Development of a quantitative optical biochip based on a double

Development of a quantitative optical biochip based on a double integrating sphere system that determines absolute photon number in bioluminescent solution: application to quantum yield scale realization Ramiz Daniel,1,* Ronen Almog,1 Yelena Sverdlov,1 Sharon Yagurkroll,2 Shimshon Belkin,2 and Yosi Shacham-Diamand1 1

Department of Physical Electronic, Electrical Engineering faculty, Tel Aviv University, Ramat Aviv 69978, Israel 2

Institute of Life Sciences, The Hebrew University of Jerusalem , Jerusalem 91904, Israel *Corresponding author: [email protected] Received 2 January 2009; revised 12 March 2009; accepted ; posted 6 May 2009 (Doc. ID 105826); published 5 June 2009

We report a new design of an optical biochip based on a double integrating sphere system to quantify the absolute number of the emitted photons or the total photon flux by a whole cell bioluminescent biosensor, for water toxicity detection, based on genetically engineered Escherichia coli bacteria carrying a recA::luxCDABE promoter–reporter fusion. The new design of the double integrating sphere system does not require any external standard light source for calibration of the tested bioluminescent solution and allows a direct determination of the total photon flux of the bioluminescent solution. In our design, we required that the two spheres are symmetric (have the same radius and reflectance) with a surface area larger than the cut cap area between the spheres. © 2009 Optical Society of America OCIS codes: 080.2720, 170.2945.

1. Introduction

Whole cell biosensors consist of living cells, such as genetically engineered bacteria, integrated with an electronic component to yield a measurable signal [1,2]. Whole cell biosensors can be used for real time environmental detection of hazardous chemicals [3–5]. A variety of detection methods have been developed for molecular detection of the reporting activity of the sensor cells, including colorimetric, fluorescent, bioluminescent, and electrochemical detection [1]. In this study we focus on bioluminescent detection, widely used due to its high sensitivity, broad dynamic range, and relatively inexpensive instrumentation [6–8]. In our system, we use 0003-6935/09/173216-09$15.00/0 © 2009 Optical Society of America 3216

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Escherichia coli bacteria genetically engineered to respond to the presence of DNA damaging (genotoxic) agents by a dose-dependent bioluminescent signal [9–12]. In most published reports, bioluminescence signals are presented as relative units (RLU) that depend on the optical properties of the biochip and the electrical or optical detector parameters. A quantitative assessment of the total bioluminescence photon flux has a significant influence on diverse biosensor engineering aspects, such as quantum yield (the probability of photon emission via the number of reacted substrate molecules [13]), geometrical light collection efficiency, and minimum detectable signal. Previous reports estimated the total flux and quantum yield of different luminescence systems such as bacteria [14,15], firefly [16], and luminol [17,18] by calibration with a secondary standard

light source (the flux of the light source under test is performed by comparison to a standard light source of known flux). Calibration with a secondary standard light source adds a complexity in the experimental setup and causes inaccuracy in the results. Recently, Ando et al. [13,19] have developed a quantitative bioluminescence spectrometer for estimating the absolute number of emitted photons at each wavelength. However, the estimation of the total flux of an omnidirectional light source, such as a lamp or a bioluminescent solution, by an integrating sphere [20,21] may be considered to be the simplest experimental setup; it yields very accurate results due to the symmetry of the measurement system and the uniformity of the diffuse coating material at viewing angles up to approximately 45° [22]. In this approach, total photon flux is usually measured by the integrating sphere photometer system in comparison with a secondary standard lamp [21]. A variation of this concept is used for measuring optical properties of live tissues by a double integrating sphere system [23,24]. In the present work, we report the development of a new double integrating sphere design for measuring total luminescence flux without using a secondary standard light source. The only requirements of our design are that the two spheres be symmetrical (have the same radii and same diffuse reflectance) and that the surface area of the exit port be equal to the surface area of the cut cap connecting the two spheres. Several problems and errors may be associated with total flux measurement by comparing the test light source with a secondary standard light source. These include geometric errors that may arise from differences in the physical dimensions, wavelength and flux distribution of an electrical source such as a laser diode, and a strong dependency on reaction conditions when a calibrated secondary bio/chemiluminescent source such as a luminol solution is used. In Section 2 we describe the theory of the method for measuring the total flux using a double integrating sphere system, and we present the mathematical model of the system by the matrix method [22,25]. The experimental setup and the design of the system are described in Section 3. The experimental results are presented in Section 4, and we discuss the results and present the conclusions in Section 5. 2. Theory of the Method

