Investigation of Bioluminescence at an Optical Fiber End for a High-Sensitive ATP Detection System

ATP detection detection limit of 10 -9 M is with the previous results with the APD-type detector limit us detect the absolute ATP number of 10 -14 mol in a 10 μ l is two orders of magnitude larger than the one. the we have performed measurements with high time resolution and analyses of data by using an enzyme reaction model including inhibitors to individulally obtain an activation ratio and a reaction rate of the immobilied luciferase. As the results, the reaction rate of 0.61 s -1 and the activation ratio of 1 % have been obtained and these results have the of two orders of magnitude higher than the expected one. For reducing the detection limit more, it is necessary to improve the activtion ratio of the immobilized luciferase on the optical fiber end as well as the enlargement of the effective area of the cut surface based on a surface asperity the increase of the FOM with the optimal values of a core diameter ϕ 0 , a numerical aperture and the coupling


Introduction
In biological research, the luminescience from fluorescent proteins or luminescent enzymes is widely applied for monitoring a change of environment at a cell. Biomolecules used as the probe, such as Green Fluorescence Protein (GFP) or luciferase molecules found in fireflies can respond to the existence of specific molecules or ions and subsequently emit a photon. The detection of a specific molecule can then be confirmed by detecting the emitted photons efficiently with a photon detector. A highly efficient detection of the luminescence is normally essential to realization of a high sensitivity to the specific molecules or ions and an improvement of the sensitivity can upgrade the capability of detection in a low concentration of sample solution. Therefore, there are many efforts to improve the efficiency of the collection of emitted photons and of the optical coupling to the photon detector.
A straightforward method is to directly detect the luminescence from the sample solution in a test tube with a single photon detector via simple coupling optics as shown in Fig. 1. This detection system is very simple and easy-operational, so that it has been widely used for various applications so far. To realize high efficiency detection, however, this method needs a single photon detector with the wide photon-sensitive area, which is ideally larger than a photon-emission area in the test tube. Here, we are introducing an alternative method, where the luminescent biomolecules are immobilized at an optical fiber end and the luminescence is detected by a photon detector which is optically coupled to the other optical fiber end. Fig. 2 illustrates the optical fiber-based system. This method has been investigated for application to a fiberoptic biosensor, which is constructed by immobilizing either an enzyme or an antibody. A review of this method is given in reference [1] and [2].  This method has three merits. The first one is to permit a local detection within the sample solution, because the optical fiber end functions as a needle-like probe. Meanwhile, the method as shown in Fig. 1 is suitable for detecting the luminescence from a large area in the sample solution. The second one is that the detection scheme does not require that the photon detection is very close to the sample solution. This feature makes it easier to mount the sensing parts in integrated bioengineering, such as μ -TAS. The third merit is that single photon detectors with a small sensitive area are also available, because the photon-emission area, which is almost identical to the cross section of the core part in the optical fiber, is small. In general, single photon detectors have lower dark counts for smaller sensitive area. Low noise is very important, because it essentially gives the upper limit of the sensitivity of photon detection. Recently, single photon detectors using avalanche photo diodes (APDs) have become widely available with good performance, but their sensitive area is small and typically 0.1 mm. The luminescence detection with the optical-fiber based system allows us to fully use the merits of compactness, high quantum efficiency, and low noise of these APD detectors. On the other hand, single photon detectors using compact and cooled photomultiplier tubes (PMTs) are also available, since they have the characteristics of low noise and much larger sensitive area than the APD's. Moreover, their quantum efficiency is not so much lower than the APD's. The fiber-based system with these PMT detectors can make a fully use of the merits of large sensitive area and low noise of the PMTs, although the quantum efficiency is lower than the APD's.
We have built detection systems of bioluminescence at an optical fiber end and investigated the sensitivity of Adenosine triphosphate (ATP) detections by using an APD-type as described in [3] and [4]. In this chapter, results with a PMT-type detector are presented in comparison with the results by using the APD-type photon detector. We also discuss the reason of limiting the present sensitivity in the system with the PMT-type detector. ATP is a good indicator of biochemical reaction or life activity, since ATP is considered as the universal currency of biological energy for all living things. Therefore, there are many efforts to develop ATP-sensing techniques for compact and efficient ATP detection in reference [5][6][7]. In particular, highsensitivity detection of ATP can indicate the existence of microorganisms even in low numbers. Thus, a compact, simple, and easy-operational system with extremely high sensitivity has been desirable.
One well-known and powerful method for highly sensitive ATP detection is to use the chemical reaction involved in thebioluminescence, the luciferin-luciferase reaction in reference [8]. In this reaction, after one ATP molecule and one luciferin molecule are bound to one luciferase molecule, and the luciferin molecule is oxidized using the energy of ATP. As consequence, one photon is emitted during the transition from the excited state to the ground state of the oxidized luciferin molecule bound to the luciferase molecule. The emission of one photon indicates the use of the energy of one ATP molecule. In the method using the luciferin-luciferase reaction, the efficient detection of the bioluminescence is essential for high-sensitivity detection of ATP.
The oxidation of luciferin is catalysed by the enzyme luciferase, so that the immobilization of luciferase molecules on solid probes of various sizes allows highly sensitive and local detection of ATP. Three types of immobilization have been used: firstly attachment to the cell surface in [9], secondary attachment to small particles, such as nanoparticles in [10], glass beads and rods in [11], thirdly attachment to extended objects with a size in the centimeter range, such as strips in [12] and [13], and films in [14]. For the ATP-detection on the intermediate scale below 1 millimeter, a fiberoptic probe employing immobilized luciferase in [2] as well as microchips in [15] and [16] is utilizable. Therefore, the efficient detection system of bioluminescence at an optical fiber end can achieve the local detection of ATP. The realization of highly sensitive detection of ATP potentically provides the local detection of extremely low number of microorganisms. Thus, it is desirable to construct a highly efficient detection system of the bioluminescence at an optical fiber end and to evaluate the detection limit with the system. In order to explore possibilities for improving the detection limit, moreover, it is also necessary to investigate the bioluminescent reaction at an optical fiber end.
The rest of this chapter is organized as follows. In sec. 2, we describe a concept for the construction of the optical fiber-based system and show how to construct the detection systems by using the PMT detector. In sec. 3, we describe the sensitivity test with the constructed system and show the results are consisitent with the APD, but also show that the sensitivity can not reach the expected detection limit. In sec. 4, we present the results of kinetic properties obtained from experimental data on the bioluminescence and clarify a dominant reason of restricting the detection limit.Sec. 5 summarizes prensent results and future problems.

