J. theor. Biol. (1976) 63, 99-110
Kinetics of Multiple Transport Systems for Amino Acids in the Brain F. DENIZEAU, J. WYSE AND T. L. SOURKES Departments of Biochemistry and Psychiatry (F. D. and T.L.S.), Faculty of Medicine, McGill University, and Royal Victoria Hospital Montreal, Quebec, Canada (Received27 May 1975,and in revisedform 20 January 1976)
Two models of multiple transport systems (one comprising diffusion plus a Michaelis-Menten component, and the other comprising two Michaelis-Menten components) were tested for their fit to a series of experimental data. A statistical method (linearization technique) wasused to get estimates for the appropriate parameters of each model. In most casesthe data could be fitted to either model. Techniques for discriminating between the two models were sought and are discussed.
1. IntroductioIl
Transport of amino acids across membranes may occur by means of more than one agency and, when two or more transport mechanisms function simultaneously as happens in a wide variety of organisms, kinetic analysis of the phenomenon presents special problems. Christensen (1969) has recently discussed this subject. In brain tissue high and low affinity systems have been described for various amines and amino acids thought to act as neurotransmitters. High aflinity transport mechanisms have been postulated to play a role in terminating the action of such neurotransmitters after their release from nerve endings (see review by Iversen, 1974). Furthermore, high and low affinity systems have been described for tryptophan (Belin & Pujol, 1973 ; Knapp & Mandell, 1973; Baumann, Bourgoin, Penda, Glowinski &z Hamon, 1974), the precursor of serotonin. In this case, the high afhnity system has been associated with the control of tryptophan concentrations within the serotonin-containing neurons; this in turn affects the rate of serotonin synthesis (Tagliamonte, Biggio, Vargiu & Gessa, 1973 ; Fernstrom & Wurtman, 1971). In addition to these proposals about tryptophan transport 99
100
F.
DENIZEAU,
J.
WYSE
AND
T.
L.
SOURKES
in the brain, a model comprising an active component, conforming to Michaelis-Menten kinetics (Michaelis & Menten, 1913) and a diffusional component has been adduced by Kiely & Sourkes (1972). In most of these reports the selection of a particular model of multiple transport, namely one comprising high and low affinity components, has been made a priori by formal application of the method of Lineweaver & Burk (1934) to the data. The aim of this paper is twofold: first to study in detail two common transport models by utilizing well established mathematical techniques. In most kinetic studies such techniques are applied, if at all, only empirically, and it was now thought desirable to examine the implications of these techniques for the theory of transport phenomena. Secondly, we wished to explore the bases for the choice between the two models for a given set of observational data. 2. Theoretical
Considerations
Carrier transport systems are characterized by kinetic constants analogous to those appropriate to the analysis of enzyme kinetic data. On the basis of the Michaelis-Menten hypothesis the initial velocity of uptake or transport of an amino acid (or other substance) is given by the expression 11= vs(K+s)-l (1) in which V = the theoretical maximal velocity attained at saturation of the transport carrier with substrate; K has the quality of the Michaelis constant, and characterizes the transport process; and S is the initial concentration of the substrate in the external medium. In transport studies, a non-linear relationship in the plot of 0-l versus S-’ provides evidence that more than one transport system is operating. The simplest models that can then be postulated are of the forms: Model I: u = --!?t K,+S
+ w K,+S
(2)
Model II:
It can be demonstrated
that for Model I, 1
limv-‘(S-l-O)=--v1+
v*
MULTIPLE
TRANSPORT
101
MODELS
and for Model II, lim v-Q-’ -+ 0) = 0 (5) Thus, in the case of model II the plot of u-l versus S” curves downwards as S becomes very great, and ultimately the line passes through the origin. For model I the intercept on the ordinate is equal to (Vl + V,)-‘. In practice, for observed data the experimental error makes it extremely difticult to decide between the two models on the basis of extrapolation of the curve to the ordinate passing through S- ’ = 0. Both models I and II are non-linear, and convenient methods have been sought to estimate the respective sets of parameters by the method of least squares in order to obtain the best possible fit to the data. The application of these methods, with derivation of the appropriate equations and of the programmes in Basic to be used for the two models, is described in Appendix A. 3. Applications (A) STATISTICALLY
FITTED
PARAMETERS
VERSUS
ESTIMATES
Previously Kiely & Sourkes (1972) showed that transport of tryptophan into slices of rat brain cortex occurs by complex kinetics that can be reduced to components involving carrier movement and diffusion. The graphical methods used in that paper to calculate the parameters were applied in the present work to provide the initial estimates (see Appendix A). The data have now been revaluated, and the parameters are presented in Table 1. The analysis of the data according to model II shows that the estimate of K made by graphic means is lower than the value obtained by the least squares fit. Moreover, the value of V estimated by computer programme is only about half as large as previously reported. The diffusion constant is essentially the same by both methods. Although the EMS (error mean square) was somewhat smaller for model I, that model yielded very high values for Kz and V,, and these were considered to be without physiological significance. 1 Tryptophan transport into rat-brain cortex t (Model IZ) TABLE
Parameters Estimates
made
K mm01 l-1
by:
0.829
Graphical method? Least squares fitting t Data
of Kiely
1.15 & Sourkes
(1972).
V mm01
1-l
& min-’
0402 O-203
min-a
0.065 0.06
102
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(B) EXF’FRIMENTAL
J.
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AND BETWEEN
T.
L.
