Rational Design of Digital Assays - ACS Publications - American

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Analytical Chemistry

Rational design of digital assays. Pawel R. Debski†,‡, Kamil Gewartowski‡, Magdalena Sulima‡, Tomasz S. Kaminski†,‡ and Piotr Garstecki†,‡,* † ‡

Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warsaw, Poland Curiosity Diagnostics Sp. z o.o. Kasprzaka 44/52, 01-224 Warsaw, Poland

KEYWORDS: PCR · digital techniques · assay · DNA · diagnostics

ABSTRACT: Optimum algorithm for digital assays treats chemical compartments as bits of probabilistic information and arranges these bits in a fractional positional system. Maximization of information gain reduces―by orders of magnitude―the number of partitions required to achieve the requested dynamic range and precision of the assay. The method simplifies the execution of digital analytical methods providing for more accessible use of absolute quantization in research and in diagnostics.

The standard 'analogue' method to quantify concentration is to measure an amplitude of a physical quantity (e.g. absorbance of light) and to calculate an estimate of as a calibrated function . In a fundamentally new concept, digital assays split the sample into a large number of partitions and estimate from the fraction of end-point signals. The concept dates back to 1915 when McCrady[1] introduced the limiting-dilution assay and the most probable number method for quantization of bacteria. In spite of its common use over decades this method has never been optimized for the information gain as a function of the design of the assay. In 1992 Sykes et al. described the method of quantisation of initial targets present in the sample using limited dilution polymerase chain reaction (PCR)[2-5] assays. Sykes’ design comprised 6 sets of 10 compartments. The consecutive sets were diluted three fold so the numbers of added DNA copies created a geometric sequence. Then, in 1999 Vogelstein and Kinzler proposed digital PCR assay using identical partitions of the sample and binary stochastic signals adopting either a positive ( 1 or negative 0 value if the partition contains at least one target molecule ( 1) or none ( 0). Digital assays provide absolute, highly sensitive and precise[5-9] estimates of and eliminate the need for calibration. As the method requires strong amplification of the presence of the analyte, its key applications are in quantitative PCR[6-9] and ELISA[10,11]. Digital methods have also important limitations. Their precision and dynamic range cannot be tuned independently. As a result the assays typically require massively large numbers of partitions: thousands to millions, elevating their cost and narrowing the range of applications. The problem stems from the coding scheme: as all compartments are identical, the maximum number of molecules that can be encoded is proportional to the number of partitions (Fig. 1a). One solution is

by combining classical digital schemes with different dynamic ranges[13,14]. The solution relies on a simultaneously performed assays on multiple sets of identical compartments[15-19] containing dilutions of the sample to increase the dynamic range. These approaches, although highly effective, have not been optimized for the information gain (the breath of the dynamic range and the tightness of the precision of the estimate of ) against the design of the assay (the number, volume and dilution factors of the compartments). The method that we describe here minimizes the number of partitions and provides explicit formulas for the design of the assay for the requested dynamic range and precision of the estimate. This optimisation is achieved via appropriate tuning of the expected number of molecules of the analyte (i.e. DNA molecules) in the partitions. The algorithms here described explicitly teach how to adjust the modification factor (i.e. the product of volume and dilution of the sample in each of the compartments) [see ESI sec. 1.1-1.4 for details].

Figure 1. (a) In a standard digital assay the number of target molecules is quantified with the count of positive signals (b) In binary coding bits allow to code numbers as high as 2 . (c) Chemical bits carry probabilistic information (probability

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function that represent either positive or negative signal) on the hypotheses that is larger (smaller) then a characteristic value ∗.

