ANALYTICAL CHEMISTRY, VOL. 51, NO. 11, SEPTEMBER 1979
mismatch to zero may be due to unknown impurities or a mechanism other than the classical competitive type. From the l / V o vs. [I] replot (Figure 12), a good estimate of Ki -0.45 mmol/L can be obtained which agrees well with KmVp a t [I] = 0 (Figure 13). Note, the plot is nearly perfect. Only one line, corresponding to 0.50 mmol/L substrate, is not confluent with the other 20 lines. The above are only a few examples of how response surfaces, coupled with traditional plots and replots, can be used to gain an understanding of enzyme catalyzed reactions.
CONCLUSIONS Many kinds of chemical experimentation can be greatly improved by the use of the appropriate instrumentation and computers on-line to the experimental apparatus. These systems can provide many times the quantity and quality of data obtainable by manual methods. In addition this apparatus enables the experimenter to perform experiments that are otherwise impossible or extremely difficult and time consuming. Further improvements in experimental capabilities are obtained when interactive graphics are used on-line as an aid for interpretation of the vast quantities of data that are generated by an automated system. We have described an apparatus with these attributes. The system is being used to study enzyme mechanisms, optimize analytical methods, and develop new experimental and control techniques. Data obtained from the alkaline phosphatase enzyme-catalyzed reaction was used to show the versatility and power of apparatus and evolving graphic techniques.
ACKNOWLEDGMENT The authors acknowledge helpful discussions with L. B.
1747
Rogers, E. Clifford Toren, Jr., and Thomas A. Brubaker.
LITERATURE CITED (1) Deming, S. N.; Pardue, H. L. Anal. Chem. 1971, 43, 192. (2) Eggert, A. A.; Hicks, G. P.; Davis, J. E. Anal. Chem. 1971, 43, 736. (3) Renoe, B. W.; O'Keefe, K. R.; Malmstadt, H. V. Anal. Chem. 1976, 48, 661-666. (4) Hicks, G. P.; Eggert. A. A.; Toren, E. C. Anal. Chem. 1970, 42, 729. (5) Weinberg, G. M. "An Introduction to General Systems Thinking"; John Wiley and Sons: New York, 1975. (6) Frazer, J. W. Mater. Res. Stand. 1972, 72, 8. (7) Frazer, J. W. Am. Lab. 1973, 2 . (8) Frazer. J. W.; Kunz, F. W. "Computerized Laboratmy Systems", A S M 5 7 8 . 1975. (9) Eykhoff, P. "System Identification"; Wiley: New York, 1974. (IO) Pomernacki, C. L.; Brubaker, T. A,; Brand, H. R.; Frazer, J. W. "Characterization of the Flow Dynamics of an Enzyme Reaction System", Lawrence Livermore Laboratory, Rept. UCRL- 79995. Submitted for publication in Anal. Chim. Acta, (11) Savitizky, A,; Golay, M. J. E. Anal. Chem. 1984. 3 6 , 1627. (12) Boyer, P.D. "The Enzymes", Voi IV, Third Edition; Academic hess: New York, 1971; p 417-447. (13) Segel, I.H. "Enzyme Kinetics", Wiley: New York, 1975. (14) Lee, H. J.; Wilson, I.B. Biochim. Biophys. Acta 1971, 242, 519. ( 1 5 ) Michaelis, L.; Menter, M. L. Biochem 2. 1913, 49, 333. (16) Lineweaver, H.; Burk, 0. J . Am. Chem. SOC.1939, 56, 658. (17) Box, G. E. P. Biometrics 1954, 70, 16. (18) Dixon, M. Biochem J . 1953, 55, 170. (19) Zeffren, E.; Hail, P. L. "The Study of Enzyme Mechanisms"; Wiley: New York; Chapter 6.
RECEIVED for review January 29,1979. Accepted May 15,1979. This work was performed under the auspices of the U S . Department of Energy by Lawrence Livermore Laboratory under contract No. W-7405-Eng-48. Reference to a company or product name does not imply approval or recommendation of the product by the University of California or the U S . Department of Energy to the exclusion of others that may be suitable.
