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Biotechnology and Bioengineering Fluorescence modeling in a multicomponent system
Fluorescence modeling in a multicomponent system
Nam Sun Wang, Michael B. SimmonsBạn thích cuốn sách này tới mức nào?
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Tập:
38
Năm:
1991
Ngôn ngữ:
english
Trang:
16
DOI:
10.1002/bit.260380812
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PDF, 1,27 MB
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Fluorescence Modeling in a Multicomponent System Nam Sun Wang* and Michael B. Simmons Department of Chemical Engineering, University of Maryland, College Park, Maryland 20742 Received March 6, 199lIAccepted March 8, 1991 An extensive fluorescence database for binary tyrosinetryptophan mixtures utilizing 280 nm excitation was collected. The database spanned three orders of magnitude (106M103M) and covered all compositions within this range. A generalized model for describing the multicomponent fluorescence signals as a function of emission wavelength, excitation wavelength, and sample composition was derived. A geometric integral that contained all the geometric factors affecting fluorescence was introduced; thus, the model was applicable to various configurations, including the three used in this study: an NADH probe, a backscatter laserinduced fluorescence setup, and a commercial spectrofluorometer. A correction factor was proposed that allowed linearization of the fluorescence signals with respect to fluorophore concentrations. The effect of the water Rarnan on fluorescence spectra was also modeled. The model contains only two wavelengthdependent parameters for each of the components present in a sample, one specifying absorption of the excitation energy and t h e other specifying the species' fluorescence tendency. These wavelengthdependent parameters were correlated with polynomials. The average prediction error at each wavelength was 1020%. a major portion of which was attributed to experimental uncertainties. Key words: fluorescence monitoring innerfilter effect biosensor tryptophan tyrosine  INTRODUCTION Optical techniques such as absorption, lightscattering, and fluorescence have traditionally been used by chemists and biologists, and they have particular advantages which make them wellsuited for online measurements in a biological system. These advantages include instantaneous response, selectivity of response, sensitivity, and nondestructiveness. Among the optical techniques availa; ble, fluorescence is perhaps the most sensitive for molecular measurements. Although it requires a more expensive setup than the absorption measurement, it is a proven, routine analytical technique for clean, pure component systems. Because many chemical and biological compounds are fluorescent, it is potentially a powerful method for measuring compositions in a complex mixture. This is a desirable feature especially in a bioreactor where there are many fluorescent biological compounds, including amino acids, cofactors, vitamins, antibiotics, and proteins. Furthermore, the power of fluorescence lies in the wide variety of measurements * To whom all correspondence should be addressed. Biotechnology and Bioengineering, Vol. 38, Pp. 907922 (1991) 0 1991 John Wiley & Sons, Inc. one can make. Changing the excitation and detection wavelengths, measuring the fluorescence decay in time or phase shift, and measuring the fluorescence polarization provide many independent dimensions from which the parameters of interest can be estimated. Despite the advantages and power, fluorescence is unfortunately not widely used in industry due to the great complexity of the information given by fluorescence techniques. Interactions between components and sensitivity to environmental conditions have thus far limited its use mainly to simple, clean systems containing a single known fluorescent species in offline analytical laboratories, and this apparent limitation has indeed raised some doubts about the applicability of fluorescence to the analysis of complex mixtures. For the fluorescence technique to be generally useful as an online monitoring tool, one must sort out the interactions. This can be accomplished by making full use of the aforementioned additional dimensions fluorescence measurements provide, in contrast to restricting oneself to only a pair of fixed excitation and emission wavelengths. This article proposes a generalized fluorescence model for predicting the fluorescence spectra of a multicomponent system, thereby advancing the fundamental understanding of the factors that affect the instrument response. Specifically, the fluorescence spectra of emission scans with a fixed excitation wavelength are derived. Because the fluorescence signal measured by a detector is strongly dependent on the measurement configuration, a generalized fluorescence model that is applicable to many configurations is first derived. In subsequent sections, the resulting model is applied to data obtained from a laser setup operated in a backscatter mode and a spectrofluorometer operated in a rightangle mode. The fluorescence model previously derived for an NADH (reduced nicotinamide adenine dinucleotide) probe' can also be shown to be a special case of the generalized model. MATERIALS AND METHODS lnstrumentation Three distinct instruments were used in this study to collect fluorescence data: (1) a commercial NADH fluo CCC 00063592/91/08090716$04.00 rescence probe, (2) a laserinduced fluorescence setup, and (3) a commercial spectrofluorometer. The last two setups were made available through the courtesy of the National Institute of Standards and Technology (Gaithersburg, MD). The first device is a steamsterilizable NADH fluorescence probe (BioChem Technology, Malvern, PA) that can be directly inserted into a bioreactor. Its notable feature compared to the other two setups considered in this article is its fixed pair of excitation and emission wavelengths. The second one is an Nd :YAG pulsed dye laser operated in the backscatter mode, also generally referred to as frontsurface illumination. The block diagram of this custombuilt ensemble is shown in Figure 1. Tunable excitation light is generated by a dye laser containing Rhodamine which is pumped by a pulsed Nd:YAG laser (Quantel YG581C). The excitation light is directed by a series of prisms toward a 1cmsquare quartz cuvette containing sample solutions. After passing through a scanning spectrometer (EV700 GCA/McPherson), a small fraction of the fluorescent light is detected at an angle of 10" relative to the incidence by a photomultiplier (R955 Hamamatsu), whose