Anal. Chem. 2003, 75, 5062-5070
Cross-Correlation Continuous Capillary Electrophoresis: Resolution, Processing Time, and Concentration Effects Martin E. Schimpf† and Semen N. Semenov*,‡
Boise State University, MS-1500, Boise, Idaho 83725, and Institute of Biochemical Physics, RAS, Kosygin Street 4, Moscow, Russia
An expression is derived that relates the signal-to-noise (S/N) ratio in continuous-mode electrophoresis to system parameters. In this process, the time-based cross-correlation function (TCCF) is computed from signals caused by individual analyte passage through two separate detectors. As a characteristic of the noise, the cross-variance of the TCCF is proposed, so that the S/N ratio is defined as the TCCF divided by the square root of the crossvariance. From the derived expressions, the important parameters for method optimization are identified. The dependence of the S/N ratio on system parameters varies with analyte size due to a contribution from the variance in the flight time, which is affected by particle diffusion. For all molecules, the S/N increases with the processing time and the rate at which particles pass through the detectors. The latter increases with particle concentration and migration velocity. Lower sample concentrations require a longer processing time to achieve a specified S/N ratio. For smaller analytes, the residence time in the detector becomes a factor and, therefore, so does detector geometry. Electrophoresis is used to separate a wide variety of species, from low-molecular ions to cells and colloidal particles.1-3 In zone electrophoresis, the probe that contains the sample to be analyzed is injected as a plug into a separation conduit containing the electrical field. The plug is separated into bands of different components according to differences in their electrophoretic mobilities. Unfortunately, perturbations in the electric and hydrodynamic properties of the system by the analyte limit optimization in these systems. These limitations are particularly severe in the analysis of macromolecules with an extensive charge distribution. Distortions in the electric field and electroosmotic flow often cause significant peak spreading in phoretograms, an effect referred to as electrokinetic dispersion or electromigration dispersion.4 Al* Corresponding author. E-mail:
[email protected]. † Boise State University. ‡ Institute of Biochemical Physics. (1) St. Claire, R. L., III. Anal. Chem. 1996, 68, 569R. (2) Freifelder, D. Physical Biochemistry; Mir: Moscow, 1980. (3) Wu, Ch.; Siems, W. F.; Asbury, G. R.; Hill, H. H., Jr. Anal. Chem. 1998, 70, 4929-4938. (4) Poppe, H., Theory of Capillary Zone Electrophoresis; Advances in Chromatography 38; Dekker: New York, 1998; pp 233-300.
5062 Analytical Chemistry, Vol. 75, No. 19, October 1, 2003
though such distortions can be attenuated by reducing the sample concentration, the required concentration is often too low for reliable detection. As a result, these effects are often the limiting factor in minimizing uncertainties in the determination of electrophoretic mobilities. When zone spreading by electrokinetic and electromigration dispersion is sufficiently reduced, resolution becomes limited by the dimension of the introduced sample (the so-called injection probe length). For example, the smallest ions studied by capillary electrophoresis (CE) have a diffusion coefficient on the order of D ≈ 10 - 5 cm2/s. With an analysis time ta of 103 s, diffusional peak dispersion σD ) (2Dta)1/2 is only 10-1 cm. For larger particles such as cells, with D ≈ 10-9 cm2/s, diffusional dispersion is on the order of 10-3 cm. The injection probe length for a standard CE capillary is typically ∼10-1 cm for a capillary with internal diameter 10-2 cm and probe volume 10 nL. Thus, the injection probe length can dominate peak dispersion with large particles and be comparable to diffusional peak dispersion for smaller ions. For shorter analysis times, the role of the injection probe length in CE resolution becomes even more important. Due to these limitations, CE never reaches its theoretical resolution limit based on longitudinal diffusion, with experimental peak widths typically being up to 2 orders of magnitude wider than predicted. The operation of electrophoretic analyses in a continuous mode using correlation techniques has the potential to eliminate these disadvantages and contribute to an improvement in system performance.5 The potential of correlation techniques to enhance signal-to-noise (S/N) ratios is well documented. Early applications focused on continuous-mode chromatography,6-9 where correlation techniques are applied to repetitive injections of sample in a randomized sequence. In 1995, van der Moolen and co-workers10 demonstrated the application of correlation techniques to capillary zone electrophoresis. Around the same time, Castro and Shera11 demonstrated the ability to measure electrophoretic velocities of (5) Felinger, A. Data Analysis and Signal Processing in Chromatography; Elsevier: Publishing: Amsterdam, 1998. (6) Smit, H. C. Chromatographia 1970, 3, 515-518. (7) Annino, R.; Bullock, L. E. Anal. Chem. 1973, 45, 1221-1227. (8) Villalanti, D. C.; Burke, M. F.; Phillips, J. B. Anal. Chem. 1979, 51, 22222225. (9) Yang, M. J.; Pawliszyn, B.; Pawliszyn, J. J. Chromatogr. Sci. 1992, 30, 306314. (10) van der Moolen, J. N.; Louwerse, D. J.; Poppe, H.; Smit, H. C. Chromatographia 1995, 40, 368-374. (11) Castro, A.; Shera, E. B. Anal. Chem. 1995, 67, 3181-3186. 10.1021/ac0300183 CCC: $25.00
© 2003 American Chemical Society Published on Web 09/03/2003