The outcome of a digital assay (i.e. the collection of positive signals from a subset of partitions of the sample) codes the concentration of the analyte (i.e. the number of target molecules in the sample volume). The most natural way to code numbers is with equally valued bits; e.g. the number 157 can be represented by 157 dots. With some sophistication stemming from the Poisson statistics, this is the scheme of coding the number of molecules (i.e. concentration ) with the number of positive signals in a classic digital assay comprising identical partitions of the sample. As a result, the classical approach requires large number of partitions to target large dynamic range of concentration of the analyte. The requirement of large number of bits to express large values is readily remedied by positional systems. For example, in binary coding one needs only 8 digits to code the value of 157. In a chemical assay, the value of the bit (compartment) reflects the likelihood that it contains at least one molecule of analyte. The Boolean character of the signals suggests the binary positional system, yet the signals in a digital PCR assay are stochastic variables of , the volume () and the dilution factor ( /). Here, it is assumed that the

efficiency of a single-molecule amplification is 100%, regardless of the volume and/or dilution ratio of any particular compartment. Therefore, the probability of obtaining a particular signal, is based on Poisson distribution and equals !!" 1| 1 $ % &' , or (!" 0| % &' , where is the expected number of molecules in the compartment. As a result, the signal carries only a probabilistic information about the unknown concentration of the analyte.

Figure 2. (a) Probability 1| of finding at least one target molecule in volume . Bayesian formalism translates the signal into probabilistic information )| 1 on . (b) The binary entropy function *+ for a single Bernoulli (positive/negative) trial achieves maximum at ∗ : the signal provides most information about ~ ∗ . (c) Information from multiple compartments

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can be combined. For the set of identical compartments, the information gain is large in a very narrow range of concentrations, while (d) for geometrical sequence of volumes of compartments the information gain is almost constant in a wide dynamic range. (e) The geometric sequence of compartments allows to tune the amount of information retrieved from the assay (by tuning the common ratio - for a wide range of concentrations - a feature that cannot be achieved with identical compartments.

This information can be expressed with the Bayesian formalism as the probability ) that a given value of has caused the recorded outcome. The lack of prior knowledge of can be encoded with an initially constant density of probability ). 1// of finding any particular value of between 0 and an arbitrary upper bound / . Then the information about gained from a single signal ( 1 or 0) is )| 1 1 $ % &01" ⁄/ $ &2 and )| 0 % &01" (Fig. 2a). Importantly, the probability distribution ) comprises all the information extracted from the signal. The density of this information is not evenly distributed along the axis of . Shannon entropy *4 ∑! ! log ! , where 9 counts the two possible outcomes, measures the information density and achieves maximum at ∗ ln2⁄. The signal provides most information about concentrations near ∗ , for which 1| ∗ 1⁄2 (Fig. 2b), while it carries little information on much smaller or much larger concentrations. In an assay, the information on combines the inputs from multiple partitions. The probability of observing a particular combination of signals ;2 , = … ?―that we call a microstate @―is given by a product of individual probabilities [ESI 2.4]: ;2 , = … ? @| ∏ B2 |. The full information / on is then given by )|@ @|DC. @| . For identical compartments, the probability of obtaining exactly positive and negative signals can be expressed with the Bernoulli distribution: EF ; $ H @IJ K L M1|N O M0|NP . The information about gained from the outcome of a classic digital assay has the same functional form as the information from a single compartment (Fig. 2b,c). The precision of the estimate from the classic assay is not constant within the dynamic range and obtaining reasonable estimates about high concentrations requires very large number of partitions. In order to provide a constant level of information on within the requested dynamic range, the compartments (i.e. their ∗ values) should be spread along the axis of . As typically the precision of assays is parameterized by a fractional precision (expressed in the percentage of the estimated value), the compartments should be distributed uniformly along the logarithmic scale of : the values of !∗ should follow a geometric sequence. What remains to be determined is the common ratio, the value of the first term, and the number of compartments for a given requested dynamic range and precision of the estimate of . To find the optimum positional system, we start with an arbitrary, known input concentration !Q and centre the assay at a compartment . . ln2 /!Q (i.e. set .∗ !Q ). We supplement this central compartment with ∆ compartments on each ‘side’: ! ! . . - ! , 9 ∈ $∆, ∆ , i.e.