The description of the system for measuring the total flux of the bioluminescence solution is presented in Fig. 1 and includes the following: (1) two integrating spheres with radii R1 , R2 , surface areas A1 ¼ 4πR21, A2 ¼ 4πR22 , and reflectances ρ1, ρ2 (we assumed perfectly diffusive). The two spheres are connected by an open window with radius RW and area AW ¼ πR2W , (2) two open ports with equal radius Rd and area Ad ¼ πR2d for measuring the flux, (3) baffles, and (4) a tested light source (bioluminescence solution) located in the center of the first sphere. The theory

Fig. 1. Schematic description of the double integrating sphere system.

of the integrating sphere was previously presented [20,21,26]. In the current study we have applied the matrix method [22,25] for analyzing the dependency of the measured flux of the two detectors on the light source. In our paper we define H as the total irradiance that includes the irradiance arising from the interreflections and the input irradiance H 0 [25]. The total irradiance H in an ideal Lambertian integrating sphere is described by an integral equation [22]: Z HðΩÞ ¼ H 0 ðΩÞ þ

A

HðΩ0 ÞρðΩ0 ÞGðΩ; Ω0 ÞdA0 ;

ð1Þ

where Ω and Ω0 are vectors describing points on the sphere surface, ρðΩ0 Þ is the reflectance at Ω0 , H 0 ðΩÞ is the source irradiance at Ω, and the integral is performed over the entire surface area. The function GðΩ; Ω0 Þ describes the radiance exchange between two points, and can be expressed as follows: GðΩ; Ω0 Þ ¼

ð−s · n0 Þðs · nÞ ; πjsj4

ð2Þ

where s ¼ Ω − Ω0 is the distance vector between the two points Ω and Ω0 , and n and n0 are the outward pointing normal units vectors at the points Ω and Ω0 . The matrix method assumes that all the surfaces are Lambertian reflectors; each surface on the interior sphere produces uniform irradiance on every other surface on the same sphere. The sphere surface can be divided into different regions, so that every region (j) has uniform irradiance and reflectance. Thus Eq. (1) can be expressed as follows: HðΩÞ ¼ H 0 ðΩÞ þ

Z X H j ρj GðΩ; Ω0 ÞdA0j : j

Aj

ð3Þ

The configuration factors for radiative exchange between Ai and Aj are defined as [27,28]: F i−j

1 ¼ 0 Ai

Z Z A0i

Aj

GðΩ; Ω0 ÞdAj dA0i :

ð4Þ

The configuration factors are defined only for surfaces whose radiances are spatially uniform 10 June 2009 / Vol. 48, No. 17 / APPLIED OPTICS

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and describe the fraction of flux radiated by surface Ai that is incident on Aj . An important property that they have is their reciprocity relation (Ai F i−j ¼ R Aj F j−i ). For each zone i, Ai HðΩÞdAi ≈ H i Ai , and thus by integrating Eq. (3) over the unprimed area Ai 0 0 and invoking the symmetry R Þ R¼ GðΩ ; ΩÞ,0 we P GðΩ;1 Ω obtain H i ðΩÞ ¼ H 0;i ðΩÞ þ j H j ρj Ai Ai Aj GðΩ; Ω ÞdAj dA0i , which is defined only for surfaces whose radiance is spatially uniform. The combination of this term with configuration factors definition (4) can transform the irradiance integral equation for matrix form as H i ¼ H 0;i þ M ij H j ;

ð5Þ

where the elements of the matrix M are given by M ij ¼ ρj F i−j :

ð6Þ

If H i are the elements of the irradiance vector H, and H o;i are the elements of the source irradiance vector H0, the solution for irradiance in the sphere is given by H ¼ ðI − MÞ−1 H0 ;

ð7Þ

where I is the identity matrix. We analyze the double integrating sphere system in two steps. In the first step we calculate matrix M for one sphere with radius R1 (surface area A1 ) and diffuse reflectance ρ1 , containing two exit ports with surface area Ad and AC . The light source is located at the center of the sphere with flux ϕ0 . The total irradiance of such a simple system was analyzed and presented in several works [25]: H¼

ϕ0 A1

1 − ρ1

1 A1 −Ad −AC A1

:

