WO2008157689A2 - Methods of nucleic acid amplification analysis - Google Patents

Abstract

A method for quantifying nucleic acid sequences in amplification or reverse transcription reactions, involving the use of real-time detection (RTD) polymerase chain reaction (PCR) systems. The invention includes formulating mathematical models, based on kinetic descriptions of PCR for different RTD systems, and development of application algorithms for these models to quantitatively analyze RTD-PCR experiments. The invention also includes quantitative analysis of the effects of non-specific (secondary reactions) amplification on the efficiency of specific target amplification. The invention also includes quantitative analysis of multiple specific concurrent amplifications (multiplex reactions) including internal control relative quantification experiments.

Description

METHODS OF NUCLEIC ACID AMPLIFICATION ANALYSIS CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of the filing date of U.S. Continuation-in-Part Patent Application No. 11/820,652, filed June 19, 2007. In its entirety, this application is hereby incorporated by reference.

FIELD OF THE INVENTION

The disclosed invention is generally in the field of biotechnology, more specifically to the field of nucleic acid amplification analysis.

BACKGROUND OF THE INVENTION

The polymerase chain reaction ("PCR") is a known biotechnological tool. It is often used as a preliminary step in deoxyribonucleic acids ("DNA") and ribonucleic acids (RNA) technologies to amplify a few copies of a target sequence to amounts adequate for typical analytical methods. Real-time detection quantitative PCR ("RTD-QPCR") evolved from PCR technology to monitor amplification of a specific target sequence during the progression of PCR. Real-time detection systems for PCR monitoring can be divided into two groups: sequence-independent (non-specific) systems and sequence- dependent (specific) systems.

Examples of non-specific systems are fluorophores that can bind to double-stranded sequences of nucleic acids based on common structural features of all nucleic acids. The dye is an intercalating fluorescent dye, or a minor groove binding dye, which exhibits fluorescence upon binding to a double-stranded amplification product on direct excitation with light. The dye has different fluorescence efficiency (quantum yields of fluorescence) depending on the bound or unbound (free) state of a fluorophore. A dye is introduced into the PCR reaction volume, with the primers, and other reaction components, prior to reaction. The dye then binds to the double stranded amplification products as they are formed in solution and the characteristic signal of the dye is measured as the reaction progresses. Examples of dyes used for this purpose include SYBR GREEN I^® (Molecular Probes, Oregon) and ethidium bromide.

Various methods have been developed for detecting and quantifying specific sequences of nucleic acids (e.g., DNA and RNA) in the context of polymerase chain reactions. Some of the most sensitive methods involve the use of fluorescence resonance energy transfer ("FRET"). A common approach is to employ FRET techniques with probe technology to detect and monitor DNA amplification. Fluorescent dye labels typically are small organic dye molecules, such as fluorescein, Texas red, or rhodamine, which can be readily conjugated to probe-type molecules. The fluorescent molecules (fluorophores) can be detected by illumination with light of an appropriate frequency. Light excites the fluorophores and produces a resultant emission spectrum that can be detected by electro-optical sensors or light microscopy.

FRET occurs between a donor fluorophore and an acceptor dye, which may be a fluorophore, when the donor fluorophore has an emission spectrum that overlaps the absorption spectrum of the acceptor dye, and the donor fluorophore and acceptor dye are in sufficiently close physical proximity. When light excites the donor fluorophore, there is then produced an emission of light that may be absorbed and quenched by the acceptor molecule. When quenching occurs, the intensity of the donor fluorophore's emission appears to be lessened. Where the acceptor is also a fluorophore, the intensity of its fluorescence may be enhanced. The efficiency of energy transfer is highly dependent on the distance between the donor and acceptor, and equations predicting these relationships have been developed by Forster. FRET is a function of the distance between the donor and acceptor molecules. A discussion of these relationships and Forster - type equations is found Parkhurst and Parkhurst, Donor-acceptor distance distributions in a double-labeled fluorescent oligonucleotide both as a single strand and in duplexes, Biochemistry, 34: 293-300 (1995), the contents of the entirety of which are incorporated by this reference.

One such approach, known as the TAQMAN^® assay (Applied Biosystems, Foster City, California; Roche Molecular Systems, Alameda, California), uses a hydrolysis probe labeled with donor fluorophore and acceptor dye, which is then cleaved by the 5' to 3' exonuclease activity of the enzyme Taq polymerase to cause an increase in the intensity of the donor fluorophore. The probe is labeled with both donor and acceptor, and (prior to the attachment of the probe to a DNA strand) the fluorescence of the donor is quenched by the acceptor. During PCR, the probe is hybridized to the DNA strand to be amplified. As the DNA polymerase acts, the 5' to 3' exonuclease activity of the polymerase causes cleavage of the probe, separating the donor and acceptor and resulting in an increase in intensity of the fluorescence of the donor fluorophore. See, for example, Woo et ai, Identification of pathogenic Leptospira by TaqMan probe in a LightCycler, Anal. Biochem., 256: 132-134 (1998), the contents of the entirety of which are incorporated by this reference.

Another method for detecting amplification products is the "molecular beacon probe." This method uses oligonucleotide hybridization probes that form hairpin structures, with the donor fluorophore on the 5' end, and the acceptor molecule on the 3^* end of the hybridization probe. When the probe is in a hairpin conformation, the donor and acceptor are in close proximity, and the fluorescence of the donor fluorophore is quenched. During PCR, the molecular beacon probe hybridizes to one of the strands of the PCR product, and is in "open conformation" such that the donor fluorophore and acceptor dye are separated and the fluorescence intensity of the donor increases to a level that can be detected. See Tyagi and Kramer, Molecular beacons: probes that fluoresce upon hybridization. Nat. Biotechnol, 14: 303-308 (1996), the contents of the entirety of which are incorporated by this reference.

In a further method, PCR may be carried out with a primer labeled with the fluorophore CY5 in the presence of a fluorescein-labeled probe. The CY5 primer is attached to the target sequence and is used to form an extension product. The fluorescein-labeled probe then hybridizes to the extension product, the CY5 strand. When the labeled probe hybridizes to the extension product, the fluorophores of the primer and probe are in close proximity, and resonance energy transfer occurs between the fluorophores, increasing the fluorescence of the CY5. In this case, observation of fluorescence and, in particular, the FRET signal, allows monitoring of the hybridization of probe to target, and melting of the probe away from the target, during melting curve analysis of the probe away from the target. See Lay and Wittwer, Real-time fluorescence genotyping of factor V Leiden during rapid cycle PCR, CHn. Chem., 43: 2262-2267 (1997), the contents of the entirety of which are incorporated by this reference.

The methods described above rely upon the efficiency of the hybridization of the probe to the target. If the probe does not efficiently hybridize to the target sequence, the intensity of the generated signal, used to measure the quantity of the amplification product, is affected. Additionally, the probes may interfere with the nucleic acid amplification process. When a probe binds to the template strand of nucleic acid, it converts a piece of single-stranded template into double-stranded helix. The probe blocks the nucleic acid polymerase from completing translocation along the nucleic acid strand halting replication until the probe is removed, either by melting or enzyme action.