Characterizing Chemical Systems with On-Line Computers and Graphics: Alkaline Phosphatase Catalyzed Reaction Jack W. Frazer," Lester P. Rigdon, Hal R. Brand, and Charles L. Pomernacki Lawrence Livermore Laboratory, University of California, Livermore, California 94550
Thomas A. Brubaker Electrical Engineering Department, Colorado State University, Fort Collins, Colorado 80523
Computer automation of experimental apparatus can aid in the global characterlzatlonof complex systems by rapidly providing large quantities of accurate data. The combination of fast data collection and manipulation provides a means for understanding complex chemical systems that first appear to be medlum number systems. We have applied graphical techniques to the study of the alkaline phosphatase-catalyzedhydrolysis of p-nkrophenylphosphate. Three dimensional Box-type surfaces gave a quick visual summary of the reaction. Model response surfaces were also generated and compared with experimental surfaces to evaluate the model and analytical techniques. Residual-surface graphic techniques that compare experlmental and model response surfaces aid in the identlflcatlon of modeling or experlmentation errors. Reaction rate surfaces, which are three-dimensional plots at constant reaction rate, are a new type of graphical representation of use in presenting complex nonlinear systems. Traditional plots and replots are also generated on-line. 0003-2700/79/0351-1747$01.00/0
Traditional methods of characterizing complex chemical systems are difficult and time consuming. In the previous paper ( I ) , we pointed out that many of the problems associated with traditional experimental techniques can be solved by taking advantage of computer control. A computer-controlled apparatus allows the experimenter to obtain the required chemical information using on-line instrumentation. The entire experiment can be fully controlled by the computer, with data acquired, manipulated, and plotted as needed. Chemical systems often appear to be medium number systems. A medium number system yields data that have too much order to be treated statistically, but are so complex that precise numerical analyses cannot be readily obtained (2). Heterogeneous catalysis used by the petroleum industry in refining crude oil is an example of a medium number system. Here the chemical system involves not only the catalyst and its properties but also many hundreds of chemical reactions. The goal of the petroleum industry in studying the chemistry 0 1979 American Chemical Society
1748
ANALYTICAL CHEMISTRY, VOL. 51, NO. 11, SEPTEMBER 1979
of the total system is to relate catalyst composition and structure (many variables) to the production of the maximum quantity of desired compounds. At the present stage of our understanding, such systems are far too complex to be numerically described by a model containing analytical expressions for all the separate properties. However, such a system has considerable order (organized complexity) and, therefore, can be treated as a medium number system. The use of on-line instrumentation, computers, and graphics can accelerate model development and understanding of these complex systems. In addition, decision tree logic such as used for sequential chemical testing (3)can often be developed to automatically accumulate more data when initial information does not appear to conform to a proposed model. At the onset of experimentation, many chemical systems appear to be medium number systems. Actually, these systems usually can be numerically modeled and accurate formulas derived, provided the experimenter can obtain enough accurate data and build the tools and techniques that will aid him in the interpretation and analyses of the data set. On-line computer-aided graphics is one such tool. Much chemical experimentation is limited by the difficulty in characterizing a complex system over wide ranges of several variables. The measurement accuracies, large number of data required, and development of appropriate models often deter such studies. Given a computer-aided apparatus, such as used in this work, one can quickly and readily perform more global characterizations, compute required constants, and produce meaningful graphic representations. For enzyme-catalyzed reactions, the more global approach can be a great aid in determining mechanisms and developing accurate analytical methods. In this paper we discuss techniques that can help experimenters interpret complex chemical systems. We describe how we have applied Box-type response surfaces, replots, modeling, and residual-surface techniques to the alkaline phosphatase-catalyzed hydrolysis of p-nitrophenylphosphate.
EXPERIMENTAL The apparatus and computer system used for this work were described earlier ( I ) . Analysis of the alkaline phosphatasecatalyzed reaction, alkaline
p-nitrophenylphosphatepho8G8m8ep-nitrophenylate +
phosphate was used for all work reported here. The enzyme, alkaline phosphatase (Enzyme Comm. No. 3.1.3.1), was a commercial preparation obtained from calf intestinal mucosa Type 1. It was of rather low activity, generally in the range of 1 to 2 U/mg. The chemicals and procedures were identical to those reported in the previous paper ( I ) .
