Method for determining the concentration of a fluorescent and/or fluorescence-labeled analyte, and calibration method for preparing such determination

Abstract

A method for determining the concentration of a fluorescent and/or fluorescence-labeled analyte, and to a calibration method for preparing such determination, for use in the field of biological and environmental analysis in order to improve the accuracy of concentration determination, comprising the following steps: performing fluorescence measurements for calibration samples that have predetermined concentrations of a plurality of fluorescent and/or fluorescence-labeled reference analytes R.sub.j that differ from each other by values m of a diffusion measure characterizing the diffusion of the reference analyte, in order to determine the values i of a concentration-dependent parameter I; establishing functions F.sub.j(c)=i which describe the dependence of the parameter I on the concentration; determining the values of a slope parameter a for the respective reference analyte as a derivative of the respective function at c=0; determining the values m.sub.j of the diffusion measure for the reference analytes; establishing the dependence of the slope parameter a on the diffusion measure by a function E(m)=a; determining the value a.sub.sample specific to the analyte using the value m.sub.sample of the diffusion measure and the function E(m)=a; establishing an analyte-specific function F.sub.sample(c)=i; performing fluorescence measurements for the analyte and determining the concentration of the analyte using the value i of the concentration-dependent parameter I and the inverse function F.sup.−1.sub.sample(c).

Claims

1. A calibration method suitable for preparing determination of the concentration of a fluorescent and/or fluorescence-labeled analyte of a sample P.sub.sample, comprising the steps of: providing a plurality N of fluorescent and/or fluorescence-labeled reference analytes R.sub.j (j=1, . . . , N) which differ from one another by values m.sub.j of a diffusion measure M increasing with j; providing at least one calibration sample P.sub.j,v (v=1, . . . , N.sub.j) for each one of the reference analytes R.sub.j, each having a predefined value c.sub.j,v of concentration C of the reference analyte R.sub.j; performing fluorescence measurements for the calibration samples P.sub.j,v and determining a respective value i.sub.j,v of a concentration-dependent parameter I from the fluorescence measurement for the respective calibration sample P.sub.j,v; establishing a dependence of the values i of the concentration-dependent parameter I on the values c of the respective concentration C in the calibration samples P.sub.j,v by a respective calibration function that is obtained by adequate fitting to the value pairs (i.sub.j,v, c.sub.j,v) for the respective reference analyte R.sub.j; wherein within a predetermined range of c, the calibration functions can be represented as monotone increasing or decreasing functions F.sub.j(c) with values a.sub.j, uniquely associated to j, of a slope parameter a corresponding to a respective derivative of the function F.sub.j(c) at c=0; determining the value m.sub.j of the diffusion measure M for the respective reference analyte R.sub.j, which influences and/or characterizes the diffusion of the reference analyte R.sub.j occurring during the fluorescence measurement; and establishing the dependence of the values of the slope parameter a on the values m of the diffusion measure M by a function E(m)=a that is obtained by adequate fitting to the value pairs (a.sub.j, m.sub.j); wherein the function E(m)=a can be used in the determination of the concentration of a fluorescent and/or fluorescence-labeled analyte, wherein for the purpose of this determination subsequent to the calibration method, the value a.sub.sample specific to the analyte is determined from the function E(m)=a using the value m.sub.sample of the diffusion measure M specific to said analyte, and a function F.sub.sample(c)=i whose derivative at c=0 gives the value a.sub.sample is obtained by adequate fitting to the functions F.sub.j(c)=i, and wherein the concentration c.sub.sample of the sample P.sub.sample is determined from the function F.sub.sample(c)=i for the value a.sub.sample and from the value i.sub.sample as determined from the fluorescence measurement for the sample P.sub.sample of said analyte by forming the inverse function F.sup.−1.sub.sample(c) or by a suitable approximation technique.

2. The calibration method of claim 1, wherein the same value i.sub.j is set for c.sub.j=0 for all functions F.sub.j(c)=i in the fitting.

3. The calibration method of claim 1, wherein at least one of the functions F.sub.j(c) can be represented as a polynomial F.sub.j(c)=const..sub.j+a.sub.j.Math.c+b.sub.j.Math.c.sup.2+ . . . .

4. The calibration method of claim 3, wherein the absolute values of the factors b.sub.j, . . . of higher degrees than two of the at least one polynomial F.sub.j(c) are small relative to the absolute value of a.sub.j.

5. The calibration method of claim 3, wherein the at least one polynomial F.sub.j(c) is a second degree polynomial.

6. The calibration method of claim 3, wherein the at least one polynomial F.sub.j(c) is a first degree polynomial.

7. The calibration method as claimed in claim 1, wherein the functions F.sub.j(c) differ only in the values a.sub.j, wherein for the purpose of determination of the concentration of a fluorescent and/or fluorescence-labeled analyte subsequent to the calibration method, the value a.sub.sample specific to the analyte is inserted into the function F.sub.j(c)=i instead of the value a.sub.j.