SOURKES
MODELS
I AND
II
In order to determine the contribution of carrier transport to the observed rate of movement of a substance into intracellular water most authors test the effect of reduced temperature on the process. Generally, carrier transport then appears to be diminished (Vis greatly reduced), but not totally inhibited. If the residual rate is attributed to diffusion, then K, is overestimated. When the erroneously estimated contribution of diffusion is subtracted from the uptake values observed at 37”C, the remaining values will result in incorrect estimates of K and V, or lead to an interpretation of the presence of two carrier systems, even if this is not the case. Data for the transport of tryptophan in rat hypothalamus at 37°C have been collected in this laboratory; the methodology involved in these measurements will be described elsewhere (Denizeau & Sourkes, 1975). When the data were subjected to a reciprocal plot, it appeared evident that a multiple transport system was operating. The data for the initial rate of uptake of 14C-tryptophan ( u) , tr eated as a function of the concentration of the amino acid in the external medium (O-05 to 1 mM) at 37°C conformed well to both models I and II. These values for the various constants were obtained by an iterative computer programme (Appendix A) and these are presented in Table 2. It can be seen that the EMS would favour model II slightly, although a choice cannot be made purely on this basis. Further experiments were then carried out with this tissue at 4”, and also at 37°C in the presence of metabolic inhibitors. The data for transport at 4°C were fitted by the least squares method to models I and II, and also to model III, in which v = &S
(6)
There was a poor fit to model III, whereas model II appeared to fit very satisfactorily. With the values then obtained for K, K, and V, it was calculated that at 4”C, for S = 0.01 mM, the contribution of active uptake is about six times as great as that of diffusion. Hence, it cannot be assumed that carrier transport has been abolished at this temperature. Slices of hypothalamic tissue were then incubated with iodoacetate (0.5 mu) and dinitrophenol (0.1 mM) at 37°C. These metabolic inhibitors block the production of ATP in glycolysis and of oxidative phosphorylation, respectively. Under these conditions, active uptake is very much reduced (K,” greatly increased) but does not seem to be totally negligible. Since it appeared almost impossible to eliminate entirely the contribution of carrier transport by means of conventional methods, the first experiment was repeated with a wider range of substrate concentrations in the medium.
-
-
1.09
0.16
-
2.14 15560
mM
Ka
0.01 0.54
KI -__
- Signifies that values did not converge. t Unpublished data of Denizeau & Sourkes. $ Error mean square X log. 5 Iodoacetate and dinitrophenol (see text).
0~10-10mM
Incubation at 37°C Concentration range: 0.0% 1 mM O*lO-10 rnM Incubation at 4°C Concentration range: 0.05- 1 nw Incubation at 37°C in presence of inhibitors9 Concentration range: 0.05- 1 mM
Conditions
-
-
0.02
0.08 0.92
-
-
oa
140 11.80
Model I Vl VZ mM min-1
-
-
0.07
2.51 112.30
EMSI
1.76 1.75
0.04
O-06 0.16
mKM
0.07 0436
0.03
0.36 0.09
0.25 0.25
0.01
0.18 0.38
V mM min-’
Model II
t
Kd min-’
TABLE 2 Tryptophan transport: slices of rat hypothalamus
0.54 6.13
0.06
244 20.70
EMS
0.22 OGJ
0.04
-
Kr,
0.37 10.11
0.08
-
EMS
Model III
104
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T.
L.
SOURKES
Our reasoning in doing so was as follows. If the highest substrate concentration used is significantly smaller than K2, then (7) where & has the character of a diffusion constant, and equations (2) and (3) become identical. Thus, with low substrate concentrations it is impossible to distinguish between models I and II. For the new control experiment the highest concentration of tryptophan used (10 mM) was well above the value estimated for & in Table 2 (i.e. 2.14 mM) after the first analysis of the data. When estimates of the parameters for models I and II are then made on the basis of this new range of higher concentrations, the following possibilities emerge. If model I is appropriate to the transport mechanism, then K, and V, should remain the same as with the other range of concentrations tested. On the other hand, if model II holds true the values estimated for K,,,, V,,, and Kd in both sets of experiments should be very close; furthermore, when the attempt is made to fit the data to model I the estimates of the parameters, especially those for Kz and V,, should keep on increasing, paripassu, as the upper range of concentration is extended. The figures obtained in this example are shown in Table 2. It can be seen that K2 increased substantially when high substrate concentrations were used. This result does not favour model I. Moreover, in this case a much larger EMS was associated with model I than with model II. This evidence in favour of a diffusional component in tryptophan uptake by the hypothalamus eliminated model I from further consideration. (C)
GLUTAMATE
TRANSPORT
IN
CEREBRAL
CORTEX
Logan & Snyder (1972) have shown that the transport of glutamate in rat cerebral cortex follows complex kinetics that conform well to model I. Their data were fitted by means of Cleland’s programme (Cleland, 1967). We have estimated, from the mean values in Fig. 4 of their paper, the parameters for models I and II using our programmes. The results in Table 3 show that either model fits the observations satisfactorily. For model I there is good agreement between the affinity constants calculated by either Cleland’s programme or the present one. (D)
TRYPTOPHAN
UPTAKE
INTO
SYNAPTOSOMES
A similar exercise was performed with data taken from the paper of Belin 8c Pujol(l973; cf. their Fig. 2). These authors have described discontinuities in the curve for the velocity of uptake of tryptophan into synaptosomes of
MULTIPLE
L-Glutamate Model
I
105
MODELS
TABLE 3 uptake: homogenates of rat cerebral cortex? Constant
Method of calculation Cleland (1967) Present programs
Kl
2=0X10-‘M 1.1 x10-3 n4.t n.g. ng.