output is measured by a Boxcar Average (model No. 250, Stanford Research Systems) and logged by an IBM personal computer. In the third setup, the fluorescence emission spectra were obtained with a commercially available spectro fluorometer (SLM Instruments, model 8000C) equipped with a 450W xenon arc lamp, a photoncounting detector, and a double excitation monochromator and a single emission monochromator, both using holographic gratings. The excitation and emission slits were set to 2 and 4 nm, respectively. The sample signal was divided by the signal obtained from a reference solution of Rhodamine B in ethylene glycol, and was otherwise uncorrected. The schematic of this spectrofluorometer is shown in Figure 2. The instrument operated in the rightangle mode. Database Generation Table I lists the location of the excitation and emission peaks of various biologically important fluorophores based on the uncorrected fluorescence spectra obtained from the SLM 8000C spectrofluorometer. In this study, the application of the model will be demonstrated with the binary aromatic amino acid mixture of tyrosine and tryptophan with a wide range of concentrations: 106M103M. Aqueous mixtures of tyrosine and tryptophan were made fresh daily by dissolving appropriate amounts of the reagent grade chemicals (Sigma) in deionized water. Since these amino acids' excitation peaks are nearby, it is possible to obtain signals from both compounds by exciting the mixture at a fixed wave __.... Figure 1. Block diagram of the Nd:YAG laserinduced fluorescence setup used for data collection. 908 BIOTECHNOLOGY AND BIOENGINEERING, VOL. 38, NO. 8, OCTOBER 20, 1991 length and by scanning across the emission wavelengths. Figure 3 shows how this mode of scanning cuts across the two peaks positioned in the excitationemission space. With the spectrofluorometer setup, the mixtures were analyzed in a 1cm quartz cuvette at a 90” angle by exciting at 280 nm and measuring the emission from 250 to 450 nm in 0.3nm increments with an integration time of 1.0 s. Typical emission spectra obtained with this setup for 0.00, 0.50, 0.75, 0.90, and 1.00 tyrosine mole fraction solutions are plotted in Figure 4. The tyrosine peak is clearly centered around 308 nm, and the tryptophan peak is centered around 352 nm. PREVIOUS MODELING WORK In the past, Rollefson and Dodged developed a fluorescence model that accounted for both primary and secondary innerfiltering as well as selfquenching by the fluorescent molecules for a pure component in an inline configuration, where the light source and the detector were at opposite sides of the sample cuvette. (Primary innerfiltering is the partial blockage of excitation light by absorption; secondary innerfiltering is the similar blockage of emission light.) Without demonstrating the theoretical basis, Parker and Barnes’ presented a correction factor based on absorbance data from a rightangle configuration when only primary innerfilteringwas significant. The correction factor was treated as empirical,” until Holland et aL3 later provided a theoretical basis for it. Leese4 derived a model of innerfiltering by a quencher for the backscatter configuration as well as the rightangle configuration with equal path lengths for excitation and emission; the model also considered polychromatic effects. Christmann et a1.l.’ utilized a cellshift method, in which the path lengths of excitation and emission light could be independently varied, to correct fluorescence measurements without explicitly measuring the sample’s absorbance. In summary, fluorescence modeling to date has focused on correcting fluorescence data to account for light absorption from predominantly two sources: (1) a single f luorophore exhibiting primary innerfiltering due to its high concentration and (2) a second component, usually a quencher, exhibiting secondary innerfiltering. Unfortunately, there is no model that can be used to predict the fluorescence signal of a mixture of absorbing compounds and fluorescent compounds additional fluorescent compounds beyond a pure component system had not been properly considered in previous modeling efforts, and absorbing compounds had entered only in the form of empirical correction factors. Furthermore, each model was applicable to only a specific configuration. It is this missing model that is to be developed in the following section. MODEL DEVELOPMENT Generalized Fluorescence Equation First, a generalized model will be developed that is independent of the specific geometric configuration of the measurement setup. We shall first assume that the excitation light intensity ZAx at a point within the sample is proportional to that at the source. zAx(x,y, 2) = IA,(O)g(x,y, 2, Prepared Solution cuvcrtc Chamber Figure 2. Schematic of spectrofluorometer (SLM SOOOC). Table I. Excitation and emission peaks of predominant biological fluorophores, based on uncorrected spectra from SLM 8OOOC, rounded to the nearest 5 nm. A, NADH, NADPH FMN, FAD, riboflavin Folic acid Tryptophan Tyrosine Pyridoxine (nm) A, (nm) 260,340 260,315,450 375 280 215 325 460 525 460 350 310 395 (1) where ZAI(0)refers to the intensity of excitation at wavelength A, at an arbitrarily chosen reference point that can be considered as the source; usually this point is where excitation light first enters the sample cuvette. Because the excitation intensity diminishes as light passes through the sample due to absorbance at the excitation wavelength, proportionality g in the above equation is a function of both the physical location in the sample, as expressed by the threedimensional coordinates ( x , y , z ) , and the sample’s absorbance at the excitation wavelength, AAx.The amount of excitation light absorbed by the ith component in a small volume y, z ) , is proelement of the sample dV, denoted dZi,Ax(x, portional to both its concentration ci and the intensity of the excitation light at that point: dZt,A,(x,y,z) = ai,AIciIAx(x,y,z)dV = ai,AxcilAx(0)g(x, y, 2, A”) dV WANG AND SIMMONS: FLUORESCENCE MODELING (2) 909  385.15 373.00  360.85   348.70 336.55   324.40  312.25   Excitation Wavelength (nm) Figure 3. Emission scans of tyrosinetryptohan mixtures. Fluorescence intensities are identified by the contour lines, and the peaks for tyrosine and tryptophan can be clearly seen. 1 0.8 .0 ffl C a, c C 8 a, 5: 0.6 C 2 0.4 0 3 ii 0.2 0. 