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Estimation of parameters in double uptake systems
ANALYTICAL
BIOCHEMISTRY
Estimation
68, 617-625
(1975)
of Parameters
in Double
H. T. HUTCHISON Division
AND
Uptake
Systems
B. HABER
Department of Human Biological Chemistry and Genetics, and of Comparative Neurobiology of the Marine Biomedical Institute, University of Texas Medical Branch, Gahaeston, Texas 77550
Received April 4, 1975: accepted May 20. I975 A method is presented for estimating the kinetic parameters describing the cellular uptake of a single molecular species mediated by two independent transport processes. Two cases are considered: first, uptake by two carrier-mediated transport systems, and second, uptake by a single carrier-mediated system and diffusion. Unweighted and weighted least-squares estimations are described: the unweighted estimation is noniterative and provides excellent first approximations for the iterative weighted estimation procedure.
Some neurotransmitters may be taken up by the cells of the nervous system by two processes acting on the same molecular species (1-7). Several approaches have been made toward the study of one of these processes in the presence of the other, among these has been the computer-aided estimation of the kinetic parameters describing each transport system (l-3). This approach takes advantage of the saturation of carrier-mediated transport systems which occurs at high (relative to the K, for that system) substrate concentrations. Cleland (8) has described a general iterative scheme for estimating kinetic parameters from this kind of data: the method described here is an extension of Cleland’s method. The transport equations have been rewritten to provide noniterative least-squares estimations of the kinetic parameters and reduce the number of variables iterated upon in the weighted least-squares procedure by two. METHODS Estimation of the Kinetic Parameters of Two Carrier-Mediated Transport Systems
The velocity of uptake (v) due to two carrier-mediated systems may be described by the equation
,,b!!&-+2
K, + s
vs K,+s’
transport
(1)
where s is the substrate concentration, V, and V, are the maximim velocities of uptake, and K, and K2 are the substrate concentrations for 617 Copyright @ 1975 by Academic Press, Inc. All rights of reproduction in any form reserved.
618
HUTCHISON
AND
HABER
which the uptake velocities for the respective systems are half-maximal (9,10), We will assign the subscripts 1 and 2 such that.K, < Z&. Equation (1) may be rewritten as ct! + py = 0,
(2)
where
cr=&+B s
and (6)
(8)
In general; experimental points will not satisfy Eq, (2) exactly. Therefore, we may define the residual f;- = ai + &Yi,
(10)
where yi is de.termined experimentally at the substrate concentrations si, and ai and@< are functions of si defined by Eqs. (4) and (5). Estimates of the constants A, B, C’, and D may be obtained which minimize the quantity R where
R = %vfi
= Cwi(ai + @&)” = Xwi [(~+B)+(r+~+DSi)yi]2.(11)
The terms wi are weights which will. be discussed below. If we assume that wi are independent of A’,B,C, and D we may take partial derivatives of R with respect to A,B,C, and D and obtain estimates of these parameters by solution of the matrix equation
KINETICS
OF
DOUBLE
UPTAKE
SYSTEMS
619
(12) ,
Estimates of the kinetic parameters K,, Kz, VI, and V, are then obtained from constants A,B,C, and D by solution of Eqs. (6)~(9). Since Eq. (12) minimizes the sum of squares ofA, the weights Wi are equal to the inverse variances of the residuals,& (8). The variances of yi may be assumed to be constant, or preferably, if replicate determinations of y are made for each S, the variances of yi may be calculated from the experimental data. The variances of the substrate concentrations, si, may be assumed to be zero; therefore, from Eq. (lo), Var V;:) = pi2 Var Cyi)
(13)
and
However, p depends not only on s but also on the parameters K, and K, which are to be estimated. Therefore, Eq. (12) cannot be solved without iteration unless an approximation for p is available. The units of p (Eq. 5) were chosen such that /? lies between 1 and 2 for substrate concentrations between K1 and K,; thus a good first approximation is to hold p constant. However, since p increases rapidly for values of s < K1 and s > Kz, points in this range will exert a disproportionate influence on the estimates of the kinetic parameters. A better approximation is to use rough estimates of K, and K2 to calculate p as a function of S. Unweighted estimates of K, and K2 obtained by solving Eq. (12) for constant p may be used for this purpose. Ideally, the approximate values of K1 and K, used to calculate p should be refined by successive iterations until they agree (within predetermined limits) with the newly obtained estimates of K1 and K2. Estimation