ð8Þ

In this model, the reflectance of the detector is assumed to be zero. The source irradiance for Lambertian surfaces is expressed as H0 ¼

ϕ0 : A1

ð9Þ

Comparing expressions (7)–(9), one can express the configuration factor of a simple sphere as F 1−1 ¼

F 1−1 and F 2−2 are calculated according to Eq. (10). The same flux falls upon a surface whether it is spherical or another shape [29]. The definition of the configuration factor as described in Eq. (4), and the properties of a sphere with Lambertian surfaces lead directly to

A1 − Ad − AC : A1

F 12 ¼

AC ; A1

ð12Þ

F 21 ¼

AC ; A2

ð13Þ

where AC is the spherical cap area cut by the second sphere, as illustrated in Fig. 2. The irradiance vector can be calculated directly by substituting the elements of M and the source irradiance vector H0 into Eq. (7). For our proposed system we have designed symmetrical double spheres with identical radii and reflectance ðR1 ¼ R2 ¼ R; A1 ¼ A2 ¼ A; ρ1 ¼ ρ2 ¼ ρÞ, and the source irradiance vector is equal to HT0 ¼ ðH 01 0Þ (the light source is located only in the center of the first sphere). For the case that A >> AC , Ad ¼ AC , and ρ < 0:995, the analytical solution of the described system is 1 H 01 ; 1 − ρ 1 − 2 AAC

ð14Þ

ρ AC A 2 H 01 ; 1 − ρ 1 − 2 AAC

ð15Þ

H1 ¼

H2 ¼

where H 1 and H 2 are the estimated irradiance in the first (containing the light source) and the second (empty) spheres, respectively. The derivation of this expression is presented in Appendix A. The total flux incident on the detector active area can be expressed as [22,30] ϕDi ¼ ρi H i Ad ;

ð10Þ

An identical result may be obtained by calculating the function GðΩ; Ω0 Þ of a symmetrical sphere. In the second step, we calculate the elements of a 2 × 2 M matrix for the double integrating sphere system as illustrated in Fig. 1: M¼ 3218

ρ1 F 1−1 ρ1 F 2−1

ρ2 F 1−2 : ρ2 F 2−2


ð11Þ Fig. 2. Double sphere system with a cap area AC .

ð16Þ

when the fraction of the sphere area within the field of view is one. The source irradiance H 01 is formed by Eq. (9). The source flux can then be expressed as ϕ2 ϕ0 ¼ D2 ; ϕD2

ð17Þ

where ϕD1 and ϕD2 are the measured detector fluxes for the first and the second spheres in a system, respectively. We have designed and modeled a quantitative optical biochip based on a double integrating sphere system that measures the total light source flux directly without using a secondary standard light source. Our design required that the two spheres be symmetrical (same radii and reflectance), that they possess a surface area larger than the cut cap surface area, and that the detector area be equal to the cut cap area. In this case, we can estimate the total source flux by a direct measurement of the detector flux of the two spheres. 3. Experimental Setup and Methods

Biological Material: The bacterial sensor used in this study was E. coli strain DPD2794 [31,9]. This strain, harboring a recA::luxCDABE fusion, was previously shown to be a sensitive reporter of genotoxicity. Cultures were kept as colonies at 4 °C or in a 50% glycerol suspension at −80 °C. Prior to the experiment, the bacteria were grown overnight with shaking at 37 °C in Luria Bertani (LB) broth containing 100 mg=liter of ampicillin. Overnight-grown cultures were diluted 200-fold in fresh LB broth (without ampicillin) and grown with shaking at 30 °C to the early exponential growth phase (optical density at 600 nm, 0.12). In our experiments, we have used nalidixic acid (NA) as a model genotoxicant. Experiments were carried out using a bacterial suspension (5 × 108 cells=mL) in LB and 16 ppm nalidixic acid concentration.

ϕ¼

1 S

Z gðλÞdλ ¼

C : S

ð18Þ

We have used a photon counting detector based on a photomultiplier tube (PMT) (Hamamatsu model 7155). Although the absolute sensitivity of the PMT was calibrated by the manufacturer labs, we recalibrated the absolute sensitivity of the PMT using a calibrated light source system (LabSphere Inc.), containing a general purpose integrating sphere (4 inch diameter), a radiometer model SC5500 with silicon photodiode head (1 mm2 ), tungsten lamp, bandpass optical filters, and neutral density (ND) glass filters to attenuate the lamp power to very low levels. The PMT head was attached to diffuser and limited its surface area for 1 mm2. The self calibration results are very similar to the manufacture calibration (for λ ¼ 490 nm we obtain an uncertainty of 8%); see Table 1. B.

