Anal. Chem. 2006, 78, 8395-8405
Improvement of Biomolecule Quantification Precision and Use of a Single-Element Aspheric Objective Lens in Fluorescence Correlation Spectroscopy Tsuyoshi Sonehara,* Takashi Anazawa, and Kenko Uchida
Central Research Laboratory, Hitachi, Ltd., 1-280 Higashi-koigakubo Kokubunji-shi, Tokyo 185-8601, Japan
We found a way to increase the precision with which biomolecules present at concentrations below 10-10 M can be quantified by fluorescence correlation spectroscopy (FCS). The effectiveness of the way was demonstrated experimentally by using a single-element aspheric objective lens, which was newly developed to reduce the cost of FCS instruments. In the first part of this paper, the relative standard deviation (RSD) of FCS-based concentration measurements is estimated theoretically by an analytical approximation assuming the detection volume profiles in FCS setups to be Gaussian and by molecular simulations in which more realistic profiles are calculated from physical parameters of the measurement setups. In a limit of infinitely bright molecules and zero background emission, the analytical approximation predicts that the RSD at a concentration is minimized when the mean number of molecules in a detection volume is ∼0.5. A detection volume of the order of 10-13 L thus gives smaller RSD values for concentrations from 10-11 to 10-10 M than does one of the order of 10-15 L, which is widely used in FCS. This prediction is supported by the molecular simulations, taking into account the finite molecule brightness and background emission. In the second part of the paper, the RSD is evaluated experimentally with an FCS setup with a detection volume of 1.1 × 10-13 L. The newly developed objective lens, serving as the bottom of the sample cell in this setup, has a large numerical aperture (0.9) without using immersion liquid. When a calibration line was made by 30-s FCS measurements of Cy3-labeled, 112-mer single-stranded DNA solutions, the RSD roughly agreed with the simulation result and was less than 0.1 for DNA concentrations from 2 × 10-11 to 10-10 M. Fluorometry is a major technique for the quantification of biomolecules, but when standard fluorescent labeling is used, molecules present at concentrations below 10-10 M cannot be readily quantified by ordinary fluorometry because of background emission. Such low concentrations are not unusual for biomolecules in cell extracts and require the use of indirect fluorometric * To whom correspondence should be addressed.
[email protected]. Fax: +81-42-327-7833. 10.1021/ac061036y CCC: $33.50 Published on Web 11/03/2006
E-mail:
© 2006 American Chemical Society
quantification with additional processing for amplification or condensation. Since additional processing increases analysis time and reduces the quantification precision, biomolecules should be quantified directly by using techniques more sensitive than ordinary fluorometry. Fluorescence correlation spectroscopy (FCS) is a promising technique because it lets us reduce the influence of the background by analyzing fluorescence intensity fluctuation,1-5 and 240-s FCS measurements have recently been used to directly quantify specific messenger RNA (mRNA) present at concentrations ranging from 10-11 to 10-10 M.6-8 Although quantification at this concentration range is insufficient for comprehensive mRNA analysis, it is useful for validation of complementary microarray analysis. The standard mRNA quantification technique, on the other hand, uses real-time polymerase chain reactions (PCR), which take hours and include enzyme reactions subject to efficiency variation that reduces the quantification precision. The relative standard deviation (RSD) of concentration measurements should be less than 0.1 (IUPAC recommendations define the quantification limit as the concentration at which the RSD is 0.19,10), and the use of enzyme reactions makes such precise measurement difficult.11,12 Although mRNA can therefore be quantified faster and more precisely by FCS than by real-time PCR, FCS has not been used as extensively as real-time PCR because it is not precise enough (1) Berne, B. J.; Pecora, R. Dynamic Light Scattering: With Applications to Chemistry, Biology, and Physics; Dover: New York, 2000; pp 105-113. (2) Rigler, R., Elson, E. S., Eds. Fluorescence Correlation Spectroscopy: Theory and Applications; Springer: Berlin, 2001. (3) Sonehara, T.; Kojima, K.; Irie, T. Anal. Chem. 2002, 74, 5121-5131. (4) Fogarty, K.; Orden, A. V. Anal. Chem. 2003, 75, 6634-6641. (5) Kannan, B.; Har, J. Y.; Liu, P.; Maruyama, I.; Ding, J. L.; Wohland, T. Anal. Chem. 2006, 78, 3444-3451. (6) Korn, K.; Gardellin, P., Liao, B.; Amacker, M.; Bergstro ¨m, A° .; Bjo ¨rkman, H.; Camacho, A.; Do¨rho ¨fer, S.; Do¨rre, K.; Enstro¨m, J.; Ericson, T.; Favez, T.; Go¨sch, M.; Honegger, A.; Jaccoud, S.; Lapczyna, M.; Litborn, E.; Thyberg, P.; Winter, H.; Rigler, R. Nucleic Acids Res. 2003, 31, e89. (7) Camacho, A.; Korn, K.; Damond, M.; Cajot, J.-F.; Litborn, E.; Liao, B.; Thyberg, P.; Winter, H.; Honegger, A.; Gardellin, P.; Rigler, R. J. Biotechnol. 2004, 107, 107-114. (8) Winter, H.; Korn, K.; Rigler, R. Curr. Pharm. Biotechnol. 2004, 5, 191197. (9) Currie, L. A. Pure Appl. Chem. 1995, 67, 1699-1723. (10) Currie, L. A. Appl. Radiat. Isot. 2004, 61, 145-149. (11) Ståhlberg, A.; Håkansson, J.; Xian, X.; Semb, H.; Kubista, M. Clin. Chem. 2004, 50, 509-515. (12) Roche. Amplicor HIV-1 Monitoroˆ test, p 29.