A method that relies not upon probes, but only upon applying fluorescent dye labels to a single oligonucleotide primer molecule in a hairpin structure has been described as an approach for detecting the presence of a target nucleic acid sequence and the quantity of such nucleic acid sequence in a sample. In this approach, the primer is designed to have two dye labels on the stem of its hairpin structure. One label is a fluorophore donor, and the other is a quencher that absorbs energy emitted by the donor. When the primer molecule is in the hairpin conformation, the fluorophore donor and acceptor are in close proximity, so that the fluorescence of the donor is substantially quenched by the acceptor. Once the primer is attached to the target sequence and replication occurs, the hairpin structure is linearized, a complementary strand is synthesized, and the primer with its fluorescent labels is incorporated into the amplification product. Once the primer is opened and incorporated into the product, the fluorophore donor and acceptor become widely separated, reducing the quenching effect. An advantage of this method is that the fluorescent signal is generated by the product itself, and not through the use of a probe. The use of a hairpin primer typically involves several difficulties, including design of an awkward, long primer having a hairpin configuration, which may not be easily "read" by the polymerase, and the need to place two labels, donor and acceptor, on the hairpin primer in specific locations. The hairpin primer may present other disadvantages, including competition of formation of the hairpin with formation of double-stranded DNA (resulting in lower sensitivity in detection of the FRET signal) and potential formation of primer-dimers, which may interfere with detection of the product signal. Also, the stability of the hairpin has to be low enough to allow enzymatic read through, but high enough to reduce background fluorescence, which creates an inherent contradiction that may reflect on sensitivity of the assay. In a further development, the hairpin primers may be designed to have only one fluorescent tag in the proximity of 3 'end using the quenching effect of adjacent G moieties. See Nazarenko et ai, Effect of primary and secondary structure of oligodeoxyribonucleotides on the fluorescent properties of conjugated dyes, Nucleic Acids Res., 30, 2089-2095 (2002), the contents of the entirety of which are incorporated by this reference (Invitrogen Corp., Carlsbad, CA). This design has several advantages; however various problems, outlined above for stem-loop primers are still exhibited within this development.

Development of different systems for real-time detection PCR ("RTD-PCR") helped to increase specificity and sensitivity of RTD-PCR and widen the scope of practical applications. However, the analytical approaches, aimed at interpreting the experimental data, have not changed significantly for the past several years.

One of the current methods of quantitative analysis of RTD-PCR is the "crossing point" or "cycle threshold" method (Applied Biosystems, Foster City, CA). If a certain level of fluorescence intensity is chosen, then the fluorescence signals, generated by RTD, reach this level at different time points (crossing points) during PCR, depending on the initial amount of the target template in the reaction volume. There are several drawbacks in this approach: the correlation between the signal and the amount of DNA is not defined, and is presumed to be similar for all detection systems, which is not always a viable assumption. Efficiency of PCR is presumed constant for all duration of the reaction for all samples, which is not a precise assumption. Using one crossing level for all samples ignores possible variability between the samples, with the risk of not detecting the sample with slow growth of fluorescence. There are also instrumental effects on fluorescence: sensitivity of detector, electronic noise, etc. which are not taken into account. The amount of information available as a result of this analysis is very limited and the quality of the data (precision and accuracy of quantification) is often poor.

Another common approach is based on numerical derivatization of the fluorescence signal over time (amplification cycle number) (LCDA 3.0, Roche Diagnostics). In this method, threshold cycle is defined as a time point corresponding to the maximal value of the second derivative, i.e., at the point where the fluorescent signal starts growing from the baseline fluorescence. In the further development of this approach, numerical derivatization is substituted by derivatization of the analytical fit (spline function) to experimental data (LCDA 3.5, Roche Diagnostics), which results in improved accuracy of quantitative results. Although this approach is more objective, and, consequently, more stable than the crossing point method, it still carries the same drawbacks as the latter. It also requires significant editing of the primary fluorescent data to reduce background noise and "smoothen" experimental curves.

Another approach has been to incorporate a known internal standard into the initial sample to be amplified and analyzed. The amplification of the internal standard is used to calculate the initial quantity of an unknown target. However, it is required that the internal standard and the unknown target be amplified with essentially the same efficiency. See Eyre et al, Real-time gene quantification with internal standards, U.S. Patent Application Publication No. 2003/0104438, published June 5, 2003, filed Aug. 29, 2002, the contents of the entirety of which are incorporated by this reference.

One of the problems with development of new analysis approaches is the lack of general theoretical description of the processes during PCR and effects of these processes on the observed detection signal. Another problem is the lack of the well-defined correlations between the detected signals and the amounts of different species of nucleic acids in RTD-

PCR. These two problems are closely related to one another. BRIEF SUMMARY OF THE INVENTION

Embodiments of the invention include an apparatus for quantitatively determining an amount of nucleic acid in a sample. The apparatus includes a means for describing amplification of nucleic acids, as an iterative process, from one cycle to a next cycle in a nucleic acid amplification method.

Embodiments of the invention include a computer-readable medium programmed with the equation:

Xn+l = Xn + (l - Xn)f(x_n),

wherein x_n is a relative quantity of nucleic acid present at a given point of a first nucleic acid amplification cycle, wherein x_n+i is a relative quantity of nucleic acid present at the same given point of a second said nucleic acid amplification cycle, and wherein f(x_n) is a growth function based upon a kinetic model of a chosen method of amplification and a chosen detection chemistry. The computer-readable medium may be further programmed to receive data generated by the chosen detection chemistry and to iteratively fit the equation to the data.

Embodiments of the invention include a method for quantifying a target nucleic acid in a sample. The method may include acquiring amplification data from real-time detection ("RTD") of amplification of the target nucleic acid. A first iterative growth function may be fit to the acquired amplification data. The fitted first iterative growth function may be reconstructed to the beginning of the amplification reaction to calculate a relative initial target quantity in the sample.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates raw fluorescence data collected during real-time detection polymerase chain reaction ("RTD-PCR") of thirteen different samples.

FIG. 2A illustrates raw primer-dimer data from two samples.

FIG. 2B illustrates the primer-dimer data of FIG. 2 A after the data has been cut and scaled.

FIG. 3 A in the upper illustrates the raw data of FIG. 1 prior to being cut and scaled. FIG. 3 A in the lower panel illustrates the raw data of FIG. 1 after one embodiment of cutting and scaling the raw data. FIG. 3B illustrates one embodiment of functions fit to the cut and scaled data of the lower panel of FIG. 3 A.

FIGS. 3 C and 3D illustrate the first and second derivative, respectively, of the functions illustrated in FIG. 3B.

FlG. 4A illustrates the functions which serves as the polynomial basis for the spline functions of FIG. 3B.

FIGS. 4B and 4C illustrate the first and second derivative, respectively, of the functions illustrated in FIG. 4A.

FIGS. 4D and 4E illustrate embodiments of filtering processes used to identify data that did not show significant growth.

FIG. 4F illustrates the fit target data of FIG. 4E after one embodiment of scaling the filtered target data.

FIG. 5 illustrates one embodiment of system specific scaling for the SYBR GREEN I^® detection system.