General concept
For the construction of an efficient detection of a bioluminescence, it is necessay to consider a collection efficiency of the luminescence at the optical fiber end and a coupling efficiency between the other optical fiber end and a photon detector as described in [4]. Using the optical fiber with a core diameter ϕ 0 and a numerical aperture NA 0 , the collection efficiency of the luminescence η at the optical fiber end depends only on NA 0 as shown in Fig.3.

Current Developments in Optical Fiber Technology
where n w is the refraction index of the substance surrounding the optical fiber end. In immersing this optical fiber end into water, its value should be identical to the one of water, which is about 1.33.The caluculated values of the collection efficiency η(NA 0 ) using Eq. (1) at n w = 1.33 are shown in Fig.4 as a function of NA 0 . It is easy to see that η(NA 0 ) monotonically increases with NA 0 . In the following, let us consider the situation where the other optical fiber end is optically coupled to a photon detector with a detection window of diameter ϕ 4 and with a circular sensitive area having a diameter ϕ 3 and a numerical aperture NA 3 . The coupling efficiency ε between the optical fiber end and the photon detector depends on ϕ 0 , NA 0 of the optical fiber and ϕ 3 , NA 3 , ϕ 4 of the photon detector used.The number of emitted photons is proportional to the square of ϕ 0 and η(NA 0 ) monotonically increases with NA 0 , so that the number of the transmitted photon to the other optical fiber end is proportional to ϕ 0 2 and η(NA 0 ). On the other hand, the coupling efficiency ε generally decreases as ϕ 0 or η(NA 0 ) increases for the fixed It should be noted that the coupling efficiency ε can be ideally 100 % under the condition of ϕ 0 ≤ ϕ 3 and NA 0 ≤ NA 3 .In many cases, the condition NA 0 ≤ NA 3 is satisfied when using the typical photon detector. Under the condition NA 0 ≤ NA 3 , thus, we can categorize into two cases: case (1) is ϕ 0 ≤ ϕ 3 and case (2) is ϕ 0 > ϕ 3 . In the case (1), ε is constant and can be ideally 100 %, so that the detection efficiency excluding the quantum efficiency of the photon detector is only limited to η(NA 0 ). Therefore, the optimization of ϕ 0 and NA 0 is not necessary. The conditions of ϕ 0 = ϕ 3 and NA 0 = NA 3 both maximize the FOM and the sensitivity becomes highest. In the case (2), however, the optimization of ϕ 0 and NA 0 and a specific design of the coupling optics are necessary, because the coupling efficiency ε decreases as ϕ 0 or NA 0 increases.