K Kd V
-
& Vl V2 EMS I1
TRANSPORT
EMS
-
1.35X10-‘M 0.78x10-3 800x10-’ 8.21x10-? 5.34x lo- I6 1.87x IO-’ M 3.53x10-4 9.24x10-’ 752x10-la
t Data of Logan & Snyder(1972).Valuesin the first columnwere taken from that paper. Other calculations,usingthe presentprograms,were basedupon meanvalues estimatedfrom Fig. 4 in that article,asdescribed in the text. $ Not given.
the rat mesencephalon as a function of the concentration of tryptophan in the external medium, and they have interpreted these breaks in delimiting different uptake systems. Their kinetic constants were derived by treating three portions of the curve as though they were kinetically independent, and by applying the method of Lineweaver & Burk (1934) to each portion. In this way they obtained three sets of values for K and V. Working from their mean values for velocity at various tryptophan concentrations it could be shown that the data fit well to model II, but they did not converge (see Appendix A). (E) CLEARANCE OF 5-HYDROXYINDOLBACETIC ACID (5-HIAA) PROM THECEREBROSPINAL PLIEDDURINGSUBARACHNOID PERFUSION
Despite the current favoring of model I to describe multiple transport systems, Wolfson, Katzman 8~ Escriva (1974) adopted model II to interpret their experiments on the clearance of SHIAA from the cerebrospinal fluid. The choice in this case was based on past evidence that active transport as well as a passive component (nonspecific drainage or bulk flow at the arachnoid villi) both contribute to this process (Katzman & Pappius, 1973), and the constants were estimated by empirical methods. These values may be compared in Table 4 with those estimated by our programme for model II, based once again on mean values for clearance rates at various perfusing concentrations of SHIAA (Wolfson et al., 1974; their Fig. 5).
106
F. DENIZEAU,
J. WYSE AND T. L. SOURKES TABLE 4
Constants derivedfrom the data of Wofion et al. (1974) Estimates according to :
Kt
Kd
Wolfsonet al. (1974) 3-O x lo- 4 g/ml Presentcalculationst 1.23~10-~
0.010 0.017
Vm*x
1.14Xlo- 5 g/min 0.62~10-~
t The Error Mean Squaresof fit to modelsI and II were approximatelyequal, For modelI, K1 = 7*26~10-~g/ml; Ka = 1.21x lObag/ml.
4. Discussion The study of transport and uptake processes, especially in the nervous system, has revealed that many substances move across the neuronal or synaptosomal membranes by complex mechanisms, i.e. mechanisms that do not conform to transport by a single carrier [equation (I)] or by simple diffusion [equation (6)]. It is possible to conceive of other complex models than those described in this paper, for example, a system of two saturable components as well as diffusion. However, the mathematical analysis for these becomes increasingly complex, and the sample size (number of data points) must be considerably increased. In this work we have selected the two simplest of the multiple systems and have tried to establish mathematical and experimental criteria for choosing the one of best fit to the data. However, this does not exclude the possibility that transport of tryptophan into brain slices, on more detailed and extensive examination, would prove to take place by a different system than model II. In the example of the data of Kiely & Sourkes (1972), the ultimate choice of model II over model I could be made on the grounds of lack of physiological significance to the values obtained for V, and K,. In regard to tryptophan transport into slices of rat hypothalamus (Table 2), additional experiments had to be performed to aid in the decision between the two models. It was these experiments that now provide the evidence in favour of a significant diffusional component. Currently model I is very much favoured in describing multiple transport systems for amino acids and amines. This choice is based upon non-linearity of the reciprocal plot of u versus S. However, the question remains as to whether there are one or two “carriers” functioning in the membrane, and whether diffusion plays a role in movement of the substrate across that membrane. In order to illustrate this point we estimated the parameters according to model II for typical data in the recent literature for transport of glutamate into the cerebral cortex of the rat (Logan & Snyder, 1972) and of tryptophan into synaptosomes of rat mesencephalon (Belin & Pujol,
MULTIPLE
TRANSPORT
MODELS
107
1973). It is clear that in these two cases the data, assumed by the respective authors to correspond to high and low aflinity systems, could also be readily fitted to model II (Table 3). Obviously, where alternative biologically sound models are available, the observed data should be tested for their fit to these. The present study indicates that it is often possible to fit data satisfactorily to either of two models of multiple transport. However, mathematical fit does not necessarily correspond to physiological process, and the decision between two models will ultimately have to be made on the basis of additional experiments. REFERENCES A., BOURGOIN, S., PFNOA, P., GLO~INSKI,
J. & HAMON, M. (1974). Bruin Res. 66,253. BELIN, M. F. & PUJOL, F. L. (1973). Experientiu 29, 411. CHRISTENSEN, H. N. (1969). A&. Enzymol. 32, 1. CLELAND, W. W. (1967). A&. Enzymol. 29, 1. DENUEAU, F. SOURKES, T. L. (1975). In preparation. DRAPER, N. Kc !knx, H. (1969). Applied Regression Analysis, Ch. 10, New York: John Wiley and Sons. FERNSTROM, J. D. & WURTMAN, R. J. (1971). Science, N.Y. 173, 149. HARTLEY, H. 0. (1961). Technometrics 3, 269. IVERSEN, L. (1974). Biochem. Pharmacol. 23, 1927. KATZMAN, R. & Pm~rus, H. M. (1973). Brain Electrolytes and Fluid Metabolism. Baltimore: Williams and Wilkins Company. KIELY, M. & Soufuuq T. L. (1972). J. Neurochem. 19, 2863. KNAPP, S. & MANDELL, A. J. (1973). Science, N. Y. 180, 645. LINEWEAVER, H. & BARK, D. (1934). J. Am. them. Sot. 56, 658. LOOAN, E. J. & SNYDER, S. H. (1972). Bruin Res. 42, 413. MICHAELIS, L. & MENTEN, M. L. (1913). Biochem. 2. 49, 333. NEAL, J. L. (1972). J. theor. Biol. 35, 113. SMITH, S. E. (1967). J. Neurochem. 14, 291. TAGLIAMONTE, A., BIGGIO, G., VARGIU, J. & GESSA, G. L. (1973). Life Sci. 12, 277. WOLFSON, L. I., KATZMAN, R. & EXRIVA, A. (1974). Neurology 24, 772. BAUMANN,
Appendix A
Estimation of the constants in equations (2) and (3) is difficult owing to the fact that both models are non-linear. Smith (1967) has already presented the equations to be used in the case of model II. In the present work a Taylor series (for linearization) (Draper & Smith, 1966) has been used for model I, with modifications (Hartley, 1961). A computer programme embodying these concepts was written in Basic by one of us (J.W.) and data were processed on a Hewlett-Packard 9830A Programmable Calculator with 6K words of read-write memory.