8. The calibration method of claim 7, wherein starting from a function F.sub.k(c), (k∈{1, . . . , N}), the other functions F.sub.j(c) can be generated by rotation of the function F.sub.k(c)=i around the point (i=const..sub.k, c=0) in the coordinate plane defined by i and c.

9. A method for determining the concentration of a sample P.sub.sample of a fluorescent and/or fluorescence-labeled analyte, wherein the method includes a calibration method according to claim 1 and comprises the further steps of: providing a sample P.sub.sample of a fluorescent and/or fluorescence-labeled analyte for which a value c.sub.sample of concentration C is to be determined; performing a fluorescence measurement for the sample P.sub.sample and determining the value i.sub.sample of a concentration-dependent parameter I from the fluorescence measurement; determining the value m.sub.sample of a diffusion measure M which influences and/or characterizes the diffusion of the analyte occurring during the fluorescence measurement; and determining the value a.sub.sample of the slope parameter a specific to the analyte on the basis of the function E(m)=a using the value m.sub.sample of the diffusion measure M specific to the analyte, establishing a function F.sub.sample(c)=i whose derivative at c=0 gives the value a.sub.sample by adequate fitting to the functions F.sub.j(c)=i, and determining the concentration c.sub.sample of the sample P.sub.sample from the function F.sub.sample(c)=i for the value a.sub.sample and from the value i.sub.sample as determined from the fluorescence measurement for the sample P.sub.sample, by forming the inverse function F.sup.−1.sub.sample(c) or by a suitable approximation technique.

10. A method for determining the concentration of a sample P.sub.sample of a fluorescent and/or fluorescence-labeled analyte using the function E(m)=a that can be determined by a calibration method according to claim 1 and using the functions F.sub.j(c)=i for the values a.sub.j of slope parameter a, comprising the steps of: providing a sample P.sub.sample of a fluorescent and/or fluorescence-labeled analyte for which a value c.sub.sample of concentration C is to be determined; performing a fluorescence measurement for the sample P.sub.sample and determining the value i.sub.sample of a concentration-dependent parameter I from the fluorescence measurement; determining the value m.sub.sample of a diffusion measure M which influences and/or characterizes the diffusion of the analyte occurring during the fluorescence measurement; and determining the value a.sub.sample of the slope parameter a specific to the analyte on the basis of the function E(m)=a using the value m.sub.sample of the diffusion measure M specific to the analyte, establishing a function F.sub.sample(c)=i whose derivative at c=0 gives the value a.sub.sample by adequate fitting to the functions F.sub.j(c)=i, and determining the concentration c.sub.sample of the sample P.sub.sample from the function F.sub.sample(c)=i for the value a.sub.sample and from the value i.sub.sample as determined from the fluorescence measurement for the sample P.sub.sample, by forming the inverse function F.sup.−1.sub.sample(c) or by a suitable approximation technique.

11. The method as claimed in claim 1, wherein the values i of the concentration-dependent parameter I are determined from fluorescence measurements using a fluorimeter, wherein the concentration-dependent parameter I is fluorescence intensity.

12. The method as claimed in claim 1, wherein the values i of the concentration-dependent parameter I are determined from fluorescence measurements using a technique based on fluorescence microscopy, in particular confocal fluorescence microscopy or multi-photon fluorescence microscopy, wherein the concentration-dependent parameter I is one of mean fluorescence intensity, mean particle count in an observation volume, and reciprocal of the coefficient of fluorescence intensity variance.

13. The method as claimed in claim 1, wherein the analyte and the reference analytes R.sub.j comprise macromolecules, in particular polymers, preferably polynucleotides and/or oligonucleotides and/or proteins.

14. The method as claimed in claim 1, wherein the analyte and the reference analytes R.sub.j comprise double-stranded polynucleotide fragments, wherein the respective reference analytes R.sub.j are each distinguished by a predetermined mean fragment length or number of base pairs in the polynucleotide fragments and differ from each other by said mean fragment length.

15. The method as claimed in claim 1, wherein the fluorescent and/or fluorescence-labeled analyte and the fluorescent and/or fluorescence-labeled reference analytes include the same fluorophores or fluorophores with the same quantum yield.

16. The method as claimed in claim 1, wherein the fluorescence-labeled analyte and the fluorescence-labeled reference analytes are labeled with a fluorescent dye which preferably labels in a sequence-independent manner.

17. The method as claimed in claim 1, wherein the diffusion measure M used is diffusion time τ.sub.D, diffusion coefficient D, or mean particle size.

18. The method as claimed in claim 1, wherein the values m of the diffusion measure M for the analyte or the reference analytes are determined by correlation spectroscopy.