Photon detectors
To construct the optical fiber-based system, a choice of a single photon detector is very important. Photon detectors generally have two significant factors contributing to the sensitivity of detection for weak light: the efficiency and the dark counts of the detector. Recently, two types of single photon detectors, which are a cooled APD and a small size of cooled PMT, are available. The cooled APD which can detect for single photons is mostly used because of the high quantum efficiency and the low dark counts. The sensitive area must be very small ( ϕ 0 ~0.2mm ), but the quantum efficiency is several times larger than that of a typical PMT.
Furthermore, it has the useful characteristics of compactness, easy operation, and durability compared to a typical PMT detector. On the other hands, the compact size of cooled PMT is also useful for the optical fiber-based system. The quantum efficiency is typically lower than the APD, but the sensitive area is roughly 10 times larger than the APD's in spite of the same dark counts as the cooled APD. Therefore, it is very easy to construct a coupling optics to the sensitive area of the detector with the high sensitivity.
We selected the PMT counting head (H7421) manufactured by Hamamatsu Photonics K.K. for this system. Its characteristics are summarized in Tab. 1. For comparison, we also present the characteristics of the APD-type photon counting module (SPCM-AQR-14) provided by Perkin Elmer. Ltd. It has already been verified that this APD is applicable to the fiber-based system by our previous investigation as discussed in [3], [4].  Table 1. Characteristics of photon detectors in reference [17] and [18] Current Developments in Optical Fiber Technology The value of NA 3 can be calculated from the geometrical structure between the sensitive area and the photon detection window.

Coupling efficiency
As showing in section 2.1, under the condition of ϕ 0 > ϕ 3 , it is necessary to optimize ϕ 0 , NA 0 and to design the coupling optics for maximal sensitivity. In the use of the PMT detector, it is not absolutely necessary to optimize ϕ 0 , NA 0 , because the characteristics of ϕ 3 = 5.0mm sufficiently satisfies the condition of ϕ 0 < ϕ 3 for the typical optical fiber. For easy construction, here, the optical fiber end was directly connected to the attachment of the PMT counting head without the additional coupling optics as shown in Fig. 5. The coupling efficiency ε(ϕ 0 , NA 0 ) in such geometrical structure can be obtained by the statistical method with matrix formalism in paraxial optics, which can describe the propagation of light. The light at the initial state (r i , r' i ), where r i is a distance from an optical axis and r' i is a slope of the light direction, can be transferred by some matrices. In the case of Fig. 5, the matrix expressing free-space propagation can transfer the initial state (r i , r' i ) at the optical fiber end to the final state (r f , r' f ) at the PMT sensitive area. We obtained the value of ε(ϕ 0 , NA 0 ) by counting the number of the final states inside the sensitive area for many initial states selected with random numbers, which were generated by using the software package based on algorithm of Mersenne Twister in reference [19]. The calculated results are shown as a function of NA 0 in Fig. 6. Since the value of ϕ 3 is approximately 5 times as large as one of ϕ 0 , the coupling efficiency ε(ϕ 0 , NA 0 ) is independent of ϕ 0 , but monotonically decreasing with NA 0 . Therefore, we calculated the FOM for confirming the existence of optimal conditions for ϕ 0 , NA 0 . Fig. 7 shows  On the other hands, in the use of the APD, the optimization of NA 0 = 0.37, ϕ 0 = 0.6 mm and the design of the optimal coupling optics are absolutely necessary. A simple optically coupling way is shown in Fig. 8. With the above APD, the condition of FOM × 100 and ε(ϕ 0 , NA 0 ) maximizes the FOM and the value of η(NA 0 ) is 0.234. The determined design parameters of the coupling optics give the maximum ε of 33.3% and NA 0 of 1.97%. The detailed description about the determination of the design parameters is given in [3] and [4].  The relative sensitivity for the bioluminescence detection can be compared by using a product of FOM and the quantum efficiency η qe of the detector as an indicator. For the PMT, the value of (FOM ⋅ η qe ) × 100 is 0.168 and 0.129 for the optimal values of the above APD.In such direct connection, the sensitivity with the PMT is almost same as with the APD. For the system with the PMT, furthermore, it should be noted that the coupling efficiency ε at NA 0 = 0.37 and ϕ 0 = 1.5 mm can be 100% using the additional coupling optics as shown in Fig. 8. These values give (FOM ⋅ η qe ) × 100 = 1.77, which is about 10 times larger than the present value.