FIG. 6A illustrates raw data collected using a hybridization probe detection system.

FIG. 6B illustrates the hybridization probe raw data after one embodiment of cutting and scaling the data.

FIG. 6C illustrates one embodiment of determining relative quantitative amounts of target DNA ("x_n") based on the cut and scaled data of FIG. 6B.

FIGS. 6D and 6E illustrate the derivatives of the data of FIG. 6C with respect to the fluorescence signal ("d_n") and with respect to the scaling factor ("s"), respectively, after a single iteration.

FIG. 6F illustrates one embodiment of a calculated relationship between d_n and x_n.

FIG. 7 illustrates one embodiment of estimated growth functions based on calculated relative initial quantities of target DNA.

FIG. 8 A illustrates one embodiment of estimated growth curves for the primer- dimer data of FIG. 2 A.

FIG. 8B illustrates one embodiment of estimated growth curves for target DNA data of FIG. 1.

FIG. 8C illustrates an estimated target DNA ("template") growth, estimated primer-dimer growth, estimated target DNA efficiency, and estimated primer-dimer efficiency. FIG. 9A illustrates a calibration curve generated using the SYBR GREEN I^® detection system and iterative growth functions.

FIG. 9B illustrates estimated target DNA growth and efficiency using the SYBR GREEN I^® data of FIG. 9A.

FIG. 9C illustrates the relative initial quantities used to generate the calibration curve ofFIG. 9A.

FIG. 1 OA illustrates a calibration curve generated using a prior art second derivative maximum method and the same SYBR GREEN I^® data used to generate the calibration curve of FlG. 9A.

FIG. 1 OB illustrates quantification verification of the second derivative maximum method data.

The upper left panel of FIG. 1 IA illustrates the raw data illustrated in FIG. 6 A. The upper right panel illustrates the scaled data illustrated in FIG. 6B. The bottom panel illustrates raw SYBR GREEN I^® negative data.

The upper left panel of FIG. 1 IB illustrates estimated target DNA growth based on the data of FIG. 1 IA. The upper right panel illustrates estimated primer-dimer growth based upon the data of FIG. 1 IA. The bottom left panel illustrates estimated target DNA efficiency over the course of amplification. The bottom right panel illustrates estimated primer-dimer efficiency over the course of amplification.

The upper left panel of FIG. 11C illustrates the calculated relative initial quantities that were used with known initial quantities to generate the calibration curve illustrated in the upper right panel. The bottom left and right panels illustrate respectively quantification verification and results using iteratively fit growth functions.

FIG. 12 illustrates in the upper left panel the same raw data illustrated in FIG. 1 IA. FIG. 12 illustrates in the upper right panel a calibration curve generated using a prior art second derivative maximum method. The bottom left and right panels illustrate respectively quantification verification and results using the second derivative maximum method data.

FIG. 13, in the upper left panel, illustrates raw fluorescence data from RTD-PCR using TAQMAN^® probes and lambda DNA with an internal control. The upper right panel illustrates just the fluorescence data from the lambda DNA growth. The bottom left and right panels illustrate respectively calculated target DNA efficiency and internal control efficiency. FIG. 14 illustrates a calibration curve generated using the data illustrated in the upper left panel of FIG. 13 and the direct method. FIG. 14 also illustrates the data used to generate the calibration curve and quantification verification and results based on that data.

FIG. 15 illustrates a comparison between one embodiment of a direct method and one embodiment of an indirect method using the data illustrated in the upper left panel of FIG. 13.

FIG. 16 illustrates a calibration curve generated using the indirect method. FIG. 16 also illustrates the data used to generate the calibration curve and quantification verification and results based on that data.

DETAILED DESCRIPTION OF THE INVENTION

Embodiments of the invention may apply to quantitative analysis of target nucleic acid amplification. In certain embodiments of the invention, each amplification reaction is presumed to be a multiplex reaction. Even with a single pair of primers there are at least two concurrent reactions: specific reaction (e.g., target nucleic acid amplification) and non-specific reaction (e.g., primer-dimer formation). Additionally, there may be a competitive nucleic acid (e.g., an internal control) competing with the target nucleic acid for primers. Every reaction in each amplification cycle, at least partially, competes with all other reactions (e.g., competition for primers, nucleotides, enzymes, Mg, hydration, hybridization, etc.). Competition is manifested in the amplification efficiency, even if one or more reactions are "detection silent" (i.e., increased competition reduces target nucleic acid amplification and thereby reduces the amplification efficiency). This, in general, results in varying amplification efficiency for each reaction in each amplification cycle.

Embodiments of the invention include acquiring amplification data from real-time detection ("RTD") of the amplification method. Hereinafter, Polymerase Chain Reaction ("PCR") will be primarily discussed as the amplification method. However, any discrete- time (e.g., cyclic) amplification method may be used, such as strand displacement amplification ("SDA"), single primer amplification ("SPA"), or helicase-dependent amplification ("HAD").

Amplification data, for the sake of example only, may include primary fluorescence data. For the sake of example only, amplification data may be generated with intercalating dyes, with labeled primers, labeled primers and probes, hybridization probes, or hydrolysis probes.

Embodiments of the invention are not limited by the type of nucleic acid amplified. Hereinafter, deoxyribonucleic acids ("DNA") are primarily referred to as the target nucleic acids; however, any nucleic acid that may be detectably amplified may be analyzed with embodiments of the invention. For example, RNA may be reversibly transcribed to form complementary DNA ("cDNA") and then the cDNA amplified.

Embodiments of the invention may include a means for describing amplification of nucleic acids, as an iterative process, from one cycle to a next cycle in a nucleic acid amplification method. The means for describing amplification of nucleic acids may take into consideration both target nucleic acid amplification and specific or non-specific amplifications that compete with target nucleic acid amplification.

The means for describing amplification of nucleic acids may include the equation:

wherein X_n is a relative quantity of nucleic acid present at a given point of a first nucleic acid amplification cycle, wherein x_n+i is a relative quantity of nucleic acid present at the same given point of a second said nucleic acid amplification cycle, and wherein f(x_n) is a growth function based upon a kinetic model of a chosen method of amplification and a chosen detection chemistry. The growth functions may be based upon a kinetic model for the particular amplification method used and the particular detection chemistry used. The growth functions may be iteratively fit to the amplification data. Relative initial target nucleic acid amounts may then be calculated using the fitted growth functions. The efficiency of target nucleic acid amplification and competing amplification may also be determined from the fitted growth functions.

The iterative growth functions may also be used to generate calibration curves. Known initial quantities of target nucleic acids may be amplified using RTD-PCR. Iterative growth functions may be fit to the RTD-PCR data and then relative initial quantities of the target DNA calculated. Calibration curves may be generated that are specific to a particular type of PCR apparatus and even specific to individual units. Additionally, calibration curves may be generated that are specific to particular primers and target DNA. In certain embodiments, it may be desirable to scale the raw amplification data prior to further analyzing the amplification data. Scaling may be accomplished by setting the maximum fluorescence intensity point of any of the samples (e.g., the top of the curve) equal to 1 and the minimum fluorescence intensity equal to zero. The fluorescence data may then be scaled between the maximum and minimum points.