19. The method as claimed in claim 1, wherein the values m of the diffusion measure M for the analyte or the reference analytes are determined by a fluorescence-based technique, in particular by a technique based on fluorescence microscopy, preferably confocal fluorescence microscopy or multi-photon fluorescence microscopy.

20. The method as claimed in claim 1, wherein the values m of the diffusion measure M for the respective analyte or reference analyte are determined by fluorescence correlation spectroscopy (FCS), in particular by autocorrelation.

21. The method as claimed in claim 1, wherein the values m of the diffusion measure M for the analyte and/or the reference analytes are determined by techniques, in particular separation techniques, based on the particle size and/or particle shape of the analyte or the reference analyte.

22. The method as claimed in claim 17, wherein the dependence of the values d.sub.j of the diffusion coefficient D of a number of reference analytes R.sub.j having different mean fragment lengths bp.sub.j on the mean fragment lengths bp is established by a function f(bp)=d that is obtained by fitting to the value pairs (d.sub.j, bp.sub.j), wherein the function f(bp)=d can be used to determine the value d.sub.sample for an analyte that comprises double-stranded polynucleotide fragments with a known mean fragment length.

23. The method as claimed in claim 1, wherein the sample P.sub.sample and P.sub.j,v contain a solution of the analyte or reference analyte in a dilution medium.

24. A kit for determining the concentration of fluorescent and/or fluorescence-labeled analytes using the method as claimed in claim 1, wherein the kit comprises a plurality of fluorescent and/or fluorescence-labeled reference analytes and/or reference analytes that can be fluorescence labeled, which differ from each other by values m of a diffusion measure M.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) Further details and advantages of the invention will become apparent below from the description of preferred exemplary embodiments, without limiting the scope of the invention, and with reference to the accompanying drawings, wherein:

(2) FIG. 1 shows the schematic setup of a measurement apparatus;

(3) FIG. 2 schematically illustrates a fluorescent or fluorescence-labeled particle diffusing during the fluorescence measurements;

(4) FIG. 3 is a graphical representation showing exemplary autocorrelation data from fluorescence measurements for labeled 50 bp DNA fragments with the autocorrelation function fitted to the data;

(5) FIG. 4 is a graphical representation illustrating the dependence of the values of the diffusion coefficient D [μm.sup.2 s] of double-stranded DNA fragments of different mean fragment lengths on the mean fragment lengths [bp] as determined according to a partial aspect of the method according to the invention;

(6) FIG. 5 is a graphical representation of the calibration functions for double-stranded DNA fragments of different mean fragment lengths (50 bp, 200 bp, 500 bp, 1 kbp, 2 kbp, 3 kbp, 6 kbp, 10 kbp) in the coordinate plane defined by the i-axis and c-axis according to one embodiment of the invention, wherein the calibration functions for 200 bp, 500 bp, 1 kbp, 2 kbp, 3 kbp, 6 kbp and 10 kbp DNA fragments were fitted by rotation of the quadratic calibration function for 50 bp DNA fragments around the i-axis intersection, where i denotes the mean fluorescence intensity [kCps], and C denotes the concentration, [pg/μl];

(7) FIG. 6 is a graphical representation of the function E(τ.sub.D)=a according to one embodiment of the invention, describing the dependence of the values of slope parameter a on the diffusion time τ.sub.D [s], where a [kilocounts.Math.s.sup.−1.Math.pg.sup.−1.Math.μl] corresponds to the slopes of the curve tangents of the graphs of FIG. 5 at c=0 or derivatives of the respective functions at c=0;

(8) FIG. 7 is a graphical representation of the calibration functions as established by a linear fit for double-stranded DNA fragments of different mean fragment lengths (50 bp, 200 bp, 500 bp, 1 kbp, 2 kbp, 3 kbp, 6 kbp, 10 kbp) in the coordinate plane defined by the i-axis and the c-axis according to one embodiment of the invention, where i stands for the values of mean fluorescence intensity [kilocounts/s], and C for the concentration, [pg.Math.μl.sup.−1];

(9) FIG. 8 is a graphical representation of the function E(τ.sub.D)=a according to one embodiment of the invention, describing the dependence of slope parameter a [kilocounts.Math.s.sup.−1.Math.pg.sup.−1.Math.μl] of the calibration functions or of the slopes of the graphs of FIG. 7 on diffusion time τ.sub.D [s];

(10) FIG. 9 is a graphical representation of the calibration functions as established by a linear fit for double-stranded DNA fragments of different mean fragment lengths (50 bp, 200 bp, 500 bp, 1 kbp, 2 kbp, 3 kbp, 6 kbp, 10 kbp) in the coordinate plane defined by the <N>-axis and the c-axis according to one embodiment of the invention, where <N> stands for the values of mean particle count in the observation volume, and C for the concentration, [pg.Math.μl.sup.−1];