single molecules from their transit times between two focused laser beams, while Brinkmeier and Rigler12 used the time-based cross-correlation function (TCCF) of signals coming from spatially separated detectors to compute the migration velocity. Both these techniques were applied to continuous sample streams, thus eliminating the sample plug injection step and associated imprecision. Van Orden and Keller13 combined CE and fluorescence correlation spectroscopy for the quantitative analysis of binary mixtures at the picomolar level using a single detector without separation. Analysis times were as low as 10 s. In 1999, Brinkmeier and co-workers14 introduced two-beam fluorescence cross-correlation spectroscopy (FCCS) for the simultaneous measurement of both directed flow and diffusional properties at the nanomolar level. Flow velocities from 0.1 to 100 mm/s were measured with continuous analyte flowstreams in 10-100 s. More recently, LeCaptain and Van Orden15 demonstrated the application of twobeam FCCS to bound and unbound fractions of fluorescently labeled DNA in a DNA-protein complex. The relative concentrations were determined at the nanomolar level in 10-15 s without prior knowledge of the pure-component cross-correlation functions. Correlation techniques have also been used recently to increase the resolution of nuclear magnetic resonance spectra in disordered solids.16 In 1998, Semenov17 introduced the concept of incorporating an additional signal-processing step prior to computing the TCCF of signals from spatially separated detectors, whereby the mean signal is subtracted from the values of the individual signals. The magnitude of the resulting TCCF increases proportionally to particle concentration, rather than reaching a plateau value. As we outline below, the method allows relative fluctuations in particle concentration as low as 10-4 to be detected, or particle concentrations as high as 0.1 M, depending on the precision of the analogto-digital converter. We refer to the application of cross-correlation techniques to continuous-mode CE as continuous cross-correlation capillary electrophoresis (C4E). Advantages of the method include enhanced sensitivity, high-speed analysis, and elimination of the sample plug injection step, all of which are desirable for the development of miniaturized devices. A schematic diagram of the measurement process in C4E is contained in Figure 1. A homogeneous analyte mixture is introduced continuously into a capillary having two detectors separated by distance l. The mixture is introduced from a reservoir and can be recirculated. Parameter l is analogous to the separation length in standard CE, and the longitudinal dimension of the detector windows l0 is analogous to the injection probe length. In computing the TCCF, the time dependencies of the signal due to particle passage through the first and second detectors φ1(t) and φ2(t), respectively, are used, where N(T)
φ1(t) )
∑ f (t) n
(1)
Figure 1. Schematic diagram of C4E measurement process: (1) analyte-containing vessel; (2) intermediate service vessel; (3) capillary for electrophoresis; (4) light sources; (5) detectors; (6) interface for signal storage and processing; (7) dc current source.
passage through the detector and N(T) is the number of particles passed through a detector during the observation (processing) time T. The signal described by eq 1 is the sum of the individual signals that arise during the passage of N particles through the first detector. In a C4E experiment, the detector signal is accumulated in successive time bins with a sampling interval equal the total processing time T divided by the number of the time bins.15 Sampling times can be as short as 10 µs. Once collected, the entire data set is used to compute the TCCF. We will consider the most typical situation in CE analysis where particles do not appear or disappear, or undergo mutual transformations, between detectors. Thus, the loss of particles due to interactions with the other particles (e.g., in immunoassays), or with the capillary wall, is assumed to be absent, although in principle such interactions could be accounted for in future refinements of the model. The signal from the second detector is the sum of the delayed signals from the first detector, where the delay (flight) times can be different for particles with the same electrophoretic velocity due to thermal movement and other stochastic or dynamic processes: N(T)
φ2(t) )
∑ f (t - τ ) n
n
(2)
n)1
although the magnitudes of φ1(t) and φ2(t) can be affected by local variations in the buffer or capillary walls, such effects will not lead to a significant change in the results obtained, as we demonstrate below. What is important in the computation of the TCCF is the signal shape as a function of time, which depends on the particle velocity and, when optical detection is used, the profile of the light intensity. Thus, the signal shape over time is affected by the light intensity distribution. Although this shape
n)1
Here, fn(t) is the detector signal from the nth particle due to its (12) Brinkmeier, M.; Rigler, R. Exp. Tech. Phys. 1995, 41, 205-210. (13) Van Orden, A.; Keller, R. A. Anal. Chem. 1998, 70, 4463-4471.
(14) Brinkmeier, M.; Do ¨rre, K.; Stephan, J.; Eigen, M. Anal. Chem. 1999, 71, 609-616. (15) LeCaptain, D. J.; Van Orden, A. Anal. Chem. 2002, 74, 1171-1176. (16) Sakellariou, D.; Brown, S. P.; Lesage, A.; Hediger, S.; Bardet, M.; Meriles, C. A.; Pines, A.; Emsley, L. J. Am. Chem. Soc. 1949, 71, S 2003, 125, 4376-4380. (17) Semenov, S. N. Anal. Commun. 1998, 35, 301-305.