Fluorescence intensity may be correlated with the concentration of the target nucleic acids, and within a broad range of concentrations (early-to-mid cycles of PCR) this correlation is monotonic. Thus, the scaling may result in a transformation of the detected fluorescence intensity into relative concentrations of amplified target nucleic acid if the maximum fluorescence intensity corresponds to the maximum target nucleic acid concentration. After scaling, the growth functions may then be iteratively fit to the scaled data.

In certain embodiments, the means for describing amplification of target nucleic acids may account for the effects of competition for target nucleic acid amplification with the amplification of non-specific products, such as primer-dimers. Accounting for the effects of competition may involve amplifying a non-specific material in the absence of target nucleic acids, wherein the non-specific material is ordinarily present when the target nucleic acid is present in a sample. For example, the primers to be used with the target nucleic acid, may be run through RTD-PCR without any target nucleic acid present. Therefore, primer-dimers would be amplified.

Next, non-specific amplification data from real-time detection ("RTD") of amplification of the non-specific material may be acquired (e.g., fluorescence data collected). Growth functions may then be iteratively fit to the acquired non-specific amplification data. The fitted growth functions may be reconstructed to calculate a relative initial quantity of non-specific material. The fitted non-specific growth functions may be used with fitted target nucleic acid growth functions (i.e., growth functions fit to target nucleic acid amplification data) to construct new target nucleic acid growth functions and new non-specific material growth functions. The new target nucleic acid growth functions may be included in the means for describing amplification of the target nucleic acids.

Embodiments of the invention may include amplifying a specific competing nucleic acid of known initial quantity while amplifying the target nucleic acid. In certain embodiments, the means for describing amplification of target nucleic acids may account for the effects of competition for target nucleic acid amplification with the amplification of competing nucleic acids, such as internal controls. Accounting for the effects of competition may involve amplifying a competing nucleic acid in the absence of target nucleic acids. For example, an internal control to be used with the target nucleic acid, may be run through RTD-PCR without any target nucleic acid present. Therefore, the internal control would be amplified.

Next, amplification data from real-time detection RTD of amplification of the competing nucleic acid may be acquired (e.g., fluorescence data collected). Growth functions may then be iteratively fit to the acquired competing nucleic acid amplification data. The fitted growth functions may be reconstructed to calculate a relative initial quantity of the competing nucleic acid. The fitted competing nucleic acid functions may be used with fitted target nucleic acid growth functions (i.e., growth functions fit to target nucleic acid amplification data) to construct new target nucleic acid growth functions and competing nucleic acid growth functions. The new target nucleic acid growth functions may be included in the means for describing amplification of the target nucleic acids.

Internal controls may be useful in medical diagnostic applications where it may be undesirable to use external calibration. The use of internal controls may increase the accuracy of calibration curves. Internal controls may be used in non-medical diagnsostic applications as well. The internal control may be a known amount of a nucleic acid that will compete with the target nucleic acid for amplification.

Internal control data may be used in a "direct" method or an "indirect" method. The direct method utilizes signals from both target DNA and internal control during target DNA amplification. The indirect method only uses signals from the internal control during target DNA amplification. The direct and indirect method may also be used with primer-dimer growth data or any other competing growth data.

In certain embodiments, the means for describing amplification of target nucleic acids may be used to analyze fluorescence intensity data from RTD-PCR. RTD-PCR may use fluorescent detection to identify when amplification has occurred. As the amount of target nucleic acid is increased by amplification, the intensity of fluorescence increases. The fluorescence is tracked cycle by cycle. Generally, RTD-PCR proceeds for about 40 cycles. After about 40 cycles the amount of DNA may have stopped amplifying, so that more cycles do not result in substantially more DNA. The fluorescence per cycle (hereinafter the "raw amplification data") may be analyzed with embodiments of the invention.

The raw amplification data output by the RTD-PCR system may be in terms of fluorescent intensity per cycle. Numerous samples of a given target nucleic acid may be amplified. Each sample may be individually amplified in a separate RTD-PCR tube.

The raw amplification data of each sample may be analyzed at the same time with embodiments of the invention. FIG. 1 illustrates the raw fluorescence data collected during RTD-PCR of thirteen different samples. FIG. 1 represents the raw amplification data of each sample as a curved line for convenience. However, it should be understood that each sample may be represented as a collection of points, without interconnecting lines between the points. It should also be understood that different detection chemistries generate amplification data at different points during each cycle. For example, when an intercalating dye is used, then fluorescence intensity will be greatest when target nucleic acids are hybridized as double-stranded DNA. Thus, fluorescence intensity will be greatest at the end of each cycle (if the cycle starts with anneal step this is not true). In contrast, when hydrolysis probes, such as TAQMAN ^® are used, then fluorescence intensity will be greatest at the end of the primer extension phase. FIG. 1 represents the peak fluorescence per cycle for each of the samples. The increase in fluorescence per cycle is proportional to the increase in the amount of target nucleic acids in each of the samples. The principal difference between TAQMAN ^® and the rest of the chemistries is that in the former case the signal is accumulated from cycle to cycle (i.e., the signal is "indestructible" because hydrolysis of the probes is irreversible), while for the latter cases the signal is generated in each cycle de novo (i.e., starting with zero at the melt point).

Embodiments of the invention may be incorporated in an apparatus for either conducting RTD-PCR or for analyzing data from RTD-PCR. Embodiments of the invention may be included in a software program for analyzing RTD-PCR data. Embodiments of the invention may also be used as an improvement or upgrade to existing RTD-PCR analysis software. Embodiments of the invention include a method of doing business where RTD-PCR data analysis is performed as a service.

In one embodiment, the growth functions are second degree polynomials. The second degree polynomials may then be fit to the amplification data. Examples 1 and 2 provide a theoretical basis for fitting second degree polynomials as the growth functions. EXAMPLES Example 1

In this example, it was assumed that when primer binds with a single-stranded target DNA (hereinafter "ss target DNA"), a double-strand target DNA (hereinafter "ds target DNA") is always produced by the end of the cycle. It was also assumed that the only thing prohibiting primer binding to ss target DNA is the reannealing of complementary ss target DNA to each other.

