Evaluating droplet digital PCR for the quantification of human genomic DNA: converting copies per nanoliter to nanograms nuclear DNA per microliter

Abstract

The highly multiplexed polymerase chain reaction (PCR) assays used for forensic human identification perform best when used with an accurately determined quantity of input DNA. To help ensure the reliable performance of these assays, we are developing a certified reference material (CRM) for calibrating human genomic DNA working standards. To enable sharing information over time and place, CRMs must provide accurate and stable values that are metrologically traceable to a common reference. We have shown that droplet digital PCR (ddPCR) limiting dilution end-point measurements of the concentration of DNA copies per volume of sample can be traceably linked to the International System of Units (SI). Unlike values assigned using conventional relationships between ultraviolet absorbance and DNA mass concentration, entity-based ddPCR measurements are expected to be stable over time. However, the forensic community expects DNA quantity to be stated in terms of mass concentration rather than entity concentration. The transformation can be accomplished given SI-traceable values and uncertainties for the number of nucleotide bases per human haploid genome equivalent (HHGE) and the average molar mass of a nucleotide monomer in the DNA polymer. This report presents the considerations required to establish the metrological traceability of ddPCR-based mass concentration estimates of human nuclear DNA.

Introduction

Polymerase chain reaction (PCR) assays perform best when used with an accurately determined quantity of input DNA. This is particularly true for highly multiplexed assays, such as those used for forensic human identification [1].

We have shown that droplet digital PCR (ddPCR) limiting dilution end-point [2] measurements of the concentration of DNA copies per volume sample can be traceably linked to the International System of Units (SI) when several basic assumptions are shown to be justified [3]. Unlike values assigned using a conventional relationship between ultraviolet absorbance at 260 nm and DNA mass concentration [4], entity-based ddPCR values can be expected to be stable over time and provide results that are traceable [5] to the International System of Units (SI).

The linkage between ddPCR measurements and the SI is through the calibrated measurements of mean droplet volume (V), volume fraction of sample in the reaction mixture (F), and counts of the number of total and negative droplets (N_tot and N_neg) where negative droplets are defined as those that do not provide an above-threshold fluorescence signal at the endpoint. Poisson transformation of the ratio of negative to total droplets, λ = –ln(N_neg/N_tot), yields the mean number of copies of the PCR target in the ddPCR reaction mixture per droplet. Dividing by the total volumetric dilution factor yields the mean number of copies in the sample per droplet. The concentration of copies in the sample is then λ/(FV). The sources of bias and imprecision that contribute to the uncertainty in λ and F, u(λ) and u(F), have been discussed by others [6–9], as have the those associated with measurement of V, u(V) [9–13].

We are using ddPCR measurements to certify the concentrations of human nuclear DNA (nDNA) in a certified reference material (CRM) designed primarily for use by the forensic community. However, this community expects DNA quantity to be stated in terms of mass concentration rather than entity concentration. The entity concentration, λ/(FV) copies per nanoliter, can be transformed to mass concentration, [nDNA] nanogram per microliter, via the relationship:

$$\begin{gathered} [\text{nDNA}]\frac{\text{ng}}{\mathrm{\mu }\mathrm{L}}=\left(\frac{\lambda \text{copies}\text{of}\text{target}}{\text{droplet}}\right)\left(\frac{\mathrm{\mu }\mathrm{L}\text{mixture}}{F\mathrm{\mu }\mathrm{L}\text{sample}}\right)\left(\frac{\text{droplet}}{V\text{mixture}}\right)\left(\frac{\text{HHGE}}{r\text{target}}\right) \\ \left(\frac{n\text{base}\text{pairs}}{\text{HHGE}}\right)\left(\frac{\overline{w}\mathrm{g}}{\text{mol}\text{base}\text{pairs}}\right)\left(\frac{\text{mol}\text{base}\text{pairs}}{6.022{10}^{23}\text{base}\text{pairs}}\right)\left(\frac{{10}^3\text{nL}}{\mathrm{\mu }\mathrm{L}}\right)\left(\frac{{10}^9\text{ng}}{\mathrm{g}}\right) \end{gathered}$$ (1)

where r is the number of assay targets per human haploid genome equivalents (HHGE), n is the number of nucleotide base pairs (bp) per double-stranded HHGE, and w̄ is the average molar mass of a bp in the DNA polymer.