Analytical Chemistry, Vol. 75, No. 19, October 1, 2003
5063
can differ between two detectors, especially if the velocity of the particle changes, such changes are not typical for CE separations and will not be considered in this work. As outlined in ref 17, it is necessary to form signals φ ˜ 1(t) and φ ˜ 2(t) with zero mean values, to ensure that the TCCF is proportional to particle concentration; this is achieved by subtracting the mean value φ j 1,2 ) 1/T ∫T0 φ1,2(t) dt from φ1(t) and φ2(t). The resulting TCCF is written as
Gc(t) )
1 T
∫ φ (t )φ˜ (t T
1
0
1
2
1
+ t) dt1
(3)
Combining eqs 1-3 we obtain
Gc(t) )
1
N(T)
∑ ∫ ˜f (t )f˜ (t
T n)1
T
0
n
1
n
1
+ t - τn) dt1
(4)
Equation 4 is also relevant to situations in which only one of the two detector signals is processed in this way, which may be preferred for technical reasons. Certain problems related to low S/N ratio are eliminated in electrophoretic measurements with this method. For example, when the noise from two detectors is statistically independent, that component of the TCCF will be zero, even when the particle component of the signal has an inherent S/N < 1. This feature was not apparent in previous reports12,15 that utilized laser-induced fluorescence because the noise levels encountered were extremely low. However, this feature will be important for systems with higher background noise, e.g., when absorbance detection is used in the presence of an electrolyte. Although there is no direct experimental demonstration of this property, the ability of crosscorrelation approaches in CE analysis to improve S/N ratio was shown in ref 12, where instead of measuring signals due to the continuous passage of particles, pseudorandom pulse sequences were arranged at the input of the CE system by a computer. In those experiments, improvements in S/N ratio scaled with the square root of the number of pulses formed in the system. Because the number of particles passing through the detectors in C4E can be made much larger that the number of pulses formed in correlation electrophoresis, one can expect a much faster increase in S/N. Furthermore, the formation of the pseudorandom pulse sequence introduces problems that, while characteristic of the standard CE process, are eliminated by C4E. Although additional (unknown) sources of peak broadening were reported in correlation electrophoresis,10 the ability of cross-correlation techniques to suppress noise in CE systems was demonstrated nevertheless. In this report, we study the parameters of the C4E process that are important for optimizing the method. One of the more important questions is how processing time affects the S/N ratio. Although an increase in S/N ratio with processing time can be expected, the specific dependence of noise on processing time and other parameters requires further characterization. The required time will depend on the passage of an adequate number of particles through the detectors, which is equivalent to the number of processed signals. As with correlation electrophoresis, one can expect S/N to be proportional to N(T)1/ 2, which increases with the particle concentration in the capillary c0. The concentration of charged species may vary widely. In the 5064
Analytical Chemistry, Vol. 75, No. 19, October 1, 2003
analysis of low-molecular-weight ions, for example, concentrations can achieve 10-5 M,10 or 1019 particles/cm3. In cell cultures, the concentration can be as low as 108 cells/cm3.18 In an ideal situation, the minimal processing time is the mean flight time for the slowest moving analyte particle, and the concentration of that analyte will govern the required processing time. Stated another way, the minimum required concentration depends on a specified minimal processing time, which is the time necessary to resolve all peak patterns in the correlogram. Thus, the number of the particles passed through the detectors is the main parameter governing S/N ratio; lower concentrations require a longer processing time. In principle, even a single particle could be analyzed by driving it through the system enough times. To obtain exact information on system performance, we first derive an equation that describes the relationship between processing time, particle concentration, S/N ratio, and other system parameters. In doing so, we establish a criterion for the S/N ratio required to achieve a specified resolution, which we designateRS/N. Thus, the main objective of this work is to define parameter RS/N as a function of system parameters and processing time. To facilitate this process, we first review the definitions and major results of TCCF shape calculations already reported. When the processing time is such that the number of particles passed through the detectors is large enough to neglect fluctuations in N(T), one can write
N(T) ) ω jT
(5)
where ω j is the average number of particles passing through detector per unit time, which is related to the mean particle velocity u and the mean numeric particle concentration c0 as follows:
ω j ) uc0S
(6)
Here S is the area of the detection spot, which is equal to the area of the capillary cross section if all particles that pass through are detected. Introducing the normalized distribution of delay (flight) times as W(τ) we obtain
Gc(t) ) c0uST
∫ g(t - τ)W(τ) dτ ∞
0
(7)
where the notation
g(t - τ) )
1 T
∫ ˜f (t )f˜(t T
0
1
1
+ t - τ) dt1
(8)