108
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L.
SOURKES
If the estimation of parameters by the method of least squares employing a Taylor series is applied to model I, the following equations result: Sf - v,si’ Av1Z~K1+s~)‘+AK1C(K~+SJ3+Avz~(K~+SJ(K*+SJ - v,s: +AKzZ(Kl +SI)(K, +S~ L\V,/,c ~-i-f& -+AK,&!?%+Av,c (Kl +Si13 (K I + Sij4
+AK2Z(K,
s: L {Ui-fi} = ‘Kl +Si
(8)
- v,s: _._.___~~__ (Kl +Si)‘(K2+Si)
Vl v2s: -v,si {Ui-fi} + Si)2(K2 + Si)2 = ’ (K, + Si)’
S: - v,s: Av1Z(K~+S~)(K~+S~)‘AK1~~~+S~)2(K~+S~)’Av2~(K~+S~)2 - v,s: -A-‘AKzZ(K,+Si)3 =ZK,+Si - v,s: v, v,s; ____-.-. Av1Z~~~~~K2+Sj)z+AK1C~+S,)2(K2+Si)Z+AV2Z(K2+Si)3 vzs2 - V2Si +AK,E ’ (K,~~34=r(K2+Si)2
(9)
Si”
(u.-ff} ‘
(10)
- v,s;
(“i-h}
Cl11
where n is the number of observations; S, u, K,, K,, VI and V, are as described previously; and AK,, AK2, AV, and AV, are the correction factors for the last four symbols, respectively. In equations (8)-(1 I), fi can be calculated from :
If initial estimates of K,, K,, VI and V, can be made, equations (8)-(11) can be reduced to four equations in four unknowns and hence can be solved for AK,, AK,, AV, and AV,. The solution, in turn, allows the initial estimates to be modified as follows: K; Kj Vi V5 These revised estimates are now whole process is repeated.
= K, +AK, = K2 + AK2 = VI +AVl =V,+AV,
(13) substituted in equations (8)-(12) and the
MULTIPLE
TRANSPORT
MODELS
109
At each stage the error mean square (EMS) is calculated by dividing the sum of squares by the degrees of freedom.
(14) Where P = the number of parameters being estimated (four in this example). This iterative procedure continues until the solution converges, i.e. the fractional change in the EMS and/or the fractional changes in all the parameters in successive iterations reach some specified value. For example, convergence may be accepted as having taken place if after n iterations. EMS,-EMS,-l < s EMS,-1 ’ where 6 is very small, say, O*OOOl. For the estimation of the parameters of model II by the same technique, the equations presented by Smith (1967) were used. The technique outlined above is usually by itself not sufficient to ensure a solution (i.e. the EMS and/or parameter values may not converge). For this reason, and in order to achieve convergence with the fewest number of iterations possible, the method suggested by Hartley (1961) was appended to the Taylor series procedure. This modified technique, along with various subroutines which enabled the calculation of the initial estimates of the parameters, formed the basis of the computer programme developed now. The programme prints out the parameters, correction factors, and EMS at each iteration stage, and plots the observed data, along with final curve of best fit. Initial estimates of the parameters are easily obtained for both models by graphical techniques. For model I, they are obtained from the slopes and intercepts of straight-line segments drawn tangent to the curve produced by a reciprocal plot (Lineweaver & Burk, 1934), at low and high values of S, by applying the equations published by Neal (1972). Estimates for model II are obtained as follows: (a) On a plot with o as ordinate and S as abscissa,
Kd = the slope of the line at high values of S. That is, Kd=
dv 0iG ,yhrn
(b) On a plot with S/v as ordinate and S as abscissa, K is given approximately by the intercept on the abscissa of the line tangent to the curve for small values of S.
110
F.
DENIZEAU,
I.
WYSE
AND
T.
L.
SOURKES
(c) On a double reciprocal plot (Lineweaver & Burk, 1934) it can be shown that the curve of u-r versus S-r is asymptotic to a line whose slope is given by
The plot of u versus S obtained with the initial estimates was checked for its fit to the data before these estimates were used as starting values in the iterative procedure.