Analytical Chemistry, Vol. 78, No. 24, December 15, 2006 8395
when mRNA is present at subnanomolar concentrations. Although mRNA concentrations less than 10-10 M were quantified in the FCS measurements reported in refs 6-8, the 240-s data collection time used in that work is rather long and the RSD obtained was larger than 0.1. The precision of FCS measurements has been investigated by using analytical calculations,13-16 computer simulations,17 and experiments,18 but methods to optimize the quantification precision for low concentrations has not been studied in detail. Another challenge is to reduce the high cost of FCS instruments, a large proportion of which is the cost of the high-end microscope objectives used to obtain high resolution and a large numerical aperture (NA). These objectives are expensive because they are precise compounds of many single lenses, and most of them are somewhat inconvenient because they are liquid immersion lenses. On the other hand, single-element aspheric dry objectives with diffraction-limited resolution and a NA close to 1 are used in optical storage,19,20 and numerical apertures larger than 1 have been obtained by combining these dry objectives with solid immersion lenses (SIL).21,22 This kind of lens can be produced at low cost by the molding technique but has not been used as an objective for FCS. Although FCS using a dry objective in combination with a SIL was reported, that objective was a compound lens.23 This paper has two topics: (1) a theoretical analysis focusing on increasing FCS-based quantification precision at low concentrations; (2) FCS experiments using a single-element objective lens that reduces the cost of FCS instruments. These topics are conceptually independent but share a common theoretical basis, and the experimental results obtained using the single-element lens confirm the results of the theoretical analysis. The theoretical analysis is described in the first part of this article (Theory, Preliminary Discussions on RSD, and Molecular Simulation), and the experiments are described in the second part (Experimental Section and Results and Discussion). In the first part of this paper, the RSD in FCS-based concentration measurements is estimated by both analytical approximation and molecular simulations. After FCS theory is restructured to adapt it to quantification experiments and to construct a realistic and convenient simulation model, a method of optimizing FCS setups to a particular concentration is provided by analytical approximation. This theoretical analysis predicts that a detection volume larger than the typical one in current FCS gives smaller RSD values at concentrations ranging from 10-11 to 10-10 M. The second part of the paper reports the results of FCS experiments using a newly developed single-element aspheric (13) Koppel, D. E. Phys. Rev. A 1974, 10, 1938-1945. (14) Qian, H. Biophys. Chem. 1990, 38, 49-57. (15) Kask, P.; Gu ¨ nther, R.; Axhausen, P. Eur. Biophys. J. 1997, 25, 163-169. (16) Saffarian, S.; Elson, E. L. Biophys. J. 2003, 84, 2030-2042. (17) Wohland, T.; Rigler, R.; Vogel, H. Biophys. J. 2001, 80, 2987-2999. (18) Starchev, K.; Ricka, J.; Buffle, J. J. Colloid Interface Sci. 2001, 233, 50-55. (19) Ariyoshi, T.; Shimano, T.; Maruyama, K. Jpn. J. Appl. Phys. 2002, 41, 18421843. (20) Itonaga, M.; Ito, F.; Saito, R.; Saito, T.; Ohira, T. International Symposium on Optical Memory 2004, Technical Digest, 2004; pp 260-261. (21) Mizuno, T.; Yamada, T.; Sakakibara, H.; Kawakita, S.; Ueda, H.; Watanabe, K. Jpn. J. Appl. Phys. 2002, 41, 617-623. (22) Zijp, F.; Mark, M. B.; Lee, J. I.; Verschuren, C. A.; Hendriks, B. H. W.; Balistreki, M. L. M.; Urbach, H. P.; Aa, M. A. H.; Padiy, A. V. Proc. SPIE 2004, 5380, 209-223. (23) Serov, A.; Rao, R.; Go ¨sch, M.; Anhut, T.; Martin, D.; Brunner, R.; Rigler, R.; Lasser, T., Biosens. Bioelectron. 2004, 20, 431-435.