*L= -kfτ-k₂τ\ (1)

^ - -k PT (2)

— = -kfT+ kJ¹ (3) dt

In equations (1) - (3), T is ss target DNA, P is primer, and Il is product (ds target DNA). Here reannealing is at a rate proportional to Ti, which is valid if the replication of target DNA is symmetric (i.e., T= T*, where T* is the complementary ss target DNA strand). Equations (1) and (2) may be rewritten as:

dT . T — = 1 + a— , (4) dP P

In equation (4), a = ki l

Equation (4) is a linear differential equation, and hence solvable with an integrating factor, yielding:

T = KP³ + — (5)

\ - a

TQ and PQ are the initial amounts of ss target DNA and primer at lhe beginning of a given cycle, respectively. Assuming the reaction runs to completion for that cycle, K may be solved from equation (5) yielding: (6)

At the end of the cycle, the amount of primer left (Pl) satisfies:

P" P₁

Da -)+ (7)

M) 1-fl 1-fl

Assuming the overall conservation law OfT₊P ₊ II = IIo (which is only true for symmetric reactions) to determine T], which is the amount of ss target DNA at the beginning of the next cycle (since II = 0 at the beginning of every cycle, although the beginning point of the cycles may be changed), then:

T_{1 +}P₁=II₀, T_{0 +} P₀ = II₀ (8)

and therefore:

oJ (9)

Equation (9) may also be written as:

j-ⁱ-o-*⁾! l.l + ( ifl- *2)Λb J r ^(IO)

In Equation (10), x = Tl FI₀. Equation (10) may also be written as:

X₁= ^X ₀ +(1-^X ₀) /(^X ₀) (H) where f(x) is 1 - ((I - x) I (1 + (α - 2)x))^1/((H) and f(0) = 0. A plot of Xi as a function of x₀ for a in the range of 10 gives curves similar to the SYBR GREEN I^® amplification data used in Examples 3 and 4. Example 2

The following is another example of deriving a growth function based upon a theoretical model. In this example, it was assumed that there is an additional reaction between the primers in Example 1 and a probe, and that this reaction precludes other reactions during a given cycle.

— = -k^PT- k^ - k^RT, (12) dt dP ÷ ^U dt = -KPT, (13)

— = -k_tPT+ k_}T². (15) dt ' ²

In equations (12) and (14) R is probe. It is assumed that fluorescent signal is directly proportional to the amount of bound probe, so that signal (S) is

≤ .yr (16) dt

and so that S + R = RQ. Equations (12) - (16) may be viewed as functions of primers:

^ - 6 * . (18) dP P

In equation (17), a = k^ I k_\ as in Example 1 and b — k$ I k\. Remarkably, this system of equations may be solved exactly, such that: T = K,P° - K₂P", (19) a - 1 a - b

R = K,P^b (20)

The constants K_\ and .K₂ may be determined from initial data according to:

Substituting equation (21) into equations (19) and (20) results in:

o ₌ (ΔV _{(rβ +} .A. ₊ ^)-Λ__^ΓΔY ₍₂₂₎

( f P \^b λ ύ, - K₀ 1 - — ^■ . ^j;

At the beginning of each cycle there is no ds target DNA, so for a symmetric reaction (which does not occur if probe only binds to one of two complementary ss target DNA strands). Therefore, T_\ + P₁ = IT₀, T₀ + P₀ = U₀, and

ώ C_| _- Λ r,₀ 1 I - I 1 - *1 ] I .

In equations (24) and (25), x = Tl U₀ and /c = (b(a - 1) / (α - fc))*(i?o / U₀ ) A: and K may be used to show the effects of competition. If b = 1, which is not a bad approximation when the primers and the probes are about the same length, then:

where β = (1 / (1 + K)) > 1, and the signal is:

The signal is an indicator of ss target DNA at the beginning of a given cycle. In the case that b = 1, the equations have the form of:

X₁ = X_{0 +} (I - xo)f(βx_o), S₁ = R_of(βxo), (28)

where f(x) = 1 - ((I - x) I (1 + (a - 2) x))^{m ' υ}. This is exactly the same function found for simple growth without competition. In equation (28), β - (IIo / ( IT₀ + Ro)) > 1 β shows the relationship between competition and growth and shows the effect of competition by the probe.

If b ≠l, then the solution is not as elegant. In that case:

0 - p(βxoW - ' + (1 - /?)(**- '- 1) - g(βx₀), (29)

S₁ = R₀ (I - X") . (30)

where X= (1 - X₁) I (1 - ^₀), p(x) = 1 + (α - 2) *), and q(x) = \ -x. A' may be solved for by finding:

[ p(βxo) P(βxo))

where:

so that: X₁ = X_{0 +} (I - X₀)F [³^r , -¹T^-) S, = R_n(I - X^b). (33)

K P(βXo) P(βxo))

The solution when b ≠\ is not as elegant as the solution when 6 = 1. Equation (33) may be iteratively fit to a target DNA growth function to solve for X. For example only, if after fitting equation (33) to the target DNA growth function the following function results,

Xn ₊ 1 = X₁, + ^X»Q- - X_n)O - ax_*), (34)

with 0 < α< 1. For this growth curve, fix) defined above is simply:

f(x) = x(\ - ax), (35)

so that the signal "S" is given by (take b = 1 and β =1), and then:

S = R₀(I - x(l - ax)). (36)

For 0 < a < Vi, this is a monotone increasing function of x\ however, for Vi < a < 1 this function is biphasic, increasing for small x and decreasing for large x. Applying this function to PCR amplification, the signal will increase during the initial cycles of target DNA amplification and decrease as the reaction nears saturation. Thus, this function may be used to describe to PCR. Thus, a theoretical justification, based upon the kinetics of competition, has been established for fitting a second degree polynomial to the signals generated during RTD-PCR.

Equations (10) and (26) are only examples of kinetic models including growth functions. Examples 1 and 2 include the assumptions discussed above. Variations in assumptions will result in different growth functions. Additionally, variations in detection chemistry may result in different growth functions. Different detection systems generate different functional dependences of fluorescence intensity versus target concentrations. These functions are combinations of amplification efficiencies and detection efficiencies (hence they are detection system specific). Data generation and processing with internal controls is similar to when primer- dimer competition is being analyzed. Kinetic models and growth functions may be generated, similar to that in Examples 1 and 2, which account for competition between the internal control and the target DNA. The primer-dimer kinetic equations in Examples 1 and 2 may be replaced with internal control equations. The resulting growth functions are still second degree polynomials, the same as when primer-dimer kinetic equations are used. Therefore, there is a theoretical basis for fitting a second degree polynomial to amplification data when internal controls are present. Therefore, equation (11) using an iteratively fit second degree polynomial as the growth function may be validly used to calculate initial quantities of target DNA when an internal control is also present.

It should be understood that primer-dimers may still form during target DNA and/or internal control amplification. The competition from primer-dimer formation may be lumped in with the competition from the internal control. The data collected from RTD-PCR with only the internal control may be considered to show both internal control DNA amplification as well as primer-dimer amplification. Thus, the growth curves iteratively fit to the internal control data may also be used to take into account the effects of primer-dimer amplification on target DNA amplification. Likewise, any other competing factor during target DNA amplification may be lumped into the competing effects of either internal control amplification or primer-dimer amplification.

The above models may form the basis for quantification analysis. Additionally, other competing reactions may also be added to or substituted into the model in a similar manner.

The invention is further described and explained with the aid of the following illustrative Examples. Example 3

Lambda DNA, approximately 150 base pairs in length, was amplified using a RTD-PCR Roche Applied Science Lightcycler^® 2.0. Amplifications were conducted using SYBR GREEN I^®, hybridization probes, and TAQMAN^® probe detection systems. Each sample was amplified in a separate PCR reaction using the same type of primers. 3.1 Primer-dimer analysis

First, primer-dimer amplification data (hereinafter "pd data") was analyzed for eventual use in accounting for the effects of competition between primer-dimer amplification and lambda DNA amplification. Primer-dimers, a non-specific material, were amplified in the absence of the target DNA. The pd data includes data from two samples run in individual experiments. Forty PCR cycles were conducted with primers present, but without any Lambda DNA present, in the PCR tube. Primer-dimer formation was monitored cycle-by-cycle by detecting fluorescence of SYBR GREEN I^®. FIG. 2A depicts raw pd data from the two samples.