System Design and Fabrication

The double integrating sphere system was designed using Solid Design Software and was fabricated by a computer numerical controlled (CNC) machine. The integrating sphere wall was coated with BaSO4 on an aluminum substrate (Labsphere Inc). The BaSO4 is an excellent diffuser [22] with high bidirectional reflectance (>0:98). The design of the double integrating sphere system is presented in Fig. 3(a), and the system dimensions are presented in Table 2. In radiometer measurements, a baffle is placed between the light source and the detector port to prevent direct illumination by the light source. The baffles are coated with the same material (BaSO4 ) as the integrating sphere wall. Baffles, however, may cause inaccuracies because the integrating sphere is no longer a perfect sphere. The number of baffles used in the sphere design should thus be kept to a minimum, and their size should be small enough to minimize the interference but large enough to prevent the

A. Calibration of Absolute Sensitivity of Photodetector

The double integrating sphere system can be modified and used for measuring diverse measurements including photometric, radiometric, or spectrum radiometric. In this work we measure the power of the bioluminescent radiant source, which is described in units of watts. In radiometer measurements, it is preferred to use a detector with a wide flat response. Conventional bioluminescent detection for low light levels is based on a photon counting detector, due to its low dark current. The readout R signal in counts=s units can be expressed as C ¼ gðλÞdλ, where gðλÞ is the spectrum of the bioluminescent measurement in counts=s · nm units. The power of the radiant light is calculated respectively to the absolute sensitivityR of the photon counting detector SðλÞð1=W·sÞ∶ϕ ¼ gðλÞ=SðλÞ dλ, and for a detector with a wide flat response SðλÞ ¼ S, the power in watt units is calculated by

Fig. 3. (a) Double integrating sphere system design; (b) layout and cross section of the baffle design. 10 June 2009 / Vol. 48, No. 17 / APPLIED OPTICS

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Table 1.

Manufacture Calibration Results: the Absolute Sensitivity of the Photodetector

Wavelength (nm)

Absolute Sensitivity (pW−1 s−1 )

400–420 500–520 600–620

2:7 × 105 2:2 × 105 2:1 × 104

light source from illuminating the detector port. We used a circular baffle as described in Fig. 3(b). The field of view (FOV), which was not considered in the theoretical model, could be neglected due to the full symmetry of the system (the distance between the detectors and the sphere wall is equal to the distance between the two spheres wall L ¼ D, and the radii of the exit ports in the two spheres are equal to the window radius between the two spheres RD ¼ RC , as illustrated in Fig. 3(a) and Table 2). When the detectors used with the integrating sphere do not have full hemispherical FOV, the exchange factor between the detector and the sphere wall is limited and equal to αFOV AD =A (where 0 ≤ αFOV ≤ 1 presents the fraction of the sphere area within the detector FOV, and ϕD1 ¼ H 1 Ad αFOV [22]). Similarly, the exchange factor between the two spheres is also limited by the FOV of the cut cap and is equal to αFOV AC =A (ϕD2 ¼ H 2 Ad αFOV 2 ). Thus the total source flux ϕ0 is independent of the field of view. There are two important considerations for placing the detector. It should not be directly illuminated by the light source, nor should it face a portion of the sphere wall that is directly illuminated by the light source. In most cases this means that the FOV of the detector needs to be limited by placing a diffuser or satellite sphere on the detector surface. In our case, we need to connect a diffuser between both spheres to conserve the system symmetry. However, this solution adds complexity to the experimental setup and can be problematic for very low light levels. We thus adopt a new approach, based on a limited FOV cell [Fig. 4(a)]. The limited FOV cell is cylindrical in shape with radius RS and height hS (Table 2). The limited FOV cell has only one side view toward the baffle and the detector; the other side views are limited and are coated by BaSO4. The tested cell also has a cylindrical shape, is made from glass, and is transTable 2.