8396
Analytical Chemistry, Vol. 78, No. 24, December 15, 2006
objective lens. We present its detailed specification and show how it is used in a new FCS setup used to obtain a large NA without immersion liquid. Results confirming the optimal detection volume determined theoretically were obtained when using the setup to evaluate Cy3-labeled single-stranded (ss) DNA solutions at concentrations from 5 × 10-12 to 10-10 M. THEORY We assume a sample solution composed of a continuum solvent and fluorescent analyte molecules in Brownian motion in the solvent. The fluorescent analyte molecules (hereafter referred to simply as “molecules”) are assumed to be isotropic point sources. General Theory of Fluorescence Detection. The detection volume in a fluorescence detection system is characterized by the spatial profile of molecule detection efficiency (MDE). The MDE at position r ) (x, y, z), here designated by φ(r), is defined by
φ(r) ≡ CEF(r)I(r)
(1)
where I(r) is the fluorescence excitation intensity profile expressed as the photon flux density of the excitation beam and CEF(r) is the collection efficiency function defined as the probability of photon arrival at the detector from a point source positioned at r. Hereafter we use the term “detection volume” to mean a value expressing the size of that volume, which is defined by
V≡
∫ ∫φ (r) dr
( φ(r) dr)2 2
(2)
We introduce the emission cross section σ by
σ≡
Q ln10 10NA
(3)
where Q and are respectively the fluorescence quantum yield and molar extinction coefficient of the molecule and NA is the Avogadro constant. When we neglect photobleaching and the duration of excited states, the count rate of photons emitted from a molecule at r and detected is given by pdσφ(r), where pd is the photon detection efficiency of the detector. The photocount rate at time t is given by
i(t) ) ib + pd
∫σF(r,t)φ(r) dr
(4)
where ib is the background count rate and F(r, t) is the number concentration of the molecules. This photocount rate i(t) corresponds to the classical fluorescence intensity ignoring the quantum nature of photons and is actually unobservable. The observable photocount in finite dwell time T is a Poisson random number whose mean value is ∫t+T i(t) dt. If we assume changes of i(t) t during T are negligible, then ∫t+T i(t) dt ) i(t)T. t Consider the sample solution in a cubic cell of edge length L, outside of which φ(r) can be regarded as zero. It contains N ≡ FjL3 molecules, where Fj is the time-averaged molecule concentration defined by Fj ≡ 〈F(r,t)〉. If rj(t) denotes the position of the jth
I(r) )
{
}
λex 2P 2(x2 + y2) exp 2 ch πω (z) ω2(z)
(9)
where ω(z) ) ω0x1+(λexz/πnω02)2, c is the light speed in vacuum, h is Planck’s constant, and n is the refractive index of the sample solution. Fluorescence emitted from molecules is collected by a lens system with an objective numerical aperture of NA and a magnification of M and is detected by a photoncounting detector through a pinhole of radius r0 whose center is positioned at the conjugate point of the xyz-coordinate origin. When we use an XY coordinate system on the pinhole plane, CEF(r) is given by
CEF(r) )
{ x ( )}∫
tr 12
NA n
1-
2
Tr(R)PSF(R,r) dR (10)
Figure 1. Schematic diagram of a FCS setup.
