Peer Reviewed: Hadamard Transform CE - Analytical Chemistry (ACS

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Takashi Kaneta Kyushu University (Japan)

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A N A LY T I C A L C H E M I S T R Y / O C T O B E R 1 , 2 0 0 1

A possible solution to the poor sensitivity of CE. espite all the advantages of CE, the technique’s application is limited, in part, because of poor detection limits. One way to improve sensitivity is with multiplexing techniques, which are powerful mathematical approaches for detecting small amounts of molecules in very dilute solutions or improving the resolution of spectroscopic methods. Incorporating Fourier transform (FT), for example, with techniques such as NMR, IR absorption, and ion cyclotron resonance MS has revolutionized chemical analysis. However, in the separation sciences arena, there are only a few examples of multiplexing methods: cross-correlation GC (1), Shah convolution FT electrophoresis (2), and cross-correlation electrophoresis (3, 4). In this article, we introduce Hadamard transform CE, an approach that offers improved sensitivity.

O C T O B E R 1 , 2 0 0 1 / A N A LY T I C A L C H E M I S T R Y

541 A

We describe the experimental setup, basic principles, performance, and potential applications of this promising technique. These multiplexing techniques encode analytical signals so that many are measured simultaneously. Then, the transform technique recovers the desired result with better resolution or S/N. Hadamard transform is a multiplexing technique that is similar to FT and now used with several chemical analyses. Whereas FT is based on sine and cosine functions, only multiplication and addition are used in a Hadamard transform. Thus, the calculation processes for a Hadamard transform are much simpler. Hadamard transform has been successfully applied to IR spectroscopy (5–7), time-of-flight MS (8), and imaging techniques (5, 9). For example, Hadamard transform IR spectroscopy basically measures the intensity of the spectral components as combined groups, rather than determining the intensity at each wavelength. Because Hadamard transform spectroscopy observes all the wavelengths in a spectrum simultaneously, it has the multiplex advantage in S/N over a conventional analysis. Hadamard transform can also be applied to separation techniques by using multiple injections, which increase the overall injection volume and thereby lower the concentration limits of detection. Without multiplexing techniques, CE has low detection limits because injection volumes are typically limited to the nanoliter range. Moreover, nanoliter injections are difficult to prepare and manipulate. Using microliter volumes would be advantageous provided that there is sufficient sample, but the direct injection of a larger sample volume severely degrades the separation resolution. For this reason, the analyte must be concentrated into a smaller volume first, which, in turn, requires additional sample handling. Using Hadamard transform CE eliminates the need for preconcentration procedures. Although a single-injection volume is small, the total injection volume can be increased by multiple injections in a pseudorandom sequence.

Hadamard transform CE apparatus Injection methods that rely on pressure, electrokinetic phenomena, or gravity are typically used with CE. For Hadamard transform CE, a sample or buffer solution must be injected over a precisely constant time period. Electrokinetic injection is one of the methods that can be used to randomly inject the sample and buffer solution. For example, this multiplex injection method has been demonstrated with cross-correlation CE using microchip technology (3, 4). In addition, electrokinetic injection can be applied to several detection methods—absorbance, conventional fluorescence, and electrochemical. However, the technique requires a complicated injection device, which uses an interface with a capillary column that injects the sample and buffer solution without any interruption of the electric current.

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Controller Beam splitter

Shutter

Personal computer

Argon ion laser Reservoir Gating beam 80% Probe beam 6%

Flow

Lens Photomultiplier Capillary Power supply

Lens

Lens Spatial filter

Reservoir Notch filter

Personal computer

FIGURE 1. Experimental setup for Hadamard transform CE.