parent from all directions [Fig. 4(b)]. The measured volume of the bioluminescence solution can be defined by calculating the cell volume (V ¼ hS · πR2S ). The glass substrate is used for keeping the chamber in the middle of the integrating sphere and for canceling the partial reflectance of the light (light loss) at the air–glass and glass–solution interfaces [13]. A schematic drawing of the bioluminescence experiential setup is shown in Fig. 5. The procedure consists two steps. The first involves filling the limited FOV cell with a bioluminescence solution, placing it in the middle of the integrating sphere and measuring the signal from the detector mounted on the same sphere ϕFOV1 (Fig. 5(a)). Then, the same bioluminescence solution (and the same limited FOV cell) is measured by the detector mounted on the

Double Integrating Sphere System Dimensions

Symbol

Size (mm)

R Rb RD RC

15 1.5 1 1

L

1

Lb D RS hS

7.5 1 1.5 1.5

3220

Fig. 4. (Color online) Structure and dimensions of (a) the limited FOV cell and (b) the tested cell.

Description Integrating sphere radius Baffle radius Exit port radius Window radius (between the double spheres) Distance between the detector and the sphere Position of the baffle Distance between the two spheres Chamber radius Chamber height


Fig. 5. (Color online) Schematic drawing of the bioluminescence experiential setup based on a double integrating sphere system.

second sphere ϕFOV2 [Fig. 5(b)]. The ratio between the two measured signals η ¼ ϕFOV1 =ϕFOV2 (system coefficient) is independent of the light source, as shown in Appendix A. In the next step, the tested cell is filled with a bioluminescence solution, placed in the center of the integrating sphere, and the signal from the detector mounted on the second sphere is measured ϕD2 [Fig. 5(c)]. Reorganizing Eq. (17), we obtain that total source flux estimated by ϕD1 2 · ϕD2 ¼ η2 ϕD2 : ϕ0 ¼ ϕD2

ð19Þ

Theoretically, we can express the system coefficient as η ¼ ð1 − ραÞ=ρβ, when αFOV ¼ 1 and the expected value of 10 ≤ η ≤ 100 (R ¼ 15 mm, Rd ¼ 1 mm). 4. Experimental Results

Formula (19) describes a novel method to estimate the total flux of the light source by a double integrating sphere system without using a secondary standard light source. However, our method may function only under specific conditions. The most difficult condition to achieve is that both of the spheres have the same reflectance coefficients ðρ1 ¼ ρ2 Þ. To address this, new measured coefficients were defined: 1. Asymmetric reflectance coefficient: If the two spheres have different reflectances ρ1 and ρ2 respectively, then, one can obtain that (Appendix A)

Fig. 7. (Color online) Simulation results of asymmetric reflectance coefficient versus reflectance. Sphere radius is 15 mm, and the detector radius is 1 mm.

η ϕD2 2

1 − αρ2 ρ · 1 ϕ0 ; ¼ 1 − αρ1 ρ2

ð20Þ

where α ¼ ðA − Ad − AC Þ=A. We can note that in the case that the reflectance coefficients of the spheres are different, we cannot estimate the total flux of the light source by Eq. (19), and a fitting coefficient must be considered. The fitting coefficient can be estimated by measuring the signals ϕD11 and ϕD12 for each sphere separately, when the cut cap between the spheres is closed by cylinder with radius 1 mm and is coated with BaSO4 . The same light source is used for both. The ratio between the two signals is obtained by the ratio between the irradiances of both spheres and can be described as κ¼

ϕD11 ¼ ϕD12

1 − αρ2 1 − αρ1

ρ · 1 : ρ2

ð21Þ

Therefore the total flux of the light source can be estimated by ϕ0 ¼

η2 ϕD : κ

ð22Þ

The coefficient κ, the asymmetric reflectance coefficient, is measured by the experiment described in Fig. 6(a). Simulation results of κ are presented in Fig. 7 with expected values 1 ≤ κ ≤ 5 (R ¼ 15 mm, Rd ¼ 1 mm). 2. The dimension coefficient is defined as 2 ϕD11 ρβ γ¼ ¼1− : ϕD1 1 − ρα Table 3.

Fig. 6. (a) Reflectance asymmetric coefficients experimental setup; the detectors are connected to a diffuser. (b) Coefficient experimental setup; the detector are connected to a diffuser. Radii of both spheres is R ¼ 15 mm, the exit ports are RD ¼ 1 mm, and the same bioluminescence light source is used.