molecule at time t, F(r, t) is given by N-1
F(r,t) )
∑ δ(r - r (t)) j
(5)
where tr is the total transmittance of the detection optics, Tr(R) is the transmittance of the pinhole at position R ) (X, Y), and PSF(R, r) is the point spread function of a point source at r on the pinhole plane, which is normalized so that ∫PSF(R,r) dR ) 1. If we define the disk function by
j)0
circ(u,v) ≡ Substituting eq 5 into eq 4, we obtain
{
1 (xu2 + v2 e 1)
(11)
0 (xu2 + v2 > 1)
N-1
i(t) ) ib + pd
∑ σφ(r (t)) j
(6)
and assume
j)0
Tr(R) ) circ Equation 6, where a single-component system is assumed, can be easily extended to multicomponent systems. If σk denotes the emission cross section of the kth component, the detector photocount rate for a K-component system is given by
∑∑
σkφ(rj(t))
circ PSF(R,r) )
(7)
k)0 j)0
Finally, for convenience, we give the relation between the number concentration Fj and molar concentration C h:
Fj ) 1000NAC h
(8)
MDE in Confocal Systems. FCS is generally performed with confocal systems detecting laser-induced fluorescence. We calculate the MDE in confocal systems based on a model similar to that in refs 24 and 25. Consider the confocal system shown in Figure 1 and the xyz coordinate system with its origin at the center of the excitation beam waist and its z-axis along the direction of beam propagation. We assume a paraxial Gaussian excitation beam of vacuum wavelength λex, 1/e2 radius at the beam waist ω0, and power P. The excitation intensity profile is then given by (24) Qian, H.; Elson, E. L. Appl. Opt. 1991, 30, 1185-1195.
XY , r0 r0
(12)
we can write
K-1Nk-1
i(t) ) ib + pd
( )
(
)
X - Mx Y - My , R(z) R(z)
(13)
2
πR (z)
where
R(z) ≡
x{
( )} + (MR )
zM2 NA tan arcsin n M
2
2
0
Here R0 is the resolution limit of the lens system. If the lens system is aplanatic, the resolution limit is given by
λem R0 ) 0.61 NA
(14)
where λem is the wavelength of the fluorescence emission. Equation 13 is a natural extension of PSF(r, R) in refs 24 and 25. Then, the integral in eq 10 can be calculated analytically as the overlap area of two circles: Analytical Chemistry, Vol. 78, No. 24, December 15, 2006
8397
CEF(r)
{ x ( )}
{
tr 12
NA n
1-
( ) {( ) 2
R(z) r0 1 π R(z)
2
(d e r0 - R(z))
1
r0
Definitions of Basic Quantities in FCS and Their Estimators. We define the autocorrelation function of photocount rate fluctuation at lag time τ by
)
G(τ) ≡ T 2〈δi(t)δi(t + τ)〉 ) T 2(〈i(t)i(t + τ)〉 - 〈i(t)〉2) (20)
(d e R(z) - r0)
2
(θ - sin θ cos θ) + φ - sin φ cos φ
}
(|r0 - R(z)|< d e r0 + R(z))
0
(r0 + R(z) < d) (15)
The square of dwell time, T 2, is used as a multiplier in order to let G(τ) be dimensionless. If the data collection time and the photocount between kT and (k + 1)T are respectively denoted by U and Fk, the estimator of G(lT), G ˆ l, is given by15
G ˆ0 ) where
(
)
(
)
R2(z) + d2 - r02 2R(z)d
Substituting eqs 9 and 15 into eq 1, we can calculate the MDE by using only elementary functions and four arithmetic rules. This is crucial to constructing a practical simulation model because direct calculation of the multiple integration in eq 10 takes modern personal computers (PCs) considerable time, which results in impractically long simulation times. According to an extrapolation of ref 24, if
Mω0 er0 e
πω02M2 NA tan arcsin 2λex M
( )
K-1
K
k)0
∑F
(16)
2
k
G ˆl )
r02 + d2 - R2(z) , d ≡ x(Mx)2 + (My)2,θ ≡ arccos 2r0d φ ≡ arccos
1
- Fˆ 2 - Fˆ , K-l-1
1 K-l
∑FF k)0
Fˆ ≡
(
ω02
-
2z2 ωz2
)
ωz ) z0
x
2A
k
k)0
)
(l g 1) (21)
1
K-1
K
k)0
∑F
(22)
k
V ) π3/2ω02ωz
(23)
and
(
G(0)
1+
4Dτ ω02
)x
1+
4Dτ ωz2
(17)
The 1/e2 radius of the MDE profile in the z direction is given by
x(A + B)2 + 4(e2 - B)A - (A + B)
∑F K-l
For a Gaussian MDE in eq 17, V and G(τ) are given by
the MDE profile can be approximated by a Gaussian:
2(x2 + y2)
(
2
K-l-1
1
where K ≡ U/T and Fˆ is the estimator of the mean photocount 〈i(t)T〉, which is defined by
G(τ) )
φ(r) ) φ(0) exp -
-
k k+l
where D is the diffusion coefficient of the analyte molecules. In the case of ω0 , ωz, so-called 2D Gaussian, G(τ) is approximated by
G(τ) ) (18)
G(0) 1 + τ/τd
(24)
where τd is the half-decay time given by where 2