Kinetics of Multiple Transport Systems for Amino Acids in the Brain F. DENIZEAU, J. WYSE AND T. L. SOURKES Departments of Biochemistry and Psychiatry (F. D. and T.L.S.), Faculty of Medicine, McGill University, and Royal Victoria Hospital Montreal, Quebec, Canada (Received27 May 1975,and in revisedform 20 January 1976)
Two models of multiple transport systems (one comprising diffusion plus a Michaelis-Menten component, and the other comprising two Michaelis-Menten components) were tested for their fit to a series of experimental data. A statistical method (linearization technique) wasused to get estimates for the appropriate parameters of each model. In most casesthe data could be fitted to either model. Techniques for discriminating between the two models were sought and are discussed.
1. IntroductioIl
Transport of amino acids across membranes may occur by means of more than one agency and, when two or more transport mechanisms function simultaneously as happens in a wide variety of organisms, kinetic analysis of the phenomenon presents special problems. Christensen (1969) has recently discussed this subject. In brain tissue high and low affinity systems have been described for various amines and amino acids thought to act as neurotransmitters. High aflinity transport mechanisms have been postulated to play a role in terminating the action of such neurotransmitters after their release from nerve endings (see review by Iversen, 1974). Furthermore, high and low affinity systems have been described for tryptophan (Belin & Pujol, 1973 ; Knapp & Mandell, 1973; Baumann, Bourgoin, Penda, Glowinski &z Hamon, 1974), the precursor of serotonin. In this case, the high afhnity system has been associated with the control of tryptophan concentrations within the serotonin-containing neurons; this in turn affects the rate of serotonin synthesis (Tagliamonte, Biggio, Vargiu & Gessa, 1973 ; Fernstrom & Wurtman, 1971). In addition to these proposals about tryptophan transport 99
100
F.
DENIZEAU,
J.
WYSE
AND
T.
L.
SOURKES
in the brain, a model comprising an active component, conforming to Michaelis-Menten kinetics (Michaelis & Menten, 1913) and a diffusional component has been adduced by Kiely & Sourkes (1972). In most of these reports the selection of a particular model of multiple transport, namely one comprising high and low affinity components, has been made a priori by formal application of the method of Lineweaver & Burk (1934) to the data. The aim of this paper is twofold: first to study in detail two common transport models by utilizing well established mathematical techniques. In most kinetic studies such techniques are applied, if at all, only empirically, and it was now thought desirable to examine the implications of these techniques for the theory of transport phenomena. Secondly, we wished to explore the bases for the choice between the two models for a given set of observational data. 2. Theoretical
Considerations
Carrier transport systems are characterized by kinetic constants analogous to those appropriate to the analysis of enzyme kinetic data. On the basis of the Michaelis-Menten hypothesis the initial velocity of uptake or transport of an amino acid (or other substance) is given by the expression 11= vs(K+s)-l (1) in which V = the theoretical maximal velocity attained at saturation of the transport carrier with substrate; K has the quality of the Michaelis constant, and characterizes the transport process; and S is the initial concentration of the substrate in the external medium. In transport studies, a non-linear relationship in the plot of 0-l versus S-’ provides evidence that more than one transport system is operating. The simplest models that can then be postulated are of the forms: Model I: u = --!?t K,+S
+ w K,+S
(2)
Model II:
It can be demonstrated
that for Model I, 1
limv-‘(S-l-O)=--v1+
v*
MULTIPLE
TRANSPORT
101
MODELS
and for Model II, lim v-Q-’ -+ 0) = 0 (5) Thus, in the case of model II the plot of u-l versus S” curves downwards as S becomes very great, and ultimately the line passes through the origin. For model I the intercept on the ordinate is equal to (Vl + V,)-‘. In practice, for observed data the experimental error makes it extremely difticult to decide between the two models on the basis of extrapolation of the curve to the ordinate passing through S- ’ = 0. Both models I and II are non-linear, and convenient methods have been sought to estimate the respective sets of parameters by the method of least squares in order to obtain the best possible fit to the data. The application of these methods, with derivation of the appropriate equations and of the programmes in Basic to be used for the two models, is described in Appendix A. 3. Applications (A) STATISTICALLY
FITTED
PARAMETERS
VERSUS
ESTIMATES
Previously Kiely & Sourkes (1972) showed that transport of tryptophan into slices of rat brain cortex occurs by complex kinetics that can be reduced to components involving carrier movement and diffusion. The graphical methods used in that paper to calculate the parameters were applied in the present work to provide the initial estimates (see Appendix A). The data have now been revaluated, and the parameters are presented in Table 1. The analysis of the data according to model II shows that the estimate of K made by graphic means is lower than the value obtained by the least squares fit. Moreover, the value of V estimated by computer programme is only about half as large as previously reported. The diffusion constant is essentially the same by both methods. Although the EMS (error mean square) was somewhat smaller for model I, that model yielded very high values for Kz and V,, and these were considered to be without physiological significance. 1 Tryptophan transport into rat-brain cortex t (Model IZ) TABLE
Parameters Estimates
made
K mm01 l-1
by:
0.829
Graphical method? Least squares fitting t Data
of Kiely
1.15 & Sourkes
(1972).
V mm01
1-l
& min-’
0402 O-203
min-a
0.065 0.06
102
F.
DENIZEAU,
(B) EXF’FRIMENTAL
J.
WYSE
DECISION
AND BETWEEN
T.
L.