Luciferin-luciferase reaction
Bioluminescence in living organisms, such as fireflies and some marine bacteria, typically occurs due to the optical transition from the excited state to the ground state of oxidized luciferin molecules produced by the luciferin-luciferase reaction under the catalytic activity of luciferase molecules. This reaction can be expressed by the following squential of reaction steps: where E indicates luciferase, S ATP, LH 2 luciferin, PP i pyrophoric acid, C 1 is an enzymesubstrate compound E ⋅ LH 2 -AMP, AMP adenosine monophosphate, P oxyluciferin ( oxidized luciferin ), C 2 a luciferase-oxyluciferin compound, and γ a photon in reference [20]. The emission of one photon at the position of luciferase molecule corresponds to the use of the energy of one ATP molecule. Against the second reaction which is a rate-limiting reaction in the above reaction chain, it is known that the following reaction is competitive, where L -AMP represents dehydroluciferyl-adenylate, C 3 is an enyzme-substrate compond E ⋅ L-AMP as described in [21]. This reaction does not induce the photon emission. Dehydroluciferyl-adenylate L -AMP as well as oxyluciferin is known as a competitive inhibitor to the equilibrium reaction in [22].
In the presence of enough luciferin molecules, the immobilization of luciferase molecules at the optical fiber end allows us to sense the present of ATP around the fiber end using single photon counts. For this purpose, we used a compound protein containing a silica-binding protein ( SBP ) molecule and a luciferase molecule ( SBP-luciferase ), which were recently synthesized by Taniguchi and co-workers in [23]. This protein makes it possible to immobilize a lucuferase molecule on the optical fiber end through a SBP molecule retaining its activity. The spectrum of the emitted photons shows a central wavelength of 550 nm and a width of about 100 nm in reference [24] and [25]. Both photon detectors of the APD and the PMT have the large quantum efficiency at 550 nm, the photon counting detectors are suitable for ATP sensing.

