We were motivated by three novel technologies, which exemplify a new design paradigm in high throughput genomics: nanostring TM, DNA‐mediated Annealing, Selection, extension, and Ligation DASL TM, and multiplex real‐time quantitative polymerase chain reaction (QPCR). All three are solution hybridization based, and all three employ on 10–1000 DNA sequence probes in a small volume, each probe specific for a particular sequence in a different human gene. nanostring TM uses 50‐mer, DASL and multiplex QPCR use ∼20‐mer probes. Assuming a 1‐nM probe concentration in a 1 μL volume, there are 10− 9 × 10− 9 × 6.23 × 1023 or 6.23 × 105 molecules of each probe present in the reaction compared to 10–1000 target molecules. Excess probe drives the sensitivity of the reaction. We are interested in the limits of multiplexing, i.e., the probability that in such a design a particular probe would bind to any other, sequence‐related probe rather than the intended, specific target. If this were to happen with appreciable frequency, this would result in much reduced sensitivity and potential failure of this design. We established upper and lower bounds for the probability that in a multiplex assay at least one probe would bind to another sequence‐related probe rather than its cognate target. These bounds are reassuring, because for reasonable degrees of multiplexing (103 probes) the probability for such an event is practically negligible. As the degree of multiplexing increases to ∼106 probes, our theoretical boundaries gain practical importance and establish a principal upper limit for the use of highly multiplexed solution‐based assays vis‐‐a‐vis solid‐support anchored designs. WIREs Comput Stat 2015, 7:394–399. doi: 10.1002/wics.1364 This article is categorized under: Applications of Computational Statistics > Genomics/Proteomics/Genetics Data: Types and Structure > Microarrays

Conceptual illustrations of binding possibility for (a) solid‐state anchored micro array, (b) multiplex solution array. Solid arrows indicate probes, the dotted arrow the correct target. The blue arrow refers to a probe or oligonucleotide of length d, which is desired, perfectly complementary to the target mRNA (dashed arrow). The red arrow refers to an oligonucleotide of length d, which is similar to the target mRNA and thus can bind the blue probe except for m mismatches. The blue‐red interactions are possible due to sequence complementarity.

Bounds on the probability of no m‐mismatched d = 40‐mers in subsets with different sizes: Extension of Figure to a range of different values of s (the size of the subset). This continues to show large probability for small m (more so for small s), for both lower (Panels A, C, and E) and upper (Panels B, D, and F) bounds. The bound remain close, indicating good approximation quality, over a range of different CG ratios, p = 0.5 (A and B), 0.28 (C and D), and 0.2 (E and F).

Upper (solid) and lower (dashed) bounds on the probability that there exist no m‐mismatched d‐mers in a subset under the different CG ratio: Show an absolute probability scale in the left panels. To depict closeness to 1 at higher resolution, the right panels are plotted on the log10 (1 − probability) scale. Columns compare affect of the CG ratio p. Upper and lower bounds are equally very closed. Dashed red and pink lines give specific interesting comparison.