5.25
a t 10.1, a response surface for the effect of enzyme activity and substrate concentration on rate of product formation can be generated for the selected set of fixed variables (see Figure 1). Similarly, the interaction of any two variables can be represented by a response surface when all but the two independent variables of interest are held constant. When an experiment is run for several variables, a large set of response surfaces is required to present the results. One partial set for the four variables discussed above is shown in Figure 2. These surfaces help the experimenter to better understand the system under study by giving him a quick visual summary of the reaction kinetics. In this experiment, the typical bell shape that indicates two successive ionizations (5) can be seen on all surfaces incorporating variable pH. The use of graphics allows the experimenter to quickly evaluate these data with regard to theory. As an example, on-line graphics quickly showed that the Michaelis F p H function (6)
and the corresponding rate equation
KO
u =
RESULTS AND DISCUSSION Box-Type Response Surface. As reported earlier ( I ) , Box-type ( 4 ) response surfaces can simultaneously represent two independent and one dependent variable. For alkaline phosphatase-catalyzed reactions the dependent variable is the rate of product (p-nitrophenol) formation in micromoles per liter per minute. The independent variables are temperature, pH, enzyme activity, and the concentrations of substrate, activator, inhibitor, and buffer. Any two independent variables together with the dependent variable can be used to generate a response surface. An examination of a few response surfaces will illustrate their utility. Consider experiments where buffer concentration (0.3 M 2-amino-2-methyl-lpropanol (2-AMP)) and temperature (30.0 "C) are kept constant, and only the enzyme activity, substrate, inhibitor, and p H are varied. If no inhibitor is present and p H is kept
0.25
3.00
Figure 1. Response surface for enzyme activity varied from 3 to 18 U/L in 0.75 U/L steps and substrate varied from 0.25 to 5.25 mmol/L in 0.25 mmol/L steps with pH held constant at 10.1
K2
[H+l 1+-+Ki
[H']
where KO,K1, and K 2 are ionization constants, do not yield a tight fit to the experimental data of Figure 2d. With enzyme at 12 U/L, no inhibitor, and a 5.85 mmol/L concentration of substrate, the following ionization constants are obtained: KO = 37.4, K1 = 2.5.10-1°, and K 2 = 0.6.10-'0. The rate equation: 37.4
u =
1+
[H+l 2.5
X
lo-''
+ 0.6 x 10-lo [H+l
as plotted in Figure 3a, is not a good fit t o the data.
(3)
ANALYTICAL CHEMISTRY, VOL. 51, NO. 11, SEPTEMBER 1979
1749
20 43
ooo
18 00
1800
3.00 -6.00
3.00
11.00
16 42 Rate
052 0 00
Rate
5 a5 I
0 25 I nh i b itor
,
0 15
Substrate
Substrate
n
Y
\
600
,-" Y
525
fin
uu
PH
1 1 10
025
Flgure 2. Response surfaces showing the effect of different variables on the reaction rate of alkaline phosphatase: (a) enzyme activity varied from 3 to 18 U/L in 0.75 U/L steps, phosphate inhibitor concentration varied from 0 to 6 mmol/L in 0.3 mmol/L steps with pH and substrate concentration held constant at 10.1 and 3 mmol/L respectively; (b) enzyme activity varied from 3 to 18 U/L in 0.75 U/L steps, pH varied from 9 to 11 in 0.1 steps with substrate and inhibitor concentration held constant at 3 mmol/L and 0 mmol/L respectively; (c) substrate concentration varied from 0.25 to 5.25mmol/L in 0.25 mmol/L steps and phosphate inhibitor concentration varied from 0 to 6 mmol/L in 0.3 mmol/L steps with pH and enzyme activity held constant at 10.1 and 12 U/L respectively; (d) substrate concentration varied from 0.25 to 5.25 mmol/L in 0.25 mmol/L steps and pH varied from 9 to 11 in 0.1 steps with enzyme activity and inhibitor concentration held constant at 12 U / L on 0 mmol/L, respectively
The poor fit of the Michaelis function suggested a somewhat more complex model: K
k,
'2
v =
+p
+ s ~ E N +s - E~N + ~ 4tK'3
H'
+ K',
ko
EN S Z E N S - - - t EN
+P
K',
H'
+ EN-^
+
K5 k* s~$'-~s+EN-~
+
p
When substrate concentration and enzyme activity are held constant the rate equation reduces to the form, u=
c1 + cm+1 C,/[H+I 1 + C,[H+] + C,/[H+] -
(4)
where C1 = k&,, C2 = klK2K3,C3= k2K5/K4,C4 = K2K3,and C b = K5/K4. For the experimental conditions under which the response surface shown in Figure 2d was obtained and a t a constant substrate concentration of 5.85 mmol/L, the rate equation is given by 43.5 u =
1
+ 4.0 X
10+lOIH+] - 1.5 X lO-'O/[H+]
+ 0.84 x 10+lo[H+]+ 0.61 x 10-l0/[H+]
The fit to the data is shown in Figure 3b.