(11) FIG. 10 is a graphical representation of the function E(τ.sub.D)=a according to one embodiment of the invention, describing the dependence of the slope parameter a [pg.sup.−1.Math.μl] of the calibration functions or of the slopes of the graphs of FIG. 9 on diffusion time τ.sub.D [s];

(12) FIG. 11 is a graphical representation of the calibration functions as established by a quadratic fit for double-stranded DNA fragments with different mean fragment lengths (50 bp, 200 bp, 500 bp, 1 kbp, 2 kbp, 3 kbp, 6 kbp, 10 kbp) in the coordinate plane defined by the 1/VarK-axis and the c-axis according to one embodiment of the invention, wherein 1/VarK is the reciprocal of the coefficient of fluorescence intensity variance, and C is the concentration, [pg/μl];

(13) FIG. 12 is a graphical representation of the function E(τ.sub.D)=a according to one embodiment of the invention, describing the dependence of slope parameter a [pg.sup.−1.Math.μl] on the diffusion time τ.sub.D, [s], wherein a corresponds to the slopes of the curve tangents of the graphs of FIG. 11 at c=0 or derivatives of the corresponding functions at c=0 here;

(14) FIG. 13 shows a comparison of the results of the concentration determinations for samples of the analytes comprising different predefined mixtures of double-stranded DNA fragments, which were performed in each case according to the prior art and according to an exemplary embodiment of the method of the invention according to FIGS. 5, 6.

DETAILED DESCRIPTION

(15) Eight reference analytes were provided, each comprising double-stranded DNA fragments and differing in their mean fragment length. DNA fragments having mean fragment lengths of 50 bp, 200 bp, 500 bp, 1000 bp, 2000 bp, 3000 bp, 6000 bp, and 10,000 bp were used. The mean fragment length influences the diffusion rate of the DNA fragments and can be considered as a diffusion measure. Depending on the mean fragment length, DNA fragments of different lengths are distinguished by different diffusion coefficients D and exhibit different diffusion times τ.sub.D.

(16) A respective series of eleven calibration samples P.sub.j,v with different amounts of the respective DNA fragments was prepared for the DNA fragments of each of the respective fragment lengths. The calibration samples of the respective calibration series had the following DNA mass concentrations: 1, 10, 20, 40, 50, 100, 200, 400, 600, 800, and 1000 pg.Math.μl.sup.−1. Thus, in this example, the dilution series were prepared with eleven samples P.sub.j,v for DNA fragments of each of the fragment lengths 50 bp, 200 bp, 500 bp, 1000 bp, 2000 bp, 3000 bp, 6000 bp, and 10,000 bp. The calibration samples P.sub.j,v were prepared as solutions of the DNA fragments. The employed dilution medium was water with dimethyl sulfoxide (DMSO) in a proportion of 75% water, 25% DMSO. The samples P.sub.sample of the analytes for which the concentration was to be determined were likewise prepared as solutions in 75% water, 25% DMSO. DMSO was added to the water to prevent any formation of tertiary structures. Water without DMSO addition may be used as well.

(17) In one example, the DNA fragments in the calibration samples were labeled with the fluorescent dye RiboGreen® (Thermo Fisher Scientific). This fluorescent dye binds double-stranded polynucleotides in a sequence-independent manner, i.e. irrespectively of the specific nucleotide sequence of the polynucleotides. Since the labelling agent is used in excess in the labeling of polynucleotides, all binding sites along the entire length of the DNA fragments are occupied by fluorophores, so that the number of bound fluorophores is proportional to the mass concentration of DNA. The fluorescence intensity of the dye in the bound state is several orders of magnitude higher than in the unbound state. As a result, the background value of fluorescence of the dilution medium with the fluorescent dye and without addition of the DNA fragments is relatively low. In the present example, the values of the concentration-dependent parameter were averaged from the fluorescence measurements for samples that each contained 1 pkg.Math.μl.sup.−1 of DNA of the reference analytes with different DNA fragment lengths and were assumed as the value of the concentration-dependent parameter at c=0 for all fluorescence measurements, since this value was not above the background value.

(18) The samples P.sub.sample for which the concentration was to be determined were similarly labeled with the fluorescent dye RiboGreen®.

(19) Fluorescence measurements were performed at about 22° C. using a confocal optical fluorescence microscope and laser light as the excitation light, with the following configuration:

(20) TABLE-US-00001 TABLE 1 Example of a configuration of the measurement apparatus for fluorescence measurements Laser Lasos LDM F series, 90 mW, 486 nm Filters Linus 1% neutral filter, BrightLine ® fluorescence filter 535/40 Dichroic Linus 500 LP mirror Objective Zeiss LD Plan-NEOFLUAR 63x/0.75 korr, ∞/0-1.5 Detector Avalanche photodiode from Micro Photon Devices, PDM series 100 μm, cooled Pinhole Diameter 100 μm

(21) FIG. 1 is a schematic diagram of the confocal configuration of a fluorescence microscope.