Symbol Notes κ η γ

ð23Þ

Measured Characterized System Coefficients

Value

1.55 ϕD11 =ϕD12 ϕFOV1 =ϕFOV2 37.5 0.98 ϕD11 =ϕD1

Description Asymmetric reflectance coefficient System coefficient Dimension coefficients

10 June 2009 / Vol. 48, No. 17 / APPLIED OPTICS

3221

Table 4.

Measured Flux and Estimation Geometrical Light: Collection Efficiency of the Limited FOV Cell

Symbol

Units

Notes

Value

Description

ϕD2 ϕ0 ϕFOV2 ϕFOV0 ηCell

counts=s pW counts=s pW

Measured η2 κ ϕD2 Measured η2 κ ϕFOV2

6131 25 1624 6.7 26.5%

Second sphere tested cell signal Bioluminescence total flux estimation Second sphere limited FOV cell signal Bioluminescence limited FOV cell total flux Geometrical light-collection efficiency

ϕFOV0 ϕ0

The coefficient γ indicates the “effective” ratio between the area of the sphere surface and the area of the cut cap surface, and it is measured by the experiment that is described in Fig. 6(b), with expected values close to 1 (R ¼ 15 mm, Rd ¼ 1 mm). The experimental results of the measured characterized system coefficients are presented in Table 3. For Rd ¼ L, we can assume that αFOV ¼ 0:5 and thus 0:97 ≤ ρ1 , ρ2 ≤ 0:99. The ratio κ=η2 is defined as the geometrical–collection efficiency of the double integrating sphere system and it is equal to 0.1%. The results of the measured signals and the estimated total flux of the bioluminescence solution are presented in Table 4. Raw data of the measured signal C (number of counts/second) were obtained by the photon counting detector, and the conversion to picowatt units was performed by dividing the raw data by the absolute sensitivity of the photodetector S using Eq. (18). The bioluminescence spectra has a sharp peak at 490 nm [12], and the photodetector has wide flat response. We thus assumed that absolute sensitivity of the photodetector is constant and is equal to Sðλ ¼ 490 nmÞ. In our case, the geometrical light collection efficiency of the limited FOV cell is calculated by ηcell ¼ ϕFOV0 =ϕ0 and is equal to 26.5%, which is considered a high value due to its special design. Figure 8 describes the normalized measured bioluminescent signal (the signal was divided by the detector area) as a function of the time for different toxin concentrations (0, 12, and 16 ppm) with chamber volume 10 μL and using a double integrating sphere system compared to a double-plate cell system [13,19]. The ratio between the measured signals is equal to the ratio between the geometrical

Fig. 8. (Color online) Experimental results of bioluminescent signal kinetics for different toxin concentrations (0, 12, and 16 ppm) using the double integrating sphere system and the double-plate cell system [13,19]. 3222


light collection efficiency of the both systems ð1:1 × 10−3 =2:6 × 10−3 Þ. 5.

Discussion

In this work we have developed a new method to measure the total flux of a bioluminescence system in watt units (radiometry), without using a secondary standard light source. The developed method is based on a double integrating sphere system, when the two spheres are connected by an open window. In our design, we required that the two spheres be symmetric (having the same radius and reflectance) with a surface area larger than the cut cap area, and that the detector area be equal to the cut cap area. Under such conditions we can estimate the total source flux by measuring the flux of both spheres. The double integrating sphere system can be modified and used for any type of measurement: photometric, radiometric, or spectrum radiometric, depending upon the spectra response of the photodetector. For ultralow light levels and spectrum-radiometric measurements, it is useful to use a solid state device such as cooled charge-coupled device detector. Recently, a single photon avalanche photodiode detector was used for measuring ultralow light levels of bioluminescence solution [9]. In this work we used a photon counting based photomultiplier tube with a wide flat response. The estimated bioluminescence total flux results of 25 pW (for volume of 10 μL) and the absolute number of the total emitted photons was about 6 × 107 photons per second at a wavelength of 495 nm. The quantum yield is obtained by dividing the total emitted photons by the number of reacted molecules; in our case the toxin is considered as the reacted molecular (for a concentration of 16 ppm, the number of NA molecules is 4 × 1014, and we obtain very low quantum yield). The double integrating sphere method has three significant advantages over other methods for measuring ultralow light levels: (1) no assumptions concerning the type of the light source (uniformity and isotropically) are made, (2) there is no need for a secondary standard light source, and (3) geometrical light collection efficiency of the double integrating sphere system can be improved by increasing the cut cap area. In our current design the geometrical light collection efficiency of the double sphere system is 0.1%, of the same order of other quantitative total flux measured systems [13] . The disadvantages of this method are similar to those of other integrating sphere systems: (1) screening and light loss effects caused by baffles,