For substrate a t 0.25 mmol/L, the fit shown in Figure 3c is obtained and the rate equation is given by
(5)
27.9 1
+ 4.7 X lO+'O[H+]
-
1.7
X
10-'O/[H+] --
+ 0.84 x lo+lOIH+]+ 5.5 x 10-lO/'[H+]
(6)
This more complex model fits the data much better than the Michaelis F pH function. When all variables but two are held constant, only one response surface is required to represent the results. If a third variable is added to the factorial design, one response surface is required for every value of the third variable if a complete representation is required. As an example, consider the set of response surfaces for substrate concentration vs. pH, where the substrate concentration is varied from 0.25 to 5.85 mmol/L, p H is varied from 9 to 11,and we develop a surface for each of the inhibitor concentrations 0 , 2 , 4 , and 6 mmol/L. All other variables remain fixed a t 12 U / L enzyme activity, zero activator, 0.3 M 2-AMP buffer, and a temperature of 30.0 "C. The set of four surfaces in Figure 4 shows the effect of pH and inhibitor on the reaction kinetics. The data from runs a t 0,2, and 4 mmol/L inhibitor are of the expected accuracy. However, the response surface for 6 mmol/L of inhibitor is not smooth, which indicates experimental difficulties. Graphic-aided digital filtering can be used, if desired, to smooth the data, thus improving the estimate of any given point. In general such filtering is not necessary because most of the traditional plotting and calculational routines, such as the Lineweaver-Burk, Michaelis-Menten, and replots used to determine constants of interest, also smooth the data via nonlinear least-squares fitting routines. Figures 4a through c are typical of the results usually obtained from alkaline phosphatase experiments.
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ANALYTICAL CHEMISTRY, VOL. 51, NO. 11, SEPTEMBER 1979
.-C
._C E
._C E
E
W
E
5
E
C
.$
0
U W
PH
LT
m
PH
a
PH
LT
Figure 3. The fit of two pH models to data from Figure 2d: (a) Michaelis F- pH function fit to data at a substrate concentration of 5.85 mmol/L; (b) the more complex model fit to data at a substrate concentration of 5.85 mmol/L; and (c)the more complex model at a substrate concentration of 0.25 mmol/L Multiple Lineweaver-Burk plot taken from substrate concentration vs pH response surface. PH 91
I
-1
l5 9 00
Substrate
\// 0 2 5 11 10
'
Subsirate
025
11 10
I
I
I
11.0
I
2
3
6
PH
\/
I
7l
009
9 00
PH
I
I
5
3 c
4
2 1
E l
0 -5
-4
-2
-3
EO 025
11 10
025
1110
Figure 4. The set of four response surfaces required to represent substrate vs. pH at the four inhibitor concentrations: all four response surfaces were obtained by varying substrate concentration from 0.25 to 5.85 mmol/L in 0.4 mmol/L steps and pH from 9 to 11.1 in 0.15 teps with enzyme activity constant at 12 U/L, and phosphate inhibitor concentration at (a) 0.0 mmol/L; (b) 2 mmol/L; (c) 4 mmol/L; and (d) 6 mmol/L Traditional Replots. On-line graphics can use data obtained to develop response surfaces and generate the traditional ?lots and replots ( I , 7). As an example, the data from the substrate-vs.-pH response surface (Figure 2d) can be used to generate the multiple Lineweaver-Burk plot (Figure 5 ) . Using the Michaelis constant (K,),slope, l/Vo-axis intercept, the V,, values obtained from these and other multiple Lineweaver-Burk plots, the data can be replotted in many ways to analyze the data and identify possible reaction mechanisms. For example, the plot of v,, vs. enzyme activity shows that the data fit a simple first-order equation with an intercept at zero. These are the expected results if the instrumentation and analytical methodology is correct. Furthermore, they indicate that if an impurity is present it is not acting as a noncompetitive or uncompetitive inhibitor both of which result in a decrease in V,,,. The plot of K, vs. enzyme concentration (Figure 6) displays a slight positive slope indicating the presence of a trace impurity, which is acting as a c.jmpetitive inhibitor. All such K, replots show the same negative slope with our alkaline phosphatase. Similarly, the replots using the 1/ V , axis-intercept and slope values can be quickly generated. They also provide the investigator with additional quantitative information. When such plots can be obtained rapidly with on-line graphic techniques and quickly and quantitatively compared, the investigator has the capability to rapidly understand the
-1 0 1iSubstrate
1
Figure 5. Multiple Lineweaver-Burk plot using data taken from the response surface plot Figure 2d; pH is varied from 9 to 11 in steps of 0.1 pH