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with larger and smaller compartments, all arranged into a geometric series with the common ratio -. The probability of reading a positive signal depends on the modulation factor of T the compartment: ! 1 $ % &0TPOUV1W"WX . Compartments much smaller or much larger than the central one will have their probabilities either very close to zero or to unity. Thus the relevant information will come from a finite set (an ‘active stripe’) of compartments around the central one (Fig. 3a). For any fixed value of -, increasing the number 2∆ Y 1 of partitions improves the , yet only to a limit: compartments with their ! ! values much larger or much smaller than . . progressively carry less and less information about ∗ . Indeed, the standard deviation Z of falls with increasing ∆ to the limit lim∆→/ Z- that is a function of - only [ESI 2.7]. This limit can be closely approximated by a simple algebraic fit: lim∆→/ Z- _1 $ -4 . Substituting the requested precision ZX for lim∆→/ Z- and inverting this equation allows us to find the common ratio - of the geometric sequence of compartments as an explicit function of the b requested precision: - 1 $ ` ∙ ZX [ESI 2.7].

Then, in order to find the optimum number ∆X of compartments in the active stripe for the calculated value of - we find the lowest integer value of ∆ at which the derivative of Z with respect to ∆ is less than 1/1000. This arbitrary condition ensures that further addition of compartments to the active stripe does not appreciably increase the precision of the assay. We found that ∆X can also be closely approximated by a simple algebraic expression ∆X e ∙ 1 $ -&f [ESI 2.7]. The two equations for - and for ∆X define the active stripe as a function of the requested precision of the assay. The assay must also be designed for the requested dynamic range of concentrations within which the precision is to be at least the requested value. As the equations for the active stripe are true for any value of . . (i.e. for any !Q ) and since the geometric progression is self similar, in order to guarantee the same precision within a required range ∈ & , g , it suffices to span the assay, keeping the required - and the required 'margins' of compartments outside & & , g g , with g/& g/& ln2 / g/& . Thus, the assay can be completely designed with the use of the following equations that use the requested dynamic range Ω g / & and maximum allowed standard deviation ZX of the estimate of as an explicit input: b

- 1 $ ` ∙ ZX

∆ i∆X j ke ∙ 1 $ -&f l 2 ∙ ∆ Y ilog X 1/Ωj

Figure 3. (a) For any !Q only one compartment (here . ln2 /!Q ) addresses this concentration most closely. The progressively larger/smaller compartments carry less information as the probability of obtaining a signal from these compartments approaches unity/zero. The active stripe (grey area) comprises a set of volumes with probability of yielding a given signal significantly different than zero or one. (b) An active stripe yields stochastic combinations of signals, or microstates (@! ), in every run of the assay. Each microstate yields )|@! that provides more information on than that retrieved from the sole number of signals c. Each of the distributions )|@! have different standard deviation. Inset (c) shows the estimated concentrations and the standard deviations based on the analysis of all possible microstates (green dots) of an active stripe comprising three compartments. The black triangles show the estimates and standard deviations retrieved from the sum of positive signals (here either d 1 or d 2). Inset (d) illustrates the same information for an active stripe comprising 15 compartments. The shaded areas indicate the range of the estimates of concentration given by 70% (95%) of most probable outcomes of the assay.