FIG. 2B depicts the pd data after the data has been cut and scaled. The first four cycles of the pd data were removed and the cycles renumbered to start from zero. The raw pd data was scaled so that the highest curve has a maximum value of 1.0. All other curves were scaled proportionally to this curve. Next, scaling factors were calculated using the first derivative. Curves that have not started growing yet were not scaled up, whereas curves that had reached plateau were scaled to 1. Curves in between were scaled up proportionally.

Primer-dimer samples (hereinafter "pd samples") that did not have any growth over the amplification cycles were removed from the pd data. The second derivative of each pd sample curve was used to determine whether the pd sample had growth or not. Pd samples having a second derivative less than 1 were considered to not have any growth and were removed from the scaled pd data.

Fake data was added to the raw pd data to test the effectiveness of the data scaling. The linear line in FIG. 2A depicts the fake data used to test the data scaling. The range of the fake data was set equal to the range of the raw pd data. FIG. 2B depicts that the fake data was effectively scaled to zero. 3.2 Analyze target nucleic acid amplification data

First, the raw target nucleic acid amplification data was converted from fluorescent intensity to relative concentrations. This was accomplished via several scaling and processing steps. First, rough scaling of target nucleic acid data (hereinafter "target data") was conducted regardless of the detection system used. Next, detection system specific scaling was conducted. Then, a final processing step was performed that was the same for all detection systems. 3.2.1 Rough Scaling

First, the first four cycles of the raw target data were removed. Raw SYBR GREEN I^® target data is depicted in the upper half of FIG. 3A. The background noise was then subtracted. For the SYBR GREEN I^® detection system it was assumed that the - background noise was zero, because the detection signal is usually very strong. For the other detection systems it was assumed that the background noise was the average of the negative samples. The cut target data was shifted to start from 0. The cut target data was scaled so that the maximum value for the highest curve is 1 and all of the other curves were scaled proportionally to this curve. Each curve was scaled to start at 0. The cut SYBR GREEN I^® target data is depicted in the lower half of FIG. 3 A.

Next, scaled target data was filtered out that did not show growth. This was done by fitting a quintic spline function to the scaled target data. FIG. 3B depicts fit SYBR. GREEN 1^® target data ("fit"). The first derivative of the fit target data was then calculated. FIG. 3C depicts the first derivative of the fit SYBR GREEN I^® target data ("fltd"). Next, the second derivative was taken. FIG. 3D depicts the second derivative of the fit SYBR GREEN I^® target data ("fitdd"). Next, three functions were generated - phi, phid, and phidd. Phi is the polynomial basis for the quintic splines of the scaled target data and is illustrated in FIG. 4A for the SYBR GREEN I^® target data. Phid is the first derivative of phi and phidd is the second derivative of phi. Phid and Phidd are depicted for SYBR GREEN I^® target data in FIGS. 4B and 4C, respectively.

Next, "significant curvature" was looked for. A straight line approximating each fit target data curve was calculated, such as depicted in FIG. 4D for SYBR GREEN I^® target data. The baseline (i.e., fluorescence without any amplification) was adjusted using a straight line. First, the cycle for each sample where fitdd was at its maximum was identified and called J. This was the cycle where template growth was starting to show. The fluorescence value 4 cycles earlier were then used, since at this cycle, J-4, primer- dimer growth should not be visible yet. Therefore, growth should only be from target nucleic acid amplification. A straight line was then fit to the data points from cycle 1 to cycle J-4 for each sample. If the straight line fit the data well, then the sample did not show target nucleic acid growth. The deviation of each line from the fit target data was also calculated (see, e.g., FIG. 4E for SYBR GREEN I^® target data). If the deviation was < 0.001, then the curve was considered not to show template growth. The fit target data was scaled using the deviations from the baseline as a scaling factor. The fit target data was scaled between 0 and 1 depending on the stage (early, middle, late of amplification reaction).

Additionally, all negative values in the fit target data were replaced with zeros. FIG. 4F depicts the fit target data after scaling and filtering for SYBR GREEN I^® target data. 3.2.2 Detection System Specific Scaling

Rough scaled SYBR GREEN I^® target data was scaled using the maximum value of the first derivative fitd (see, e.g., FIG. 3C). The first derivate of the last cycle was compared to the cycle where the first derivative was maximal. The function bt was used to scale the target data with regard to the degree of completion of the amplification reaction. The behavior of bt and scale is illustrated in FIG. 5. If at the final cycle the curve had not yet started to grow, the data was not scaled up. If at the final cycle the curve had reached plateau, the max value was set to one. Otherwise, the data was scaled up proportionally. The scaling was used to generate a first guess for later scaling processes.

Regarding the hybridization probes detection system, FIGS. 6A and 6B illustrate hybridization probe raw data and hybridization probe rough scaled data, respectively. The following was used to determine the scaling factor between fluorescence signal and copy number:

X_n+1 = x_n + s*(l - X_n)* d_π = X_n + (1 - x_n)*f(x_n), (37)

where f(x_n) = s* d_n, X_n = copy number, d_n = fluorescence signal, and s = scaling factor. The initial guess for s was 1 , which is true when PCR amplification has reached plateau. Once s was calculated, then X_n was calculated from equation (37). Similarly, the derivatives of X_n with respect to d_n and s were also calculated, s was optimized in a bisection loop with 35 turns and determined to equal 0.1541. FIG. 6C illustrates X_n after the first turn of the loop. FIG. 6D illustrates the derivative of x_n with respect to d_n after the first loop. FIG. 6E illustrates the derivative of X_n with respect to s after the first loop. Next, another loop with 200 turns was entered. The purpose of this loop was to further optimize s, which will be used for scaling each cycle. Result from the last rum in the second loop resulted in s = 0.1541. Therefore, s and thereby X_n had not changed since the first loop. FIG. 6F illustrates the calculated relationship between d_n and x_n.

For target data from the TAQMAN^® detection system, no further processing was performed after the rough scaling as there is a linear relationship between target nucleic acid amplification and detection signal. Therefore, x_n was immediately set equal to d_n. 3.2.3 Final Scaling After the detection-system specific processing was done, then the following processing was made: First all negative numbers were replaced by small nonzero numbers, (i.e., 1E-5). Next, the quintic spline used during rough scaling was used to get a new fit of the data for the next step. Small values were assumed to be unreliable and were filtered out (i.e., values less than 0.1). 3.3 Polynomial Fit and Scaling Factor Determination

Polynomial fit and scaling factor determination were the same for all detection systems. First, a check was made to make sure that each sample had a value that exceeded noise. If this was not the case, then the process was stopped and all initial values were set to 0. If at least one sample had values that exceed noise, then the fitting process was continued. Variables were initiated according to the following:

X₁₁₊I = X_n + Xn (1 - S* X_n)G(S* X_n) (38)

where G is the quotient of two polynomials: g_n/g_d, where g_n is a second degree polynomial and gd is a zero degree polynomial. Therefore, gd was set to 1, which means G equals g_n. At first, in the calculations the scaling factor "s" was kept constant while the coefficients of g_n were being optimized. Therefore,

gn = A*(l - X₁)² + B*(l - X₁)* x, + C* x, ². (39)

The efficiency of the PCR reaction was also calculated.