Michaelis-Menten formula
In the solution containing nonlocalized homogeneously dispersed luciferase and ATP, the Michaelis-Menten formula is applicable to the enzyme reaction as descibed in [26]. In the presence of sufficient luciferin molecules in the solution, a rate of emitted photons at steady state v γ can be expressed as the Michaelis-Menten formula, where V max is a maximum reaction rate which is equivalent to a product of a concentration of luciferase molecules and a reaction constant h 1 from C 1 to C 2 , K M is the Michaelis constant, and s is the ATP concentration. In the fiber-based system for sensing dispersed ATP molecules, on the other hands, an ATP-flow generated by a gradient of ATP concentration around the luciferase-terminated fiber end carries ATP molecules to the vicinity of immobilized luciferase molecules. One of ATP molecules is bound to one immobilized luciferase molecule near them and subsequently used for the luciferin-luciferase reaction at this fiber end. By solving reactiondiffusion equations, as described in [4], we have confirmed that an ATP diffusion rate is not a rate-limiting process when the number of immobilized luciferase molecules is 10 11 order and the ATP diffusion constant is D = 0.5 × 10 -3 mm 2 / s given in [27]. Therefore, the Michaelis-Menten formula Eq. (3) can be also applied for the fiber-based system without change. Fig.9 shows the experimental setup for the sensitvity test and the investigation of bioluminescence at the optical fiber end. One optical fiber end was optically connected to the PMT as describing in the previous section. On the other fiber end, the luciferase molecules were immobilized via SBP molecules and the bioluminescent reaction occurs by immersing the luciferase-terminated fiber end into a sample solution. The emitted photons were transmitted to the PMT through the optical fiber and TTL pulses outputted from the PMT were counted by a PC card installed in a personal computer(PC). To reducing the background light, the whole system was put into the dark box.

Experimental setup
For the observation of the bioluminescence rising, a test tube containing the sample solution was fixed on the Z-stage, which is a motorized stage and externally controllable. By raising the test tube for immersing the luciferase-terminated fiber end after starting the data acquisition system, the photon counts rising from the background level and subsequently reaching a maximum were observed with time. The numbers of detected photons during 0.0298 s or during 0.1 s were recorded every 0.0321s or every 0.1s by the PC, respectively. These values were obtained from a calibration test of data acquisition system. The details on the experimental setup and the measurements are described in reference [28].

Immobilization of luciferase
Before immobilizing luciferase molecules, we cut the optical fiber and cleaned the cut surface with ethanol and Tris buffer (0.25mM Tris-HCl with 0.15 M NaCl). Different from the previous experiments with the APD, we cleaved the optical fiber for making a flat surface on the fiber end, which can reproduce the number of immobilized luciferase. The flat surface also allows us to indivisually evaluate the number of immobilized luciferase molecules by using element analysis, although the sensitivity with the flat surface is about 10 times lower than the one with the appropriately irregular surface cut without the cleaving technique as described in [29]. The cut surface was direcly observed by using the fiber scope and checked its flattness by eyes. After cleaning, the surface was immersed in a solultion of SBP-luciferase and was left at a temperature of 3℃ to 6℃ for a period of about two hours.
For evaluating the number of immobilized luciferase molecules, element analysis to the fiber end was carried out by using Xray Photoemission Spectroscopy (XPS). We measured a spectrum including peaks from nitrogen in the SBP-luciferase and from silicon on the surface of the fiber end which made with the silica and obtained the ratio of the area of the nitrogenpeak to the one of the silicon-peak. Utilizing the absolute number of silicon on the surface of the fiber end, the surface density of immobilized luciferase molecules was determined to be 2.0 × 10 10 mm -2 . By repeating the same measurement and analysis, the error of the surface density was estimated to be 15 %.

Sample solutions
The samples were a 1:4:4:31 mixture of 20 mM D-luciferin solution, Tris buffer solution( 250 mM Tris-HCl mixed with 50 mM MgCl 2 ), ATP solution, and distilled water. Several solutions of ATP with different ATP concentrations were made by diluting the ATP standard in ATP Bioluminescence Assay Kit CLS II manufactured by Roche Co. Ltd. A series of sample solutions with different ATP concentrations were prepared in advance. To obtain a background before the ATP measurements, an additional sample without ATP was also produced by mixing distilled water instead of the ATP solution.