. mn2 ∙ - &∆ 1/ & with `, o, e and p being positive constants [ESI 1.1-1.4, 2.7]. The above analytical expressions are derived from the approximations and demonstrate the relation between the precision of the assessment and design of the active stripe. The model can be used over a wide range of requested precision ZX q 0.89. The assay can be designed in the following steps: (i) for a given ZX , & and g calculate , ∆ and -, (ii) using - and ∆ calculate the volume . of the first compartment in the sequence and (iii) create the sequence of partitions of volume ! ! . . - ! for 9 1 … $ 1. This recipe provides a powerful analytical tool. The dynamic range can be tuned independently of precision (Fig. S7A). For given, requested, Ω and ZX the assays require orders of magnitude less compartments than the classic digital assays. For example, in classical assays in order to address u 10v or u 10w one needs 2 ∙ 10x or 2 ∙ 10y compartments respectively. If ZX 50% is satisfactory, the same us can be assessed with as few as 35 or 47 compartments. More precise estimates (e.g. ZX 25% or ZX 10%) require 140 and 795, or 192 and 1120 test-volumes respectively. The method that we describe can be used to assess i) a wide range with good precision (e.g. Ω 10| with ZX 50% with 65), ii) a wide range with high precision (e.g. Ω 10| , ZX 10%, 1615) or iii) a narrow range with high precision and again ultra small (e.g. Ω 10x offering ZX 30% can be realized with 80 and offering ZX 10% with 631) [20]. We note that the logarithmic distance between the compartments (-) is typically much smaller than the standard



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deviation of : - &2 $ 1 ≪ Z. One may thus expect that the precision of the estimate roots more in the number of compartments falling into a given active stripe than from the fine gradation of their ∗ values. As generation of fine gradations (e.g. - 0.99) may be technically challenging, we consider assays built with larger common ratios (i.e. smaller values of -) with copies of each compartment, similarity to the multivolume approach of Ismagilov et al [8-10], yet again, reducing the number of compartments needed to achieve the requested precision within the requested range of concentrations. Formally, each partition characterized by the dilution factor ! and volume ! can be prepared ’! times, yielding ∑&2 !B. ′! partitions in the assay. Then, the microstate of the assay is no longer a set of binary values but a set of real numbers proportional to the number of positive compartments belonging to the 9-th family ! ! . We derived equations that allow to freely exchange the value of - into the number of copies of compartments [ESI 1.2, 1.4, 2.8]. For example, Ω 10y and ZX 10%, can be realized with 1, - 0.986 and 1123, or with 10, - 0.869 and 1211, or 100, - 0.2 and 1400. Again the assay can be completely designed from the input of Ω, ZX and the maximum allowed value of - [ESI 1.2, 1.4], with the limitation that ZX q 0.89 and - ∈ 0,1.

Figure 4. Block diagram for effective design of the assay depending on the method of analysis and technical means of partitioning of the sample. If it is possible to construct a device that can track all the compartments during amplification and readout (assays designed for practical purposes can comprise up to hundreds of compartments), the optimal assay would be designed using instructions for identifiable compartments [ESI 1.1]. Otherwise, the user should use the protocols that describe the assays comprising non-tractable compartments [ESI 1.3]. Although assays with the smallest number of compartments are designed using instructions 1.1 (tracking) and 1.3 (no tracking), they may be challenging in execution of the fine gradation of volume or dilution factors of compartments (the common ratio of the sequence of volume or dilution of compartments is close to unity). To overcome this difficulty, the user can set the finest (optimum) gradation achievable using available equipment (i.e. set the maximum value of the

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common ratio of the sequence of volume or dilution of compartments) and enhance the precision of the assessment by making sets of copies of each compartment (libraries) [ESI 1.2 and 1.4].