Intensity Capillary

Electroosmotic flow Injection point

0 1 0 1 1 1 0 0 1 0 1 1 1

Migration time Electropherogram [X]

Multiple injections

Detection point

An alternative method is optically gated injection (10), which is a straightforward and reliable technique for accomplishing multiple injections. In this approach, fluorescent components are continuously introduced into the capillary column. A highpower gating laser photodegrades the fluorescent components and renders the material undetectable to a fluorescence detector, which is located further along the capillary column. A sample zone is generated by simply blocking the laser light. The injection time is easily controlled by opening and closing a shutter located at the pathway of a gating laser beam. Thus, it is possible to use this technique with a standard capillary column and without an injection device, which represents a distinct advantage for applying this approach with other separation methods such as capillary gel elecrophoresis, capillary electrochromatography, and capillary LC. Figure 1 is an illustration of a Hadamard transform CE apparatus developed by our group (11). An Ar+ laser is split into two for use as gating and probe beams. The gating beam is passed through or blocked by an optical shutter, which is modulated by a controller that is interfaced with a personal computer. The capillary and reservoirs are filled with a sample solution containing fluorescent analytes. When the shutter is opened, the fluorescent analyte is optically decomposed by strong irradiation with visible light, resulting in a sample zone that has no signal at the detection point. This is the equivalent of not injecting a sample. Thus, the sample is injected only when the shutter is closed. Repeatedly closing and opening the shutter according to a sequence based on a cyclic S matrix, which is a Hadamard matrix consisting of 1s and 0s, permits the multiplex injection of the analyte. It is important to satisfy a condition in which the time of data sampling is exactly equal to the injection time. The personal computer easily accomplishes this by controlling the times for the opening and closing of the shutter and the data sampling. Our experimental apparatus uses a capillary with a relatively small diameter (25 µm i.d.) to achieve the efficient photodestruction of analytes by the gating beam. The effective length, which is determined by the distance from the gating and probe beams, is typically adjusted to 4 cm to achieve rapid separation.

Intensity

Encoding and decoding the electropherogram

Migration time Encoded electropherogram [Y] FIGURE 2. Schematic of sample zones formed by a multiplex injection method and the encoding of the electropherogram.

The Hadamard transform CE electropherogram is obtained by multiple conventional injections. A sample or separation buffer solution is injected according to a pseudorandom sequence based on the cyclic S matrix (5, 6). A Hadamard matrix of order n, Hn, is an n ⫻ n matrix of +1s and –1s satisfying Hn H nT = nIn

(1)


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in which HnT is the transpose of Hn, and In is an n ⫻ n unit matrix. Hn is normalized if the first row and column contain only +1s. Deleting this row and column, we obtain an (n – 1) ⫻ (n – 1) matrix labeled Gn–1. If the +1s in Gn–1 are changed to 0s and the –1s to 1s, we obtain an (n – 1) ⫻ (n – 1) matrix of 0s and 1s, labeled an S matrix of order n – 1 (Sn–1). If the S matrix has the property that each row is obtained by cyclically shifting the previous row one place to the left, with any overflow from the left inserted on the right, the matrix is called a cyclic S matrix. Further details of the properties of H, G, and S matrices can be found in a textbook (6). Figure 2 illustrates the resulting sample zones formed by a multiplex injection method and the electropherograms obtained by single and multiple injections. Here, the 7 ⫻ 7 S matrix is used as a simple example and shown in Equation 2.

S7 =

1 1 1 0 1 0 0

1 1 0 1 0 0 1

1 0 1 0 0 1 1

0 1 0 0 1 1 1

1 0 0 1 1 1 0

0 0 1 1 1 0 1

0 1 1 1 0 1 0

1 2

(2)

Several elements in the pseudorandom sequence are determined by the order of the matrix used in the experiment. Thus, the number of elements will be 2n – 1 when the matrix is n order. The pseudorandom sequence for a 7 ⫻ 7 cyclic S matrix is illustrated in Figure 2, so the number of the elements is 2 ⫻ 7 – 1 = 13. The pseudorandom sequence is constructed by placing two copies of the first row side-by-side and omitting the last element. In the case of S7, the last element, 0, is omitted, so the pseudorandom sequence can be represented as (1110100111010). When the element is 1 or 0, the sample or buffer solutions are injected, respectively. When the sample is injected into the capillary according to the pseudorandom sequence, (1110100111010), the resulting electropherogram is encoded by the cyclic S matrix, S7. The encoded electropherogram contains numerous peaks for a single component due to the multiple injections. The encoded electropherogram matrix, [Y], is mathematically represented by [Y] = [S] [X]

(3)

in which [S] is the cyclic S matrix, and [X] is the electropherogram matrix obtained by a conventional single injection. [S] is a square matrix with order n, and [X] and [Y] are 1 – n matrices, in which the components represent elements of the electropherogram—x1, x2, x3, …xn for [X], and y1, y2, y3, …yn for 544 A