(22) Samples of 2 μl sample volume were subjected to fluorescence measurements under irradiation with approximately 100 μW of laser power injected into the objective.

(23) Fluorescence measurements for analytes and reference analytes were carried out under identical or comparable conditions, in particular in terms of temperature, duration of measurement period, and settings of the measurement apparatus.

(24) The profile of the fluorescence signal, or the number of photons emitted by the fluorophores in the observation volume during the measurement period and detected by the detector at different points in time was recorded during the measurement period of 30 s. The observation volume was about 1 fl and hence a small portion of the sample volume. Since the analyte particles diffuse into and out of the observation volume from the rest of the sample volume, as schematically illustrated in FIG. 2, the number of particles in the observation volume and thus the fluorescence signal fluctuates around the mean value during the measurement period. These fluctuations of the detected photon count or fluctuations of fluorescence intensity or the time profile of the fluorescence signal during the measurement period, the time track, was recorded and processed by fluorescence correlation spectroscopy (FCS) using the ALV correlator card. Depending on the exemplary embodiment, the processed data were used to determine the values of one of the concentration-dependent parameters on the basis of the fluorescence measurement, i.a. mean fluorescence intensity <I>, mean particle count <N> in the observation volume, and/or reciprocal of the coefficient of fluorescence intensity variation, VarK-1. Furthermore, the data processed using the ALV correlator card were used to determine the diffusion properties of the analyte in the sample, or of a diffusion measure M. Using the correlator map, the autocorrelation values G(τ) were calculated for different values of lag time τ, with 0≤τ≤duration of measurement period, according to the formula

(25) $G\left(\tau \right)=\frac{{.Math.\delta I\left(t\right).Math.\delta I\left(t+\tau \right).Math.}_t}{{.Math.I\left(t\right).Math.}_t^2}.$

(26) These autocorrelation data or value pairs (G(τ),τ) are shown in FIG. 3, by way of example, as points in the coordinate plane defined by the G-axis and the τ-axis. A model function G(τ) was fitted to the autocorrelation data to determine the values for the function parameters. In one exemplary embodiment, the model function according to the formula

(27) $G\left(\tau \right)=\frac{1}{.Math.N.Math.}\frac{1}{1+\frac{\tau }{{\tau }_D}}\frac{1}{\sqrt{1+\frac{r_0^2}{z_0^2}\frac{\tau }{{\tau }_D}}}$
was used, which includes the diffusion time τ.sub.D as a parameter. With the fitting, the diffusion time τ.sub.D for the analyte was determined as the diffusion measure. In one exemplary embodiment, the corresponding diffusion coefficient D was calculated as the characteristic parameter for the diffusion behavior or as the diffusion measure, from the value τ.sub.D according to the formula

(28) $D=\frac{r_0^2}{4{\tau }_D}.$

(29) The size and shape of the observation volume or the dimensions of the observation volume in the direction of the light beam of the excitation light, 2z.sub.0, and perpendicular to this direction, 2r.sub.0, were determined by fluorescence measurements according to the FCS technique for a sample with a known concentration of a substance with a known diffusion coefficient D, for example with the fluorescent dye Alexa 488 (Thermo Fisher Scientific) that has a diffusion coefficient of 435 μm.sup.2 s.sup.−1 previously known from literature (e.g. from Petrášek and Schwille, Biophysical J., 94(4): 1437-1448, 2008).

(30) The values of the diffusion coefficient D determined by the FCS for DNA fragments of different fragment lengths were used to establish the dependence between the different diffusion measures, particle size—here the fragment length—and diffusion coefficient D. The model function D=f(bp)=k.sub.1×bp.sup.k2 was fitted to the value pairs (D, bp) and the following relationship was found for DNA fragments in aqueous solutions at 22° C.: D=494.065×bp.sup.−0.567, where bp is the fragment length or number of base pairs. FIG. 4 shows a graphical representation of the fitting result. The determined ratio enables to determine the diffusion coefficients D for DNA fragments for which the fragment lengths are known or were determined by other techniques such as by analysis of DNA mixtures by particle size-based separation methods or analysis methods which allow to determine the particle size, such as, e.g., electrophoresis or size-exclusion chromatography. From the value D, the value of the diffusion time τ.sub.D can be calculated according to the formula

(31) 0${\tau }_D=\frac{r_0^2}{4D}$
if the shape and size of the observation volume were determined by fluorescence measurements for a sample with a known concentration of a substance for which the diffusion coefficient D is known.

