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of Michaelis and Menten (1913) describes the facil itated diffusion of substrates through the blood brain barrier (Crone, 1965). The facilitated trans port is usually ...
Journal of Cerebral Blood Flow and Metabolism 4:241-249 © 1984 Raven Press, New York

Estimates of Michaelis-Menten Constants for the Two Membranes of the Brain Endothelium

Albert Gjedde and Ove Christensen Medical Physiology Department A, The Panum Institute, Copenhagen University, Copenhagen, Denmark

Summary: Tracer studies on facilitated diffusion across

mined in tracer experiments describe facilitated diffusion

the blood-brain barrier lead to the calculation of Mi­

in the steady state only if the two membranes have similar

chaelis-Menten constants that describe the rate of trans­

transport properties. As an application of this observa­

port. However, the barrier consists of two endothelial cell

tion, we have examined three experimental studies that

membranes, and the relevance of single Michaelis­

measure glucose transport in the steady state and show

Menten constants in relation to the two cell membranes

that the Michaelis-Menten constants for glucose transport

is unknown. We have formulated a model of two endo­

calculated from the tracer experiments are equal to the

thelial cell membranes and show that the measured Mi­

constants calculated from the steady-state experiments.

chaelis-Menten constants are simple functions of the

We conclude that the luminal and abluminal membranes

properties of the individual membranes when transport across the endothelium is rapid (P > 10-6 cm S-I). We I also show that the Michaelis-Menten constants deter-

of brain capillary endothelial cells have equal glucose

transport properties. Key Words: Blood-brain barrier­ Brain endothelial membranes-Facilitated diffusion.

It is a fundamental observation that the equation of Michaelis and Menten (1913) describes the facil­ itated diffusion of substrates through the blood­ brain barrier (Crone, 1965). The facilitated trans­ port is usually represented by the constants T max and Kt (for maximal transport rate and half-satura­ tion concentration, respectively), despite the ana­ tomical reality of two separate endothelial mem­ branes through which transport must occur. The question of whether it is permissible to represent the transport properties of two separate membranes by a single set of Michaelis-Menten constants has not previously been examined for the blood-brain barrier. Pappenheimer and Setchell (1973) took into ac­ count the existence of two membranes when they considered glucose transport across the brain en­ dothelium. However, they did not analyze possible differences in the transport properties of the two membranes. Although they developed an equation for the relationship between plasma and brain glu-

cose levels, and the maximal transport capacity and half-saturation constants for each membrane, they regarded the membranes as equal with respect to glucose transport in order to solve the equation for T max and Kt. Pappenheimer and Setchell (1973) con­ sidered it quite likely that the membranes have identical transport properties on the grounds that the same membrane continues around the cell. In view of the well-established polarity of epithelial tissue, this assumption is, a priori, not very cred­ ible. More recently, Lund-Andersen (1979) concluded that an "exact evaluation of [the] problem presup­ poses . . . an analysis of non-steady-state tracer flux data by aid of a capillary transport model in which the capillary wall consists of two series-cou­ pled saturable transport mechanisms. . . . [Such a model] is not available in the literature at present. " The endothelium is a space between the blood­ stream and the tissue. Because of its small volume, it qualifies as a rapidly equilibrating space. Patlak et al. (1983) and Blasberg et al. (1983) recently fo­ cused on the role of rapidly equilibrating spaces of the blood-brain barrier in tracer transport. In a general form, Patlak et al. (1983) showed that uni-

Address correspondence and reprint requests to Dr. Gjedde at Medical Physiology Department A, The Panum Institute, Co­ penhagen University, Copenhagen, Denmark 2200.

241

242

A. GJEDDE AND O. CHRISTENSEN

directional "steps" could be measured without re­ course to specific models. The term "step" indi­ cates transport or binding that irreversibly traps the tracer during the experimental period. Thus, trans­ port across the second of the two endothelial cell membranes is unidirectional. However, the cell equlibrates so quickly that the transport properties of the two membranes cannot be assessed sepa­ rately without recourse to a simplifying model. In the following treatment we formulate a model of the two membranes of the endothelium and ex­ amine the flux of a tracer undergoing facilitated dif­ fusion through the endothelium. The model readily yields the time constant for tracer transport through the rapidly equilibrating space represented by the endothelial cells. After arriving at an approximation of the speed of equilibration, we proceed to derive the exact relationship between the Michaelis­ Menten constants of the individual membranes and the Michaelis-Menten constants calculated for the ] endothelium. THE PREVIOUS ONE-MEMBRANE MODEL