represents the autocorrelation function of an individual signal with time shift τ corresponding to the particle flight time. The distribution function W(τ) reflects physicochemical processes that occur between detectors. For the most desirable situations in which only particle diffusion and electrophoretic drift at constant velocity occur, the following expression for the distribution function W(τ)is obtained19
e-(τ - 〈t〉) /2σ x2πσ 2
W(τ) )
2
(9)
Here l is the distance between detectors, assumed to be much larger than the characteristic longitudinal size of the detector window l0; parameter 〈t〉 ) l/u is the mean particle flight time between detectors and σ2 ) 2Dl/u3 ) 2〈t〉2/Pe is the variance in flight time due to longitudinal diffusion, where D is the particle diffusion coefficient. In situations where a temperature gradient exists in the capillary, an additional skew is expected in the distribution of flight times, due to the dependence of D on temperature. However, the analysis of such secondary effects is beyond the scope of this work. The Peclet number is equivalent to twice the number of theoretical plates used to denote chromatographic separating power. Equation 9 holds when the Peclet number Pe ) ul/D is large, a condition that is fulfilled in most situations. For example, when u ) 10-2 cm/s (a common ion velocity in electrophoresis), l ) 1 cm (a minimal capillary length in CE systems), and D ) 10-5 cm2/s (a maximum value in liquids), then Pe ) 100. In experiments that use laser-induced fluorescence,12,15 however, the distance between detectors is 5-10 µm, and Pe could potentially be as low as 10. In this case, the distribution function takes more complex form.19 Equation 9 corresponds to a simple model of the CE process and neglects thermal peak broadening and particle interactions. At this stage, however, we are interested only in formulating the main principles of the method, to examine the potential of the system. Further refinements will be made later, using models established previously.19 FORMULATION OF CRITERION FOR S/N RATIO We follow the same approach that has been applied previously to fluorescence correlation spectroscopy and dynamic light scattering,20,21 but instead of the variance obtained for individual detectors, we consider the cross-variance of the signals from two detectors. Thus, the following function is used to maximize the S/N ratio:
RS/N )
Gc(t)
xcross varGpc(t)
(10)
where cross var indicates cross-variance. The necessity of the replacement of variance with cross-variance can be explained by considering the standard variance function. Following refs 2022, but using the fact that the mean value of the TCCF is equal to zero, the variance of the TCCF is defined as
varGc(t) )
1 T
∫ G (t )G (t T
c
0
1
c
1
+ t) dt1
(11)
while the cross-variance can be defined as
cross varGc(t) )
1 T
∫
T
0
Gc(t1)Gc*(t1 + t) dt1
(12)
(18) Armstrong, D. W.; Schulte, G.; Schneiderheinze, J. M.; Westenberg, D. J. Anal. Chem. 1999, 71, 5465-5469. (19) Semenov, S. N. Russ. J. Phys. Chem. 1995, 69 (11), 1884-1886. (20) Koppel, D. E. Phys. Rev. A 1972, 10, 1938-1945.
where
Gc*(t) ) c0uST
∫ W(τ)g(t - 2τ) dτ ∞
0
(13)
contains flight times multiplied by two. Equations 11 and 12 give correlation functions of the signal in a form that allows us to evaluate the stochastic variance as a function of time. The analyzed signal in this case is the TCCF itself. Using eqs 1 and 2, we can recast eqs 11 and 12 as follows:
varGc(t) ) N(T)h(t) cross varGc(t) ) N(T)
(14)
∫ W(τ)h(t - τ) dτ ∞
0
(15)
where
h(t) )
1 T
∫ g(t T
0
1
- τ)g(t1 + t - τ) dt1 )
1 T
∫ g(t )g × T
0
1
(t1 + t) dt1 (16) The function h(t) may be determined as the second-order autocorrelation function of individual signals, that is, the autocorrelation function of the autocorrelation function, where the signal contains stochastic noise. The time domain over which the function h(t) changes significantly is about equal to the residence time of a particle in a detector, but much less than the mean flight time 〈t〉 where the peak in the TCCF will occur. Thus, in a reasonable measurement regime, the width of the TCCF peak should be much smaller than 〈t〉. However, the fact that functions h(t) and Gc(t) have maximums at t ) 0 and t ) 〈t〉, respectively, which are placed far one from another, means that the variance expressed by eq 14 will approach a very small value in the time regime of interest. The form expressed in eq 10 will be important when the noise component of the signal changes with time. For example, if the noise is higher in the time domain surrounding the peak of the TCCF, the standard method that uses variance will overestimate the S/N ratio. Thus, the signal-to-noise parameter defined by eq 10 can be expressed as
RS/N
∫ W(τ)g(t - τ) dτ ) ) xN(T) x|cross varG (t)| x|∫ W(τ)h(t - τ) dτ| ∞
Gc(t)
0
c
∞
0
(17)
Equation 17 is a generalization of the standard criterion for the case of a TCCF. This newly derived criterion that includes crossvariance in estimations of the TCCF variance is analogous to the use of standard variance in evaluating conventional (singledetector) time correlation functions. TCCF AND S/N RATIO WITH OPTICAL DETECTORS The use of correlation techniques for spectroscopic detection in analytical separations originated from a desire to lower the (21) Jakeman, E. Correlation of photons. In Photon Correlation and Light Beating Spectroscopy; Cummings, H. Z., Pike, E. R., Eds.; Plenum: New York, 1974. (22) Qian, H. Biophys. Chem. 1990, 38, 49-57.