So far we used the microstates, i.e. the combination of signals from identifiable compartments. The classic digital assays use the sum of positive signals, as it does not require tracking the identity of the compartments during amplification. The method that we propose here can also be used with nonidentifiable compartments. The loss of information associated with the lack of knowledge from which compartment the signal comes from is small. This follows from the observation that the probability function that a given concentration caused the recorded sum of the signals comprises inputs from analogous functions for the microstates. This summed probability function is dominated by the most probable microstates (in the simple example shown in Fig. 3b the most probable microstate contributes more than 60% to the stochastic outcome). As a result, the precision of an assay operated on the sum of signals is only slightly smaller than the precision of the estimate based on the microstate. We approximate [ESI 2.9] that the standard deviation Z of the estimate based on the sum of signals is the following function of ZX , as we used this parameter in the derivation of the assay based on the microstates: Z 1.023 ∙ ZX Y 0.021 (linear part of the plot in Fig, S8b, ZX 0.55. In other words, when designing an assay based on the sum of signals it is sufficient to replace the requested maximum standard deviation ZX in the equations with the expression (0.978 ∙ ZX $ 0.02. We have also listed the guideline on how to construct the assay for protocols that do not track the identity of compartments in ESI 1.3 and 1.4. The algorithms for the design of the assay use as an input the dynamic range within which the assay should return an estimate of concentration of the analyte with the requested precision. Depending on the method of analysis (identifiable and non-identifiable compartments) and technical means for generation of the partitions of the sample (i.e. limitation in the minimum dilution factor), the assays can be divided into four groups, each having a different design protocol (Fig. 4). In short, the first decision is whether to execute the assay with tracking of the compartments during amplification or not. Tracking of the identity allows to extract full information from the outcome of the assay, yet tracking might be experimentally difficult. The second decision concerns the modulation factor. If the optimum value of the modulation factor is too high (i.e. too close to unity), one can input the maximum allowed value of the modulation factor to obtain the design of the assay with slightly higher number of compartments, yet one that can be easily realized in practice.


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overestimates the dynamic range and underestimates the precision in a fragment of the dynamic range. The distribution of precision is not flat (it is equal to 8% in the middle of the declared dynamic range and more than twice that at both ends). Our model allows to design an assay that covers the same declared dynamic range (Ω 7650) and precision (Z _- 17%) using nearly twice less compartments (350) with a flat profile of precision, exactly as requested. The design of ref [10] can also be improved by using the same number of 640 compartments and providing a uniformly precise estimate (Z _- 11%) over the entire requested dynamic range.

The assays designed according to the method here described satisfy the requirements and compare well to the existing methods [8-10]. Fig. 5 b-d shows a set of outcomes of the assays designed to deliver the precision ZX 17% over the dynamic range of Ω 7650, the same as in an exemplary assay comprising 640 compartments designed by Kreutz et al [10] (Fig. 5a). The assays designed according the method here described deliver the requested precision uniformly over the whole dynamic range, while using less compartments ( 348, 360 or 350, depending on the design). When allowed to use the same number of compartments 640 it is possible to improve the precision within the range to ZX 11% (Fig. 5 e and f). We also tested the assays experimentally. In one experiment we designed a set of 16 (Fig. 6), 32, 48 and 96 [ESI 4.1] compartment assays that cover the dynamic range Ω 10v with precision ZX varying from 70% to 30%. Additionally we verified the assays with Monte Carlo simulations, including four assays designed to cover dynamic range Ω 10y and Ω 10| with precision ZX 15% and ZX 50% [ESI 5]. All these results agree with the analytical predictions.

Figure 5. Comparison of the performance of assays designed using the described protocol and state-of-art method (Kreutz et al. Slip Chip design [8-10]). The comparison was done via grand canonical Monte Carlo simulations. The assay of ref [10] slightly

Figure 6. Experimental verification of a 16-compartment digital assay. (a) The estimated concentration of DNA as a function of known input concentration. Each data point is averaged over 12 runs of the assay and error bars indicate the standard deviation of the estimates. The solid line is a power fit to the results. The grey lines indicate the region of expected outputs of the digital assay, given by nominal input concentration plus/minus one standard deviation. The green region marks the dynamic range. The circles mark the output of the digital assay when all the signals were either positive, or negative indicating the concentration that with 95% confidence is higher (lower) than !Q . (b) Standard deviation of the estimates with the solid line showing the expected



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value within the dynamic range. The assays shown in Fig. 6 and in Fig. S11 (ESI 4.1) are designed to cover the same dynamic range with different precision of the assessment. This feature is accomplished by changing (i.e. increasing, for better precision) the number of copies of each compartment. Fig. 4, shows the performance of an assay that has only one copy of each compartment, while fig. S11 shows performance of the assays comprising 2, 3 or 6 copies of each compartment. The common ratio of the sequence of products of volume and dilution ratios are the same.