A tracer study allows the determination of clear­ ance, i.e., the ratio between the amount of tracer present in a selected part of the brain and the time integral of the arterial tracer concentration (integral method; Ohno et aI., 1978; Gjedde et aI., 1980; Blas­ berg et al., 1983): (1)

The term K] symbolizes the clearance, M*(D the tracer amount in the brain, and C�(t) the tracer con­ centration in the arterial circulation (source fluid, i.e., plasma). The time integral of the arterial tracer concentration multiplied by the source fluid flow F (i.e., the plasma flow) is the total amount of tracer offered to the brain. Calculation of the clearance is complicated by the presence of tracer in the bloodstream and the en­ dothelium. We assume that we can measure and subtract the amount of tracer remaining in the bloodstream by means of a vascular tracer. We also assume that the experiment can be performed in so short a time that tracer which has passed from the endothelium into brain tissue is trapped in a large volume of distribution. In that case, K] is the clear­ ance of unidirectional transport across the endothe-

lium. Because of the unidirectionality, K] is a model-independent index of the rate of transendo­ thelial flux under given experimental circum­ stances. It is not a physical property of the mem­ brane or barrier and not directly comparable to clearances obtained under other circumstances. A simple single-membrane, single-capillary model of the brain permits further calculations of additional parameters of facilitated diffusion. The first is the calculation of permeability from the clearance. Permeability is a specific property of a membrane subjected to nonmediated diffusion but, by analogy, the "apparent" permeability of the membrane can be calculated for substances under­ going facilitated diffusion. The apparent perme­ ability represents the permeability of the membrane if transport proceeded at the same velocity by non­ mediated diffusion. When the perfusion rate of the source fluid (F) is known, the ratio K/F (the ex­ traction fraction) is the basis for calculation of the apparent single-membrane permeability (p]) (Crone, 1963):

P]

=

_

F In S

(

1

_

K] F

)

(2)

where S is the surface area of a given section of the capillary bed and F is the perfusion. K] also refers to this section. The second calculation is the estimation of Mi­ chaelis-Menten constants. It is a fundamental ob­ servation in facilitated diffusion that the apparent permeability decreases with an increasing concen­ tration of the tracer's parent substance in the plasma (the source fluid). This dependence ofp] on the plasma concentration is the basis for the second calculation, estimation of the Michaelis-Menten constants for the endothelium from the value of PIS at various concentrations of native substrate in the arterial source fluid (Gjedde, 1980):

PIS

==

T max Kt + C a

(3)

where C a is the arterial concentration of native sub­ strate in the source fluid. When Eq. 3 is rearranged in linear form:

p]SC a

=

-KtP]S

+

T max

(4)

corresponding values of the product p]SC a (flux) and the product P]S (permeability) yield Kt and T max (Eadie, 1942; Hofstee, 1954-1956). THE TWO-MEMBRANE MODEL

1 Following the text and Appendix is a Glossary in which terms used in the equations are defined.

J

Cereh Blood Flow Metabol, Vol, 4, No.2, 1984

The model described below yields an expression of the apparent permeability based on the Mi-

TWO MEMBRANES OF BRAIN ENDOTHELIUM chaelis-Menten constants of two membranes. We consider a capillary with an internal diameter of 5 ILm and endothelial cell thickness of 0.1 ILm. The volume of blood in capillaries is close to 1 ml hg - 1 (Gjedde, 198 1). Hence, the volume of the endothe­ lial cells, Vi' is only 8% of the blood volume, or 80 ILl hg-l. Now we consider the two-membrane, three-com­ partment model consisting of vascular, cell, and tissue compartments as shown in Fig. 1. The native substrate concentrations in the three compartments are constant in time and place, and the tissue com­ partment is "infinitely" large. Tracer transport into the tissue compartment is unidirectional because the tracer concentration remains zero in the infi­ nitely large compartment. The transport of tracer into any portion of the endothelial compartment is given by the product Pa S and the source fluid concentration. Tracer flux out of the endothelial compartment is given by P -a s + PbS multiplied by the endothelial cell tracer concentration. The subscripts refer to mem­ brane a (the luminal membrane) and membrane b (the abluminal membrane). Thus, tracer flux is bi­ directional across the luminal membrane and effec­ tively unidirectional across the abluminal mem­ brane. The product Pa S is the local clearance in ml hg-1 ENDOTHELIAL CELLS membrane

membrane

b VASCULAR

TISSUE

COMPARTMENT

COMPARTMENT

243

min - 1. When the transport across each membrane obeys Michaelis-Menten kinetics, the relationships between the Michaelis-Menten constants and the permeabilities are

Pa S

J �x a Kf + C 1

(5a)

J �x a Kf + C i b J max PbS= b K t + C,

P -s a

(5b) (5c)

where the subscripts refer to the two membranes.