For independent multiplicative factors such as these, the combined relative uncertainty of their product can be estimated from the square root of the sum-of-squares of the individual relative uncertainties [14, Section 5.1.6]:

$$\frac{u([\text{nDNA}])}{[\text{nDNA}]}=\sqrt{{\left(\frac{u(\lambda )}{\lambda }\right)}^2+{\left(\frac{u(F)}{F}\right)}^2+{\left(\frac{u(V)}{V}\right)}^2+{\left(\frac{u(r)}{r}\right)}^2+{\left(\frac{u(n)}{n}\right)}^2+{\left(\frac{u(\overline{w})}{\overline{w}}\right)}^2}$$ (2)

The essential elements of these relationships are provided in Griffiths et al. [6, Table 1], but without addressing all assumptions, values, and uncertainties. This report presents the considerations required to establish the metrological traceability of ddPCR-based mass concentration estimates of nDNA.

Table 1.

NIST-Developed human genomic assays

Assay Target	Chromosome Band Accession #	Primers and Probe a	Amplicon Length, bp
NEIF	Chr 2	Fgccaaacttcagccttctcttc	67
Gene EIF5B	p11.1-q11.1	Rctctggcaacatttcacactaca
	NC_000002.12	PB+tcatgcagttgtcagaagctg
2PR4	Chr 2	Fcgggtttgggttcaggtctt	97
Gene RPS27A	p16	Rtgctacaatgaaaacattcagaagtct
	NC_000002.12	PBtttgtctaccacttgcaaagctggccttt
POTP	Chr 2	Fccaccttcctctgcttcacttt	60
STR TPOX	p25.3	Racatgggtttttgcctttgg
	NC_000002.12	PTcaccaactgaaatatg
NR4Q	Chr 4	Ftggtgggaatgttcttcagatga	83
Gene DCK	q13.3-q21.1	Rtcgactgagacaggcatatgtt
	NC_000004.12	PB+tgtatgagaaacctgaacgatggt
D5	Chr 5	Fttcatacaggcaagcaatgcat	75
STR D5S2500	q11.2	Rcttaaagggtaaatgtttgcagtaatagat
	NC_000005.10	PTataatatcagggtaaacaggg
ND6	Chr 6	Fgcatggctgagtctaaagttcaaag	82
STR D6S474	q21–22	Rgcagcctcagggttctcaa
	NC_000006.12	PB+cccagaaccaaggaagatggt
D9	Chr 9	Fggctttgctgggtactgctt	60
STR D9S2157	q34.2	Rggaccacagcacatcagtcact
	NC_000009.12	PTcagggcacatgaat
HBB1	Chr 11	Fgctgagggtttgaagtccaactc	76
Gene HBB	p15.5	Rggtctaagtgatgacagccgtacct
	NC_000011.10	PTagccagtgccagaagagccaagga
ND14	Chr 14	Ftccaccactgggttctatagttc	109
STR D14S1434	q32.13C	Rggctgggaagtcccacaatc
	NC_000014.9	PB+tcagactgaatcacaccatcag
22C3	Chr 22	Fcccctaagaggtctgttgtgttg	78
Gene PMM1	q13.2	Raggtctggtggcttctccaat
	NC_000022.10	PBcaaatcacctgaggtcaaggccagaaca

^aF: Forward primer, R: Reverse primer,

P^B : Blackhole quencher probe, P^B+ : Blackhole Plus quencher probe, P^T : Taqman MGB probe

Materials and Methods

ddPCR System

The measurements discussed here were performed using a Bio-Rad QX200 Droplet Digital PCR System (Hercules, CA) system. The manufacturer’s QuantaSoft version 1.7.4.0917 software was used to determine the number of positive and negative droplets at the end of 60 cycles using assay-specific intensity thresholds. These results were exported into a spreadsheet for further manipulation.