280 300 320 340 360 380 Wavelength (nm) 400 420 440 Figure 4. Typical emission spectra obtained for tyrosinetryptophan mixtures at 280 nm excitation. Total amino acid concentration: 3.0 x 105M. 910 BIOTECHNOLOGY AND BIOENGINEERING, VOL. 38, NO. 8, OCTOBER 20, 1991 where the proportionality constant u,,A, is the extinction coefficient for the ith component, an intrinsic property of the ith component independent of the physical configuration. Note that fluorescent species, as well as nonfluorescent species and solvent, may absorb at the excitation wavelength; therefore, AAx= ELl u , , ~ , cis , the absorbance from all N species, not just from the ith fluorescent component. The total amount of fluorescent light emitted at wavelength A, by the ith component within the differential volume is proportional to the amount of excitation light absorbed. where the proportionality constant q i , A X + A , , , is known as the fluorescence yield. The subscript "total" indicates that the above equation expresses the total amount of light emitted within the volume element; however, not all of it impinges on the detector and subsequently contributes to the measurement. First, because emission is directed at all angles, only a small fraction of the emitted light is geometrically aligned with the detector, the fraction being generally positiondependent. Secondly, because fluorescent light at the emission wavelength can be reabsorbed by the solution as it travels through the sample, only a fraction of the light in line with the detector actually reaches the detector. This overall fraction, represented by h in the following equation, is a function of both the differential volume's position within the sample an the sample's absorbance at the emission wavelength A,,,: Thus, the total amount of emitted light measured by the detector, AF, is obtained by integrating the above equation over the sample volume and by summing over all N components. where "A" signifies that base line signals are subtracted from the raw signals. For generality, all N components present in a sample, whether fluorescent or not, are included in the above equation. Nonfluorescent but absorbing components as well as fluorescent components that are not excited at A, can be treated by setting ( P f , A ,  r A , = 0. The above equation is difficult to apply in its orginal form because it depends on the geometric configuration of the system, in addition to the fluorophore concentrations. A new function r that depends only on the system geometry and the sample's optical density is defined below to isolate these effects: T(AAx, AAm) = I g(x, y, z, AAx)h(x, y, z, A A mdV ) (6) V One can obtain the corrected fluorescence A F ' by dividing the measured fluorescence by C The principal motivation for doing so is that A F ' is now linear in the fluorophore concentrations: At this point, well established linear theories can be applied to the above equation to yield various extended results. For example, the corrected fluorescence for a multicomponent system can be obtained by simply superimposing the corrected spectra of the constituent pure components. To eliminate the unknown factors I ~ x ( 0 ) ~ i , ~ x +(i~ m = u1,. i ,.~. ,xN ) from the above equation for a specific setup , fluorescence spectra, AF,,, are recorded for each pure solution of component i of known concentrations, cl0. The correction factors CO(Ard;, A:),) are also evaluated for each of the singlecomponent solutions. Substituting the above equation in eq. (7), we obtain the following form: Thus, if the corrected fluorescence is obtained for each pure component present in the system, the corrected fluorescence of a multicomponent system can be easily predicted. One problem that remains to be solved is the determination of the integral factor T(AAZ, A',), which doubles as the fluorescence correction factor. This factor as a function of the sample absorbance can be determined through either (1) rigorous theoretical analysis of the geometry, (2) empirical experimental correlation, or (3) a judicious combination of the above two approaches. The last approach is demonstrated in this article; a theoretical analysis is first derived below for a simplified, manageable version of the real system to yield the functional form of T(AAr, AAm), followed by experimental determination of the model parameters. WANG AND SIMMONS: FLUORESCENCE MODELING 911 Analytical Expression for Geometric Integral Case I. NADH Probe The analytical approach is demonstrated for the backscatter configuration of a commercial NADH probe inserted into an openended sample. The local monitoring efficiency representing the fraction of the emitted light aimed at the detector is a function of the distance x from the detector and can be approximated by fo exp( xS), where fo and S are constants. As a result, the following equation applies: L r(AAx,AA) = f o exp[x(S 0 + A A+~~ ~  ) l d x the large distance between the sample and the detector, the monitoring efficiency, in contrast to the laser setup, is independent of the position of the volume element. Furthermore, the factor exp( xAAm)represents the fraction of emitted light that passes through the sample back toward the detector without being absorbed. Substituting the above expressions for g and h into eq. (6),an expression for r is obtained: (laser setup) (14) Integration of this equation yields: (NADH probe) (10) Integrating the above equation fromx = 0 t o n = L, L being the dimension of the sample container, we obtain: fo r(AAx,AA)= i &  cos e x x (1  exp[L(S + AAx+ A”)]} (11) The above function has been used to correlate fluorescence measured at fixed excitationemission wavelength with both the fluorophore concentration and the path length.’ The same expression is valid for a fluorescence probe based on bifurcated optical fibers where excitation light is brought to the sample through one leg and emission light travels to the detector through the second leg. Geometric integrals for various detector configurations and optical nonidealities in a fluorescence probe have been derived.’,* Case 11. Laser Setup The derivation of the geometric integral r for the laser setup shown in Figure 1 can be carried out in a manner similar to that for the NADH probe because they both operate in the backscatter mode. Emission was measured at an angle 8 relative to the incidence of the excitation, which was less than lo” when refraction of the laser beam was considered. The light path from the cuvette to the detector is aligned perpendicular to the incident cuvette surface. The laser beam can be considered as monochromatic and nondivergent; thus, y and z coordinates are not significant in the integration of the volume, and x , the distance into the sample solution measured from the incident cuvette surface, varies from 0 to L. As a result, the proportionality functions g and h are: { + Ah 1  exp[ L(& (13) where fo is the