glass substrates, and limited FOV cells, which can be estimated as 1–2% [21]. Quantitative evaluation of the light loss effects by computer simulation [32] or numerical calculation is possible and yields very accurate results. However, it is not presently applicable for estimating the geometrical light collection efficiency of a biochip. (2) A disadvantage is the dependency of the white coating material on the wavelength; however, the BaSO4 material has a very weak dependency on the wavelength in the visible light range (0.5% over 400–600 nm). The investigation of the quantitative measurements of the total flux of biosensor based bioluminescence has a significant influence on the determination of diverse factors essential for biosensor engineering, such as quantum yield, geometrical light collection efficiency of a biochip, and minimum detectable signal. Appendix A. Irradiance Solution for a Double Integrating Sphere System

H2 ¼

ρ1 β H 01 : ð1 − ρ1 αÞð1 − ρ2 αÞ − ρ1 ρ2 β2

For the case that the sphere surface area is much larger than the cap surface area (A ≫ AC ) and ρ < 0:995, we obtain that the irradiance for the two spheres is expressed as 1 H ; ð1 − ρ1 αÞ 01

ðA7Þ

ρ1 β H : ð1 − ρ1 αÞð1 − ρ2 αÞ 01

ðA8Þ

H1 ¼

H2 ¼

The irradiance input vector is equal to

The irradiance in the form matrix is described by −1

H ¼ ðI − MÞ H0 :

H0 ¼

ðA1Þ

The M matrix of the symmetric double integrating sphere system and different reflectance is expressed as follows: ρ1 α ρ2 β M¼ ; ðA2Þ ρ1 β ρ2 α α¼

A − Ad − AC ; A

ϕ 0

A

0

:

ϕDi ¼ ρi H i Ad :

2

The inverse of the matrix ðI − MÞ is expressed as ðI − MÞ

1 ¼ ð1 − ρ1 αÞð1 − ρ2 αÞ − ρ1 ρ2 β2 1 − ρ2 α ρ2 β : × ρ1 β 1 − ρ1 α

H0 ¼

ϕD1 1 − ρ2 α ; ¼ ϕD2 ρ2 β

H1 H2

ðA3Þ

H 01 : 0

¼

H 1 ¼ ð1−ρ

1 ð1−ρ1 αÞð1−ρ2 αÞ−ρ1 ρ2 β2

ðA4Þ

1−ρ2 α

1 αÞð1−ρ2 αÞ−ρ1 ρ2 β

2

H 01 ;

1 − ρ2 α ρ1 β

ρ2 β

1 − ρ1 α

ðA12Þ

and the ratio is ϕ2D1 ¼ ϕD2

H 01 0

1 − ρ2 α ρ · 1 ϕ0 : ρ2 1 − ρ1 α

ðA13Þ

For the case that the two spheres have equal reflectance (ρ1 ¼ ρ2 ): ϕ2D1 ¼ ϕ0 : ϕD2

Thus, substituting Eqs. (A3) and (A4) into Eq. (A1), one obtains

ðA11Þ

The ratio between the fluxes is equal,

The irradiance input vector is equal to

ðA10Þ

Substituting the Eqs. (A7)–(A9) into (A10), and using the relation β ¼ AC =A ¼ Ad =A (Ad ¼ AC ), one can express the flux incident on the two detectors as

ρ2 β ϕ0 : ϕD2 ¼ ð1−ρρ1 1αÞð1−ρ 2 αÞ

A β ¼ C: A

ðA9Þ

By definition, the total flux incident on the detector active area can be expressed as

ρ1 β ϕD1 ¼ ð1−ρ ϕ0 ; 1 αÞ

−1

ðA6Þ

ðA14Þ

References

; ðA5Þ

1. P. D. Patel, “Biosensors for measurement of analytes implicated in food safety: a review,” Trends Anal. Chem. 21, 96–115 (2002). 2. S. Daunert, G. Barrett, J. S. Feliciano, R. S. Shetty, S. Shrestha and W. S. Spencer, “Genetically engineered whole-cell sensing 10 June 2009 / Vol. 48, No. 17 / APPLIED OPTICS

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