Sample Materials

Three human genomic DNA extracts were investigated, all components of a candidate CRM. These extracts were prepared from the buffy coat fraction of anticoagulated blood from anonymous donors. The component labeled “A” was derived from a single-source male, “B” from a single-source female, and “C” a mixture of a single-source male and a single-source female.

Each donor buffy coat bag was aliquoted (5 mL per aliquot) into sterile 50 mL conical tubes then stored at 4 °C prior to extraction. A modified salt-out manual extraction protocol was performed for each of the individual buffy coat samples [15], with rehydration of the DNA in 10 mmol/L tris(hydroxymethyl)aminomethane HCl, 0.1 mmol/L disodium ethylenediaminetetraacetic acid, pH 8.0 buffer (TE⁻⁴). All components were solubilized from the air-dried state in TE⁻⁴ buffer and diluted to a working concentration of ≈50 ng/μL based upon double-stranded DNA absorbance at 260 nm [4].

Since there is no consensus on the infectious status of extracted DNA, all solutions were handled as biosafety level 1 materials capable of transmitting disease [16].

ddPCR Measurements

Ten PCR assays developed at the National Institute of Standards and Technology (NIST) were used in this study. Each assay was confirmed to target one locus per HHGE using the National Center for Biotechnology Information’s BLAST/ blastn system [17]. The targets for these assays are located on eight chromosomes; three of the assays target widely separated loci on chromosome 2. Table 1 lists the genomic targets, primers and probes for these assays.

Measurements with all ten assays were made on five independently prepared aliquots of each of the three components. Each set of three aliquots was processed on one 96-well plate. The five sets were processed over four days by the same analyst using the same equipment and materials. Two technical replicates were collected for each assay for each component along with non-template controls (NTCs) for each assay. On average 17,000 droplets were counted per sample (technical replicates and NTCs). There was a maximum of three positive droplets in any of the NTCs, averaging less than one positive per NTC. Of the (3 × 5 × 10 × 2 sample + 5 × 10 × 2 NTC) = 400 ddPCR measurements, three were rejected as technical failures based on the manufacturer’s software diagnostics.

The independent aliquots were prepared as 1 volume of the ≈50 ng/μL CRM material to 3 volumes of TE⁻⁴ buffer. For each 25 μL reaction, 2.5 μL DNA of this 1:4 diluted solution was added to 22.5 μL of PCR mastermix for a total dilution factor of 1/40. For all three materials, this dilution yielded λ ≈ 0.3 copies per droplet and dilution-adjusted λ/F ≈ 12 copies per droplet. Use of this rather low λ value with unfragmented human nDNA helps to minimize assay bias at the cost of a modest increase in technical replicate variability: see the Electronic Supplementary Material (ESM) section on Poisson Sampling. The 22.5 μL of mastermix consisted of: 12.5 μL of Supermix for Probes (no dUTPs), 1.88 μL each of 5 μmol forward and reverse primers, 1.25 μL of 5 μmol probe, and 5.0 μL nuclease-free water.

For each replicate assessment, 22 μL of the mastermix solution were loaded into a 96-well plate, heat-sealed with foil, and placed on Bio-Rad Automated Droplet Generator (AutoDG). The AutoDG generated droplets and all formed droplets were loaded into a new 96-well plate. That plate was heat-sealed with foil and PCR amplified on a Pro-Flex PCR system (Applied Biosystems, Foster City CA). The amplification protocol was: 95 °C for 10 min, followed by 60 cycles of 15 s at 95 °C and 1 min at 61 °C with ramp rate set to 2.5 °C/s between temperatures. After the 60th cycle there was a 98 °C hold for 10 min followed by a 4 °C hold until the samples were removed from the thermal cycler and put onto the QX200 droplet reader.

Results and Discussion

There are two autosomal targets per diploid genome, one from each chromosome pair. Since all ten of the assays we use target autosomal loci, for simplicity the following discussions focus on the 1-to-1 relationship between a single ddPCR target and a HHGE. Except for the minor complication of X and Y sex chromosomes in males, this is identical to the 2-to-2 relationship between two ddPCR targets and the full diploid genome.