5.05
-
. . -
c"
5.00
c
*
>O
4.95 4.90
r r
$
4.85
X
y'
4.80 e
4.75
Alkaline phosphatase - U/Iiter
Figure 6. Replot using data obtained from Lineweaver-Burk plot of K, vs. enzyme activity with a positive slope indicating the presence of a trace impurity that acts as a competitive inhibitor experimental data. With such capability he can not only rapidly analyze results but also can quickly develop new design strategies and execate experiments to elucidate a complex chemical mechanism. Modeling. The various plotting and replotting routines discussed above are very useful techniques. They provide both analytical information and visual aids crucial to an understanding of enzyme kinetics and mechanisms. They also aid
ANALYTICAL CHEMISTRY, VOL. 51, NO. 11, SEPTEMBER 1979
1751
where L' = initial velocity (d[P]/dt or -d[S]/dt) a t the given substrate concentration and Eo = total enzyme concentration in all forms,
K , is also equivalent to the substrate concentration required for half maximal velocity. Note that Equation 7 simplifies to the well known Michaelis-Menten rate expression if kl >> k,. Under this condition the steady-state E S concentration is very nearly equivalent to the equilibrium concentration, herice k , becomes simply k l / k I . Equation 7 can be rewritten in terms of the reciprocal velocity
5.25
giving an equation of the general form y = C1X1
+ czx,
(10)
where the independent variables are: 24 29
The constants then become:
k,
Rate
5.25
0.25
Figure 7. Response surfaces for variable enzyme activity and substrate concentration with pH constant at 10.1, inhibitor at zero concentration, and temperature at 30.0 O C . (a) original data, (b) model-generated response surface
in the development and optimization of analytical methods. However, the power of our system is not limited by traditional techniques. The computational power of the computer can also be used to more completely model reactions. That is, the experimental data set can be used to determine the coefficients of a model that represents a proposed reaction mechanism. Data from the alkaline phosphatase-catalyzed reaction at constant p H was used to produce a response surface (Figure 7a). For this experiment, the pH was held constant a t 10.1, the 2-AMP buffer at 0.3 M, temperature a t 30.0 f 0.1 "C, and inhibitor a t zero concentration. The enzyme activity was varied from 3 to 18 U / L and the PNPP substrate concentration varied from 0.25 to 5.25 mmol/L. We assume that, a t a constant pH, the reaction can be represented by the simplified scheme kl
kP
E+ S=ES-E+P k-I
where E is enzyme, S is substrate, P is product, and ES is the enzyme-substrate complex. Assuming steady-state conditions for the enzyme-substrate complex ( E S ) ,Briggs and Haldane (8)derived a general velocity equation for this system,
k,[El,[Sl u =
c1
u=
, .
3.00
1 -
K m + [SI
c2
and K , = -
(12)
c1
Performing a linear regression analysis of the 21 X 21 (441) point data set (used to generate the enzyme vs. substrate response surface, Figure 7a) gives the following estimate of the parameters: k , = 1.5 and K , = 0.47. lJsing these estimates in the Briggs and Haldane equation gives:
1 46
18 00
=
(7)
1 . 5 [ ~ [SI 1~ 0.47 +
[K
(13)
Response surfaces can now be modeled with Equation 13, see Figure 7b. Residual Surface Technique. Given a model response surface, one can assess the model accuracy by comparing the computed data to the experimental data set. Conventional techniques using analytical functions such as the sum of the squares of the residuals (SSQ) are generally used. However, graphic techniques, which are better aids for determining the source of errors, can often be developed. The residual surface graphic technique makes use of the differences between the data generated by the model and the experimental data set. Figure 8a is a residual surface showing the difference between the experimental data set used to generate the enzyme and substrate response surface shown in Figure 7a and the model data shown in Figure 7b. If the model were an exact fit to the experimental data, the residual surface would be a flat plane with the only variations being those due to noise. Note, on the experimental response surface (Figure 7a) at a substrate concentration of 2.75 mmol/L and enzyme concentrations between 14.25 and 18.00 U / L the data appear to be slightly irregular. This irregularity can also be seen cm the residual surface (Figure 8a). One of the values of computer graphics is that the operator can magnify a display to any required power. Thus, residual surfaces can be magnified to show up small irregularities that result from instrument malfunction, model inadequacies, or inappropriate analytical techniques. Figure 8b is a tenfold magnification of the residual surface. Note that, in general, within the system analytical accuracy of 0.9% standard