DISCUSSION The proposed design of an assay is based on three assumptions. First, we use Bayesian approach to derive probability distribution of . Second, we use the most probable microstate, i.e. the most probable output state of an assay for a given concentration to determine the precision of the assessment as the standard deviation of the probability distribution of . Other approaches have also been tried, for example, Kreutz et al.[10] determine the concentration using the most probable number method, and precision of the assessment by the amount of Fisher information assuming the upper limit of precision given by the Cramer-Rao inequality. As the existence of such an ideal estimator cannot be rigorously assumed, we rested our method simply on the most probable precision of the estimate. The two methods are similar in that they both use a collection of compartments of different volumes to construct the assay. The differences include i) the architecture of the solution and the assumptions accepted in the derivation, ii) the actual number of compartments needed to realize the requested functionality (our model provides for a smaller number of compartments) and iii) explicit mathematical formulation of the recipe for designing the assay. Finally, we assumed that the efficiency of the single-molecule amplification is 100% and does not depend on the volume and/or dilution ratio of a compartment. However, for real samples, the efficiency of amplification may depend on surface-to-volume ratio and on the extent of dilution of the sample. In such a case, running a series of control experiments is advised. In conclusion, we have shown that it is possible to extract full information from stochastic realizations of digital assays. The optimized use of information [21] allows to reduce the number of compartments needed to run the assay with respect to the multi-volume and multi-dilution approaches [13,22]. The algorithm that we described allows for rational and simple design of an assay with independently tuned dynamic range and precision, requiring orders of magnitude less partitions of the sample than the classic, single volume digital schemes. It is further possible to construct assays that provide different required precision in different ranges of concentration. This could be useful whenever e.g. at low and high concentrations precision can be sacrificed for the dynamic range, while in the intermediate range of concentration the exact estimate is clinically informative [ESI 3.2]. The drastic reduction of the number of partitions minimizes the technical requirements for running digital assays. These features may be important in bringing the reference-free, absolute, digital quantization of DNA to multiple new formats and in increasing the availability of digital PCR and digital immunoassays. The assays proposed here can offer same precision and sensitivity as the classic digital methods. Because the assays use small number of partitions, these are likely to be

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large and thus not providing the advantage of increased specificity of PCR [23]. The dramatic reduction of the number of partitions qualitatively changes the technical requirements to run the assays and could perhaps lead to the development of an alternative technology for the RT-PCR to provide absolutely accurate estimates at requested precision and at the throughput of modern real-time devices. Finally, for the same reason of decreased technical requirements to execute the assays they may find use in quantitative identification of viral and microbial pathogens in the point-of-care applications.

ASSOCIATED CONTENT Supporting Information. Practical guideline how to design a digital assay that provides the required dynamic range and precision of the assessment, Derivation of the model, Other possible applications of the algorithms, Experimental verification of the algorithms, Numerical verification of the algorithms. This material is available free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION Corresponding Author * E-mail:[email protected]

Author Contributions PRD and PG developed the algorithms, analyzed data and wrote the manuscript; TSK designed the experiments and, together with PRD and PG, the implementation of the algorithms for diagnostic assays; KG and MS executed the experiments.

Notes PRD, TSK and PG declare competing financial interest in possession of shares of Curiosity Diagnostics Sp. z o.o. that seeks commercial implementation of the method.

ACKNOWLEDGMENT The Authors thank Curiosity Diagnostics Sp. z o. o. for financing the research. Curiosity Diagnostics Sp. z o. o. acknowledges the E’8042 OPTIGENS grant provided by the National Centre for Research and Development within the Eureka Initiative.

ABBREVIATIONS PCR Polymerase Chain Reaction.

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