(32) Fluorescence measurements were performed five times for each sample P.sub.j,v, P.sub.sample, a median value was calculated and was assumed as the measurement result of the concentration-dependent parameter for the respective sample. The measurement results for the respective calibration series comprising calibration samples with different values c of concentration C are shown in FIGS. 5, 7, 9, and 11. In the figures, the fluorescence measurement results for the calibration series for the DNA fragments of different fragment lengths are shown as points, and the graphs of the functions that were fitted to the measurement results and which establish the dependence of the corresponding concentration-dependent parameter on the concentration are shown as lines. FIGS. 5 and 7 show the mean fluorescence intensity <I>, FIG. 9 shows the mean particle count <N> in the observation volume, and FIG. 11 shows the reciprocal of the coefficient of fluorescence intensity variation, VarK.sup.−1, from the FCS, as a function of the concentration in each case.

(33) In an exemplary embodiment according to FIG. 5, the calibration functions were established as described below. First, a function F.sub.50bp (c) was established for the 50 bp DNA fragments by fitting a quadratic function or a second degree polynomial
F.sub.50bp(c)=const.a.sub.50bp×c+b.sub.50bp×c.sup.2=i
to the value pairs of the 50 bp calibration series, i.e. to the measurement results for the mean fluorescence intensity <I> from FCS, and values a.sub.50bp=2.68 and b.sub.50bp=−0.0006 were determined. The value const.=16.417 was calculated in this exemplary embodiment as a mean value from fluorescence measurements for samples with 1 pg.Math.μl.sup.−1 DNA of different fragment lengths. The value a.sub.50bp of the slope parameter a is the derivative of the function F.sub.50bp(c) at c=0. The functions F.sub.j(c) for the other calibration series were fitted to the respective value pairs for 200 bp, 500 bp, etc. DNA fragments by rotation of the function F.sub.50bp(c) in the coordinate plane defined by the i-axis and the c-axis around the i-axis intersection (i=const, c=0). The parameter whose values were continuously changed by the rotation and determined as a result of the fitting was the rotation angle θ between the graph of the respective function F.sub.j(c) and the graph of F.sub.50bp (c).

(34) In order to be able to perform such a fitting, first the function

(35) $F_j\left(C\right)=-\frac{1}{4b_{50\mathrm{bp}}}\csc {\theta }_j\left(-\cos {\theta }_j+a_{50\mathrm{bp}}\sin {\theta }_j+\sqrt{-4b_{50\mathrm{bp}}C\sin {\theta }_j+{\left(\cos {\theta }_j-a_{50\mathrm{bp}}\sin {\theta }_j\right)}^2}\right)\times \left(a_{50\mathrm{bp}}\cos {\theta }_j+\cos {\theta }_j\cot {\theta }_j+2\sin {\theta }_j-\cot {\theta }_j\sqrt{-4b_{50\mathrm{bp}}C\sin {\theta }_j+{\left(\cos {\theta }_j-a_{50\mathrm{bp}}\sin {\theta }_j\right)}^2}\right)+\mathrm{const}$
was established, which includes the rotation angle θ as the parameter to be fitted, with arctan a.sub.k<θ<π/2−arctan a.sub.k. The fitting value θ.sub.j for the rotation angle θ was determined for each of the calibration series by fitting the function F.sub.j(c) to the value pairs of the calibration series. The respective values of the factor a.sub.j which represents the derivative of the function F.sub.j(c) with θ=θ.sub.j at c=0 were calculated from the values θ.sub.j of the rotation angle θ according to the formula

(36) $a_j=\frac{\tan {\theta }_j+a_{50\mathrm{bp}}}{1-a_{50\mathrm{bp}}\tan {\theta }_j}.$

(37) In the present exemplary embodiment, the values a.sub.j of the slope parameter a as determined for the DNA fragments of different fragment length were associated with the values of diffusion time τ.sub.D for the respective DNA fragments. The respective value pairs (a.sub.j, τ.sub.Dj) are illustrated in FIG. 6 by points in the coordinate plane defined by the a-axis and the τ.sub.D-axis.

(38) In order to establish the dependence of the slope parameter a on the diffusion time τ.sub.D, a probability-based approach was chosen, according to which, first, the mean number of fluorescence events N.sub.f was represented as a function of the diffusion time τ.sub.D according to the formula

(39) $N_f\left({\tau }_D\right)=\frac{k_f}{k_{\mathrm{bl}}}\left(1-e^{-k_{\mathrm{bl}}{\tau }_D}\right),$
where k.sub.f [s.sup.−1] is the fluorescence rate and k.sub.bl [s.sup.−1] is the bleaching rate. The relationship k.sub.f/k.sub.bl together with the dimensional prefactor k.sub.dim [counts.Math.pg.sup.−1.Math.μl] was expressed as a coefficient k.sub.int [counts.Math.pg.sup.−1 μl]: k.sub.int=k.sub.dim.Math.k.sub.f/k.sub.bl. Furthermore, the term was divided by τ.sub.D to obtain the number of fluorescence photons as a function of the diffusion time τ.sub.D, and a constant was added, since even for great values of diffusion time the function values E(τ.sub.D) must be greater than zero. From this follows the model function