Efficiency = 1 + Pj* g_n / g_d, (40)

Where P_j = 1-x,. Therefore,

x(j+l) = x(J) + X(J)-O - S-X(J))* gn / gd (41)

= x(j) + xj*Pj* g_n / gd (42)

= xj*Eff (44) This means that Efficiency = x(j+l)/xj. This allows the efficiency to vary while g_n and the coefficients of g_n vary from cycle to cycle during the PCR reaction. The variation in efficiency from cycle to cycle was also calculated. Next, steps were introduced into the coefficients of gj and the coefficients iteratively optimized to fit the x_n data. The coefficients of g_d were held constant. The step was designed to not start varying until the third iteration. 1500 iterations were run.

Next, appropriate scaling factors were calculated. g_n was kept constant while the scaling factors were optimized to fit the x _n data.

Next, the starting point for each data set was calculated (i.e., the relative initial concentration of target DNA in each sample) using bisection, gn and the efficiency of amplification were calculated using equations (39) and (40) and also:

dgnP = 2*A*(1 - Xi) + B* x; (45) dgnx = B*(l - Xj) + 2*C* X₁ (46)

where dgnP is the derivative of gn with respect to P, which equals (1 - Xi) in this case, and dgnx is the derivative of g_n with respect to x,.

The estimated growth for each target DNA sample was calculated for each cycle, using the previously calculated relative initial quantity of target DNA at cycle 0 as the starting point. The relative copy number (i.e., relative number of target DNA copies) was then calculated one cycle at a time using equations (39), (40), (45), and (46). For the hybridization probe data, the result for target DNA was stored in the variable z and the result for the primer-dimers was stored in the variable zp, both illustrated in FIG. 7. The fit of the calculated z data to the x_n data was then calculated. 3.4 Account for the effects of competition

It was assumed that a portion of the fluorescent signal detected during PCR was due to primer-dimer amplification and not the result of target DNA amplification. Therefore, the initial quantity of target DNA calculated based upon the raw data depicted in FIG. 3A would be skewed. The primer-dimer data collected during PCR of just the primers (see FIGS. 2A and 2B) was available; however, this signal could not just be subtracted from the raw data signal. When only primers are present during PCR, then there is not competition between primers and target DNA for primers. Therefore, the growth of primer-dimers may be artificially high when target DNA is not present during amplification.

First, estimated growth curves were calculated for the primer-dimer data depicted in FIG. 2A. Next, estimated growth curves were calculated for the raw data depicted in the upper panel of FIG. 3 A. The estimated growth curves are illustrated in FIGS. 8A and 8B, respectively. Initial quantities for the estimated growth curves were calculated using bisection.

Based on the calculated initial quantities of primer and target DNA, primer-dimer growth (yj) and target DNA growth (xj) in each cycle were then calculated using equations (39), (40), (45), and (46). The total growth was contained in variable z. For primer dimer growth, Pj = 1-zj-xj. For target DNA growth, Pj = 1-yj-xj. FIG. 8C depicts the estimated template ("target DNA") growth, estimated primer-dimer growth, estimated template efficiency, and estimated primer-dimer efficiency for the hybridization probe data.

Table 1 lists the calculated initial quantities of target DNA per sample. Before optimization is the calculated initial quantity prior to taking into account the effects of competition. After optimization is the calculated initial quantity after such calculations.

Table 1

Example 4

Lambda DNA was amplified using a RTD-PCR Roche Applied Science Lightcycler ^®. Amplifications were conducted using SYBR GREEN I^®, hybridization probe, and TAQMAN ^® probe detection systems. This example is similar to Example 3. One of the differences between this example and Example 3 is that a SYBR GREEN I^® positive sample was added as a reference for the initial raw scaling of primer-dimer negative samples, when a detection system other than SYBR GREEN I^® was used. The primer-dimer analysis was similar to that of Example 3. Fake data was added to the raw pd data to test the effectiveness of the data scaling. However, the range of the fake data was changed for detection systems other than SYBR GREEN I^®. The range was set equal to the range of the SYBR GREEN I positive samples that were included in the experiment setup. Example 5

This example illustrates advantages of embodiments of the present invention over prior art quantitative analysis methods. FIG. 9A illustrates a calibration curve generated using SYBR GREEN I^® target data and iterative growth functions. FIG. 9B illustrates estimated target DNA growth and efficiency using the SYBR GREEN I^® detection system and a competition model. FIG. 9C illustrates the relative initial quantities calculated with primer-dimer correction that were used to generate the calibration curve of FIG. 9A. FIG. 9C also illustrates the relative initial quantities calculated without primer-dimer correction {i.e., without accounting for the effects of primer-dimer competition). FIG. 1OA illustrates a calibration curve generated using the prior art second derivative maximum method and the same SYBR GREEN I^® data used to generate the calibration curve of FIG. 9A. FIG. 1OB illustrates quantification verification of the second derivative maximum method data.

Certain embodiments of the invention maybe used to generate calibration curves that are more precise at lower target nucleic acid concentrations than with prior art analysis methods. For example, with the prior art second derivative maximum method of the SYBR GREEN I^® data depicted in FIG. 1OA, it would be difficult to distinguish between initial samples sizes that are 10⁵ or smaller. In contrast, using the same data, a calibration curve generated using embodiments of the invention allows for initial sample sizes to be distinguishable to a much lower range.

The upper left panel of FIG. 1 IA illustrates the raw data illustrated in FIG. 6A generated using a hybridization probe detection system. The upper right panel illustrates the scaled data illustrated in FIG. 6B. The bottom panel illustrates raw SYBR GREEN I^® negative data (i.e., raw primer-dimer data). FIG. 11 B illustrates estimated target DNA growth and efficiency and estimated primer-dimer growth and efficiency using iteratively fit growth functions. The upper right panel of FIG. 11C illustrates a calibration curve generated using iterative growth functions fit to the hybridization probe data. The upper left panel of FIG. 11C illustrates the calculated initial quantities that were used to generate the calibration curve. The bottom panels illustrate quantification verification and results using iteratively fit growth functions. FIG. 12 illustrates a calibration curve generated using the raw hybridization probe data illustrated in FIG. 1 IA and the second derivative maximum method. FIG. 12 also illustrates quantification verification and results using the second derivative maximum method. Example 6

This example uses analysis of competitive nucleic acids rather than analysis of primer-dimers. FIGS. 13-16 illustrate the use of internal controls during target DNA amplification. FIG. 13 illustrates use of the direct method. FIG. 13, in the upper left panel, illustrates raw fluorescence data from RTD-PCR of lambda DNA ("target DNA") with an internal control. TAQMAN^® hydrolysis probes were used for detecting target DNA and internal control amplification in a two color experiment. The target DNA signal was at a distinguishable wavelength from the internal control signal. The upper right panel illustrates just the fluorescence data from the target DNA growth. FIG. 13 also illustrates calculated target DNA efficiency and internal control efficiency. FIG. 14 illustrates a calibration curve generated using the data illustrated in FIG. 13 and the direct method. FIG. 14 also illustrates the data used to generate the calibration curve and quantification verification and results based on that data.