SOURKES
MODELS
I AND
II
In order to determine the contribution of carrier transport to the observed rate of movement of a substance into intracellular water most authors test the effect of reduced temperature on the process. Generally, carrier transport then appears to be diminished (Vis greatly reduced), but not totally inhibited. If the residual rate is attributed to diffusion, then K, is overestimated. When the erroneously estimated contribution of diffusion is subtracted from the uptake values observed at 37”C, the remaining values will result in incorrect estimates of K and V, or lead to an interpretation of the presence of two carrier systems, even if this is not the case. Data for the transport of tryptophan in rat hypothalamus at 37°C have been collected in this laboratory; the methodology involved in these measurements will be described elsewhere (Denizeau & Sourkes, 1975). When the data were subjected to a reciprocal plot, it appeared evident that a multiple transport system was operating. The data for the initial rate of uptake of 14C-tryptophan ( u) , tr eated as a function of the concentration of the amino acid in the external medium (O-05 to 1 mM) at 37°C conformed well to both models I and II. These values for the various constants were obtained by an iterative computer programme (Appendix A) and these are presented in Table 2. It can be seen that the EMS would favour model II slightly, although a choice cannot be made purely on this basis. Further experiments were then carried out with this tissue at 4”, and also at 37°C in the presence of metabolic inhibitors. The data for transport at 4°C were fitted by the least squares method to models I and II, and also to model III, in which v = &S
(6)
There was a poor fit to model III, whereas model II appeared to fit very satisfactorily. With the values then obtained for K, K, and V, it was calculated that at 4”C, for S = 0.01 mM, the contribution of active uptake is about six times as great as that of diffusion. Hence, it cannot be assumed that carrier transport has been abolished at this temperature. Slices of hypothalamic tissue were then incubated with iodoacetate (0.5 mu) and dinitrophenol (0.1 mM) at 37°C. These metabolic inhibitors block the production of ATP in glycolysis and of oxidative phosphorylation, respectively. Under these conditions, active uptake is very much reduced (K,” greatly increased) but does not seem to be totally negligible. Since it appeared almost impossible to eliminate entirely the contribution of carrier transport by means of conventional methods, the first experiment was repeated with a wider range of substrate concentrations in the medium.
-
-
1.09
0.16
-
2.14 15560
mM
Ka
0.01 0.54
KI -__
- Signifies that values did not converge. t Unpublished data of Denizeau & Sourkes. $ Error mean square X log. 5 Iodoacetate and dinitrophenol (see text).
0~10-10mM
Incubation at 37°C Concentration range: 0.0% 1 mM O*lO-10 rnM Incubation at 4°C Concentration range: 0.05- 1 nw Incubation at 37°C in presence of inhibitors9 Concentration range: 0.05- 1 mM
Conditions
-
-
0.02
0.08 0.92
-
-
oa
140 11.80
Model I Vl VZ mM min-1
-
-
0.07
2.51 112.30
EMSI
1.76 1.75
0.04
O-06 0.16
mKM
0.07 0436
0.03
0.36 0.09
0.25 0.25
0.01
0.18 0.38
V mM min-’
Model II
t
Kd min-’
TABLE 2 Tryptophan transport: slices of rat hypothalamus
0.54 6.13
0.06
244 20.70
EMS
0.22 OGJ
0.04
-
Kr,
0.37 10.11
0.08
-
EMS
Model III
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Our reasoning in doing so was as follows. If the highest substrate concentration used is significantly smaller than K2, then (7) where & has the character of a diffusion constant, and equations (2) and (3) become identical. Thus, with low substrate concentrations it is impossible to distinguish between models I and II. For the new control experiment the highest concentration of tryptophan used (10 mM) was well above the value estimated for & in Table 2 (i.e. 2.14 mM) after the first analysis of the data. When estimates of the parameters for models I and II are then made on the basis of this new range of higher concentrations, the following possibilities emerge. If model I is appropriate to the transport mechanism, then K, and V, should remain the same as with the other range of concentrations tested. On the other hand, if model II holds true the values estimated for K,,,, V,,, and Kd in both sets of experiments should be very close; furthermore, when the attempt is made to fit the data to model I the estimates of the parameters, especially those for Kz and V,, should keep on increasing, paripassu, as the upper range of concentration is extended. The figures obtained in this example are shown in Table 2. It can be seen that K2 increased substantially when high substrate concentrations were used. This result does not favour model I. Moreover, in this case a much larger EMS was associated with model I than with model II. This evidence in favour of a diffusional component in tryptophan uptake by the hypothalamus eliminated model I from further consideration. (C)
GLUTAMATE
TRANSPORT
IN
CEREBRAL
CORTEX
Logan & Snyder (1972) have shown that the transport of glutamate in rat cerebral cortex follows complex kinetics that conform well to model I. Their data were fitted by means of Cleland’s programme (Cleland, 1967). We have estimated, from the mean values in Fig. 4 of their paper, the parameters for models I and II using our programmes. The results in Table 3 show that either model fits the observations satisfactorily. For model I there is good agreement between the affinity constants calculated by either Cleland’s programme or the present one. (D)
TRYPTOPHAN
UPTAKE
INTO
SYNAPTOSOMES
A similar exercise was performed with data taken from the paper of Belin 8c Pujol(l973; cf. their Fig. 2). These authors have described discontinuities in the curve for the velocity of uptake of tryptophan into synaptosomes of
MULTIPLE
L-Glutamate Model
I
105
MODELS
TABLE 3 uptake: homogenates of rat cerebral cortex? Constant
Method of calculation Cleland (1967) Present programs
Kl
2=0X10-‘M 1.1 x10-3 n4.t n.g. ng.