Results
The time dependence of photon counts per 0.1-s interval were measured in immersing the luciferase-terminated fiber end into the sample solutions with various ATP concentration and converted to the values of photon counting rate. A typical result for 100 μl at the ATP concentration of 1.65 × 10 -6 M is shown in Fig. 10.  Statistical errors were estimated as one standard deviation assuming Poisson distribution.
From Fig. 11, the sensitivity in this system is limited to 1.65 × 10 -9 M, whichcorresponds to a number of ATP molecules of about 10 -14 mol in the 10 μl solution. This value is about 10 times higher than the one in the previous experiment with the APD as described in [3] and [4], although the FOM of this system is almost same as of the APD system. In this experiment, the surface flatness of the luciferase-terminated fiber end makes us identify the effective area as the cross section of the cut surface, as while the effective area of the cut surface in the previous experiment with the APD system was enlarged due to a surface asperity.On the effect of such different cutting ways, we have already confirmed that the sensitivity in the flat surface is about 10 times lower than the one in the appropriately irregular surface cut without the cleaving technique. Therefore, the above results are consistent with our previous results by using the APD.
To check the ATP concentration dependence of the photon counting rate at maximum, the average of counts in sixteen 1-s intervals around the time at which the counting rate become maximal was calculated for each ATP concentration. The results are indicated by the solid circles in Fig. 12. By the analysis of fitting data points in Fig. 12 to Eq.

Discussion
The detection limitis essentially determined by both of the parameter V max and the dark noise of the photon detector. Since the V max can be expressed as V max = ε total ⋅ h 1 ⋅ e 0 , where h 1 is a reaction rate of one luciferase molecule, e 0 is a total number of immobilized luciferase molecules on the optical fiber end and ε total is a total detection efficiency, we can calculate the value of V max with ε total = η ⋅ ε ⋅ η qe which is 0.00171 at 550 nm, ϕ 0 = 1 mm, the surface density of immobilized luciferase given by 2.0 × 10 10 mm -2 , and h 1 = 0.125 s -1 in reference [30]. The prediction of V max is 3.36 × 10 6 s -1 , which is two orders of magnitude larger than the obtained value of 7.38 × 10 4 s -1 . In our previous experiment with the APD, the predicted value of V max was also one order of magnitude rather than the obtained one. Possible reasons are a reduction of h 1 , or an existance of inactive luciferase molecules, or both of them as discussed in reference [4] and [31]. To clarify the reason, it is necessary to individually evaluate the number of active immobilized luciferase molecules e a and the reaction rate h 1 , respectively, from experimental data.
In the PMT system, it is noted that the improvement of two orders of magnitude for V max is promising by using the optimal optics coupled to the optical fiber with ϕ 0 = 1.5 mm and NA 0 = 0.37 and the optical fiber end with the appropriately irregular surface cut without the cleaving technique for immobilizing the luciferase.
Investigation of Bioluminescence at an Optical Fiber End for a High-Sensitive ATP Detection System http://dx.doi.org/10.5772/52747

Measurement of the bioluminescence with high time resolution
To obtain the reaction rate h 1 , the number of acitve luciferase molecules e a , and other kinetic parameters from experimental data, the counts of photons with high time resolution and the advaced analysis to such data are necessary. The data acquisition system has a capability of recoding the numbers of detected photons every 0.0321s. By making a full use of this specification, the time dependence of detected photons with high time-resolution at the ATP concentration of 1.65 × 10 -4 M was obtained as shown in Fig. 13.  For comparison, we also measured the time depencence of the photons with homogeniously dispersed SBP-luciferase molecules in the solution of 500 μl. The direct measurement of bioluminescence was carried out with an other type of cooled PMT detector having a huge sensitve area of 1 cm × 1 cm to detect the bioluminescence from a large area of the solution. The details on the direct measurement and the data analysis for dispersed luciferase molecules are given in reference [28].