Analytical Chemistry, Vol. 75, No. 19, October 1, 2003
5065
Table 1. Equations for Signals with a Gaussian Shape 1. individual signal with zero mean value
˜f (t) ) e-t /2σ0 - x2π(σ0/T)
2. autocorrelation function of individual signal
g(t) )
3. second-order autocorrelation function
h(t) ) 16πx2π
4. time cross-correlation function (TCCF)
Gc(t) ) 4n
detection limits. Consequently, the discussion and implementation of correlation techniques have been limited primarily to highly sensitive spectroscopic techniques such as laser-induced fluorescence. As we stated above, however, correlation techniques may be critical to the development of continuous high-speed separations in miniaturized devices for the analysis of more concentrated samples. Therefore, we now consider the implementation of correlation techniques with optical (absorbance or scattering) detection, including a discussion of the upper limit to sample concentration, which depends on the ability to distinguish between differences in large particle counts during the detection and digitization process. Equation 17 allows the characteristic correlation times of the stochastic process to be analyzed. When dispersion of the flight times is low enough, e.g., with relatively large particles having low diffusion coefficients, we can approximate the distribution function W(τ) with the Dirac δ-function W(τ) ) δ(τ - 〈t〉), so that eq 17 takes the following form:
)
g(t - 〈t〉)
x|cross varGc(t)| x|h(t - 〈t〉)|
xN(T)
(18)
This case corresponds to the situation in which the dispersion σ of the distribution function is much smaller than the residence time σ0 of the particles in the detector window, i.e., when Pe . 2〈t〉/σ0. In this regime, where diffusional peak broadening is negligible, the profile is determined by the shape (durability) of the individual signal. This optimal regime can be found in the C4E analysis of cells and colloids, provided particle-wall interactions are suppressed.18 In this regime, the residence time of the particle in the detector window is most important for the resolution of the method. For a broad distribution function with a moderately large Pe (2〈t〉/σ0 . Pe . 1), the time dependence of the S/N ratio will be determined mainly by the shape of W(τ). This situation is analogous to the CE separation of small ions with high diffusivity, where zone dispersion is so large that the injection probe length can be neglected. In contrast to CE analysis, the contribution of particle diffusion to TCCF shape becomes significant at much lower values of D because the longitudinal length of the detector 5066
( ( )(
Analytical Chemistry, Vol. 75, No. 19, October 1, 2003
)
2xσ0 σ0 2 2 σ T e-t /4σ0 - 2xπ 2π 0 T
(x
σ0 T
3
e-t
2/8σ 2 0
σ02 2
)
σ0 T
- 2x2π 2
2
2
( ) (x σ0 T
2
σ02 2
)
σ0 T
e-(t-〈t〉) /2σ0 +σ - xπ 2
2σ0 + σ
cross varGc(t) ) 32πn
5. cross-variance of TCCF
Gc(t)
2
2
4σ0 + σ
)
σ0 T
e-(t-〈t〉)2/4σ0 + σ2 - xπ 2
2
window in C4E is much smaller (∼10-4 cm with a laser beam). Thus, diffusional peak broadening is expected to play a more significant role in C4E compared to CE. A more quantitative comparison could be made using a concrete shape for the individual signals, corresponding to an actual light intensity profile, in the general equations above. For detection by fluorescence, scattering, or absorbance, the intensity of laser light illuminating the capillary has a Gaussian shape. Following the approach used in refs 21 and 22, we define the shape of an individual signal impulse by
fn(t) ) e-(t-tn) /2σ0 2
2
(19)
where tn is the time of the appearance of the nth particle in the first detector and σ0 ) l0/u is the root-mean-square particle residence time in the detector with longitudinal window length l0. When the signal shape is rectangular rather than Gaussian, one can still approximate it by a Gaussian function with an equivalent root-mean-square injection probe length.4 We will assume that particle residence time is the same for all particles of a given size, thereby neglecting differences related to stochastic processes. With a Gaussian signal shape, the integrals in eqs 3-18 can be calculated analytically. The main results are summarized in Table 1. In reality, any kind of detector can be used. For example, crosscorrelation methods have been used with flame detectors in gas chromatography to increase S/N ratio.23 To suppress very high levels of noise in such signals, the characteristic durability of a signal contained in the noise must be much shorter that the characteristic delay time in the passage of the signal (or particle) between detectors. The main requirement is that the noise in the two detectors is statistically independent. Thus, the detectors should be isolated in all possible ways, i.e., different power supplies, separate optical systems, etc. The equation for the TCCF from a Gaussian-shaped signal is
Gc(t) ) nx32π
(x
σ02 2
2σ0 + σ
2
2
2
)
σ0 T
e-(t-〈t〉) 2σ0 +σ - xπ
2
(20)
where n ) σ0N(T)/T is the average number of particles that are
observed in the detector at any time moment, which is analogous to the number of particles introduced into the capillary in CE analysis. Combining eqs 5, 6, and 20 yields
n ) σ0uc0S
(21)
Equations 20 and 21 allow us to explain certain problems that have been described10 in the TCCF of a pseudorandom pulse sequence. First, it is impossible to provide a pulse duration comparable to the passage time of a single particle. The minimal pulse duration is limited by the device used for probe introduction and appears always to be larger than several seconds. As a consequence, the pulse number that can be achieved over a comparable time is much smaller than that possible with C4E. To compare the data from ref 10 with the present theory, one must define parameter σ0 for the rectangular pulses used in that work. Following the procedure used for conventional CE in ref 4 we obtain
σ02 ) τ02/12
rectangular pulse
(22)
where τ0 is the duration of the pseudorandom rectangular pulse (∼10 s). The pulse duration can be as low as 10-3-10-2 s in C4E, and ∼0.1 s in CE. The value of σ0 for the pseudorandom pulses formed in ref 10 is 2.9 s, which is at least 300 times greater than that in used C4E. As a result, the potential for improvement in the S/N ratio is ∼300 times greater than that reported. Using eq 22 for σ0, eq 20 yields a peak dispersion σT for correlation electrophoresis of
σT ) xτ02/6 + σ2
Figure 2. Dependence of F1(p) on peak dispersion parameter p ) ( σ/ σ0)2.