of the Uncertainty
An estimate of the inherent from the expression
of Kinetic
Parameters
variability
(+- ~+%A” n-p’
of the data may be obtained
(15)
620
HUTCHISON
AND
HABER
where n is the number of experimental points and p is the number of parameters estimated from the data (in this case p = 4). The variability of the parameters A ,B,C, and D is estimated from: Var (A) [
Cov MB) Var (B) -
cov (A$) Cov (B,C) Var (C) -
cov Cov cov Var
where Inv is the inverse of the symmetric variances of the kinetic parameters K1, &, calculated from the general expression Var [g(A,B,C,D)]
= (5)”
(AD) (B,D) = cr2 x Inv, (16) (C,D) (D) I
matrix in Eq. (12). The v,, and V, may then be
Var (A) + (s)
(s)
Cov(A,B)+
* * * . (17)
It should be noted that the variances obtained from Eq. (17) involve several successive approximations and therefore should be regarded only as rough estimates of the uncertainty of the kinetic parameters. Estimation of the Kinetic Parameters of One Carrier-Mediated Transport System and Difusion The second case we will consider is that of uptake due to a single carrier-mediated transport system and to diffusion of substrate into the cell. In this case Eq. (1) becomes
where K. is a diffusion constant. Equation (18) may be rewritten in the same form as Eq. (2) where now the constantsA,B,C, and D are defined as A = I’, - K,KD
(19)
B=-KD
(20)
C = K,
(21)
D = 0.
(22)
Since D = 0, Eq. (12) reduces to a 3 x 3 matrix equation. As in the previous case, this equation cannot be solved exactly without iteration since B is a function of K,, one of the parameters to be estimated. An unweighted fit may be obtained directly by holding B constant. This results in a disproportionate weighting of points for s < K1 since 8 lies between 1 and 2 for s > K, and increases rapidly for s < K,. The
KINETICS
OF
DOUBLE
UPTAKE
SYSTEMS
621
unweighted estimate of K,, obtained from Eq. (12) by holding p constant, may be used to calculate p as a function of s, thus refining the estimation of the kinetic parameters. Again, this iterative process should be continued until successive values of K, agree within predetermined limits. These procedures have been programmed for a Wang model 600-2 desk calculator and all of the estimations reported herein were done on this instrument. A similar program has been written in FORTRAN which allows a larger computer to be used to facilitate the iterative procedures. Measurement
of Neurotransmitter
Uptake
The origin and growth of the NB41 neuroblastoma cell line and the procedures for measurement of y-aminobutyric acid (GABA) uptake were as described previously (1). The RN22 cell line is a subclone of the peripheral neurinoma line described by Pfeiffer and Wechsler (11) and was kindly provided by Dr. S. E. Pfeiffer. The uptake of 5hydroxytryptamine (5HT) was as described previously (1) except that Hanks’ BSS was used in the incubation medium in place of a modified Earle’s BSS. [3H] 5-HT was obtained from New England Nuclear (Boston, Mass.). RESULTS The Uptake
of S-Hydroxytryptamine
(5-HT)
by RN22 Neurinoma
Cells
The initial rates of uptake of 5-HT by RN22 cells were determined at several substrate concentrations and plotted as described by Hofstee (9). A curved line was obtained (Fig. 1) suggesting that 5-HT uptake in these cells is mediated by more than one mechanism ( 10,lL). The methods described above were used to estimate the kinetic parameters of a double carrier-mediated transport system (Eq. 1). Table 1 shows that the unweighted estimates differ from the corresponding weighted estimates by 10% or less. Furthermore, using the unweighted estimates of K, and K2 as initial approximations, the first weighted estimate and the final weighted estimate of each of the parameters differ by at most 1.3%. The curve defined by the unweighted estimate of the kinetic parameters is indicated by the solid line in Fig. 1; the curve defined by the weighted estimate overlies this one almost exactly and therefore is not drawn. The “goodness of fit” of these curves to the experimental points was estimated from the formula unexplained
variance =
E(Y-j)’ Iz(Y --Yj2’
where y and $ are the observed and calculated
values of y for each
622
HUTCHISON
AND
HABER
4c
9 20
0
500
1000
1500
V(pmoles/mg/min)
FIG. 1. RN22 neurinoma cells were labeled for 5 min with 0.2 &i [“HI S-HT at total S-HT concentrations of 0.3-1000 PM. Each point indicates the mean and standard error of three or four determinations: a total of 29 determinations is represented. The solid line is a least-squares curve fit for two carrier-mediated transport systems.