FIG. 15 illustrates a comparison between the direct method and the indirect method using the data illustrated in FIG. 13. The dashed lines are theoretical curves calculated with embodiments of the invention. The solid lines represent the raw data collected. FIG. 16 illustrates a calibration curve generated using the indirect method. FIG. 16 also illustrates the data used to generate the calibration curve and quantification verification and results based on that data.

Specific embodiments have been shown by way of example in the drawings and have been described in detail herein. However, it should be understood that the invention is not intended to be limited to the particular forms disclosed. Rather, the invention is to cover all modifications, equivalents, additions, and alternatives falling within the spirit and scope of the invention as defined by the following appended claims. REFERENCES

Eyre et al., Real-time gene quantification with internal standards, U.S. Patent Application Publication No. 2003/0104438, published June 5, 2003, filed Aug. 29, 2002.

Lay and Wittwer. (1997). Real-time fluorescence genotyping of factor V Leiden during rapid cycle PCR. Clin. Chem., 43: 2262-2267.

Nazarenko et al. (2002). Effect of primary and secondary structure of oligodeoxyribonucleotides on the fluorescent properties of conjugated dyes. Nucleic Acids Res., 30, 2089-2095.

Parkhurst and Parkhurst. (1995). Donor-acceptor distance distributions in a double-labeled fluorescent oligonucleotide both as a single strand and in duplexes. Biochemistry, 34: 293- 300.

Tyagi and Kramer. (1996). Molecular beacons: probes that fluoresce upon hybridization. Nat. Biotechnoi, 14: 303-308.

Woo et al. (1998). Identification of pathogenic Leptospira by TaqMan probe in a LightCycler. Anal. Biochem., 256: 132-134.

Claims

CLAIMSWhat is claimed is:

1. ^■ An apparatus for quantitatively determining an initial amount of nucleic acid in a sample, said apparatus comprising: a means for receiving amplification data from a nucleic acid amplification method; and a means for describing amplification of nucleic acids, as an iterative process, from one cycle to a next cycle in a nucleic acid amplification method.

2. The apparatus of claim 1, wherein said nucleic acid amplification method comprises polymerase chain reaction ("PCR").

3. The apparatus of claim 1 , wherein said nucleic acids comprise deoxyribonucleic acid ('ONA").

4. The apparatus of claim 1, wherein said means for describing amplification of said nucleic acids comprises means for describing amplification of target nucleic acids and at least one non-specific material.

5. The apparatus of claim 4, wherein describing amplification of at least one said non-specific material comprises describing amplification of primer-dimers.

6. The apparatus of claim 1, wherein said means for describing amplification of said nucleic acids comprises means for describing amplification of target nucleic acids and at least one specific competing nucleic acid.

7. The apparatus of claim 6, wherein said at least one specific competing nucleic acid comprises an internal control.

8. A computer-readable medium programmed with the equation.

X_n+I = X_n ⁺ O - Xn)f(Xn), wherein X_n is a relative quantity of nucleic acid present at a given point of a first nucleic acid amplification cycle, wherein x_n+ι is a relative quantity of nucleic acid present at the same given point of a second said nucleic acid amplification cycle, and wherein f(x_n) is a growth function based upon a kinetic model of a chosen method of amplification and a chosen detection chemistry.

9. The computer-readable medium of claim 8, wherein f(x_n) is a second-degree polynomial.

10. The computer-readable medium of claim 8 , further comprising programming to iteratively fit said equation to real-time detected ("RTD") amplification data.

11. The computer-readable medium of claim 8, wherein ^x_n) is 1- ((I - X_n) / (l+(α - 2) X_n))"*"^"0, wherein a = k₂/

from equations:

d\\

-k_}PT+ k₂T² (3) dt ^'

wherein T is single-stranded target nucleic acid, P is primer, and II is product double-stranded target nucleic acid.

12. The computer-readable medium of claim 8, wherein f(x_n) is 1- ((I - /3x_n)/(l + (a - 2) jSx_n))^{I/(a '} °, wherein a = k₂ /k_u wherein b = k₃/ k_it wherein /3 = (1 / (1 + K)) > 1, wherein K = (b(a-l)/(a-b))*(R₀/ U₀), from equations: 4L = -^ - Ic₂V - Ic₁RT, (12) at

^- = - k_λPT, (13) at

^- = - Ic₁ RT, (14) at

— = -k_xPT+ k₂T². (15) dt wherein T is single-stranded target nucleic acid, P is primer, II is product double- stranded target nucleic acid, R is probe, Ro is probe at the beginning of an amplification cycle, and ITo double-stranded target nucleic acid at the beginning of said amplification cycle.

13. A method for quantifying an initial target nucleic acid in a sample, the method comprising: acquiring amplification data from real-time detection ("RTD") of amplification of the target nucleic acid; fitting a first iterative growth function to the acquired amplification data; and reconstructing said fitted first iterative growth function to the beginning of the amplification reaction to calculate a relative initial target quantity in said sample.

14. The method according to claim 13, further comprising determining an actual initial target quantity in said sample using said relative initial target quantity and a calibration curve.

15. The method according to claim 13, further comprising scaling the acquired amplification data prior to fitting the iterative growth function to the acquired amplification data.

16. The method according to claim 13, further comprising calculating, using said fitted first iterative growth function, relative efficiencies of target nucleic acid amplification over the course of the amplification reaction.

17. The method according to claim 13, further comprising generating amplification data with intercalating dyes, with labeled primers, labeled primers and probes, hybridization probes, or hydrolysis probes.

18. The method according to claim 13, further comprising quantifying said target nucleic acid in multiple samples at the same time.

19. The method according claim 13, further comprising: amplifying a non-specific material in the absence of said target nucleic acid, wherein said non-specific material is ordinarily present when said target nucleic acid is present in a sample; acquiring non-specific amplification data from real-time detection ("RTD") of amplification of said non-specific material; and fitting a second iterative growth function to said acquired non-specific amplification data.

20. The method according to claim 19, further comprising constructing a target nucleic acid growth function using said first fitted iterative growth function and said second fitted iterative growth function.

21. The method according to claim 13, further comprising amplifying a specific competing nucleic acid of known initial quantity while amplifying said target nucleic acid.

22. The method according claim 21 , further comprising: amplifying a specific competing nucleic acid in the absence of said target nucleic acid; acquiring specific amplification data from real-time detection ("RTD") of amplification of said specific competing nucleic acid; and fitting a third iterative growth function to said acquired specific competing nucleic acid amplification data.

23. The method according to claim 22, further comprising constructing a target nucleic acid growth function using said first fitted iterative growth function and said third fitted iterative growth function.