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ANALYTICAL CHEMISTRY, VOL. 51, NO. 11, SEPTEMBER 1979 (a
(a)
1 1 52
Rate
4 , ,
I
Enzyme
5.25
Substrate
I
300
0 25
x
ib)
3.00
0.25
h
5.25
3.00
0.25
Figure 8. Residual surface plots showing the difference between experimental data and model generated data. The same resaual surface is shown at 1OX magnification in (b). The heavy lines represent the
region where the rate calculated from the experimental data exceeds the rate calculated from the model. The light lines are regions where the experimental rate is less deviation, most variations from the model can be attributed to noise. The small set of six data points at a substrate concentration of 2.75 mmol/L (described above) are not random. The error in these data points is 1.6 to 2.3% absolute and is probably due to an intermittent pumping irregularity. Modeling can be extended to other two-dimensional representations, such as enzyme vs. pH and substrate vs. inhibitor, to include all combinations of the variables taken two a t a time. For each combination there is a response surface generated from the experimental data set, a calculated response surface obtained from the theoretical model, and the residual surface which is the difference between the two surfaces. With these graphic representations one can, even with only two-dimensional modeling, quickly obtain a visual as well as an analytical representation of a three-, four-, or five-variable experiment. The residual surfaces quickly show the exact regions where the experimental design, instrumentation, or modeling are in error. These response and residual surfaces contain much of the data required for analytical evaluations as discussed earlier (1). Traditional plots, replots, and calculations are obtained wherever appropriate. When modeling efforts are expanded to four and more dimensions, the residual surface techniques can be used to examine selected regions of hyperspace and assess the quality of the experimental data and modeling design (9). Reaction Rate Surfaces. Response surfaces for chemical systems can be improved by incorporating additional quantitative references. A good set of references is contour
9.00
3.00
Figure 9. Response surfaces with contour lines at constant reaction
rates, showing three parameter conditions required to obtain selected rates lines connecting points of equivalent reaction rates (Figure 9a and b). Even with contouring, response-surface representations as presented on a graphics terminal can be deceiving and difficult to comprehend. Therefore, we made software provisions for rotation of the plots so they could be viewed from any angle. From the two different contoured response surfaces shown, some of the combinations of concentrations of the three variables (enzyme activity, pH, and substrate) required to produce a given reaction rate can be identified. However, over the concentration ranges studied, most of the conditions required to produce a given reaction rate cannot be identified. As an example, at a temperature of 30.0 O C , no inhibitor, and 0.3 M 2-AMP buffer, the three variables of enzyme, substrate, and hydrogen-ion concentrations can be systematically varied to produce sets of response surfaces, of which two are shown in Figure 9. As more variables are added, such as inhibitor concentration, the number of data points and response surfaces required to represent the system under study increases rapidly. For instance, in the surfaces shown there are 21 measurements in each variable. The enzyme vs. pH response surface in Figure 9b is for a substrate concentration of 2.75 mmol/L. If we want to also study reaction rates as a function of 21 substrate concentrations (0.25 to 5.25 mmol/L in steps of 0.25 mmol/L) as well as the 21 each of enzyme and hydrogen-ion concentrations, we need 2 1 response surfaces to represent the experimental data. Thus, it requires X”-2
1753
ANALYTICAL CHEMISTRY, VOL. 51, NO. 11, SEPTEMBER 1979 181
is1
R e a r t l o r r a t ? sL,rface
0 00 2 00
hlI3lt01=
Rate =
R e a c t l o i r a t e wr‘ace inhibitor = 0 00 Rate = 4 00
1b1 R e a m o n rate surface
( a/ Reaction rate surface Inhibitor = 2 00 Rate = 2 W ,/.;
inhi3itar =
-
2 00
, - />-
I 5 25
5.25
5 25
5 25
Substlate
Substrate
025 00
E
1.