C 1 and C i are the local vascular and endothelial con­

centrations of native substrate, respectively. The kinetic constants J max represent the maximal trans­ port capacities of the membranes, and Kt the half­ saturation concentrations. For simplicity, we as­ sume the individual membranes to be symmetrical K f). with respect to transport ( K I a We now consider the time variation in endothelial cell tracer concentration at an arbitrary point in the vascular bed. Let Vi denote the endothelial cell volume (ILl hg-l) and S the endothelial surface area (cm2 hg - 1) per unit weight. Consider an infinitesi­ mally small region of weight I1g including the point of interest. It encloses an endothelial cell volume of I1g Vi with a capillary surface area of I1g S. �he following equation describes the net accumulatiOn of tracer in the endothelial cell: =

dC '" I1g V I , dt

__

Pa I1g SCj(t) - ( P -a

+

Pb ) I1g sq (6)

native substrate tracer substrate

C1

Ci

Ce

C,

C· I

C e•

where the asterisk refers to the tracer substance rather than to the/parent substance. Notice that I1g cancels in Eq. 6; only the ratio V/S is important. Equation 6 describes the time course of the en­ dothelial cell tracer concentration at an arbitrary point in the vascular bed in response to a local vas­ cular tracer concentration Cj(t). It is a standard first-order differential equation. For a vascular con­ centration which is constant in time, the local steady-state concentration of cell tracer substance is found by setting the left-hand side of the equation equal to zero:

J" b

FIG. 1. Single-capillary model of the transendothelial pas­ sage. The terms Ct, Ci, and Ce refer to the local concentra­ tions of native substrate in the three compartments, and the term Vi to the volume of the endothelial cells. J; PaSCi, =

J�a = P _aSCi; Jt, = PbSCi·

C '" ,

= ( C*1

Pa P -a

+

Pb

)

(7)

This result shows that the local steady-state tracer concentration in the endothelium is proportional to the local vascular concentration. J

Cereh Blood Flow Metabol. Vol. 4, No.2, 1984

A. GJEDDE AND O. CHRISTENSEN

244

to the measured apparent permeability-surface area product (P S): I

DERIVATION OF THE APPARENT PERMEABILITY

The time constant by which the cell concentra­ tion approaches its steady-state value is V/S( P_a + Pb )' When we want a tracer steady state to occur in the endothelium during a single capillary transit (about 1 s), the time constant should be much less than this value. In the following discussion, we make a conser­ vative estimate of the time constant for a typical substrate such as glucose, i.e., an estimate of its largest possible value. If one of the membranes is much more permeable than the other, P -a or Pb will be large and the time constant small, corresponding to rapid equilibration of the cell concentration with either the vascular or the tissue tracer concentra­ tion. The time constant is lowest when the perme­ abilities are equal. On the other hand, the permea­ bilities cannot be lower than the apparent perme­ ability of the entire barrier, given by Eq. 3. Thus, a conservative estimate of the time constant can be made by setting P -as and PbS equal to T maAKt + C a ). Using the typical values C a 10 mM ' T max 400 f.Lmol hg-I min-I, and Kt 6 mM for glucose (Gjedde, 1983), we obtain an apparent perme­ ability-surface area product of 25 ml hg-1 min-I, corresponding to an apparent permeability of 10- 5 cm S-I when S equals 100 cmz g-I (Gjedde and Rasmussen, 1980). When this value is combined with an endothelial cell volume of 80 f.LI hg-I, the value of the time constant is 100 ms. For tracers with permeabilities below this value, we do not ex­ pect an endothelial steady state within a single cap­ illary transit. If C a rises to 30 mM. the PS products will fall to 10 ml hg-I min-I and the time constant will rise to 250 ms. Under normal experimental conditions, we as­ sume the variation in the arterial tracer concentra­ tion to be negligible within the period of the time constant for endothelial tracer accumulation. At any point in the vascular bed. the endothelial cell concentration must therefore be proportional to the vascular concentration, as indicated by Eq. 7. We next calculate the local flux of tracer through a capillary wall area S from the endothelial cell into the brain (Fig. 1) and show that it is proportional to the vascular tracer concentration: =

=

(9) The crucial link in the derivation is the propor­ tionality of the local tracer flux to the local vascular tracer concentration Cj (Eq. 7). It is immaterial whether or not the local tracer concentration varies with time when the variations are small with respect to the endothelial cell time constant. In this case the endothelial cell concentration reaches a quasi� steady state relative to the plasma concentration, which is proportional to the vascular plasma con­ centration. In this manner, the two membranes of the endothelial cell simulate a single barrier char­ acterized by the permeability expression in Eq. 9. The endothelial cell is like a transformer that con­ verts the vascular concentration Cj to the endothe­ lial cell concentration Cf.