(23)
This expression allows us to analyze and solve certain problems that have been raised with peak dispersion in correlation electrophoresis. For example, the correlogram peak dispersion in ref 10 is estimated as 4.0 s, while for a CE phoretogram with the same components it is 3.3s. Although specific data were not included, the contribution of injection probe length to the value of σ0 for CE can be neglected as a first approximation in the evaluation of the CE dispersion in that work, so that CE peak dispersion can be related to diffusion broadening alone. In that case, the additional peak broadening in the transition from correlation electrophoresis to C4E can be related to the durability of the pseudorandom pulses generated. Under these assumptions, the contribution of pulse durability to parameter σ0 in eq 23 for correlation electrophoresis is estimated to be 1.5 s. Considering the simplifications and assumptions that were made, the agreement between this value and 2.9 s obtained from eq 23 is satisfactory. The discrepancy may be due to an overestimation of diffusion broadening of the CE peak, which leads to an underestimation of the contribution of the initial probe length in CE. Unfortunately, we have no means for a more exact estimation of the diffusion contribution. However, we are able to make the conclusion that a significant part of the peak width in correlation electrophoresis is related to the pulse duration, which was too (23) Valentin, J. R.; Carle, G. C.; Phillips, J. B. Anal. Chem. 1985, 57, 10351039.
Figure 3. Dependence of F2(p) on peak dispersion parameter p ) (σ/σ0)2.
long in the cited work. Thus, C4E would effectively increase both the S/N ratio and the resolution compared with correlation electrophoresis. Equations from lines 4 and 5 of Table 1 allow us to express the S/N ratio in the time domain around the mean flight time 〈t〉 as follows:
RS/N )
x2πn F (p)e
T σ0
1
-(t-〈t〉)2/2σ02F2(p)
(24a)
where
x
xp + 4 p+2
F1(p) ) F2(p) ) p)
(p + 4)(p + 2) p+6
() σ σ0
2
)
()
2 l Pe l0
2
(24b) (24c) (24d)
Parameter p defines the relative contribution to TCCF shape that arises from intrinsic CE peak broadening (longitudinal diffusion in the model considered here). Plots of the functions F1(p) and F2(p) are illustrated in Figures 2 and 3. In actual CE systems, where values of Pe range from 104 for small ions (D ∼ 10-5 cm2/s) to 108 for cells (D ∼ 10-9 cm2/s), the detector window size is ∼10-2 cm, the distance between Analytical Chemistry, Vol. 75, No. 19, October 1, 2003
5067
Figure 4. Dependence of the S/N time profile (Gc(t)/(|cross varGc(t)|)1/2)/(T/σ0)(n/2π)1/2 on dimensionless time parameter ϑ ) (t - 〈t〉)2/ 2σ02 at different values of the peak dispersion parameter p ) (σ/σ0)2.