z0 ≡ (πnω0 /λex),
2
A ≡ (M z0tan arcsin(NA/M)/nr0)
and
q≡
If A . B and A . 4e2, it can be approximated as follows:
8398
enr0 2
M tan arcsin(NA/M)
(25)
FCS-Based Quantification. We define the mean photocount rate per molecule, q, by
B ≡ (MR0/r0)2
ωz ≈
ω02 τd ) 4D
2
(19)
Analytical Chemistry, Vol. 78, No. 24, December 15, 2006
〈pd
∫σF(r,t)φ(r) dr〉 FjV
∫φ (r) dr ∫φ(r) dr 2
) pdσ
Then from 〈δF(r,t)δF(r′,t)〉 ) Fjδ(r - r′) we obtain
(26)
〈i(t)T〉 ) (qT)VFj + ibT
(27)
G(0) ) (qT)2VFj
(28)
molecules present at concentrations below 10-10 M. In those cases, we can obtain m from eq 28 and the qT determined by the method described above.
and
Equation 28 provides calibration lines for FCS-based quantification. We see in eq 28 that G(0) is proportional to the concentration Fj and independent of background count rate ib. That is, a background-free measurement of Fj can be obtained by using the estimator of G(0), G ˆ 0. Thus, we regard RSD of G ˆ 0 in repetitive FCS measurements as RSD of FCS-based concentration measurements. It should be noted that the mean value of G ˆ 0 does not depend on the background but its standard deviation does. Equation 27 provides calibration lines for ordinary fluorometrybased quantification, and it may seem that background-free quantification could be performed by subtracting the background ibT from the mean photocount 〈i(t)T〉. In ordinary fluorometry, however, a background drift between measurements yields a bias commensurate with it. Equation 28, on the other hand, holds if the background simply remains constant during a single measurement. This is an advantage of FCS-based quantification. The slope of calibration lines for FCS-based quantification, (qT)2V, is determined as in ordinary fluorometry: by measuring known concentration solutions. The value of qT, however, can be determined by measuring only unknown concentration solutions. Eliminating VFj from eqs 27 and 28, we obtain
G(0) ) qT(〈i(t)T〉 - ibT)
(29)
This equation shows that points (〈i(t)T〉,G(0)) are on a line with a slope of qT independent of concentrations. Therefore, qT is determined by estimating points (〈i(t)T〉,G(0)) by (Fˆ , G ˆ 0) for a dilution series of an unknown concentration solutions and fitting a line to the estimated points. Once V is determined by using a known concentration solution, therefore, the concentration of an unknown brightness molecule can be determined without a known concentration solution of that molecule. This is another advantage of FCS-based quantification because the brightness of molecules of the same kind can change even when they are analyzed in conditions supposed to be identical. We express the mean number of analyte molecules in the detection volume as m. That is,
PRELIMINARY DISCUSSIONS ON RSD We formulate our optimization problem as follows: what MDE profile minimizes the RSD of G ˆ 0 for a given molecule concentration, data collection time, and molecule characteristics? In this section, we describe the solution based on Gaussian MDE profiles and an analytical approximation theory. Several authors have evaluated the precision of G(τ) measurements by using analytical theories,13-16 but most of their results have been obtained for only the limits of m . 1 and m f 0, neither of which minimizes the RSD for a fixed concentration. Assuming G(τ) decays exponentially, Koppel derived the standard deviation of G(τ) as a function of m and τd.13 His result, however, cannot be readily used to minimize the RSD because his derivation does not take into account the interdependence of m and τd. Although Kask et al. also gave a general expression of the RSD, it contains complex integrations and its value cannot be readily calculated from experimental parameters.15 In the limit of qT . 1, ib f 0, and ω0 , ωz, Qian gave an explicit expression of the standard deviation of G(τ) for unlimited m.14 We calculated the RSD of G ˆ 0 on the basis of Qian’s result and obtained
RSD2 )
RSD2 )
( )
ln(4DU/ω02) 8DU 1 ln + 2π1.5DUFjωz eω02 DU/ω02
V ) π2.5nω04/λex
(30)
(25) Rigler, R.; Mets, U ¨ .; Widengren, J.; Kask, P. Eur. Biophys. J. 1993, 22, 169175.