monitoring efficiency. Because the beam of emission light reaching the detector is parallel due to 912 (15) This function depends only on the absorbance, the incident angle, and the path length, which can all be readily determined. The constant fo depends on the detector sensitivity, slit width, and other factors in a particular apparatus and must be determined experimentally. Case Ill. Spectrofluorometer Although fluorescence is physically a threedimensional phenomenon, we can simplify our analysis by considering only two dimensions because only horizontally directed light will be detected as a result of the relative alignment among the light source, the detector, and the sample. See Figure 5 , where the top part is an oblique view of the cuvette and the bottom part is the top view thereof. The light source and detector are both placed sufficiently far from the sample, so that the illumination efficiency and monitoring efficiency are constant along the perpendicular paths intersecting at the center of the cuvette. These paths are narrow enough so that the volume of intersection can be considered to be a point at the center of the cuvette. The narrowness of the beam was visually confirmed. The BeerLambert Law is applied to account for absorption of both excitation light traveling through the sample to the center of the cuvette (15,) and emission light traveling from the center of the cuvette out toward the edge of the cuvette (L,,,).This leads to the following equation: r~ h(x,y, z, A”) = fo exp(xA”) + A”)]] 1 For a 1 x 1 cm cuvette centered with respect to both the excitation and emission beams, the path length in the solution traveled by excitation light and that trav BIOTECHNOLOGY AND BIOENGINEERING, VOL. 38, NO. 8, OCTOBER 20, 1991 Emission .. Ax x=Lx 3. Emission 4 x=o  analytical function is not always feasible. Alternatively, I(&, A A mcan ) be experimentally determined for a specific fluorescence arrangement by varying the sample absorbance and by fitting the results based on a phenomenological model or a blackbox model. One method by which this can be accomplished experimentally is to utilize nonfluorescent compounds which absorb only and AAmcan be varat A, or only at A,. In this way, AAX ied independently so that the model parameters can be evaluated. In one extreme rigorous modeling approach, the various assumptions used in deriving the fluorescence models can be relaxed until fluorescence spectra can be accurately predicted with only those model parameters that are truly independently evaluated, but this approach will likely be counterproductive. At the other extreme, a pure blackbox approach can be used, where entirely empirical parameter fitting is practiced without any consideration of a physical model. However, it is more fruitful to base the general functional form on an approximate model rather than choosing an arbitrary form, e.g., power expansion. A sensor model based on an idealized approximation of a given apparatus will normally contain many groups of parameters, such as the lightsource intensity, detector sensitivity, fluorescence yields of the fluorophores, extinction coefficients of the absorbing species, and various geometric dimensions. Some of the parameters may be known accurately or can be independently measured, while others, e.g., detector and sensitivity, must be determined by fitting fluorescence data. Excitation Figure 5. The top part shows an oblique view of the sample illumination obtained with SLM 8000C spectrofluorometer. Fluorescence is detected only from a narrow region in the sample where the excitation light beam intersects with the detector view line. The bottom part shows sample illumination viewed from the top of the cuvette. Excitation light travels 0.5 cm through the solution to the center of the cuvette, and emitted light travels 0.5 cm through the solution from the center toward the detector. eled by emission light are identical: L, = L , = 0.5 cm. As before, the factors and u ~ , ~in , , ,the above equation account for absorption of excitation light and emission light, respectively, and K is a constant that includes the monitoring efficiency and the instrument sensitivity at the detection wavelength. Thus, the geometric factor for a rightangle spectrofluorometer is: Case I. NA DH Probe The parameter fitting approach was first used to model the fluorescence response of an NADH probe. For a singlecomponent system, the following threeparameter equation was obtained by substituting eq. (11)into eq. (7): In the above equation, P I , identified asfoZAx(0)cpA,+AmaAr, can only be determined experimentally, since the light source intensity and the detector sensitivity are specific to a given sensor. On the other hand, p2, identified as (uAX+ u A min ) the model, consists of intrinsic constants. As an example,' the value of p2 for thioflavin S was expected to be 0.0115 cm' ppm' based on independent absorbance measurements, and that of S was expected to be approximately 1.0 cm' based on geometric dimensions. The model parameter values obtained through a regression procedure were found as follows: PI = 60.3 NFU/cm Experimental Model Parameter Evaluation Because the geometric configuration is often illdefined, AAm) as a closed expressing the geometric integral T(AAx, ppm; p2 = 0.0172 cm' ppm'; and S = 0.412 cm' (19) Note that the above values for p2 and S do not exactly match the corresponding theoretical values. The major WANG AND SIMMONS: FLUORESCENCE MODELING 913 source of discrepancy was traced to oversimplifications made in deriving eq. (18).' Nevertheless, eq. (18) was able to fit experimental data extremely well at different conditions. Although a multicomponent version of the above equation can be derived by combining eqs. (7) and (ll), deducing two or more concentrations from a single point measurement is theoretically impossible; thus, the multicomponent application of the NADH probe will not be further pursued. The model predicted expressions of q1 and q 2 can be related to p ( x l ) as: Case It. Laser Setup For cI = 103M,ql(xl) differs fromp(xl) by a maximum of 6%, and q2(x1)differs from 1  p ( x l ) by a maximum of 5%. Table I1 shows the comparison between the values of p(xl) predicted by eq. (23) and the regression values for mixtures with tyrosine mole fractions ranging from 0.25 to 0.90. The regression values were obtained by minimizing the sum of the squares of the residuals between the prediction and the experimental spectra. The agreement becomes closer as x 1 approaches unity, but discrepancies do exist. For example, the experimentally determined value ofp was 0.242 for a 50% tyrosine