UNWEICHTED
AND
TABLE 1 WEIGHTED ESTIMATES OF THE KINETIC OF 5-HT UPTAKE IN RN22 CELLS" Estimated
Iteration Unweighted Weighted 1 2 3 4 5 6
K,
PARAMETERS
parameters
VI
K,
V*
0.6727
+- 0.0602
45.18
k 2.80
788.5
2 222.8
3062 + 803
0.6344 0.6354 0.6355 0.6355 0.6355 0.6355
f k 2 + k k
43.88 43.90 43.91 43.90 43.91 43.91
k ? k + + t
882.0 869.5 871.4 871.1 871.2 871.2
+ e 2 2 k 2
3408 3365 3371 3370 3370 3370
0.0733 0.0729 0.0729 0.0729 0.0729 0.0729
(2 The unweighted estimates Methods from the data of Fig. using the unweighted estimates are listed.
3.02 3.92 3.02 3.02 3.02 3.02
341.7 325.6 328.1 327.7 327.7 327.7
-c 1247 Z!T 1188 -+ 1197 2 1196 t 11% k 11%
of the kinetic parameters were obtained as described in 1. The final weighted estimates were obtained by iteration of K, and Kz as starting values; six successive iterations
KINETICS
OF
oc 0
DOUBLE
UPTAKE
20
623
SYSTEMS
40
M
v tpmoleslmglminl FIG. 2. NB41 neuroblastoma cells were labeled for 1 min with I or 2 PCi [3H] GABA at total GABA concentrations of 0.1-10 PM. Each point indicates the mean and standard error of four or five determinations: a total of 39 determinations is represented. The solid line is a least-squares curve fit for one carrier-mediated transport system and diffusion.
experimental point and j is the mean of the observed values of y. The unexplained variance of the unweighted curve-fit is 0.00059 and of the weighted fit is 0.00076. The uptake of y-aminobutyric acid (GABA) by NB41 neuroblastoma cells in culture is illustrated in Fig . 1. The solid line in this figure represents the unweighted least-squares fit of the single carrier-mediated transport and diffusion model to the experimental points; the kinetic parameters of this fit are listed in Table 2. Table 2 also lists several iterations done to obtain a weighted estimate of these parameters. The difference between the weighted and unweighted estimates is less than 10% for any of the parameters. It is also apparent that the unweighted fit provides an excellent estimate of K, needed as a starting value for deTABLE UNWEIGHTED
AND
WEIGHTED OF GABA
Iteration Unweighted Weighted 1 2 3 4
2
ESTIMATES OF THE KINETIC UPTAKE IN NB41 CELLS
Kl
PARAMETERS
V,
KD
0.1495
2 0.0349
2.051
t 0.354
3.270
+ 0.197
0.1378 0.1384 0.1384 0.1384
‘2 t %
1.974 1.979 1.978 1.978
2 + t +
3.276 3.275 3.275 3.275
” 0.120 2 0.124 rt_ 0.124 k 0.124
0.0352 0.0353 0.0353 0.0353
0.335 0.337 0.337 0.337
n The unweighted estimates of the kinetic parameters were obtained as described in Methods from the data of Fig. 2. The final weighted estimates were obtained by iteration using the unweighted estimate of K, as a starting value; four successive iterations arc listed.