[Y]. The electropherogram, [X], can be solved by multiplying both sides of Equation 2 by the inverse matrix of [S] ([S] –1). [X] = [S]–1 [Y]

(4)

in which [S]–1 is obtained by replacing all the 0s in [S] with –1s and multiplying by 2/(n + 1),

S –17 = 2 n+1

1 1 1 1 1 1 1

1 1 1 1 1 1 1

1 1 1 1 1 1 1

1 1 1 1 1 1 1

1 1 1 1 1 1 1

1 1 1 1 1 1 1

1 1 1 1 1 1 1

1 2

(5)

The encoded electropherogram in Figure 2 contains more than seven elements, although a cyclic S matrix of order 7 is used for encoding. Thus, it is necessary to know which data group should be used for the inverse Hadamard transform. The n elements from the nth to the (2n – 1)th are always used for the inverse Hadamard transform (a group from 7th to 13th, which is surrounded by a red rectangle in Figure 2). Therefore, it is not necessary to know the migration times of the analytes prior to a run. After an inverse Hadamard transform, the resulting electropherogram automatically has a better S/N, although the analysis time is twice that of a conventional CE. With an S matrix of order n, S/N increases by a factor of (n + 1)/2n1/2. Therefore, further improvement in the S/N would only be achieved with a larger n value. The inverse Hadamard transform process also requires the reversal of the decoded electropherogram. Assuming that the sequence of the elements for [X] is represented by (0000100) in Figure 2, then the sequence of the group, which is extracted from the encoded electropherogram, can be given as (1010011). On the other hand, the sequence of [Y] should be (1100101), which is similar to the sequence of the fifth column of the S7 matrix. These results indicate that, with multiple injections, reversal of the electropherogram occurs in the encoding process. Therefore, it is necessary to reverse the sequence of the decoded or encoded electropherogram before or after multiplying the inverse matrix by the group of elements extracted from the encoded electropherogram. The calculation process of the inverse Hadamard transform includes the following steps: extraction of a group of elements, nth to (2n – 1)th, from an encoded electropherogram; multiplication of an [S]–1 by the group of elements; and sequence reversal of the elements in the group.

Fluorescence intensity (mV)

28

Improving S/N

(a)

27 26 25 24 23 22 21 20 0

25

50

75

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125

Time (s)

50

40

34

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1

26 0 22

50

60 70 Time (s)

80

Element of sequence

(b)



60

90

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20 0


5

100

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200 150 Time (s)

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(c)

4 3 2 1 0 –1 –2 –3 0

25

50 75 100 125 Time (s) FIGURE 3. Electropherograms obtained by (a) a conventional single injection, (b) multiplex injections, and (c) an inverse Hadamard transformation of (b). Sample injection time is 0.5 s for a single injection. In multiple injections, the injection time for each code is 0.5 s. Concentration of sodium fluorescein, 10 nM; buffer, 10 mM KHCO3 + NaOH (pH 9.3); voltage, 15 kV; gating beam, 120 mW; probe beam, 9 mW; wavelength, 514.5 nm.