11 00 /-
1800
9 00
1800 IC1
900
ReactIoP rate suriace
Inhibitor = 0 00 R e x 8 00
18 W
900
1800
900
I ~ Reaction ] r a t e surface Inhibitor = 2 00 R a t e = 8.W
~
5 2E 3 00
su3rrrate
3 00
Enzvme
1.00 PH
0 25
ElZ“me
I. 00
1800
900
Figure 10. Reaction rate surfaces for the three independent variables: hydrogen ion, enzyme activity, and substrate concentrations at reaction rates of (a) 2 pmol L-’ min-’, (b) 4 pmol L-’ min-’, and (c) 8 pmol L-’ min-’ all at zero inhibitor concentrations. These show the inter-relationships of three parameters on reaction rate
Figure 11. Three frames from one set of a time-varying reaction-rate surface display for the three independent variables: hydrogen ion, enzyme activity, and substrate concentrations at inhibitor concentrations of 2 rnmol/L-’ and reaction rates of (a) 2 mol/L-’/rnin-’, (b) 4 mol/ L-’/min-’, and (c) 8 mol/L-’/min-‘. The effect of product inhibition can easily be seen by comparing the surfaces at zero inhibRor concentration, Figure 10, with the above
response surfaces to completely represent the data set, where = number of levels studied in each variable and n = number of variables. In the above example, it would require 2 1 X 2 1 = 441 response surfaces to represent the data obtained from a study of the four variables, e.g., enzyme activity, substrate, hydrogen ion, and inhibitor at 21 different concentration levels of each variable. Although response surfaces can be valuable representations that yield much information, large sets of response surfaces are difficult to interpret. We are attempting to develop new graphic representations that will help us to better understand complex nonlinear systems. One such representation is the reaction rate surface (RRS),which is a surface of equal reaction rate (dependent variable) as a function of three independent variables. Consider the two response surfaces in Figure 9 where the enzyme axes of each response surface are adjacent and the reaction rate contours aligned to equal rate positions. One can then visualize a merging of the two surfaces along lines of equal reaction rates. The result is the RRS shown in Figure loa. Note, there is one RRS for each rate measured in the experiment. The reaction rate surfaces shown in Figures loa, lob, and 10c are for three different reaction rates: 2 , 4, and 8 pmol/L/s of p-nitrophenylate a t zero inhibitor concentration, and enzyme activity, hydrogen-ion, and substrate concentrations varied over the ranges shown. If sufficient data are available, or the appropriate models developed, a set of RRSs can be animated by a time varying display that systematically proceeds through the concentration range of the dependent variable encompassed by the experiment. When numerical models are used, any reaction rate of interest can be displayed. For the study of the alkaline phosphatase-catalyzed reaction, differences of 0.5 pmol/L/min of product in successively projected RRSs are appropriate for time varying displays. If each surface is displayed for 1.25 s, rates from 0.5 to 24 pmol/L/min can be shown in 1 min. A rate meter, which appears in the upper left hand corner of the CRT, gives
the actual rate for each reaction rate surface a t the time of display. Therefore, by taking advantage of dynamic display capabilities, the user has a representation for three independent and one dependent variable. Time-varying residual surfaces were also developed. They are, as before, the differences between the data set and the model and are run as a diagnostic program to find the regions where the model and data set are at variance. Often it will be desirable to obtain visual representations of the effects of four independent variables on a single dependent variable. While it is not possible to develop a stationary representation without the use of color graphics, it is possible to provide an effective display that takes advantage of the human power of “short-term” memory. For reaction-rate surfaces, the displays can be run as described above, where one full set of displays, a t a fixed fourth variable, are shown for rates proceeding from zero to the maximum encompassed by the experimental design. Each level of the fourth variable requires another full set of time-varying displays where the other three variables again proceed from zero to the maximum rate determined. As an example, suppose the fourth variable under study is inhibitor concentration and its effect was studied over the concentration range of 0 to 6 mmol/L, in increments of 1 mmol/L. For a complete representation, there would be seven sets of reaction-rate surfaces displayed, each set progressing from Lero to the maximum rate of product synthesis possible for the three independent variables. Three frames from the timevarying set of reaction rate surfaces a t 2 mmol/L inhibitor concentration are shown in Figure 11. For this representation, a second indexing register is displayed in the upper left hand corner of the CRT, showing the inhibitor concentration for each surface. Many reaction characteristics, in addition to the obvious ones such as overall rate reduction because of inhibitor, can be gleaned from these representations, directing our attention immediately to regions requiring more experimentation or extensive numerical analysis. One example is the influence of inhibitor and hydrogen-ion concentrations on the substrate
1800
X
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ANALYTICAL CHEMISTRY, VOL. 51, NO. 11, SEPTEMBER 1979
concentrations required to provide a given reaction rate. Reaction rate surfaces show that a t zero inhibitor concentration, the reaction rate in the lower pH region is greatly influenced by enzyme concentration; a small increase in enzyme concentration results in a large decrease in substrate concentration required to produce a given reaction rate. As inhibitor concentration is increased, the relationship between enzyme and substrate concentrations required to produce a given reaction rate becomes a smoother function more like the relationships in the higher p H region.