mixture; whereas, that obtained from evaluating eq. (23) with independently measured values of the extinction , ~ u2,A,) ~ = 0.177. The coefficients was p = U I , A ~ / ( U ~ + intrinsic extinction coefficients are 6 1,280 = 1280 cm' MI and ii2,280 = 5960 cm' M  ' , where, to minimize confusion, the tilde signifies that the reported values are based on common logarithm rather than natural logarithm. (Note that in this article models are derived in natural logarithm for it is the most appropriate, while numerical values are reported in common logarithm in conformance with the existing practice whereby a spectrophotometer displays the optical density or absorbance in common logarithmic units. The natural logarithmic extinction coefficients, u , and the common logarithmic counterparts, 6, differ by a factor of 2.303.) Alternatively, a more general approach allows both q 1 and q2 to vary independently without being constrained by the underlying theoretical model parameters. Also shown in Table I1 is the comparison of the model pre A similar approach is applied to the binary amino acid data collected with the laserinduced fluorescence apparatus. An approximate equation describing the binary fluorescence is first derived, and the parameters in the equation are later determined experimentally. In deriving the approximate model for the special case, the exponential term in eq. (15) can be ignored when the solution is highly concentrated. In addition, light absorption by either tryptophan or tyrosine at the emission wavelength is negligible, i.e., AAm< A". Thus, the correction factor T(AAx,AAm) expressed in eq. (15) can be further approximated as follows: Substituting c , with x , c f , r,, with l/a,,A,c~O,and r(&,A A m ) with 1 / ~ ~ I ( & A & )into c { eq. (9) for a specific binary solution, where x1 and x2 are the tyrosine and tryptophan mole fractions, respectively, and c f is the total amino acid concentration, we obtain the following expression: or where p(xl) is a nonlinear function of the tyrosine mole fraction: Table 11. Predicted and experimental molefractiondependent fluorescence coefficients for laserinduced fluorescence measurements of aqueous tyrosinetryptophan mixtures with a total amino acid concentration of lO'M. Mixture More precisely, when the exponential term in eq. (15) is not ignored when the concentrations are not high, there will be two coefficients, q1 and q2, that are nonlinear in x I . 914 XI ppred. Pev qlprcd qIe"pt q2prcd 0.25 0.25 0.50 0.50 0.75 0.75 0.90 0.90 0.90 0.067 0.067 0.177 0.177 0.392 0.392 0.659 0.659 0.659 0.164 0.154 0.242 0.250 0.456 0.467 0.655 0.711 0.673 0.071 0.071 0.188 0.188 0.412 0.412 0.685 0.685 0.685 0.053 0.092 0.220 0.205 0.365 0.393 0.592 0.670 0.622 0.933 0.933 0.823 0.823 0.606 0.606 0.330 0.330 0.330 BIOTECHNOLOGY AND BIOENGINEERING, VOL. 38, NO. 8, OCTOBER 20, 1991 4z..p,  0.671 0.753 0.725 0.684 0.408 0.424 0.251 0.230 0.251 dicted values of q1 and q2 against the experimental regression values. Shifting from eq. (22) to eq. (24), the sum of squares of the residuals of spectral prediction was reduced by an average of 10%. A majority of the residual error can be attributed to the significant noisetosignal ratio for the laserinduced apparatus. Based on eq. (24), the predicted spectrum is plotted in Figure 6 for a 50% tyrosine mixture using the regression values of q1 = 0.245 and q2 = 0.785, which are calculated from the Fourier transform smoothed pure component spectra of AFlo and AFz0. (Note that the values presented in Table 11 are based on the original, unsmoothed spectra.) Case Ill. Spectrofluorometer Derivation of correction factor based model. The binary mixture of tyrosine and tryptophan will be examined based on eq. (9): AF r(AAx, AFio rlo(AAx,~ Clo+ ci ~ m AFzo cz )rz0(AAz,AAm) Z9 slightly modified model: AF r(&, k m ) clo+ ~1 ) AF;o r20(AAx, k m c2 ) cz0 (27) where the Raman modified pure component spectra AFYo and AFYo are given by: AF;o = AFlo  rlo(AAx,AAm)AF,o A F & = AF20  T20(AAx,AAm)AF,0 (28) Note that AFWois nearly zero for all emission wavelengths except at 308 nm, at which point AFWo= 1.356 x lo’ NFU, a value corresponding to the maximum occurring near 308 nm in the pure water specfor trum. As long as Beer’s Law is followed, AAxand AAm a specific sample can be either experimentally measured with an independent spectrophotometer or theoretically calculated according to: AAx= (al,Axcl+ az,A,~d (26) where the subscript “1”again denotes tyrosine, the subscript “2” denotes tryptophan, and the subscript “w” will be reserved for water. The geometric integral r(AAX, AAm) for the rightangle configuration is given by eq. (17). Because the solvent can also contribute to absorbance or fluorescence, strictly speaking, a binary mixture is a threecomponent system with the solvent as the third component, a consideration that can sometimes be important. Because of its Raman spectrum, water should be included as a third component. With TWo(AAX,AAm) = 1, c,/cw0 = 1, and AFWoexperimentally measured with a blank water sample, we now have a AFYo rlo(AAX, k m A A m = (al.Amcl+ (29) a2,Am~2) Derivation of phenomenological model. When water is employed as the solvent, eq. (16) can be written as: A F = (Klcl + K2c2 + K,c,)10L(~’C’+~2C2+“cw) (30) 7 where the constants Ki and given by: Ki = iji for the ith species are KIA,(O)(Pi,A,A,,,zi,A, zi = + ( i i , ~ ~f i i , ~ , , , ) . (31) (32) Since the concentration of water is nearly constant, the expression can be simplified to: A F = (klcl + k2c2 + @w)10L(liici+i2Cz)(33) Prediction of Tyrosinefltyptophan Mlxture Fluorescence 6 5 4 al 0 c al 2 s0! ixwre Spectrum (Measured and Predicted) 3 3 ii 2 1 0. 