Copies per Nanoliter

Because measurements cannot be made on undiluted sample, in practice ddPCR results expressed in terms of copies per sample combine the λ and F factors. The λ/F ± u(λ/F) for a given analysis system can be characterized through repeated independent measurements of samples prepared from the same stock material. In a series of five independent determinations, three different DNA extracts were evaluated in duplicate with the ten human genomic assays listed in Table 1. For these measurements the expected relative standard uncertainty, u(λ/F)/(λ/F), for a single estimate of λ/F with a single assay is about 7.64%; see ESM Table S3 for details. This estimate is strictly appropriate only for these experiments, but may be indicative of similar processes.

Noting that the relative uncertainty for the mean, x̄, of a series of m independent determinations of some measurand X can be estimated as [14, Section 4.2.3]

$$u\left(\overline{x}\right)/\overline{x}=u(x)/\sqrt{m}$$ (3)

we estimate the relative uncertainty of the mean λ/F from m_rep = 5 single-assay measurements of independently prepared replicates as $u\left(\overline{\lambda /F}\right)/\left(\overline{\lambda /F}\right)=7.64/\sqrt{5}=3.42\%$. The λ/F values are metrologically traceable to the derived SI unit for volume, nL, and natural unit count-one (1) [5].

The mean droplet volume was measured for droplets made using our equipment with the same lot of mastermix used for the ddPCR measurements. Using the concentrated method described in Dagata et al. [12], V for this lot was 0.7349 nL with a standard relative uncertainty, u(V)/V, of 1.15%. For a given lot of mastermix and using our equipment and supplies, the value of V remains constant well within this uncertainty over at least six months (data not shown) and is metrologically traceable to the derived SI unit for volume, nL. The u(V)/V is characteristic of the volume measurement process and is not a function of the number or nature of ddPCR measurements.

Human Genomes per Target

The concentration of a given target is not necessarily the same as the concentration of HHGEs containing that target. Fragmentation, either intentional or during sample preparation, can reduce the number of amplifiable assay-specific targets in the reaction mixture and/or increase their accessibility [18]. While the ten assays used in this study are designed to amplify one and only one site per HHGE (i.e., r = 1), sample-specific mutations can reduce or prevent amplification or introduce additional binding sites. We therefore advocate using multiple assays, designed to amplify widely separated autosomal targets, to confirm that target measurements imply genome measurements.

Figure 1 displays the relative performance of the assays for three human genomic extracts in TE⁻⁴ buffer. The agreement among the assays confirms that for these assays r = 1. The relative between-assay standard uncertainty, u(r)/r, is 2.17%; see ESM Table S4 for details. This value is strictly appropriate only for these experiments, but may be indicative of similar processes.

Fig. 1.

Human Genomes per Target. The “dot&bar” represent the mean relative differences among the ten assays for the three components, where the individual differences are $\%d_{ij}=100\left({\overline{(\lambda /F)}}_{ij}-{\overline{(\lambda /F)}}_i\right)/{\overline{(\lambda /F)}}_i$, i indexes samples, j assays, and the “bar” above the symbol denotes averaging across replicates, plates, and (for ${\overline{(\lambda /F)}}_i$) assays. The solid “dots” mark the median of the posterior distribution determined by an empirical Bayesian evaluation; the vertical “bars” span the central 50%of the distribution. The green circles and lines represent the results for component “A” of the candidate CRM, red squares and lines for component “B”, and blue triangles and lines for component “C”. The thin black horizontal lines bound an approximate 95%confidence interval around the zero-difference line. See ESM Table S5 for the numerical values

By Eq. 3, the corresponding relative standard uncertainty for the mean of our m_assay = 10 independent assays, r̄, is $u(\overline{r})/\overline{r}=2.17/\sqrt{10}=0.686\%$. The r values are metrologically traceable to the natural unit ratio-one (1) [5].