624
HUTCHISON
AND
HABER
termining the weighted fit. As in the previous case the unweighted and weighted curves overlie each other almost exactly; the unexplained variances of these fits are 0.0052 and 0.0055, respectively. A reasonable fit of the data in Fig. 2 was also obtained with the double carrier-mediated transport model. However, this model was rendered less likely by showing experimentally that no apparent saturation of the low-afhnity uptake of GABA occurred at GABA concentrations as high as 10 mM (1). DISCUSSION
We have described a method for estimating the kinetic parameters of two models describing the uptake of a single molecular species by two different transport processes. Unweighted least-squares estimates of these parameters are obtained directly whereas weighted estimates require iteration. This iteration usually closes in just a few cycles for two reasons. First, the iteration involves only one or two parameters, namely the value of K, in the case of Eq. (18) or the values of K, and K, in the case of Eq. (l), and second, the unweighted estimates are usually good approximations to the final weighted estimates. Generally, we have found that the unweighted and weighted estimates agree rather closely unless there are large experimental variations in the data, the substrate concentrations are poorly chosen, or a plot of the data clearly differs in shape from the curve being fitted. The standard errors of the kinetic parameters may be used as indicators of the validity of the estimation. The values of the standard errors regarded as acceptable will depend upon the application; some problems require more accuracy than others. Care should be taken in using the standard errors to test the significance of differences between estimates of a single parameter. For example, K, (or K2) may vary over a wider range without materially changing the goodness of fit if I/, (or V,) also varies proportionally and in the same direction (8). This can be seen by comparing the weighted and unweighted estimates in Tables 1 and 2 although much more dramatic differences are seen with less adequate data. Tables 1 and 2 also illustrate that the estimates of the standard errors are influenced more by weighting the equations than are the estimates of the parameters themselves. The methods described here are readily adapted to a programmable desk calculator and require only a few seconds of computation time (the unweighted estimation illustrated in Fig. 1 required 17 set of computation time after the data was entered). ACKNOWLEDGMENTS We wish Community
to thank Mr. Harvey Bunce of the Department Health at The University of Texas Medical
of Preventive Medicine Branch at Galveston for
and many
KINETICS
OF DOUBLE
UPTAKE
SYSTEMS
625
helpful discussions. This work was supported by USPHS Grants, DHEW AA 0027103, NS 11354, and NS 11255, Grants from the Robert A. Welch Foundation (H-504 and H-536), and supporting Grants from the Lanier and Moody Foundations.
REFERENCES 1. Hutchison, H. T., Werrbach, K., Vance, C., and Haber, B. (1974) Brain Res. 66, 265-274. 2. Yamamura, H I., and Snyder, S. H. (1972) Science 178, 626-628. 3. Logan, W. J. and Snyder, S. H. (1972) Bruin Res. 42, 413-431. 4. Wofsey, A., Kuhar, M. J., and Snyder, S. H. (1971) Proc. Nat. Acad. Sci. 68, 1102-l 106. 5. Iversen, L. L. and Johnston, G. A. R. (197 1) J. Neurochem. 18, 1939-1950. 6. Johnston, G. A. R. and Iversen, L. L. (1971)J. Neurochem. 18, 1951-1961. 7. Richelson, E. and Thompson, E. J. (1973) Nature New Biol. 241, 201-204. 8. Cleland, W. W. (1967) Advan. Enzymol. 29, l-32. 9. Hofstee, B. H. J. (1952) Science 116, 329-331. 10. Dixon, M. and Webb, E. C. (1964) Enzymes, pp. 68-70, Academic Press, New York. 11. Pfeiffer, S. E. and Wechsler, W. (1972) Proc. Nut. Acad. Sci. 69, 2885-2889. 12. Borst-Pauwels, G. W. F. H. (1973) J. Theor. Biol. 40, 19-3 1.
BIOCHEMISTRY
Estimation
68, 617-625
(1975)
of Parameters
in Double
H. T. HUTCHISON Division
AND
Uptake
Systems
B. HABER
Department of Human Biological Chemistry and Genetics, and of Comparative Neurobiology of the Marine Biomedical Institute, University of Texas Medical Branch, Gahaeston, Texas 77550
Received April 4, 1975: accepted May 20. I975 A method is presented for estimating the kinetic parameters describing the cellular uptake of a single molecular species mediated by two independent transport processes. Two cases are considered: first, uptake by two carrier-mediated transport systems, and second, uptake by a single carrier-mediated system and diffusion. Unweighted and weighted least-squares estimations are described: the unweighted estimation is noniterative and provides excellent first approximations for the iterative weighted estimation procedure.
Some neurotransmitters may be taken up by the cells of the nervous system by two processes acting on the same molecular species (1-7). Several approaches have been made toward the study of one of these processes in the presence of the other, among these has been the computer-aided estimation of the kinetic parameters describing each transport system (l-3). This approach takes advantage of the saturation of carrier-mediated transport systems which occurs at high (relative to the K, for that system) substrate concentrations. Cleland (8) has described a general iterative scheme for estimating kinetic parameters from this kind of data: the method described here is an extension of Cleland’s method. The transport equations have been rewritten to provide noniterative least-squares estimations of the kinetic parameters and reduce the number of variables iterated upon in the weighted least-squares procedure by two. METHODS Estimation of the Kinetic Parameters of Two Carrier-Mediated Transport Systems
The velocity of uptake (v) due to two carrier-mediated systems may be described by the equation
,,b!!&-+2
K, + s
vs K,+s’
transport
(1)
where s is the substrate concentration, V, and V, are the maximim velocities of uptake, and K, and K2 are the substrate concentrations for 617 Copyright @ 1975 by Academic Press, Inc. All rights of reproduction in any form reserved.