detectors (which corresponds to the capillary length in CE) is ∼10 cm, and values of p range from about 10-2 for cells to 102 for small ions. This means that the S/N ratio in C4E for extreme situations will be determined by either the size of the detector window or the magnitude of the diffusion coefficient. Equations 24a-24c allow for the determination of S/N ratio at any time near the mean flight time 〈t〉, i.e., near the peak maximum position, and represent the most exact characteristic of the fluctuations of the TCCF. The plot of the “dimensionless” function corresponding to the time profile of the S/N ratio at different values of p is demonstrated in Figure 4. The time profile of the S/N ratio becomes lower and wider with increase in dimensionless peak dispersion, which is related to the particle diffusion. However, the order of magnitude remains about the same for most situations. Equation 24 combined with the equation for the TCCF from line 4 of Table 1 allows us to evaluate the resolution of C4E using the accepted “4σ” criterion. Thus, the minimum relative velocity difference between two kinds of species ∆u/u required for resolution in C4E is given by
x
∆u ) 4x2 u
2
l0 1 + 2 Pe l
(25)
In systems with very large Peclet numbers, the minimal velocity difference is 4(21/2)(l0/l). For a detector window with length 10-2 cm, and a distance 10 cm between detectors, velocity differences of ∼1% can be resolved for ions with diffusion coefficients smaller than ∼10-7 cm2/s. Thus, for large proteins, diffusional peak broadening can be ignored and the resolution of C4E will be defined by the length of the detector window l0. A parameter that is commonly used in separation science is the ratio of the peak maximum value to the root-mean-square amplitude of the noise. The corresponding value of RS/N at t ) 〈t〉, where the TCCF peak has its maximum value, is
RS/N0 )
x2πn F (p)
T σ0
1
(26)
Using data from the few experiments where the TCCF was computed in CE capillaries, we can check the suitability of eq 26 5068
Analytical Chemistry, Vol. 75, No. 19, October 1, 2003
as a particular case of the more general eq 17. These experiments were performed using laser fluorescence microscopy with a laser beam radius in the focal region of ∼1 µm and a distance between laser beams of several micrometers.. In ref 12, 7-kb DNA fragments were studied in picomolar solutions with a processing time of 6 s (see Figure 3a in ref 12). The root-mean-square particle residence time in the detector (σ0) corresponding to a laser beam radius (l0) of 1.5 µm and a particle velocity (u) of 10-2 cm/s is ∼15 ms. The detector volume required to calculate the mean particle number in the detector (n) is difficult to estimate due to the lack of information on the longitudinal length of the focal region. We will assume this length to be the radius of the laser beam in estimating a detection volume of 10-11 cm3, which corresponds to n ≈ 10-2 particles. Substituting these values into eq 26 and assuming that diffusional peak broadening can be neglected (p , 1), we get RS/N0 ≈ 20, which is in satisfactory agreement with the TCCF plot in ref 12. Note that, with an increase in concentration to 10-9 M, as made in ref 15, the required processing time would decrease by a factor of 30 with the same system parameters, which is about the same as the flight time of 80 ms between detectors. In the analysis of DNA fragments in ref 15, most of the parameters required to check eq 26 can be obtained by fitting the experimental plot of the TCCF. These result in the following values: n ≈ 6.5 particles/detector, σ0 ≈ 1.2 ms, and p ≈ 0.55 (calculated). The required processing time is T ≈ 15 s. Substituting these parameters into eq 26 yields RS/N0 ≈ 103. The TCCF plot in ref 15 demonstrates the practical absence of visible noise (S/N > 100) with processing times that exceed 100 ms, which is in qualitative agreement with the theory, while intense noise is present with processing times less that 1 ms. This short-time noise may be caused by so-called pseudoautocorrelations that arise from the overlapping of the laser beams in the system. Such processes were studied in ref 25 and found to be so intense that they dominated all other noise sources, rendering the application of the theory proposed in this paper impossible. We can also evaluate the S/N ratio in systems that contain several kinds of particles. When particles of different kinds are not correlated, the total TCCF and cross-variance can be taken as the weighted sum of the corresponding partial functions, with weighting coefficients that scale with the amplitudes of individual signals from different kinds of particles. With well-resolved peaks, the S/N ratio for the mth peak may be written as
RS/N )
Gcm(〈t〉)
x
M
|cross varGm c (t)| + Rmm′
∑ |cross varG
m′*m
m′ c (∞)|
(27)
where Rmm′ ) (am′/am),4 and parameters am and am′, are the amplitudes of the individual signals of the given species. When the different kinds of particles have similar diffusion coefficients and are at about the same concentration, the S/N ratio for a given (24) Berne, B. J., Pecora, R., Dynamic light scattering; Wiley: New York, 1976; Chapter 9. (25) Brinkmeier, M., Doerre, K., Stephan, J., Eigen, M., Anal. Chem. 1999, 71, 609-616.