K Kd V
-
& Vl V2 EMS I1
TRANSPORT
EMS
-
1.35X10-‘M 0.78x10-3 800x10-’ 8.21x10-? 5.34x lo- I6 1.87x IO-’ M 3.53x10-4 9.24x10-’ 752x10-la
t Data of Logan & Snyder(1972).Valuesin the first columnwere taken from that paper. Other calculations,usingthe presentprograms,were basedupon meanvalues estimatedfrom Fig. 4 in that article,asdescribed in the text. $ Not given.
the rat mesencephalon as a function of the concentration of tryptophan in the external medium, and they have interpreted these breaks in delimiting different uptake systems. Their kinetic constants were derived by treating three portions of the curve as though they were kinetically independent, and by applying the method of Lineweaver & Burk (1934) to each portion. In this way they obtained three sets of values for K and V. Working from their mean values for velocity at various tryptophan concentrations it could be shown that the data fit well to model II, but they did not converge (see Appendix A). (E) CLEARANCE OF 5-HYDROXYINDOLBACETIC ACID (5-HIAA) PROM THECEREBROSPINAL PLIEDDURINGSUBARACHNOID PERFUSION
Despite the current favoring of model I to describe multiple transport systems, Wolfson, Katzman 8~ Escriva (1974) adopted model II to interpret their experiments on the clearance of SHIAA from the cerebrospinal fluid. The choice in this case was based on past evidence that active transport as well as a passive component (nonspecific drainage or bulk flow at the arachnoid villi) both contribute to this process (Katzman & Pappius, 1973), and the constants were estimated by empirical methods. These values may be compared in Table 4 with those estimated by our programme for model II, based once again on mean values for clearance rates at various perfusing concentrations of SHIAA (Wolfson et al., 1974; their Fig. 5).
106
F. DENIZEAU,
J. WYSE AND T. L. SOURKES TABLE 4
Constants derivedfrom the data of Wofion et al. (1974) Estimates according to :
Kt
Kd
Wolfsonet al. (1974) 3-O x lo- 4 g/ml Presentcalculationst 1.23~10-~
0.010 0.017
Vm*x
1.14Xlo- 5 g/min 0.62~10-~
t The Error Mean Squaresof fit to modelsI and II were approximatelyequal, For modelI, K1 = 7*26~10-~g/ml; Ka = 1.21x lObag/ml.
4. Discussion The study of transport and uptake processes, especially in the nervous system, has revealed that many substances move across the neuronal or synaptosomal membranes by complex mechanisms, i.e. mechanisms that do not conform to transport by a single carrier [equation (I)] or by simple diffusion [equation (6)]. It is possible to conceive of other complex models than those described in this paper, for example, a system of two saturable components as well as diffusion. However, the mathematical analysis for these becomes increasingly complex, and the sample size (number of data points) must be considerably increased. In this work we have selected the two simplest of the multiple systems and have tried to establish mathematical and experimental criteria for choosing the one of best fit to the data. However, this does not exclude the possibility that transport of tryptophan into brain slices, on more detailed and extensive examination, would prove to take place by a different system than model II. In the example of the data of Kiely & Sourkes (1972), the ultimate choice of model II over model I could be made on the grounds of lack of physiological significance to the values obtained for V, and K,. In regard to tryptophan transport into slices of rat hypothalamus (Table 2), additional experiments had to be performed to aid in the decision between the two models. It was these experiments that now provide the evidence in favour of a significant diffusional component. Currently model I is very much favoured in describing multiple transport systems for amino acids and amines. This choice is based upon non-linearity of the reciprocal plot of u versus S. However, the question remains as to whether there are one or two “carriers” functioning in the membrane, and whether diffusion plays a role in movement of the substrate across that membrane. In order to illustrate this point we estimated the parameters according to model II for typical data in the recent literature for transport of glutamate into the cerebral cortex of the rat (Logan & Snyder, 1972) and of tryptophan into synaptosomes of rat mesencephalon (Belin & Pujol,
MULTIPLE
TRANSPORT
MODELS
107
1973). It is clear that in these two cases the data, assumed by the respective authors to correspond to high and low aflinity systems, could also be readily fitted to model II (Table 3). Obviously, where alternative biologically sound models are available, the observed data should be tested for their fit to these. The present study indicates that it is often possible to fit data satisfactorily to either of two models of multiple transport. However, mathematical fit does not necessarily correspond to physiological process, and the decision between two models will ultimately have to be made on the basis of additional experiments. REFERENCES A., BOURGOIN, S., PFNOA, P., GLO~INSKI,
J. & HAMON, M. (1974). Bruin Res. 66,253. BELIN, M. F. & PUJOL, F. L. (1973). Experientiu 29, 411. CHRISTENSEN, H. N. (1969). A&. Enzymol. 32, 1. CLELAND, W. W. (1967). A&. Enzymol. 29, 1. DENUEAU, F. SOURKES, T. L. (1975). In preparation. DRAPER, N. Kc !knx, H. (1969). Applied Regression Analysis, Ch. 10, New York: John Wiley and Sons. FERNSTROM, J. D. & WURTMAN, R. J. (1971). Science, N.Y. 173, 149. HARTLEY, H. 0. (1961). Technometrics 3, 269. IVERSEN, L. (1974). Biochem. Pharmacol. 23, 1927. KATZMAN, R. & Pm~rus, H. M. (1973). Brain Electrolytes and Fluid Metabolism. Baltimore: Williams and Wilkins Company. KIELY, M. & Soufuuq T. L. (1972). J. Neurochem. 19, 2863. KNAPP, S. & MANDELL, A. J. (1973). Science, N. Y. 180, 645. LINEWEAVER, H. & BARK, D. (1934). J. Am. them. Sot. 56, 658. LOOAN, E. J. & SNYDER, S. H. (1972). Bruin Res. 42, 413. MICHAELIS, L. & MENTEN, M. L. (1913). Biochem. 2. 49, 333. NEAL, J. L. (1972). J. theor. Biol. 35, 113. SMITH, S. E. (1967). J. Neurochem. 14, 291. TAGLIAMONTE, A., BIGGIO, G., VARGIU, J. & GESSA, G. L. (1973). Life Sci. 12, 277. WOLFSON, L. I., KATZMAN, R. & EXRIVA, A. (1974). Neurology 24, 772. BAUMANN,
Appendix A
Estimation of the constants in equations (2) and (3) is difficult owing to the fact that both models are non-linear. Smith (1967) has already presented the equations to be used in the case of model II. In the present work a Taylor series (for linearization) (Draper & Smith, 1966) has been used for model I, with modifications (Hartley, 1961). A computer programme embodying these concepts was written in Basic by one of us (J.W.) and data were processed on a Hewlett-Packard 9830A Programmable Calculator with 6K words of read-write memory.