275 300 325 350 375 400 425 450 Emission Wavelength (nm) Figure 6. Generalized parameter fitting approach applied to laserinduced fluorescence data. WANG AND SIMMONS: FLUORESCENCE MODELING 915 where k l = KllOLiwcw;k 2 = KzlOLiwcw;and Qw= K w cw are constants independent of the tyrosine and tryptophan concentrations. The above equation is valid for excitation scans, emission scans, and synchronous scans. In general, the socalled constants in eq. (33) are functions of A, and A,. These constants' dependence on A, for emission scans can be analyzed. For wavelengths greater than 300 nm, a, is approximately constant, becausea, = a,,Ax+ a,,Am = a,,Axduetoal,Ax % a,,Am and because A, is fixed. As in the laser setup, these constants can be evaluated independently from the absorbance = 1280 cmI M' and spectra at A, = 280 nm (a'1,28~ 62,280 = 5960 cm' it'). However, for wavelengths less than 330 nm, a,,A,cannot be ignored and should be retained. Overall, each a, is a weak function of A,. Although the fluorescent efficiency of water is zero, for 280 nm excitation water exhibits a sharp Raman peak at 308 nm. This Raman peak can be conveniently modeled as a form of fluorescence: @,,,(A,) = 0 except at A, = 308 nm. To identify k,'s dependence on A, for emission scans, the parameter is split into two factors, one depending only on A, and another depending only on A,. ki(A,,Am) = HA,) X klm(Am) = [ML)a,(A,) exp(  Law,A,cw)l x [K(Am)(pi,ArtA, e x ~ (  l a , ~ ~ c ~(34) )l For most situations, is independent of A, because the excited molecules quickly relax to a pseudosteadystate distribution of energized states independent of the initial excitation; fluorescence intensity's dependence on AAxmainly originates in a f luorophore's ability to absorb at AAx,whereas its dependence on A A mis caused by the distribution of the excited energy states. Since A, is held fixed and only A, is varied during an emission scan, k, can be expressed as a function of A,. These constants depend very heavily on the instrument characteristics, and they cannot be predicted without experimentation. Thus, all the parameters in eq. (33) ( k l , k2, Qw, cil, and iiz) are functions of the emission wavelength, and a different set of values can be obtained at each wavelength. Regression of model parameters. Determination of model parameters will be demonstrated for the binary system of tyrosine and tryptophan. In the first approach, ii, and ii2 based on independently acquired absorbance spectra were used to obtain corrected fluorescence spectra from measured fluorescence spectra according to the following expression: equation: A F ' = klcl (36) Note that AF' is linear in c1and c2,although AF is nonlinear in c1 and CZ. The best values of k l and k z were determined from fluorescence data within W 6 M Icf I 3 X 105M. The results are listed in Table 111 for two select wavelengths at 320 and 352 nm. Only data at low concentrations were used because small errors in 6 and iiz could be magnified by high concentrations; these errors are further enlarged exponentially in calculating the correction factor. Moreover, not all data were used for model calibration, because some were saved to gauge the model's predictive ability during the testing stage that was to follow. For solutions with cf = 104M and c, = 103M, Figure 7 compares the measured fluorescence to the regression model prediction at 320 and 352 nm with the parameter values listed in Table 111. Similar agreement was obtained at all other wavelengths as well. Likewise, Figure 8 compares the measured fluorescence to the model prediction as a function of the total amino acid concentration for solutions with a constant tyrosine mole fraction of 0.50. These figures indicate that the predicted fluorescence is too low at higher concentrations, as a consequence of iil and iiz being too large and overcompensating for the nonlinearity caused by innerfiltering, especially for mixtures containing much tryptophan. Considering the various simplifying assumptions made in model derivation, a certain degree of discrepancy at high concentrations is expected. In the second approach, instead of being assigned values obtained from absorbance measurements, iil and iiz, as well as k l , k 2 , and Q w , were all determined directly from fluorescence calibration data via nonlinear regression. Because the degree of freedom in data fitting is now greater than that of the first approach, the resulting predictions are expected to be better. To find the best values of these parameters, the following error function was minimized at each emission wavelength: (37) where M is the number of mixtures in the calibration data set. The fluorescence spectrum for the jth mixture Table 111. Linear regression of corrected tyrosinetryptophan fluorescence at two select wavelengths. (35) Subsequently, k l , kZ, and % at various emission wavelengths were estimated from the corrected fluorescence spectra via linear regression based on the following 916 + k 2 ~ 2+ Qw BIOTECHNOLOGY AND BIOENGINEERING, VOL. 38, NO. 8, OCTOBER 20, 1991 0.8 0.6 tE cu 0 PI c m J2 .In c 0.4 0) 0 c al u) 0 I 2 U 0.2 0 0.8 0.6 0.4 0.2 0. 1 Mole Fraction Tyrosine  Measured 320 nm : A 352 nm : 0 Predicted ._ 0 c In a, c  0.8 A a, c u 0 u) u 2 2 B 0.6 U 0.4 0.2 1 0. 0 0.2 0.6 0.4 0.8 1 Mole Fraction Tyrosine Figure 7. Predicted and measured fluorescence at 320.2 and 352.0 nm for solutions with a total amino acid concentration of 1.0 x 104M (top graph) 1.0 x lO%f(bottom graph). The prediction is based on linear regression of the corrected fluorescence signal. The excitation wavelength was 280 nm. The triangles (A) and the solid line () indicate the measured and the predicted emission at 320 nm, while the squares (0)and the dashed line () indicate the measured and the predicted emission at 352 nm. according to the model, AFmodei,,,is given by eq. (33): AFm0del.I = {[kl(xl,,) + kz(1 x  x ~ , , ) l ~ t+, ,Q w } 10Y[a1(x1.,)+i2(1x1.,)~~.,1 (38) wherexl,, and c ~ ,are , the tyrosine mole fraction and total amino concentration, respectively, of the jth mixture. The error function defined in eq. (37) is the rootmeansquare of the relative error. The error is calculated relative to the average of the measured and predicted error instead of the true value because the true fluorescence value is unknown. The additional term of 0.004 NFU, which represents the resolution of the measurement at the settings used, is included to prevent dividing by WANG AND SIMMONS: FLUORESCENCE MODELING 917 A 320 nrn 352nrn 0 10 .cA. v) c a, + 1 1. C a, 0 C 10 1 I I I I I I I I I I I I I I1111 I I I 1 1 1 1 1 1 I I I I I I I I I I I I I I Ill1 I I I I I I Il lI Il ~nlrl/I I/I I I I111 I I 1 1 1 1 1 1 1 I I 1 1 1 1 1 1 1 I I I Ill I I I Ill a, 0 I v) L 0 3 U  2 I , A I I , , , Pl I11111 I I I IIII 10 10 3 10 7 10 5 10 Total Concentration (M) 6 10 4 10 3 Figure 8. Predicted and measured fluorescence at 320.2 and 352.0 nm of an equimolar aqueous tyrosinetryptophan mixture as a function of total amino acid concentration. The prediction is based on linear regression of the corrected fluorescence signal. The excitation indicate the measured wavelength was 280 nm. The triangles (A) and the solid line () and the predicted emission at 320 nm, while the squares (0) and the dashed line () indicate the measured and the predicted emission at 352 nm. Table IV. Parameters for tyrosinetryptophan fluorescence for various wavelengths. The values of 81 and 82 are based on common logarithm. Am kl k2 GI (nm) (NFU/M) (NFU/M) (cm'M') 290.0 295.0 300.0 308.2 315.0 320.2 325.0 330.0 335.0 340.0 345.0 350.0 352.0 355.0 360.0 365.0 370.0 380.0 