peak will still be determined by eq 24, with the contributions to the total cross-variance from the presence of another particle type being approximately M(σ0/T), where M is the number of different particle types. Compared to the overall cross-variance. this represents a small contribution. Only for particles present as an impurity with very high diffusion coefficients will the presence of other particle kinds be an obstacle in determining their contribution to the S/N ratio in the time domain of importance. In this situation, one could neglect the cross-variance from the analyte particles and write the S/N ratio for the mth component as
RS/N )
Gm c (〈t〉)
x
Rmm′
∑ |cross varG
(28) m′ c (∞)|
m′*m
Equation 27 can be also used to evaluate the S/N ratio in cases where noise arises spontaneously in the detector itself or from the signals of other particles, such as ions in the background electrolyte. In this case, the TCCF of the signals due to such noise should be substituted into eq 27 together with the TCCF for the analyte particles. Unfortunately, we do not have a reliable physical model for detector noise. In the lower limit, for example, when the noise in different detectors is not correlated, its TCCF will be equal to zero, and the S/N ratio will still be determined by the analyte noise in the system. In practice, this theoretical limit may be achieved when the two detectors have isolated light and power sources because the correlation times for detector noise usually is much shorter than the particle flight times 〈t〉. In the evaluation of S/N ratio for a real system, it is simpler to obtain this function in experiments that record background noise and compute the respective TCCF in an “empty” capillary, i.e., with no analyte particles. In that case, eq 27 can be used for the selection of an acceptable particle concentration and processing time, once the computation of the detectors’ cross-variance has been completed. Up to this point, we have discussed the situation in which the detector signals arise primarily from analyte particles. This situation may be true in detectors that rely on fluorescence and scattering, where background electrolytes and other impurities contribute very little to the signal. In the case of absorbance, however, background electrolytes can make significant contributions to the TCCF. The movement of these ions is related to the necessity to maintain electric neutrality in the system. Time correlation functions for these systems under an applied electric field were calculated in ref 24 and correspond to purely diffusive movement with an effective diffusion coefficient that is lower than the lowest ion diffusion coefficient. In CE capillaries, for example, where an electroosmotic flow exists, background electrolyte ions yield a TCCF peak having a maximum at 〈t〉 ) l/ueof, where ueof is the electroosmotic flow velocity. This peak should be resolved from the peaks of the analyte particles. A problem could arise, however, when absorbance of the background ions is significant due to their much higher concentration compared to the analyte. Even in this situation, it is still possible to resolve the background signal because the peaks should be located in different time domains, provided the electrophoretic velocities of the analyte ions differ from that of the electroosmotic flow. However, the resolution could require the analysis time to be increased. The most
promising situation is one in which the analyte particles are moving in the opposite direction as electroosmotic flow. In this case, the background ions will give no correlation in the second detector and, therefore, have no contribution in the TCCF. In the application of C4E at high analyte concentrations, the limiting factor will be the resolution of the analog-to-digital converter. The current resolution of such transformations is ∼10-5, meaning each signal value is present as a sum of 1-105 elementary values. This means that the relative fluctuations in the signal must be significantly larger than 10-5 in order for these fluctuations to be analyzed in computing the TCCF. Using the law of large numbers, this relative fluctuation level scales to the average number of particles in the detector as 1/(n1/2). If we take the minimal acceptable fluctuation level as 10-4, then the fluctuations will be represented by 1-10 elementary digital values. Then, one has for the maximal number of particles in the detector window nmax ≈ 108. The maximum particle concentration will of course depend on the detection volume. With a laser beam diameter of 10-4 cm, the detection volume would be 10-12 cm3, and the maximum particle concentration would be 0.1 M. For a higher precision in the digital representation of the analyzed fluctuations, e.g., 102 elementary digital values, the maximum particle concentration would decrease to 10-3 M. This limitation would be acceptable for most situations, and the analog-to-digital conversion process will not be a problem in C4E experiments when laser sources with thin beams are used. For larger detector volumes, however, this limitation will be encountered. CONCLUSIONS C4E has the potential to improve the resolution of CE analysis. The theory presented here has allowed us to establish the effect of specific design parameters on the resolution and S/N ratio of the system, as well as to define the particle concentration required for a specified S/N ratio and processing time, when capillary electrophoresis is carried out in a continuous mode. In the limit where the effect of diffusional and other sample-specific broadening mechanisms becomes negligible, the resolution in CE is determined simply by the initial probe length. In C4E, the detector spot size plays an analogous role to the initial probe length. Thus, a smaller spot size leads to higher resolution, but also a higher sample concentration, to realize a specific S/N ratio. Finally, the two detectors should be placed close together, but not so close that signal overlap occurs, leading to contamination of the crosscorrelation function by an autocorrelation component. GLOSSARY φ1,2(t)
signal from detector 1 or 2
φ ˜ 1,2(t)
signal from detector 1 or 2 after the mean value has been subtracted
σ
standard deviation in flight time due to longitudinal diffusion
σ0
rms average particle residence time in detector window
σT
standard deviation of TCCF for correlation electrophoresis
τ
delay (flight) time
τ0
duration of pseudorandom rectangular pulse Analytical Chemistry, Vol. 75, No. 19, October 1, 2003
5069
ω j
average rate at which particles pass through the detector
p
relative contribution to TCCF shape from CE peak broadening (σ/σ0)2
am
relative amplitude of detector signal from a given type of particle
Pe
Peclet number (U(l/D)) signal-to-noise ratio of the TCCF
RS/N
c0
numeric particle concentration
RS/N
RS/N at TCCF maximum
D
diffusion coefficient
S
area of capillary cross section
F1,2(p)
defined in eqs 24b and 24c
t
time
fn(t)
signal of nth particle in detector
〈t〉
mean particle flight time between detectors
˜fn(t)
signal of nth particle in detector after mean value has been subtracted
T
processing (analysis) time
ta
analysis time
Gc(t)
time cross-correlation function
Gm c (t)
tn
time of appearance of nth particle in first detector
Gc(t) of mth peak in correlogram
g(t - τ)
u
mean particle velocity
autocorrelation function of individual signal with time shift τ
ueof
electrophoretic flow velocity
h(t)
second-order autocorrelation function of signal
W(τ)
normalized distribution of flight times between detectors
l
distance between detectors
0
l0
length of detector window
M
number of different particle types
N(T)
number of particles passed through detector during time T
Received for review January 7, 2003. Accepted May 31, 2003.
n
average number of particles residing in the detector volume
AC0300183
5070
Analytical Chemistry, Vol. 75, No. 19, October 1, 2003