Number of Nucleotide Bases per Haploid Human Genome Equivalent

The nucleotide base composition of the human genome is known to differ among individuals [19], but this variability is likely to be small except for rare cases (e.g., trisomies). Given current technology, determining the “exact” number of bases, n, for a personal genome is impractical. However, the Human Assembly Data web-resource maintained by The Genome Reference Consortium (GRC) provides the lengths in bp of the 22 autosomal and the X and Y sex chromosomes of the Reference Genome as estimated in the 2006 NCBI36, 2009 GRCh37, and 2013 GRCh38 assemblies (https://www.ncbi.nlm.nih.gov/grc/human/data). These chromosome lengths are identical in all “patch” releases (periodic minor updates) of these assemblies. Table 2 lists these values.

Table 2.

Lengths of chromosomes in the human reference genome, bp ^a

Chromosome	Assembly
	GRCh38	GRCh37	NCBI36
1	248,956,422	249,250,621	247,249,719
2	242,193,529	243,199,373	242,951,149
3	198,295,559	198,022,430	199,501,827
4	190,214,555	191,154,276	191,273,063
5	181,538,259	180,915,260	180,857,866
6	170,805,979	171,115,067	170,899,992
7	159,345,973	159,138,663	158,821,424
8	145,138,636	146,364,022	146,274,826
9	138,394,717	141,213,431	140,273,252
10	133,797,422	135,534,747	135,374,737
11	135,086,622	135,006,516	134,452,384
12	133,275,309	133,851,895	132,349,534
13	114,364,328	115,169,878	114,142,980
14	107,043,718	107,349,540	106,368,585
15	101,991,189	102,531,392	100,338,915
16	90,338,345	90,354,753	88,827,254
17	83,257,441	81,195,210	78,774,742
18	80,373,285	78,077,248	76,117,153
19	58,617,616	59,128,983	63,811,651
20	64,444,167	63,025,520	62,435,964
21	46,709,983	48,129,895	46,944,323
22	50,818,468	51,304,566	49,691,432
X	156,040,895	155,270,560	154,913,754
Y	57,227,415	59,373,566	57,772,954

^aas listed by The Genome Reference Consortium (GRC) (https://www.ncbi.nlm.nih.gov/grc/human/data)

The Human Assembly Data resource also provides a “Total bases” summary for every patch release of the GRCh37 and GRCh38 assemblies. These totals include bp from alternate loci (large polymorphisms in bp composition but generally not large differences in bp number) as well as un-placed and un-localized scaffolds (known bp sequences not yet positioned within the assembly) [20]. These scaffolds likely fit within the known gaps (difficult-to-sequence stretches of approximately known length) in the Reference Genome sequence and so are unlikely to have much impact on the Reference Genome bp number. The gradual increase in the number of “Total bases” thus reflects increasing knowledge of the human genome and its variability but not the bp size of the Reference genome.

Based on the GRCh38 assignment, the total length of the autosomes plus the X chromosome is 3.031 10⁹ bp and that for the autosomes plus the Y chromosome is 2.932 10⁹ bp. Without determining the sex chromosome ratio in a sample, an average gender-neutral HHGE then has ((3 × 3.031 + 1 × 2.932)/4) 10⁹ = 3.006 10⁹ bp. Using the same calculations, the GRCh37 and NCBI36 estimates are 3.012 10⁹ bp and 2.998 10⁹ bp. The standard deviation (and thus an estimate of the standard uncertainty) of the three values is 7.024 10⁶ bp.

Given the upper and lower bounds, a₊ and a₋, of a rectangular (uniform) distribution of X, in the absence of other information the usual estimate of the expected value, x, and it standard uncertainty, u(x), is [14, Section 4.3.7]:

$$x=(a_{-}+a_{+})/2;u(x)=(a_{+}-a_{-})/\sqrt{12}$$ (4)

The GRCh38 value for the n of a XX female is 3.031 10⁹ bp while for a XY male it is 2.982 10⁹ bp. Treating these values as the bounds on the bp of a sample with an undetermined sex chromosome ratio, the estimated standard uncertainty is $((3.031-2.982){10}^9)/\sqrt{12}=\left(\frac{0.049{10}^9}{3.46}\right)\cong 1.41{10}^7\text{bp}$.