390.0 400.0 410.0 420.0 430.0 440.0 4198.3 5927.4 7685.4 8303.3 7142.9 6183.5 4725.0 3544.7 2548.9 1791.9 1213.3 849.3 737.2 582.2 441.7 317.8 217.7 150.1 139.7 123.3 89.1 64.9 56.3 51.0 58.7 80.1 238.2 1479.3 4699.2 8882.4 13455.0 18100.0 22459.0 25628.3 27882.8 29039.3 29 183.2 28788.6 27816.9 25993.7 23829.9 18745.2 15577.3 11495.8 6583.7 4037.8 2628.9 1685.3 1466.2 1367.1 1356.6 1309.7 1274.2 1311.9 1266.0 1249.6 1229.6 1209.1 1198.3 1197.6 1195.5 1191.8 1197.0 1199.1 1196.3 1218.0 1227.3 1246.5 1260.0 1277.3 1317.6 1382.3 ;rz (cm'M') 8243.4 6041.3 5224.8 4757.4 4719.5 4738.4 4762.6 4751.1 4767.3 4772.4 4791.5 4806.1 4812.0 4808.0 4821.0 4829.9 4848.1 4876.3 4866.9 4894.1 4895.9 4921.5 4944.9 4948.5 @w E (NFU) (%) 0.00 20.5 0.00 17.3 0.00 12.9 0.014 10.9 0.002 12.0 0.00 12.8 0.00 12.3 0.00 13.2 0.00 13.4 0.00 13.4 0.00 12.4 0.00 12.8 0.00 13.3 0.00 12.8 0.00 13.3 0.00 14.3 0.00 12.3 0.00 14.6 0.00 16.4 0.00 16.0 0.00 17.5 0.00 18.1 0.00 17.6 0.00 18.1 numbers very close to zero. This term is negligible for all mixtures except those with cI = 106M. The results from this nonlinear regression procedure are shown in Table IV at some select wavelengths. Except for @, the model parameters obtained via nonlinear regression were further fit with polynomials of A,, since they are continuous functions of A, for emission scans. These polynomials enable the estimation of k l , k Z ,61, and Z 2 at all wavelengths through interpolation, although extrapolation should be avoided. 918 The polynomial coefficients for k l and k 2 are listed in Table V. These correlations are plotted in Figure 9, which also shows the rootmeansquare error as a funtion of the emission wavelength. The point at 320 nm in Figure 9 is significant, since it is the point at which the fluorescence constant of the two compounds at 280 nm excitation are equal. This wavelength also corresponds to the invariant region in Figure 4. Likewise, 6, and Gz were correlated with A, as shown in Table VI and Table VII, respectively. The correlations for G I and 6 2 are plotted in Figure 10. The difference between the values of u l and u 2 obtained from fluorescence data and the corresponding values obtained from absorbance data are not too severe. Parameter @, which is nonzero only near a sharp peak at 308 nm, could not be correlated to A, with a polynomial without more data around the peak. Instead, @, was estimated directly from the water spectrum, i.e., A F = CP, for pure water. Figures 11 and 12 show some typical predicted fluorescence responses at four select wavelengths (A, = 290, 308, 320, and 352 nm) as functions of the tyrosine TableV. Correlation of the parameterskl and kz with the emission wavelength, A,. for 290 nm 5 A, I440 bo = 5.747 bl = 3.985 b2 = 4.581 61 = 0.935 bq 6.961 b5 = 0.828 bb = 2.788 BIOTECHNOLOGY AND BIOENGINEERING, VOL. 38, NO. 8, OCTOBER 20, 1991 nm for 290 nm 5 A, 5 440 nm 10.157 1.217 2.026 5.494 2.038  16.768 6.733 32.664 6.388 18.493 35000 k 30000 5 L 25000 v) c s 20000 : # 15000 2 10000 LL 5000 Tryptophan r c .' 8, a L  0 22 0 48 8 18 L g W14 10 290 320 350 380 410 440 Wavelength (nm) Figure 9. Correlations of kl and kz with emission wavelength (A,,,). The correlation for log(kl)is a sixthorder polynomial in A,,,, and the correlation for log(k2) is a ninthorder polynomial in A,,,. Also shown at the bottom is the modified rootmeansquare relative error of the predictions at each wavelength Table VI. Correlation of the parameter, a ' , with the emission wavelength, A,,,.  365 nm] 75 nm for 290 nm IA,,, I440 nm = $B,( Am ,=O BO = 1195.3 cml Bl = 6.827 cm' B Z = 203.79 cml B3 = 38.870 cm' M' M' Ml M' Table VII. Correlation of the parameter, a2, with the emission wavelength, A,,,. for 290 nm IA, I320 BA = 4878.3 cm' B ; = 719.71 cm' Bi = 743.43 cm' B ; = 273.90 cm' B ; = 869.17 cm' B ; = 758.89 cm' nm M' M' M' M' M' M' for 320 nm IA,,, I 440 nm BI = 4856.7 cm' M' BY = 105.4 cm'M' BY = 11.59 cm' M  ' mole fraction for c, = 104Mand c, = 103M based on the parameters listed in Table IV. (Note that 308 nm corresponds to the maximum of the pure tyrosine fluorescence, 352 nm corresponds to the maximum of the pure tryptophan fluorescence, 320 nm corresponds to the invariant region, and 290 nm is the first scan wavelength.) Likewise, in Figure 13 the predicted fluorescence responses as functions of the total amino acid concentration for equimolar mixtures of tyrosine and tryptophan are compared with the measured values at select wavelengths. Similar agreement was also obtained at other wavelengths. It is apparent from these figures that predictions made from the second approach are superior to those made from the first approach. In the second approach, the rootmeansquare of the relative error between the model prediction and the experimental measurement was 1020% for all wavelengths above 290 nm. Through a careful error analysis, a significant portion of this error can be attributed to experimental sources: solution preparation, dilution, positioning of the sample cuvette, and reproducibility of the spectrofluorometer. Although the model was specifically demonstrated with the binary system of tyrosine and tryptophan in this article, it also performed equally well for many other binary combinations listed in Table I. CONCLUSIONS A generalized fluorescence model applicable to various measurement devices was developed to describe fluorescence in a multicomponent system that included both fluorophores and nonfluorophores. The model was applied to a commercial NADH probe, a laserinduced fluorescence setup, and a commercial spectrof luorometer. Two approaches were taken to estimate the model parameters. In the first approach, the factors that depended on the specific system geometry were lumped together into a geometrical integral, which could be de WANG AND SIMMONS: FLUORESCENCE MODELING 919 CD Empircal Correlation of Effective Absorbance Coefficients: a,, a, * ** ** Tyrosine, a , Tryptophan, +++++ \ *' *\ \ >\ * \ * \* \ * \ 't 4 320 280 360 4 00 440 Emission Wavelength ( n m ) Figure 10. Correlations of u l and a2 with emission wavelength (A,). The correlation of a1 is a cubic in A,. The correlation of a2 requires two polynomials. For 290.0 nm < A, < 320.0 nm, a fifthorder polynomial was used. For 320.0 nm < A, < 440.0 nm, a parabola was used. 0.5 1 0.3  0.2  Predicted A  290nm 308nm 0.4  1 .o Measured 2.25 ______ 0 1.75 ? 1.25  5 A Predicted  1.00 0 E! _/C 0.1  &0no hr .. 290nm 1.50 K 0.0 0.0 Measured 2.00 A A c" :WW: 2 $ G 0.75 " 7 " 7 . ;,;"y, j . ,  0022 H , ,A K, A . , , 0.8 0.7 320nm: 35znm: 06 , , , , , , , , , Measured Predicted 0  , , , 00 ,, 1 00 0 4 25 20 03 ..  