(33)
the detection volume is given by
2
where mb ≡ ib/q. We can obtain m directly from eq 30 if m . mb, but that condition is not satisfied in most measurements of
(32)
which shows that the RSD is smaller when the ωz is larger. The value of ωz, however, does not exceed (e2 - 1)1/2πnω02/λex owing to the decay of I(r) caused by diffraction-limited beam divergence. Actually, ωz is even smaller than this because CEF(r) also decays along the z-axis. We set ωz to the largest value in the range where the MDE profile is well approximated by a Gaussian. Experimentally, this is done by setting the pinhole radius r0 to the upper bound specified in eq 16. Since the largest ωz is approximated by
ωz ) πnω02/λex
Then from eqs 27 and 28 we obtain the following well-known equation13,24
(31)
for the case of U . T. Substitution of eqs 23 and 25 into eq 31 results in
m ≡ VFj
〈i(t)T〉2 (m + mb) ) m G(0)
2 ln(2U/τd) - 1 4 ln(U/τd) + FjV U/τd U/τd
(34)
Substituting eq 33 into eq 32, we obtain,
RSD2 )
λex 2π2.5DUnFjω0
ln 2
( )
ln(4DU/ω02) 8DU + (35) eω02 DU/ω02
Analytical Chemistry, Vol. 78, No. 24, December 15, 2006
8399
Figure 2. Analytically approximated RSD of FCS-based concentration measurements at fixed concentrations versus excitation beam radius ω0.
If parameters other than ω0 are considered constant, this RSD has a minimum at a certain value of ω0. The value is almost independent of D and U as long as 2π2.5nFj(DU)2 . λex (most experiments satisfies this condition) and approximated by
ω0 )
(
λex 2.5
)
2π nFj
x ) xj(kT) + x-4DT lnR1cos(2πR2) and
xj((k + 1)T) ) x - L
[Lx + 0.5]
1/4
(36)
Substitution of eq 36 into eq 34 results in m ) 0.5. That is, the mean number of molecules in the detection volume is ∼0.5 when the RSD is minimized. On the other hand, the RSD in eq 32 monotonically decreases with D, so rapidly diffusing molecules (i.e., low-molecular-weight analytes) are particularly suitable for analysis by FCS. Rhodamine 6G, for example, has been analyzed under a condition of m ) 0.025.24 This condition is fairly far from the optimized m = 0.5 described above. This discrepancy was not markedly disadvantageous because the diffusion constant of rhodamine 6G is fairly large (2.8 × 10-10 m2/s). Most physiologically important biomolecules have diffusion constants smaller than 10-10 m2/s and should therefore be analyzed under conditions closer to the optimal one. The RSD at 10-12, 10-11, 10-10, 10-9, and 10-8 M computed using eq 35 and D ) 4 × 10-11 m2/s, U ) 30 s, n ) 1.33, and λex ) 532 nm is shown in Figure 2 as a function of ω0. As shown there, a ω0 of 0.2-0.4 µm, common in FCS experiments, is adequate for concentrations of 10-8-10-9 M but not for concentrations of 10-11-10-10 M. For the latter range of concentrations, the value of ω0 should instead be ∼1 µm. If this result is expressed as a detection volume by using eq 34, a detection volume on the order of 10-13 L is better for concentrations of 10-11-10-10 M than the currently prevalent detection volume on the order of 10-15 L. MOLECULAR SIMULATION Overview. In the preceding section, we concluded that to quantify concentrations below 10-10 M by FCS, the detection volume should be increased from the currently typical 10-15 L. The assumptions used there, however (2D Gaussian MDE profiles, infinitely bright molecules, and zero background), are not realistic because the background level must increase with the detection volume. We therefore conducted molecular simulations of FCS while allowing for the existence of background and for 8400
the finite brightness of molecules. Based on the Brownian dynamics method and periodic boundary conditions,26 they were executed using a 1.8-GHz Pentium 4 (Intel) PC (MT7000, Epson). If k denotes the index of a time series, the flow of a simulation is described as follows: (1) Initial molecule positions for k ) 0 are randomly dispersed in the cubic cell (simulation box) of edge length L by using uniform random numbers. (2) The photocount rate at time kT, i(kT), is calculated on the basis of eq 6. (3) The kth photocount Fk is given as a Poisson random number whose mean value equals the product of the photocount rate and dwell time T. (4) Molecule positions in the next time instant, (k + 1)T, are calculated on the basis of Brownian dynamics and a periodic boundary condition. For example, the x coordinate of the jth molecule at time (k + 1)T, xj((k + 1)T) is given by
Analytical Chemistry, Vol. 78, No. 24, December 15, 2006
