolalike shapes with their minimum/maximum values near the center position of the probe sequence (see, e.g. [26, 28]). Washing inflates these curves (compare dashed and solid curves in part a of Figure 14) indicating that the specific sequence effect of the different nucleotides on the intensity increases after washing. In particular, the difference profiles show that the intensitydifference between cytosinerich and adeninerich probes increases with washing. This trend can be easily rationalized by the stronger base pairings formed by the C's. They cause not only larger intensities of Crich probes before washing [15], but also their stronger resistance against washing. As a consequence the basespecific effect increases with washing. The observed trend is supported by theory (see Eq. (45) in the Methodssection).
The shape and the values of the specific profiles differ from the profiles of the nonspecific ones (compare dashed curves in panel c and a of Figure 14). This change can be explained by the stronger effect of saturation on the intensities [29] and/or by bulk hybridization of the specific transcripts [30]. Interestingly, the difference profiles closely resemble that of nonspecific hybridization (compare panel d, b and a of Figure 14). These results imply that the basespecific efficiency of washing is less distorted by saturation and bulk hybridization than the binding affinity of the transcripts. This interpretation seems reasonable because saturation and possible bulk effects will not affect the removal of bound transcripts according to the respective reactions (Eq. (4)). In the Methodssection we link the sensitivity profiles, σ_{k}, and their increment upon washing, Δσ_{k}, with the base and positional dependent affinity of base pairings, ε_{k} (Eqs. (45) and (46)). According to Eq. (46), washing is indeed expected to expose the sequence effects of direct probe/target interactions, i.e. Δσ_{k}∝ ε_{k}, which are partly distorted in the original sensitivity profiles due to specific hybridization (see Eq. (45)).
In addition to the singlebase sensitivity profiles we apply the positionaldependent nearest neighbour sensitivity model to the intensity data. Part b and d of Figure 14 show the incrementalprofiles of the homocouples AA, TT, GG and CC (dashed curves). Particularly, the CCprofiles significantly exceed that of single C: Hence, two neighbouring cytosines more strongly resist washing than, e.g., two cytosines which are separated by other intervening bases.
In summary, we found that washing (i) increases the sequencespecific sensitivity of the probes for target binding and (ii) decreases the effect of saturation and bulk hybridization on the observed profiles. The resulting sensitivity profiles depend directly on the strength of the probe/target interactions and on the washing function which, in turn, is an exponential function of the probe/target interactions.