1 50 71 s 1.5 02 0.50 0.25 : I >. 125 ar 1.00 320nm Measured Predicted 0  0 , E! %_ 01 0.0 0.0 0.2 0.4 0.6 Mole Fraction Tyrosine 0.8 10 0.0 Figure 11. Predicted and measured fluorescence at 290.0, 308.2, 320.0, and 352.0 nm for solutions with a total amino acid concentration of 1.0 x 104M.The prediction is based on nonlinear regression of the parameters, k,k l , k2, u l , and u 2 . In the top graph, the predicted curves [() and ()I are plotted beside the measured emission values at 290.0 nm (A) and 308.2 nm (0). In the bottom graph, the predicted curves, )( and (), are compared with the measurements at 320.0 nm (0)and 352.0 nm (W). The excitation wavelength was 280 nm. 920 10 0.0000.00 Figure 12. Predicted and measured fluorescence at 290.0, 308.2, 320.0, and 352.0 nm for solutions with a total amino acid concentration of 1.0 x 103M. The prediction is based on nonlinear regression of the parameters, A w ,k l , kz, a l , and u2. In the top graph, the predicted curves [() and ()I are plotted beside the measured emission values at 290.0 cm (A) and 308.2 nm (0).In the bottom graph, the predicted curves [() and ()I are compared with the measurements at 320.0 nm (0) and 352.0 nm (W). The excitation wavelength was 280 nm. BIOTECHNOLOGY AND BIOENGINEERING, VOL. 38, NO. 8, OCTOBER 20, 1991 290 nrn A 0  308nm B,; B,'; B:' e A 10 ' 103 107 106 105 Total Concentration (M) 104 103 Figure 13. Predicted and measured fluorescence at 290.0, 308.2, 320.2, and 352.0 nm for solutions with a tyrosine mole fraction of 0.50. The prediction is based on nonlinear regression of the parameters, Q W , k , , k2, a ' , and a2. In the top graph, the predicted curves [() and ()I are plotted beside the measured emission values at 290.0 nm (A) and 308.2 nm (0).In the bottom graph, the predicted curves [() and ()I are compared with the measurements at 320.0 nm (0)and 352.0 nm (W). The excitation wavelength was 280 nm. termined separately from independent measurements aided by the model. In the second approach, an idealized model was derived, and fluorescence spectral data were used to determine the optimal values for model parameters, including supposedly intrinsic parameters. In each case, the model was generally capable of accurately predicting the fluorescence of binary tyrosinetryptophan mixtures, although slightly better predictions were obtained from the second approach as expected. The authors would like to acknowledge the assistance of John J. Horvath and Hratch G. Semerjian (Chemical Process Metrology Division, U.S. Department of Commerce, National Insitute of Standards and Technology, Gaithersburg, MD 20899), whose fluorescence setups were used in this study. This work is supported in part by a DuPont Young Faculty Award, AlliedSignal Corporation Foundation Faculty Support Grant, Systems Research Center under National Science Foundation grants CDR8500108 and EET8720046. NOMENCLATURE a,;& at,Ar;a~,Ar u , , A (7,.~, ~; sum of natural logarithmicbased and common logarithmicbased excitation and emission extinction coefficients (cm' M' or cm' ppm') natural logarithmicbased and common logarithmicbased specific extinction coefficient of species i at the excitation wavelength (cm' M' or cm' ppm'1 natural logarithmicbased and common logarithmicbased specific extinction coefficient of spe k, k." L M N NADH P 9 S cies i at the emission wavelength (cm'M' or cm' ppm') solution absorbance at A, or A, (cm') polynomial coefficients used to correlate kl and k2 with A, [SLM spectrofluorometer] polynomial coefficients used to correlate a1 and a2 with A, (cm'Ml) concentration of component i ( M or ppm) 2.7182818.. . fluorescence model parameter for NADH probe, monitoring efficiency, instrument sensitivity (NFU s cm*/photon) normalized fluorescence signal (NFU) corrected fluorescence signal (NFU) geometrydependent ratio of local excitation intensity to lightsource intensity geometrydependent ratio of fluorescence light reaching the detector to total fluroescence light emitted at ( x , y , z ) excitation intensity at a point ( x , y , z ) within the sample (photons/cm2 s) excitation lightsource intensity (photons/cm's) parameter in fluorescence model (NFU/M or NFU/ppm) parameter in fluorescence model (SLM spectrofluorometer) (NFU/M) excitationdependent, emissionindependent contribution to k , emissiondependent, excitationindependent contribution to k , path length (cuvette width for laser setup, onehalf of cuvette width for spectrofluorometer) (cm) number of fluorescence spectra used in estimating model parameters total number of absorbing and fluorescing components in sample reduced nicotinamide adenine dinucleotide mole fractiondependent linear fluorescence coefficient mole fractiondependent linear fluorescence coefficient exponential parameter used to model the NADH probe (cmI) illuminated volume (cm') spatial coordinates (cm) mole fraction of component i Greek letters fluorescence model powerlaw parameter parameter in fluorescence model [NADH probe] (NFU/ppm) parameter in fluorescence model [NADH probe] P2 (cm' ppm') I'(AA',AAm) geometrydependent fluorescence integral modified rootmeansquare of the relative re& siduals angle between emission and excitation light e paths (") fluorescence model parameter excitation and emission wavelengths (nm) fluorescence yield apparent fluorescence of a blank sample; water Raman peak (NFU) a PI Subscripts 1 2 1 m tyrosine tryptophan ith component in sample emission WANG AND SIMMONS: FLUORESCENCE MODELING 921 excitation total water solution containing only component i X t W i0 Accents common logarithmic based units References 1. Christmann, D.R., Crouch, S.R., Timnick, A. 1981. Anal. Chem. 53: 2040. 2. Christmann, D.R., Crouch, S.R., Timnick, A. 1981. Anal. Chem. 53: 276. 922 3. Holland, J. F., Teets, R. E., Kelly, P.M., Timnick, A. 1977. Anal. Chem. 49: 706. 4. Leese, R.A., Wehry, E. L. 1978. Anal. Chem. 50: 1193. 5. Parker, C. A., Barnes, W. J. 1957. Analyst 82: 606. 6. Rollefson, G. K., Dodgen, H.W. 1944. J. Chem. Phys. 12: 107. 7. Simmons, M. B., Wang, N. S. 1991. Characterization of an online commercial fluorescence probe. 11: Considerations in sensor design. Biotechnol. Bioeng., to appear. 8. Simmons, M. B., Wang, N. S. 1991. Characterization of an online commercial fluorescence probe. 111: Effect of divergent and polychromatic light source. Biotechnol. Bioeng., to appear. 9. Wang, N. S., Simmons, M. B. 1991. Characterization of an online commercial fluorescence probe. I: Modeling of the probe signal. Biotechnol. Bioeng., to appear. 10. Wiechelman, K. J. 1986. American Laboratory, February 49. BIOTECHNOLOGY AND BIOENGINEERING, VOL. 38, NO. 8, OCTOBER 20, 1991