618
HUTCHISON
AND
HABER
which the uptake velocities for the respective systems are half-maximal (9,10), We will assign the subscripts 1 and 2 such that.K, < Z&. Equation (1) may be rewritten as ct! + py = 0,
(2)
where
cr=&+B s
and (6)
(8)
In general; experimental points will not satisfy Eq, (2) exactly. Therefore, we may define the residual f;- = ai + &Yi,
(10)
where yi is de.termined experimentally at the substrate concentrations si, and ai and@< are functions of si defined by Eqs. (4) and (5). Estimates of the constants A, B, C’, and D may be obtained which minimize the quantity R where
R = %vfi
= Cwi(ai + @&)” = Xwi [(~+B)+(r+~+DSi)yi]2.(11)
The terms wi are weights which will. be discussed below. If we assume that wi are independent of A’,B,C, and D we may take partial derivatives of R with respect to A,B,C, and D and obtain estimates of these parameters by solution of the matrix equation
KINETICS
OF
DOUBLE
UPTAKE
SYSTEMS
619
(12) ,
Estimates of the kinetic parameters K,, Kz, VI, and V, are then obtained from constants A,B,C, and D by solution of Eqs. (6)~(9). Since Eq. (12) minimizes the sum of squares ofA, the weights Wi are equal to the inverse variances of the residuals,& (8). The variances of yi may be assumed to be constant, or preferably, if replicate determinations of y are made for each S, the variances of yi may be calculated from the experimental data. The variances of the substrate concentrations, si, may be assumed to be zero; therefore, from Eq. (lo), Var V;:) = pi2 Var Cyi)
(13)
and
However, p depends not only on s but also on the parameters K, and K, which are to be estimated. Therefore, Eq. (12) cannot be solved without iteration unless an approximation for p is available. The units of p (Eq. 5) were chosen such that /? lies between 1 and 2 for substrate concentrations between K1 and K,; thus a good first approximation is to hold p constant. However, since p increases rapidly for values of s < K1 and s > Kz, points in this range will exert a disproportionate influence on the estimates of the kinetic parameters. A better approximation is to use rough estimates of K, and K2 to calculate p as a function of S. Unweighted estimates of K, and K2 obtained by solving Eq. (12) for constant p may be used for this purpose. Ideally, the approximate values of K1 and K, used to calculate p should be refined by successive iterations until they agree (within predetermined limits) with the newly obtained estimates of K1 and K2. Estimation
of the Uncertainty
An estimate of the inherent from the expression
of Kinetic
Parameters
variability
(+- ~+%A” n-p’
of the data may be obtained
(15)
620
HUTCHISON
AND
HABER
where n is the number of experimental points and p is the number of parameters estimated from the data (in this case p = 4). The variability of the parameters A ,B,C, and D is estimated from: Var (A) [
Cov MB) Var (B) -
cov (A$) Cov (B,C) Var (C) -
cov Cov cov Var
where Inv is the inverse of the symmetric variances of the kinetic parameters K1, &, calculated from the general expression Var [g(A,B,C,D)]
= (5)”
(AD) (B,D) = cr2 x Inv, (16) (C,D) (D) I
matrix in Eq. (12). The v,, and V, may then be
Var (A) + (s)
(s)
Cov(A,B)+
* * * . (17)
It should be noted that the variances obtained from Eq. (17) involve several successive approximations and therefore should be regarded only as rough estimates of the uncertainty of the kinetic parameters. Estimation of the Kinetic Parameters of One Carrier-Mediated Transport System and Difusion The second case we will consider is that of uptake due to a single carrier-mediated transport system and to diffusion of substrate into the cell. In this case Eq. (1) becomes
where K. is a diffusion constant. Equation (18) may be rewritten in the same form as Eq. (2) where now the constantsA,B,C, and D are defined as A = I’, - K,KD
(19)
B=-KD
(20)
C = K,
(21)
D = 0.