Reaction model including inhibitors
For obtaining kinetic parameters of bioluminescient reaction, we consider the rate equations including the effects of inhibitors. In the luciferin-luciferase reaction, two kinds of products, oxyluciferin and L -AMP are strong cometitive inhibitors to substrates. Each equilibrium constant for exyluciferin K i and for L -AMP K j has been measured and the values of K i = 0.5 μM and K j = 3.8 nM are given in reference [22]. For simplicity of the model, we assume that two inhibitors contribute the competitive inhibition to the equilibrium reactions between luciferase and ATP in the presence of enough luciferin molecules. Fig. 14 shows the reaction steps of luciferin-luciferase reaction including the effects of competitive inhibitors. Here, s, e, p, n γ , c 1 , c 2 , c 3 in Fig. 14 represents a concentration of ATP, luciferase, oxyluciferin, photon, E ⋅ LH 2 -AMP, E ⋅ P, and E ⋅ L-AMP, respectively, k + , k -, k i+ , and k i-are kinetic coeffi- cients for equilibrium and h 1 and h 2 are reaction rates. Since the inhibition by L -AMP is much stronger than the ones by oxyluciferin, it can be assumed that the enzymes in the state of the enzyme-susbstrate compound do not release L -AMP molecules and concequently lose the activation in the time scale of our experiment.
In the use of the immobilized luciferase molecules for sensing dispersed ATP molecules, it is natural to consider that the reaction occures in a volume ΔV which is the vicinity of the luciferase-terminated optical fiber end. Therefore, the serise of the rate equations describing the enzyme reaction shown in Fig. 14 in the volume ΔV can be expressed as where the variable e, n γ , c 1 , c 2 , c 3 is a number of each kind of molecules or photon in the volume ΔV. The variable s and p is the concentration of ATP and oxyluciferin, respectively, and their unit is M. The N A is Avogadro number and the V 0 is a volume of the solution. The unit of k -, k i-, h 1 , h 2 is s -1 and of k + , k i+ is M -1 s -1 . The volume V 0 is explicitly utilized into Eq. (9), since the concentration of ATP in the volume of V 0 is almost same as in the volume of ΔV because of a rapid diffusion rate. The volume ΔV can be approximately considered as a cylinder with a diameter of ϕ 0 =1 mm and a height of 10 nm because the size of SBP-luciferase molecules is about several nanometer, and its value is ΔV = 7.85 × 10 -12 l.
In addition to the above rate equations, the following conditions described as, e a = e + c 1 + c 2 + c 3 (11) should be satisfied. The condition of Eq. (11) shows that the total number of active luciferase molecules e a is constant and Eq. (12) means that the total number of photons is equivalent to that of oxyluciferin molecules. The Michaelis constant K m and the equilibrium constant of oxyluciferin K i can be expressed as Using Eq. (11) To simplify the numerical calculation, we assumed that oxyluciferin molecules instantaneously move out from the volume of ΔV because of the fast diffusion rate of the oxyluciferin molecules. Therefore, the kinetic constant k i+ can be practically zero. This treatment means that the influence of the competitive inhibition by the oxyluciferin can be neglected.
In the solution containing non-localized homogenous dispersed SBP-luciferase, the volume of ΔV in the equations is replaced with the volume of the solution V 0 and the kinetic constant k i+ is treated as non-zero.

Results of analysis
The five parameters a(e a , k + , k i-, h 1 , h 2 ) were treated as fitting parameters and the data was fitted to numerical solution of n γ for obtaining the values of a. As the first step, by inputting the initial values of a to the rate equations, the emitted photon N th i (a) was calculated every time step of 0.0321 s with K i = 0.5 μM given in [22] and K m = 6.47 × 10 -5 M, which was deduced from the data analysis shown in Fig. 12. The data-set of N th i (a), the n 0 number of measurend counts per 0.0321 s N exp i , and the background counts N exp 0 enable us to calculate a chi-square χ 2 with the formula given by The chi-square χ 2 (a) is a good indicator for fitting data and its minimum gives the optimal combination of probable values in a. Solving the series of the rate equations was executed by using the software package RKSUITE based on the Runge-Kutta method [32] and the minimization of the chi-square was performed with routines of MINUIT package provided by CERN software [33].
The result of fitting the data from 0 s to 30 s is represented as a solid line in Fig. 13 (a), which is extrapolated to 60 s using the obtaind parameters. This result is not reproduced completely in the time range from 0 s to 60 s, because the effect of the competitive inhibition of oxyluciferin is not considered and the fitting fuction includes only the contribution of the deactivation process. The inhibition of the oxyluciferin weakens with time due to its diffusion process, but this diffusion effect is not considered in this analysis. In contrast, the contribution of the deactivation process, which was evaluated by fitting the data around the peak, is concequently overestimated compared to the actual contribution. Therefore, the effect of the relatively strong evaluation for the deactivation process appears in the time range after 30 s.
The parameters obtained by fitting the data are summarized in table 2 together with the results of the dispersed luciferase for comparison. The parameter r represents the activation ratio of the SBP-luciferase, which can be defined as a ratio of the number of active luciferase molecules e a to the total number of immobilized luciferase molecules e 0 , which can be calculatedfrom the surface density of immobilized SBP-luciferase or from the concentration of the dispersed SBPluciferase. In addtion, the parameter kand k i+ was derived from Eq. (13) and Eq. (14), respectively. The statistical error was 3 % at the maximum, but the systematical error was estimated to be 20 % taking account of the errors of the numerical calculation and the parameters used.