Since the measurement technologies used to construct the reference tend to contract the length of tandem repeats [21, 22], we estimate the number of bp in the reference HHGE as the largest of the currently defensible values, the GRCh37 value of 3.012 10⁹ bp.

While we are only concerned with nDNA, the widely used spectroscopic methods are insensitive to the DNA source [4]. When comparing target-specific ddPCR and non-specific spectroscopic results, the possible influence of mitochondrial DNA (mtDNA) should be considered. While highly variable, the ratio of mtDNA to nDNA copies in whole blood is about 100-to-1 [23]. Since the reference human mitochondrial genome contains 16,569 bp [24], the impact per 100 mtDNA/ nDNA is (100 × 16,569) = 1.657 10⁶ bp.

Noting that for independent additive factors, x_i, the combined standard uncertainty of their sum, y, can be estimated as the square root of the sum-of-squares of the individual standard uncertainties, u(x_i) [14, Section 5.1.2]:

$$u(y)=\sqrt{{\sum }_i{u}^2(x_i)}.$$ (5)

An estimate of the combined standard uncertainty of n is then $u(n)={10}^7\sqrt{{0.702}^2+{1.41}^2+{0.166}^2}\cong 1.58{10}^7\text{bp}$ for a relative standard uncertainty, u(n)/n, of (1.58 10⁷)/(3.012 10⁹) = 0.525%.

While the chromosome bp lengths are not provided as certified values, the GRC is an internationally recognized authority on the reference genome. The data it provides are adequate to establish n ± u(n) as metrologically traceable to the natural unit count-one (1).

Mean Molar Mass of DNA Nucleotides

The value given in relevant literature as the mean molar mass of the nucleotide bases that comprise DNA (A: deoxyadenosine monophosphate, T: deoxythymidine monophosphate, G: deoxyguanosine monophosphate, and C: deoxycytidine monophosphate) is surprisingly variable, ranging from (308 to 330) g/mol or, expressed as A-T and G-C bp, (616 to 660) g/mol [25, 26]. Other values provided by on-line resources in response to the query “What is the average molar mass of a DNA base pair?” include (649, 650, and 654) g/mol. On evaluation, the (649 and 650) g/mol values result from averaging the molar masses of the deprotonated nucleotide monomers (loss of 2 H⁺ per base), with and without rounding to integer values and ignoring the loss of water during polymerization. The 654 g/mol value is the mean of the protonated monomers and ignores the loss of water. The 616 g/mol value reflects the mean mass of the deprotonated polymeric bases (loss of 1 H⁺ per base). The 660 g/mol value is the mean mass of the sodium salt of the polymer. Table 3 details these calculations using the estimated atomic masses, uncertainties, and ranges provided in Meija et al. [27]. The atomic mass uncertainties have been estimated following Possolo et al. [28].

Table 3.

Average Molecular Masses of Bases and Base Pairs ^a,^b

Base d	Formula	Nucleotide Monomer		Polynucleotide Polymer c
		Acid e	Base−2 f	Acid e	Base−1 f	Na+ Salt
A	C10H14O6N5P	331.222(1)	329.206(1)	313.207(1)	312.199(1)	334.181(1)
T	C10H15O8N2P	322.208(1)	320.192(1)	304.193(1)	303.185(1)	325.167(1)
G	C10H14O7N5P	347.221(1)	345.206(1)	329.206(1)	328.198(1)	350.180(1)
C	C9H14O7N3P	307.197(1)	305.181(1)	289.182(1)	288.174(1)	310.156(1)
	Mean	326.962(2)	324.946(2)	308.947(2)	307.939(2)	329.921(2)
	30:30:20:20 Mean h	326.913(2)	324.897(2)	308.898(2)	307.890(2)	329.872(2)
AT g	C20H29O14N7P2	653.430(2)	649.398(2)	617.400(2)	615.384(2)	659.347(2)
GC g	C19H28O14N8P2	654.419(2)	650.387(2)	618.388(2)	616.372(2)	660.336(2)
	Mean	653.925(3)	649.893(3)	617.894(3)	615.878(3)	659.842(3)
	60:40 Mean i	653.826(3)	649.794(3)	617.795(3)	615.779(3)	659.743(3)