where [X] denotes the maximum integer that does not exceed X and Rl (l ) 1, 2) are uniform random numbers distributed between 0 and 1. (5) k ) k + 1 and steps 2-4 are repeated while kT < U. (6) The photocount fluctuation autocorrelation function is estimated from Fk by using eq 21. The algorithm used for generating Poisson random numbers is described in the Appendix. Parameters. Input and output parameters are listed in Table 1. Case 1 simulates the experiment in ref 24, and case 2 simulates a typical FCS experiment with biomolecules. The parameters for case 3 were set on the basis of the results in the preceding section to improve the quantification precision at concentrations ranging from 10-11 to 10-10 M. In cases 2 and 3, the simulation was repeated 20 times at each concentration. Parameters related to molecular brightness were the same for all cases and were determined such that the simulation results for case 1 agree with the experimental results reported in ref 24. To take into account of the background elevation caused by the detection volume expansion, we assumed the following equation:
ib ) id + pdβ
∫φ(r) dr
where id is the dark count rate of the detector and the constant β is the background emission coefficient of the solvent. We set id ) 100 s-1 and determined β using the same criterion we used to determined the parameters related to molecular brightness. The diffusion constant, data collection time, excitation power, and wavelength are the same in cases 2 and 3 for comparison of the quantification precision. Results. Figure 3 shows the simulated autocorrelation function G(τ) for case 1. The solid and dashed lines are respectively the best-fit line based on eq 24 and residual error. Like the one reported in ref 24, it decays by half in 40 µs. This similarity shows (26) Satoh, A.; Chantrell, R. W.; Coverdale, G. N. J. Colloid Interface Sci. 1999, 209, 44-59.
Table 1. Simulation Parameters type
parameter
input
Q D pd ib T U L C h λex P ω0 n tr M NA R0 r0 σ V qT
output
unit cm-1 M-1 m2 s-1 s-1 s s m pM (10-12 M) m W m
m m m2 L
case 1
case 2
case 3
0.1 6.6 × 104 2.8 × 10-10 0.55 1.3 × 103 10-5 2 1 × 10-5 10000 5.15 × 10-7 2.5 × 10-4 2.55 × 10-7 1.33 0.1 63 1.2 2.7 × 10-7 1.5 × 10-5 2.5 × 10-21 4 × 10-16 0.38
0.1 6.6 × 104 4 × 10-11 0.55 6.8 × 103 3.125 × 10-5 30 1 × 10-5 5, 10, 25, 50, 100 5.32 × 10-7 2.5 × 10-4 2.75 × 10-7 1.35 0.1 63 1.2 2.7 × 10-7 5 × 10-5 2.5 × 10-21 1.8 × 10-15 1.3
0.1 6.6 × 104 4 × 10-11 0.55 1.9 × 104 5 × 10-4 30 5 × 10-5 5, 10, 25, 50, 100 5.32 × 10-7 2.5 × 10-4 1.1 × 10-6 1.35 0.1 23 0.85 4.2 × 10-7 8.5 × 10-5 2.5 × 10-21 1.3 × 10-13 0.85
Figure 3. Photocount fluctuation autocorrelation function obtained by the molecular simulations for case 1 (parameters listed in Table 1). The solid and dashed lines are respectively the best-fit line based on eq 24 and residual error.
that the simulation model works correctly. The G(0) values computed for cases 2 and 3 are shown in Figure 4 as functions of concentration. In each case, a calibration line including the origin was obtained. This shows that G ˆ 0 can be used to obtain backgroundfree quantification. The concentration dependence of the RSD of the G ˆ 0 values obtained in the simulations and by analytical approximation (eq 32) is shown in Figure 5 for cases 2 and 3. Although the agreement between the simulation results and approximation results is not so particularly good, both give smaller RSD values for case 3 whenever the concentration is equal to or less than 10-10 M. Their disagreement is probably attributed to the simulation parameters that insufficiently meet the assumption of the approximation. Since the parameters assumed in the simulations are realistic, the advantage of case 3 shown by the simulations strongly suggests that detection volumes larger than those typical in conventional FCS are more suitable for concentrations below 10-10 M. It should be noted that the samples are assumed to be homogeneous solutions and the results obtained here may not be applicable to inhomogeneous samples with much higher background emission, such as biological cells. The molecular simulations and analytical approximation gave the same results on the order of merit of cases 2 and 3. They
Figure 4. Values of G ˆ 0 obtained by the molecular simulation versus concentration, and the corresponding calibration lines, for cases 2 (part A) and 3 (part B) (parameters listed in Table 1).