(40) $E\left({\tau }_D\right)=a=\frac{k_{\mathrm{int}}}{{\tau }_D}\left(1-e^{-k_{\mathrm{bl}}{\tau }_D}\right)+\mathrm{const}.$
representing the slope parameter a as a function of the diffusion time τ.sub.D. By fitting this function to the value pairs (a.sub.j, τ.sub.D), the following parameter values were determined: k.sub.int=0.176; k.sub.bl=19.016; constant=0.257. The result is shown in FIG. 6 as a line of the graph of the function

(41) $E\left({\tau }_D\right)=a=\frac{0.176}{{\tau }_D}\left(1-e^{-19.016{\tau }_D}\right)+0.257.$

(42) The function E(τ.sub.D)=a allows to determine the analyte-specific values of the slope parameter a for analytes with predetermined diffusion times τ.sub.D in order to use these values in the determination of the concentration of the analyte.

(43) A further exemplary embodiment shows the use of the function E(τ.sub.D)=a in the determination of the concentration of different samples P.sub.sample. To verify the method, a plurality of samples of the DNA solutions were prepared with predetermined mixtures of double-stranded DNA fragments of different fragment lengths and were subjected to the concentration determination according to an exemplary embodiment of the method according to the invention. The eleven DNA mixtures which were prepared as analytes are listed in Table 2. Each of the mixtures contained DNA fragments of two fragment lengths in the indicated mass ratio. For each of the mixtures, dilutions were prepared with four different DNA mass concentrations: 20 pg/μl, 50 pg/μl, 100 pg/μl, and 200 pg/pl. All concentrations of analytes in the samples P.sub.sample were within the range of concentrations of the calibration samples that were used to establish the functions F.sub.j(c).

(44) The DNA fragments in the samples P.sub.sample were labeled with the fluorescent dye RiboGreen® and subjected to fluorescence measurements using the confocal fluorescence microscope as described above. Fluorescence Correlation Spectroscopy (FCS) was used to determine the values of mean fluorescence intensity i.sub.sample and diffusion time τ.sub.Dsample for each of the samples P.sub.sample. Five fluorescence measurements were performed for each sample P.sub.sample, and the median value was used as the measured value. The analyte-specific values a.sub.sample corresponding to the analyte-specific diffusion times τ.sub.Dsample were determined for different samples P.sub.sample according to the formula

(45) $a_{\mathrm{sample}}=\frac{0.176}{{\tau }_{D_{\mathrm{sample}}}}\left(1-e^{-19.016{\tau }_{D_{\mathrm{sample}}}}\right)+0.257.$

(46) On the basis of the measured value i.sub.sample and the respective value a.sub.sample, the concentration c.sub.sample was calculated using the formula

(47) ${\theta }_{\mathrm{sample}}=\arctan \left(\frac{-a_{50\mathrm{bp}}+a_{\mathrm{sample}}}{1+a_{50\mathrm{bp}}{a}_{\mathrm{sample}}}\right)$
for the value of rotation angle θ.sub.sample and the formula

(48) $c_{\mathrm{sample}}=\frac{1}{4}\left(-\frac{2a_{50\mathrm{bp}}\cos {\theta }_{\mathrm{sample}}}{b_{50\mathrm{bp}}}-\frac{2\sin {\theta }_{\mathrm{sample}}}{b_{50\mathrm{bp}}}+4\mathrm{const}\tan {\theta }_{\mathrm{sample}}-4I_{\mathrm{sample}}\tan {\theta }_{\mathrm{sample}}-\frac{2a_{50\mathrm{bp}}\sin {\theta }_{\mathrm{sample}}\tan {\theta }_{\mathrm{sample}}}{b}-\frac{2\sin {\theta }_{\mathrm{sample}}{\tan }^2{\theta }_{\mathrm{sample}}}{b_{50\mathrm{bp}}}\right)-\frac{1}{b_{50\mathrm{bp}}}\sqrt{2}\sin {\theta }_{\mathrm{sample}}\times {\left\{-8b_{50\mathrm{bp}}\mathrm{const}\cot {\theta }_{\mathrm{sample}}{\csc }^5{\theta }_{\mathrm{sample}}+8b_{50\mathrm{bp}}{I}_{\mathrm{sample}}\cot {\theta }_{\mathrm{sample}}{\csc }^5{\theta }_{\mathrm{sample}}+{\csc }^6{\theta }_{\mathrm{sample}}+a_{50\mathrm{bp}}^2{\csc }^6{\theta }_{\mathrm{sample}}-\cos \left(2{\theta }_{\mathrm{sample}}\right){\csc }^6{\theta }_{\mathrm{sample}}+a_{50\mathrm{bp}}^2\cos \left(2{\theta }_{\mathrm{sample}}\right){\csc }^6{\theta }_{\mathrm{sample}}+2a_{50\mathrm{bp}}{\csc }^6{\theta }_{\mathrm{sample}}\sin \left(2{\theta }_{\mathrm{sample}}\right)\right\}}^{1\text{/}2}{\tan }^2{\theta }_{\mathrm{sample},}$
with a.sub.50bp=2.68; b.sub.50bp=−0.0006; const.=16.417. The values are summarized in Table 2.