^aIgnoring end-groups and methylation

^bMolar masses are calculated from the atomic masses, uncertainties, and ranges listed in (Meija et al., 2016) [27]. Standard uncertainties, estimated following (Possolo et al., 2017) [28], are enclosed by parentheses

^cCondensation polymer; n-1 waters are lost for every n condensed monomers

^dA: deoxyadenosine monophosphate, T: deoxythymidine monophosphate, G: deoxyguanosine monophosphate, C: deoxycytidine monophosphate

^eConjugate acid, phosphate oxygens fully protonated

^fConjugate base, phosphate oxygens fully deprotonated

^gBase pairs (bp) associated via hydrogen bonding

^hCalculated as (30A + 30 T + 20G + 20C)/100

ⁱCalculated as (60AT + 40GC)/100

The values based upon averaging the monomers are inappropriate for polymerized fragments, as are values that ignore counter ions when the polymers are in neutral to basic solution. The 660 g/mol estimate for the sodium salt more correctly reflects the chemical composition in TE⁻⁴ pH 8.0 buffer. Treating the “molar mass of DNA” as that of the negatively charged polymerized bases is analogous to treating “the molar mass of table salt” as that of chloride.

The mean (659.841 ± 0.003) g/mol value of the “Na⁺ Salt” in Table 3 is for equal proportions of AT and GC pairs. AT pairs typically outnumber GC pairs in a ratio of 60-to-40 [29]. The (659.743 ± 0.003) g/mol of the 60:40 weighted mean is a more appropriate estimate for genomic DNA.

Two factors can increase the effective mean bp mass: the addition of one water per fragment when DNA is fragmented and the variable methylation of A and C bases. The extent of fragmentation depends upon the treatment history of the material. Since the typical fragment size of the DNA in our sample materials exceeds 48,502 bp [3], the impact of adding two waters (molar mass 18.015 g/mol) per double-strand break in our materials can be conservatively estimated as 2 × 18.015/ (659.743 × 48,502) = 1.126 10⁻⁶ g/mol. Taking the maximum number of breaks to be 3.012 10⁹/48502 = 6.210 10⁴, the upper bound on water addition is (1.126 10⁻⁶)(6.210 10⁴) = 0.070 g/mol. Taking no fragmentation as the lower bound and assuming fragmentation is rectangularly distributed, by Eq.4 the additional mass due to fragmentation is 0.070/2 = 0.035 g/mol and a conservative estimate of the standard uncertainty is $\frac{0.070-0}{\sqrt{12}}=0.020\mathrm{g}/\text{mol}$.

Adenosine methylation of nDNA in eukaryotes is known but is infrequent [30]. About 1% of human DNA nucleotides are 5-methylcystine monophosphate [31], although this percentage is known to vary among individuals as well as tissue types [32]. Taking no methylation as the lower bound, 2% methylation as an upper bound, and assuming the methylation is rectangularly distributed between these limits, the upper bound on the additional mass per average bp due to methylation (molar mass 15.034 g/mol per methyl group) is (0.02 × 15.034) = 0.301 g/mol. By Eq. 4, the expected increase is 0.301/2 = 0.150 g/mol and a conservative estimate of the standard uncertainty is $(0.301-0)/\sqrt{12}=\frac{0.301}{3.46}=0.087\mathrm{g}/\text{mol}$.

Combining the (659.743 ± 0.002) g/mol of the weighted mean, the (0.035 ± 0.020) from fragmentation, and the (0.150 ± 0.087) from methylation, we estimate w̄ to be (659.743 + 0.035 + 0.150) = 659.928 g/mol and from Eq. 5 $u(\overline{w})=\sqrt{{0.002}^2+{0.020}^2+{0.087}^2}=0.089\mathrm{g}/\text{mol}$. The relative standard uncertainty, u(w̄ )/w̄, is then 0.089/659.928 = 0.013%. While there is much confusion regarding the definition of what is meant by “the mean molarmass”, there is very little uncertainty once a definition is adopted.