CONCLUSIONS Fully computer-automated apparatus yields accurate data that can be used in the rapid characterization of complex chemical systems. An ideal experimental apparatus would be one designed to not only rapidly acquire accurate data, but one that also provides tools to aid the experimenter to rapidly analyze the data and present it in forms that would be easily understood by other experimenters. We have built a computer-automated apparatus to study enzyme-catalyzed reactions with the goal of developing fully automated experimentation supported by real-time graphics and eventually self-adaptive control algorithms. In this paper we have described some of the graphic techniques being developed and used to interpret data obtained from complex reactions. Eventually these and other graphical techniques will be used in a real-time mode to interpret experimental data as it is obtained. We have shown the utility of a few traditional plots and Box-type response surfaces. However, automated experimentation produces too much data to be easily handled by traditional methods. Numerical modeling coupled with new graphic representations will often provide the means for rapid data interpretation. We have discussed graphical techniques and how they aid in advanced modeling efforts. Residual surfaces, which are the difference between the theoretical model and the data set displayed as a response surface, were shown to be useful in quickly locating areas of the experiment where the model, analytical method, or instrumentation are in error. These residual surfaces can be magnified as necessary to show deterministic errors. Reaction rate surfaces, which are surfaces a t constant reaction rate, can be used to interpret data from experiments with more than three variables. Using time varying graphics and taking advantage of human power of "short-term" memory as many as four independent and one dependent
variable can be displayed. Similarly, time-varying residual surfaces can be used to probe this space to isolate regions where modeling, analytical methodology, or instrumentation are in error. The techniques presented are useful for the rapid characterization of complex chemical systems. In addition, they are a necessary step in the development of self-adaptive experimentation where one begins with a minimal factorial design and a model. The computer-controlled apparatus gathers, in sequence, the next most needed data point required to prove or disprove the validity of the model, i.e., it is a closed-loop experiment until completion of the experiment or until the model is shown to be invalid. The experiment would revert to open-loop if the model were shown to be invalid. Graphic displays and relevant constants would be automatically presented to aid the experimenter to develop a new model and experimental design. Upon completion of such a system the experimenter would be able to request graphic displays a t any time during the closed-loop phase of the experiment. The combination of these tools and techniques aids the experimenter in the development of valid chemical models with ease and confidence.
ACKNOWLEDGMENT The authors acknowledge the helpful discussions with E. Clifford Toren, Jr., Eugene Cordes, Kelly O'Keefe, and Jack Campbell.
LITERATURE CITED Frazer, J. W.; Rigdon, L. P.; Brand, H. R.; Pomernacki, C. L. Anal. Chem. preceding paper in this issue. Weinberg, Gerald, M. "An Introduction to General Systems Thinking": John Wiley and Sons: New York. 1975. Cembrowski, G. S.;Larson, F. C.; Huntington, R. W.; Selliken, Jr., J. H.; Toren, Jr., E. C. Clin. Cbem. ( Winston-Salem, N.C.) 1978, 2 4 , 4, 555. Box, G. E. P. Biometrics 1954, 10, 6. Zeffren, E.; Hall, P. F. "The Study of Enzyme Mechanisms"; John Wiley and Sons: New York, 1973; Chapter 5. Dixon, M.; Webb, E. C. "Enzymes", 2nd ed.;Academic: New York. 1964 Chapter 4 . Segel, I. H. "Enzyme Kinetics", Wiley: New York, 1975. Briggs. G. E.; Haldane, J. B. S. Biocbern. J . 1925, 78, 246 and 253. Frazer, J. W.; Rigdon, L. P.: Brand, H. R . "Characterization of Chemical Systems: Alkaline Phosphatase Catalyzed Reaction 11" to be submitted to Anal. Chem. for publication.
RECEIVED for review January 29,1979. Accepted May 15,1979. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore Laboratory under contract No. W-7405-Eng-48 and partially funded by OBES.