also indicated similar dependency of the RSD at a fixed concentration on the excitation beam radius (data not shown). As indicated by these results, the analytical approximation is effective for roughly optimizing FCS setups. The computation time for a data point is proportional to the molecule concentration and simulation box volume L3 and is inversely proportion to the dwell time T (note that T , τd is Analytical Chemistry, Vol. 78, No. 24, December 15, 2006
8401
Figure 5. Theoretical RSD values as functions of the molecule concentration.
necessary). The simulation for 100 pM in case 3 took the longest time, ∼2 h. The simulation time should be shorter, however, when high-end PCs already available are used. As described in the Theory section, our simulation model is composed of only four arithmetic rules, elementary functions, and random number generation (the last occupies a negligible percentage of computation time). It includes neither a hidden integral operation nor a step choice for integral operations. One can conduct a simulation straightforwardly by simply setting the input parameters listed in Table 1. Moreover, the MDE profile is not assumed to be an analytic function and is calculated by using a realistic model based on semiclassical optics. Although MDE profiles more realistic than ours have been calculated by using diffraction theory, their calculation requires integral operations.27 As a result, molecular simulations based on those MDE profiles, which have not been reported, will take a very long time. We think our approach is well-balanced with regard to reality and convenience. EXPERIMENTAL SECTION Reagents. The fluorescent analyte molecule we used was a 112-mer ssDNA labeled with Cy3 at the 5′-end. The nucleotide sequence of the DNA, which was synthesized by Sigma-Aldrich Japan K. K., was TGGAGATTTGGGCGTGCCCCCGCGAGACTGCTAGCCGAGTAGTGTTGGGTTTTAAATGTGCAAGTGACCTTAAACATGAAGTGAGCTTAAACCAGAAGTGGTCTTAAACGCG. The solvent for sample solutions was a 6× SSC buffer containing 60 ppm Nonidet P-40 (Pierce). It was prepared by dilution of a commercial 20× SSC buffer (Promega Corp., Madison, WI). The DNA concentrations prepared were 0 (pure solvent), 5, 10, 20, 50, and 100 pM. Instrumentation. Figure 6 is a photograph of the singleelement aspheric objective lens whose specifications are listed in Table 2. The objective, a plastic lens designed and produced by Pentax Corp. (Tokyo, Japan), is designed to have its image side immersed in sample solutions of n ) 1.35. Acting as the bottom of a sample cell, it becomes a liquid-immersion lens with a large NA (0.9) without additional immersion liquid. Diffraction-limited performance with a wavefront aberration less than 0.07 λrms was obtained. (27) Hess, S. T.; Webb, W. W. Biophys. J. 2002, 83, 2300-2317.
8402 Analytical Chemistry, Vol. 78, No. 24, December 15, 2006
Figure 6. Photograph of the single-element objective lens. Table 2. Specifications of the Single-Element Aspheric Objective Lens design wavelength NA refractive index light-source side lens image side light-source position cover glass thickness thickness focal length working distance effective diameter outer diameter wavefront aberration transmittance
594 nm 0.9 1 1.54 1.35 ∞ 0 4.5 mm 3.68 mm 0.88 mm 4.9 mm 7.4 mm 0) { V +) ln(rand()); i++; } return i; } The function described below is mathematically equivalent to the one above but is unpractical because it easily yields overflows for values of lambda that are only slightly large (if double precision, for lambda larger than 710). int poisson_rand(double lambda) { int i ) 0; double V ) exp(lambda) * rand(); while(V > 1){ V *) rand(); i++; } return i; } ACKNOWLEDGMENT We thank Dr. T. Shimano, Dr. S. Kimura, Dr. K. Watanabe, and Mr. T. Ide for their helpful discussions.
T
dwell time for photon counting (s)
tr
transmittance of detection optics for fluorescence emission
Received for review June 6, 2006. Accepted September 28, 2006.
Tr(R)
transmittance of the pinhole at R
AC061036Y
Analytical Chemistry, Vol. 78, No. 24, December 15, 2006
8405