Analogous with the GRC value for n, the International Union of Pure and Applied Chemistry’s Inorganic Chemistry Division Committee is the recognized authority on atomic mass. The data it provides, in conjunction with literature best-estimates of minor influences, establish w̄ ± u(w̄) as metrologically traceable to the derived SI unit of mass, g, and the SI unit for amount of substance, mol.

Conclusions

Segregating the experimental λ (copies per droplet), F (dilution factor), V (droplet volume), and r (targets per HHGE) factors in Eq. 1 from the constants and our estimates for n (number of bp per HHGE) and w̄ (mean molar mass per bp), the mass concentration of nDNA per microliter sample solution can be estimated as:

$$[\text{nDNA}]\frac{\text{ng}}{\mathrm{\mu }\mathrm{L}}=\left(\frac{\lambda }{\mathit{FVr}}\frac{\text{HHGE}}{\text{nL}}\right)\left(\frac{3.012{10}^9\times 659.928\times {10}^3\times {10}^9}{6.022{10}^{23}}\frac{\text{nL}\text{ng}}{\mathrm{\mu }\mathrm{L}\text{HHGE}}\right)=3.301\frac{\lambda }{\mathit{FVr}}\frac{\text{ng}}{\mathrm{\mu }\mathrm{L}}$$ (6)

From Eq. 2, the associated relative standard uncertainty is:

$$\frac{u([\text{nDNA}])}{[\text{nDNA}]}=\sqrt{{\left(\frac{u(\lambda /F)}{\lambda /F}\right)}^2+{\left(\frac{u(V)}{V}\right)}^2+{\left(\frac{u\left(\overline{r}\right)}{\overline{r}}\right)}^2+{0.525}^2+{0.013}^2}\%$$ (7)

For this series of measurements, $\frac{u([\text{nDNA}])}{[\text{nDNA}]}=\sqrt{{3.42}^2+{1.15}^2+{0.686}^2+{0.525}^2+{0.013}^2}=\sqrt{{3.67}^2+{0.525}^2}\cong 3.71\%$. Note that the u(n)/n value of 0.525% completely swamps the u(w̄ )/w̄ of 0.013% but barely registers against the 3.67% of the combined uncertainties from the experimental factors.

A 95% confidence relative uncertainty, U₉₅([nDNA]), can be estimated as:

$$U_{95}([\text{nDNA}]=k_{95}\times u([\text{nDNA}])$$ (8)

where k₉₅ is the appropriate expansion factor for the combined degrees of freedom, v([nDNA]). When all standard uncertainties are associated with large degrees of freedom, k₉₅ = 2; otherwise v([nDNA]) can be estimated using the Welch-Satterthwaite approximation and k₉₅ from the Student’s t distribution [14, Section G4.1]. When V is determined with the method used here, v(V) is “large”. Being based on literature data, v(n) and v(w̄) can also be considered “large”. Given the number of components, assays, and independent determinations used in this study, v(λ/F) and v(r̄) are “large enough” to justify using the k₉₅ = 2 approximation.

The 95% confidence relative expanded uncertainty of the ddPCR assignment of [nDNA] for this series of measurements is then 2 × 3.71 ≅ 7.4%.

Since all factors are metrologically traceable to natural and/ or SI units, ddPCR-based values for the mass concentration of human nuclear DNA can be metrologically traceable to the SI.

Biography

From left to right: Blaza Toman, mathematical statistician with the Statistical Engineering Division; Margaret Kline, research biologist in the Applied Genetics Group of the Biomolecular Measurement Division; David Duewer, research chemist in the Chemical Sciences Division; and Erica Romsos, research biologist in the Applied Genetics Group. The team combines interests in forensic human identification, the use of digital PCR, metrology, and Bayesian Markov Chain Monte Carlo methods