(49) TABLE-US-00002 TABLE 2 Determination of mass concentration of the DNA mixtures prepared with concentrations of 20 pg/μl, 50 pg/μl, 100 pg/μl, and 200 pg/μl. Fragment lengths in the Mixing DNA concentrations employed, pg/μl mixtures, bp ratio 20 50 100 200 50 and 1000 1:1 21.01 47.77 111.38 214.21 50 and 1000 1:2 17.47 51.98 97.57 176.96 50 and 1000 1:3 20.51 49.29 92.91 210.87 50 and 1000 1:4 19.04 49.70 93.78 208.29 50 and 1000 1:5 19.62 44.80 96.71 190.54 50 and 1000 1:6 17.52 45.37 104.06 165.71 200 and 500 1:1 24.98 54.99 99.59 204.91 200 and 500 1:2 23.78 53.02 93.66 206.66 200 and 500 1:3 16.05 42.71 94.54 200.74 200 and 500 1:4 30.47 51.44 95.79 202.47 200 and 500 1:5 28.40 46.72 103.98 195.55 Mean value, pg/μl 21.71 48.89 98.54 197.90 Variance coefficient, % 23.31 7.60 5.76 7.47 Standard deviation, %  8.57 −2.22 −1.46 −1.05

(50) FIG. 13 shows a comparison of the results of the concentration determinations for samples P.sub.sample, which were performed according to the prior art using the calibration function that was established for 50 bp DNA fragments, as well as according to the exemplary embodiment of the method according to the invention described above. The values of DNA concentration as determined according to the exemplary embodiment of the method according to the invention have a higher accuracy and thus are closer to the expected value of concentration than the values obtained by the conventional method. The method of the invention takes into account the influence of diffusion of the analytes on the fluorescence measurements and thus improves the trueness concentration determination even for very low concentrations such as, e.g., 20 pg/μl.

(51) In further exemplary embodiments according to FIGS. 7 and 9, the calibration functions were established by fitting a respective linear function or first degree polynomial F.sub.j (c)=const.+a.sub.j×c to the FCS measurement results for mean fluorescence intensity <I> or mean particle count <N>. The value of ‘const.’ from the respective fitting for the different calibration series was averaged and the mean value was assumed as the same value for all functions F.sub.j(c). The values a.sub.j of slope parameter a for the respective calibration series as determined from the fitting indicate the slope of the respective calibration straight line. The calibration functions of FIGS. 7 and 9 can thus be represented as first degree polynomials with the respective values a.sub.j of slope parameter a within the entire illustrated range of concentrations C from 1 to 1000 pg/μl. Different values a.sub.j of the different calibration series were related to the values m of a diffusion measure M for the respective DNA fragments by the function E(m)=a, as shown in FIGS. 8 and 10 for the diffusion time τ.sub.D as the diffusion measure.

(52) The calibration functions of FIG. 11 were each established by fitting a quadratic function or second degree polynomial F.sub.j (c)=const.+a.sub.j×c+b.sub.j×c.sup.2 to the FCS measurement results for VarK.sup.−1. The value of ‘const.’ from the respective fitting for the different calibration series was averaged and the mean value was assumed as the same value for all functions F.sub.j(c). By the fitting, the respective values of slope parameter a.sub.j and of second degree coefficient b.sub.j were determined. Thus, within the entire illustrated range of concentrations C from 1 to 1000 pg/μl, the calibration functions of FIG. 11 can be represented as second degree polynomials with the values a.sub.j of slope parameter a. Different calibration series with different values a.sub.j of slope parameter a were related to the values m of a diffusion measure M for the respective DNA fragments by the function E(m)=a, as is shown in FIG. 12 for the diffusion time τ.sub.D as the diffusion measure, for example.

LIST OF REFERENCE NUMERALS

(53) 1 Detector 2 Pinhole 3 Lens 4 Dichroic mirror 5 Objective 6 Object level 7 Laser 8 Light beam of excitation light 9 Observation volume 10 Diffusing particle