Figure 3a shows a conventional electropherogram obtained by injecting a sample solution containing 10-nM sodium fluorescein over a 0.5-s period. A signal peak corresponding to the fluorescein ion is observed at a migration time of ~50 s, but the S/N is not acceptable. The electropherogram in Figure 3b shows an encoded electropherogram obtained by a pseudorandom injection sequence (n = 255). Both the injection and data sampling times were adjusted to 0.5 s. Many peaks appear in the electropherogram, but all can be assigned to the fluorescein ion. The order of the cyclic S matrix in Figure 3b is 255, so the pseudorandom sequence is composed of 509 elements (2n – 1). The signal is just background until 49.0 s, when the first signal appears. An irregular signal persists up to 299.5 s, at which time the sample is no longer injected, and then the signal reverts to background. An expanded portion of the electropherogram is inserted in Figure 3b. The red line indicates the elements of the pseudorandom sequence. All the peaks appearing at 1s and decreasing in intensity are observed at 0s, indicating that the sampling time is exactly equal to the injection time. All the peaks observed at 1s and that decrease in intensity are observed at 0s. A portion of the encoded electropherogram from 255 (n) to 509 (2n – 1) is extracted and then decoded by multiplying by [S]–1. Figure 3c represents the inverse Hadamard transformation for the data shown in Figure 3b. The value of n is 255; therefore, theory says that the S/N should improve by a factor of 8.02. In fact, the S/N shows an 8-fold improvement over the value in Figure 3a, which is in good agreement with theory. Of note, the peak width obtained using a Hadamard transform technique is nearly identical to that obtained by the conventional method. This indicates that the signal was not distorted by the calculation process for the Hadamard transformation. Signal intensities are equal in Figures 3a and 3c, but the background noise is reduced in Figure 3c. Therefore, the absolute value of the peak intensity remains unchanged after the inverse Hadamard transform. When the maximum power of the laser was used, the detection limit for the fluorescein solution was 500 fM, which corresponds to ~170 molecules per single injection. The calibration curve for the fluorescein ion was linear over the range 2.0 ⫻ 10–12 to 1.5 ⫻ 10–7 M. Thus, the present method is also applicable to quantitative analysis. Figure 4 compares the electropherograms obtained by conventional CE and Hadamard transform CE. A solution containing a mixture of two fluorescein isothiocyanate (FITC)-labeled amino acids was used. In Figure 4a, two components, arginine and tryptophan, are clearly separated, but the S/N is unaccept-


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Limitations Unfortunately, Hadamard transform CE suffers some of the same limitations that plague other multiplex techniques, such as the cross-correlation method, when used to determine multiple components. If a sample contains a major component and a trace component with overlapping peaks, the peak corresponding to the trace component cannot be distinguished from the major component by mathematical treatments. Our results indicate that a minor component could not be detected when the sample solution contained a 100-fold excess of the major component. Furthermore, the peak shape for the major component was degraded in the presence of the minor component. In addition, the sample injection technique used in our experimental setup is rather complicated and expensive. Furthermore, the optically gated injection is limited to fluorescent molecules that are easily photodegraded. Although these may be important problems, there are ways to overcome them. Most commercial instruments are computer-controlled, and sample injection and signal detection are typically performed automatically. Therefore, the conventional CE system can be simply modified with additional software to accommodate the Hadamard transform. Even without the optically gated injection, commercial instruments can take advantage of Hadamard transforms. Moreover, there are a few reports of multiple electrokinetic sample injections using microchip devices (3, 4). Injections are easier with these devices than with a modified conventional CE system. Finally, the Hadamard transform technique is applicable to other types of detection such as UV absorption and electrochemistry. Thus, the present approach to Hadamard using the injection system could become a universal analytical method.

Single molecules and microscale Because this technique has excellent sensitivity for determining ultradilute samples, Hadamard transform CE could conceivably detect a single molecule. For example, laser-induced fluorometry is a promising technique for the detection of single molecules (12, 13). However, almost all of the successful analyses reported thus far use samples that contain only a single component. Therefore, for a solution that contains more than two components, laser-induced fluorometry must be combined with a separation method such as CE. Several papers have re-

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11 1

(a)

10


able. Using Hadamard transform CE, two large peaks are observed at the same migration times as those obtained by a single injection method. No undesirable effects such as band-broadening and shifts in migration time were observed. Therefore, this technique is very useful for improving the S/N of single and multiple components.

2

9 8 7 2

12

(b)

1 0 –1 0

25

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Time (s) FIGURE 4. Electropherograms of a mixture of amino acids by (a) conventional and (b) Hadamard transform CE. Peak 1, FITC-labeled arginine; peak 2, FITC-labeled tryptophan, sample concentration, 150 nM for each amino acid; voltage, 18 kV; gating beam, 145 mW; probe beam, 10 mW. The other conditions are the same as Figure 3.

cently appeared reporting the use of CE or microchip electrophoresis for detecting single molecules (14–16). Although Hadamard transform CE could detect single molecules, statistical problems might well be encountered in the multiplex injection process. In the case of single-molecule detection, the detection efficiency is not necessarily 100%. The efficiency of detection depends on the properties of the molecule and the detection system such as fluorescence quantum yield, bleaching lifetime, probe volume, and molecular transit time (17 ). Thus, if detection efficiency is

Peer Reviewed: Hadamard Transform CE - Analytical Chemistry (ACS

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