Discussion
In table 2, it is easily seen that the kinetic parameters in the immobilized luciferase are almost same as in the non-localized dispersed luciferase except the reaction rate of h 1 . Since the rising time is approximately given by 1 / k + s, it is useful for checking a consistency of k + with the reference. For dispersed luciferase, the rising time of 0.29 s is obtained with k + = 1.7 × 10 4 M -1 s -1 and for immobilized luciferase, the rising time of 0.36 s is given by the value of k + = 2.1 × 10 4 M -1 s -1 . Both of them are close to the value of 0.3 s in reference [34], so that their values are consistent with the reference.
On the reaction rates, the obtained value of h 1 = 0.61 s -1 for immobilized lucifrase is about 5 times larger than the reference value 0.125s -1 given in [30], but the order of both values is the same. A more precise comparison is not meaningful, because the surrounding environment of luciferase is not exactly same as in the reference.On the other hands, a branching ratio to the deactivation process, which is given by h 2 / (h 1 + h 2 ), can be estimated to be 30 % for immobilized luciferase and 10 % for dispersed luciferase, respectively. Since the reference value of 20 % is given in [22], both of them are close to the refrence value. Thus, we can consider the obtained values of the parameters are almost consistent with the values which had been measured so far.
From table 2, the activation ratio r is 44 % for the dispersed SBP-luciferase and 1 % for the immobilized SBP-luciferase, respectively. In contrast, the reaction rate h 1 is the same order as the value we expected. Therefore, the detection limit for ATP detection results in two orders of magnitude larger than the expected one. As a concequence, the results of the sensitivity test described in Section 3 can be explained from the reduction of the activation ratio for the immobilized luciferase molecules.

Summary
We introduced a method of high-sensitivity detection of bioluminescence at an optical fiber end for an ATP detection as an efficient alternative to direct detection of bioluminescence for a sample solution. For investigation of the bioluminescence, we constructed an optical fiberbased system, where the luciferase molecules are immobilized on the optical fiber end and the other end is optically coupled to a compact size of cooled PMT-type photon counting head which has a large sensitive area. Although the sensitivity for the bioluminescence is not optimal, it is almost same as the system which had been constructed with an APD-type photon counting detector. We have evaluated the sensitivity for ATP detection and verified the detection limit of 10 -9 M which is consistent with the previous results with the APD-type detector. This detector limit allows us to detect the absolute ATP number of 10 -14 mol in a 10 μl solution, but it is two orders of magnitude larger than the expected one. For clarifying the reason, we have performed measurements with high time resolution and analyses of data by using an enzyme reaction model including inhibitors to individulally obtain an activation ratio and a reaction rate of the immobilied luciferase. As the results, the reaction rate of 0.61 s -1 and the activation ratio of 1 % have been obtained and these results have explained the reason of two orders of magnitude higher than the expected one. For reducing the detection limit more, it is necessary to improve the activtion ratio of the immobilized luciferase on the optical fiber end as well as the enlargement of the effective area of the cut surface based on a surface asperity and the increase of the FOM with the optimal values of a core diameter ϕ 0 , a numerical aperture NA 0 and parameters of the coupling optics.
Investigation of Bioluminescence at an Optical Fiber End for a High-Sensitive ATP Detection System http://dx.doi.org/10.5772/52747