108
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If the estimation of parameters by the method of least squares employing a Taylor series is applied to model I, the following equations result: Sf - v,si’ Av1Z~K1+s~)‘+AK1C(K~+SJ3+Avz~(K~+SJ(K*+SJ - v,s: +AKzZ(Kl +SI)(K, +S~ L\V,/,c ~-i-f& -+AK,&!?%+Av,c (Kl +Si13 (K I + Sij4
+AK2Z(K,
s: L {Ui-fi} = ‘Kl +Si
(8)
- v,s: _._.___~~__ (Kl +Si)‘(K2+Si)
Vl v2s: -v,si {Ui-fi} + Si)2(K2 + Si)2 = ’ (K, + Si)’
S: - v,s: Av1Z(K~+S~)(K~+S~)‘AK1~~~+S~)2(K~+S~)’Av2~(K~+S~)2 - v,s: -A-‘AKzZ(K,+Si)3 =ZK,+Si - v,s: v, v,s; ____-.-. Av1Z~~~~~K2+Sj)z+AK1C~+S,)2(K2+Si)Z+AV2Z(K2+Si)3 vzs2 - V2Si +AK,E ’ (K,~~34=r(K2+Si)2
(9)
Si”
(u.-ff} ‘
(10)
- v,s;
(“i-h}
Cl11
where n is the number of observations; S, u, K,, K,, VI and V, are as described previously; and AK,, AK2, AV, and AV, are the correction factors for the last four symbols, respectively. In equations (8)-(1 I), fi can be calculated from :
If initial estimates of K,, K,, VI and V, can be made, equations (8)-(11) can be reduced to four equations in four unknowns and hence can be solved for AK,, AK,, AV, and AV,. The solution, in turn, allows the initial estimates to be modified as follows: K; Kj Vi V5 These revised estimates are now whole process is repeated.
= K, +AK, = K2 + AK2 = VI +AVl =V,+AV,
(13) substituted in equations (8)-(12) and the
MULTIPLE
TRANSPORT
MODELS
109
At each stage the error mean square (EMS) is calculated by dividing the sum of squares by the degrees of freedom.
(14) Where P = the number of parameters being estimated (four in this example). This iterative procedure continues until the solution converges, i.e. the fractional change in the EMS and/or the fractional changes in all the parameters in successive iterations reach some specified value. For example, convergence may be accepted as having taken place if after n iterations. EMS,-EMS,-l < s EMS,-1 ’ where 6 is very small, say, O*OOOl. For the estimation of the parameters of model II by the same technique, the equations presented by Smith (1967) were used. The technique outlined above is usually by itself not sufficient to ensure a solution (i.e. the EMS and/or parameter values may not converge). For this reason, and in order to achieve convergence with the fewest number of iterations possible, the method suggested by Hartley (1961) was appended to the Taylor series procedure. This modified technique, along with various subroutines which enabled the calculation of the initial estimates of the parameters, formed the basis of the computer programme developed now. The programme prints out the parameters, correction factors, and EMS at each iteration stage, and plots the observed data, along with final curve of best fit. Initial estimates of the parameters are easily obtained for both models by graphical techniques. For model I, they are obtained from the slopes and intercepts of straight-line segments drawn tangent to the curve produced by a reciprocal plot (Lineweaver & Burk, 1934), at low and high values of S, by applying the equations published by Neal (1972). Estimates for model II are obtained as follows: (a) On a plot with o as ordinate and S as abscissa,
Kd = the slope of the line at high values of S. That is, Kd=
dv 0iG ,yhrn
(b) On a plot with S/v as ordinate and S as abscissa, K is given approximately by the intercept on the abscissa of the line tangent to the curve for small values of S.
110
F.
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L.
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(c) On a double reciprocal plot (Lineweaver & Burk, 1934) it can be shown that the curve of u-r versus S-r is asymptotic to a line whose slope is given by
The plot of u versus S obtained with the initial estimates was checked for its fit to the data before these estimates were used as starting values in the iterative procedure.