(22)
Since D = 0, Eq. (12) reduces to a 3 x 3 matrix equation. As in the previous case, this equation cannot be solved exactly without iteration since B is a function of K,, one of the parameters to be estimated. An unweighted fit may be obtained directly by holding B constant. This results in a disproportionate weighting of points for s < K1 since 8 lies between 1 and 2 for s > K, and increases rapidly for s < K,. The
KINETICS
OF
DOUBLE
UPTAKE
SYSTEMS
621
unweighted estimate of K,, obtained from Eq. (12) by holding p constant, may be used to calculate p as a function of s, thus refining the estimation of the kinetic parameters. Again, this iterative process should be continued until successive values of K, agree within predetermined limits. These procedures have been programmed for a Wang model 600-2 desk calculator and all of the estimations reported herein were done on this instrument. A similar program has been written in FORTRAN which allows a larger computer to be used to facilitate the iterative procedures. Measurement
of Neurotransmitter
Uptake
The origin and growth of the NB41 neuroblastoma cell line and the procedures for measurement of y-aminobutyric acid (GABA) uptake were as described previously (1). The RN22 cell line is a subclone of the peripheral neurinoma line described by Pfeiffer and Wechsler (11) and was kindly provided by Dr. S. E. Pfeiffer. The uptake of 5hydroxytryptamine (5HT) was as described previously (1) except that Hanks’ BSS was used in the incubation medium in place of a modified Earle’s BSS. [3H] 5-HT was obtained from New England Nuclear (Boston, Mass.). RESULTS The Uptake
of S-Hydroxytryptamine
(5-HT)
by RN22 Neurinoma
Cells
The initial rates of uptake of 5-HT by RN22 cells were determined at several substrate concentrations and plotted as described by Hofstee (9). A curved line was obtained (Fig. 1) suggesting that 5-HT uptake in these cells is mediated by more than one mechanism ( 10,lL). The methods described above were used to estimate the kinetic parameters of a double carrier-mediated transport system (Eq. 1). Table 1 shows that the unweighted estimates differ from the corresponding weighted estimates by 10% or less. Furthermore, using the unweighted estimates of K, and K2 as initial approximations, the first weighted estimate and the final weighted estimate of each of the parameters differ by at most 1.3%. The curve defined by the unweighted estimate of the kinetic parameters is indicated by the solid line in Fig. 1; the curve defined by the weighted estimate overlies this one almost exactly and therefore is not drawn. The “goodness of fit” of these curves to the experimental points was estimated from the formula unexplained
variance =
E(Y-j)’ Iz(Y --Yj2’
where y and $ are the observed and calculated
values of y for each
622
HUTCHISON
AND
HABER
4c
9 20
0
500
1000
1500
V(pmoles/mg/min)
FIG. 1. RN22 neurinoma cells were labeled for 5 min with 0.2 &i [“HI S-HT at total S-HT concentrations of 0.3-1000 PM. Each point indicates the mean and standard error of three or four determinations: a total of 29 determinations is represented. The solid line is a least-squares curve fit for two carrier-mediated transport systems.
UNWEICHTED
AND
TABLE 1 WEIGHTED ESTIMATES OF THE KINETIC OF 5-HT UPTAKE IN RN22 CELLS" Estimated
Iteration Unweighted Weighted 1 2 3 4 5 6
K,
PARAMETERS
parameters
VI
K,
V*
0.6727
+- 0.0602
45.18
k 2.80
788.5
2 222.8
3062 + 803
0.6344 0.6354 0.6355 0.6355 0.6355 0.6355
f k 2 + k k
43.88 43.90 43.91 43.90 43.91 43.91
k ? k + + t
882.0 869.5 871.4 871.1 871.2 871.2
+ e 2 2 k 2
3408 3365 3371 3370 3370 3370
0.0733 0.0729 0.0729 0.0729 0.0729 0.0729
(2 The unweighted estimates Methods from the data of Fig. using the unweighted estimates are listed.
3.02 3.92 3.02 3.02 3.02 3.02
341.7 325.6 328.1 327.7 327.7 327.7
-c 1247 Z!T 1188 -+ 1197 2 1196 t 11% k 11%
of the kinetic parameters were obtained as described in 1. The final weighted estimates were obtained by iteration of K, and Kz as starting values; six successive iterations
KINETICS
OF
oc 0
DOUBLE
UPTAKE
20
623
SYSTEMS
40
M
v tpmoleslmglminl FIG. 2. NB41 neuroblastoma cells were labeled for 1 min with I or 2 PCi [3H] GABA at total GABA concentrations of 0.1-10 PM. Each point indicates the mean and standard error of four or five determinations: a total of 39 determinations is represented. The solid line is a least-squares curve fit for one carrier-mediated transport system and diffusion.
experimental point and j is the mean of the observed values of y. The unexplained variance of the unweighted curve-fit is 0.00059 and of the weighted fit is 0.00076. The uptake of y-aminobutyric acid (GABA) by NB41 neuroblastoma cells in culture is illustrated in Fig . 1. The solid line in this figure represents the unweighted least-squares fit of the single carrier-mediated transport and diffusion model to the experimental points; the kinetic parameters of this fit are listed in Table 2. Table 2 also lists several iterations done to obtain a weighted estimate of these parameters. The difference between the weighted and unweighted estimates is less than 10% for any of the parameters. It is also apparent that the unweighted fit provides an excellent estimate of K, needed as a starting value for deTABLE UNWEIGHTED
AND
WEIGHTED OF GABA
Iteration Unweighted Weighted 1 2 3 4
2
ESTIMATES OF THE KINETIC UPTAKE IN NB41 CELLS
Kl
PARAMETERS
V,
KD
0.1495
2 0.0349
2.051
t 0.354
3.270
+ 0.197
0.1378 0.1384 0.1384 0.1384
‘2 t %
1.974 1.979 1.978 1.978
2 + t +
3.276 3.275 3.275 3.275
” 0.120 2 0.124 rt_ 0.124 k 0.124
0.0352 0.0353 0.0353 0.0353
0.335 0.337 0.337 0.337
n The unweighted estimates of the kinetic parameters were obtained as described in Methods from the data of Fig. 2. The final weighted estimates were obtained by iteration using the unweighted estimate of K, as a starting value; four successive iterations arc listed.
624
HUTCHISON
AND
HABER
termining the weighted fit. As in the previous case the unweighted and weighted curves overlie each other almost exactly; the unexplained variances of these fits are 0.0052 and 0.0055, respectively. A reasonable fit of the data in Fig. 2 was also obtained with the double carrier-mediated transport model. However, this model was rendered less likely by showing experimentally that no apparent saturation of the low-afhnity uptake of GABA occurred at GABA concentrations as high as 10 mM (1). DISCUSSION
We have described a method for estimating the kinetic parameters of two models describing the uptake of a single molecular species by two different transport processes. Unweighted least-squares estimates of these parameters are obtained directly whereas weighted estimates require iteration. This iteration usually closes in just a few cycles for two reasons. First, the iteration involves only one or two parameters, namely the value of K, in the case of Eq. (18) or the values of K, and K, in the case of Eq. (l), and second, the unweighted estimates are usually good approximations to the final weighted estimates. Generally, we have found that the unweighted and weighted estimates agree rather closely unless there are large experimental variations in the data, the substrate concentrations are poorly chosen, or a plot of the data clearly differs in shape from the curve being fitted. The standard errors of the kinetic parameters may be used as indicators of the validity of the estimation. The values of the standard errors regarded as acceptable will depend upon the application; some problems require more accuracy than others. Care should be taken in using the standard errors to test the significance of differences between estimates of a single parameter. For example, K, (or K2) may vary over a wider range without materially changing the goodness of fit if I/, (or V,) also varies proportionally and in the same direction (8). This can be seen by comparing the weighted and unweighted estimates in Tables 1 and 2 although much more dramatic differences are seen with less adequate data. Tables 1 and 2 also illustrate that the estimates of the standard errors are influenced more by weighting the equations than are the estimates of the parameters themselves. The methods described here are readily adapted to a programmable desk calculator and require only a few seconds of computation time (the unweighted estimation illustrated in Fig. 1 required 17 set of computation time after the data was entered). ACKNOWLEDGMENTS We wish Community
to thank Mr. Harvey Bunce of the Department Health at The University of Texas Medical
of Preventive Medicine Branch at Galveston for
and many
KINETICS
OF DOUBLE
UPTAKE
SYSTEMS
625
helpful discussions. This work was supported by USPHS Grants, DHEW AA 0027103, NS 11354, and NS 11255, Grants from the Robert A. Welch Foundation (H-504 and H-536), and supporting Grants from the Lanier and Moody Foundations.
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