DERIVATION OF THE MICHAELIS­ MENTEN CONSTANTS

=

J *b

=

PbSC"fi

=

PbSC*I p

-a

Pa +

Pb

(8)

Under steady-state conditions, Eq. 8 characterizes net tracer transport from the vasculature to the brain by a permeability-surface area product equal

J

Cereb Blood Flow Me/abo/, Vol. 4, No.2, 1984

We have given two independent definitions of P S, one in the form of Eq. 3, based on Eqs. 1 and I 2, and another in the form of Eq. 9, based on Eq. 5a-c. In this section, we derive Eq. 3 from Eqs. 5 and 9. To accomplish the derivation, we must know the net transport of the parent substance through the endothelium. The concentrations of the parent sub­ stance in the endothelial cell and in the interstitial space both depend on the net delivery of substrate through the endothelium, equal to the net flux through the two membranes in the steady state. The concentrations in the cells and in the brain tissue are tied to each other by the net transport rates across the two membranes:

�J

=

J� a x

�J

-

_

b J max

( (

Kf Kbt

CI

+ CI

C

Kf

Cj

+ Cj

; Cj

Ce K�

+

) )

Ce

(10)

(II)

Equation lOis used to obtain Cj as a function of CI. This value of Cj is substituted into Eq. 5a-c,

which were combined into Eq. 9. On the condition that CI C a (i.e., net transport is low compared to T max), and after extensive but trivial algebraic ma­ nipulations (see the Appendix), Eq. 9 can be put in the same form as Eq. 3 when =

TWO MEMBRANES OF BRAIN ENDOTHELIUM J�ax J�ax

IlJ(K� - IG) + ---'----- ----'---K� + J�axIG + �J(JG - �) IlJK ( � - K�) b + Jmax + ---'---K�-- �

J � ax J� ax� J�ax

+

b

c.; 00

__

(13)

From Eqs. 12 and 13 it is apparent that IlJfor the parent substance has little influence on the values of K( and T max when �J is much smaller than J�ax and lZ ax- This condition is naturally fulfilled in the case of facilitated transport of glucose into the brain because brain glucose consumption is much less than the maximum transport capacity of the blood­ brain barrier. It is also approximately true when IlJ varies as a function of C a' To discuss this point, we recalculated the relationship between P1SC a and PIS in case D in Table 1 for values of �J ranging linearly from 30 flmol hg-I min --I at C a = 2.5 mM to 70 flmol hg-I min-I at C a = 40 mM In this case, the variation in IlJ caused an imperceptible devia­ tion from the rectilinear relationship shown in Fig. 2 under the assumption of an invariant �J. There­ fore, it is possible to calculate the Michaelis­ Menten constants from the plots shown in Fig. 2 .

TABLE 1. Michaelis-Menten constants used to estimate the plots in Fig. 2 and the apparent single­ barrier Michaelis-Menten constants obtained for transendothelial transport" Single barrier

Case

Ji:tax

Kat

J!ax

J(f;

Tmax

Kt

A B C D E

600 600 600 600 1,200

10 10 10 10 10

300 600 1,200 3,000 3,000

20 10 5 0.1 0.1

189 300 406 507 867

16.3 10.0 8.3 8.3 7.1



·

'-�

�'"

o E

.

(lJ

Influence of net transport

1max and Tmax have units of ""mol hg-I min-', and umts of mM. !!J = 50 ""mol hg-I min -I.



:J. .

DISCUSSION

Membrane b

E 'OJ L

�o .­ N

These equations express the endothelial Michaelis­ Menten constants as functions of the properties of the individual membranes and of the net flux of sub­ strate through the system. We illustrate the equa­ tions by five examples given in Table 1. The Eadie-Hofstee plots for cases A-D are shown in Fig. 2, in which the net transport of sub­ strate IlJ is the same in all plots but the kinetic constants of the membranes vary as shown in Table 1.

Membrane a

o o U>

(12)

Jmax

245

Kt

has

U

(J)



. 0

p-�

00•

12.

P1S

D

36.

24.

60.

4B.

[ml/hg/minl

FIG. 2. Relationship between P ,SCa and P,S for the two­ membrane model for the four cases (A-D) listed in Table 1. The lines were calculated as described in the text. The points represent concentrations of native substrate (Ca) in the vas­ cular compartment of "0", 5, 10, 20, 30, 40, 50, and mM. "00"

without knowing the exact magnitude of �J, i.e., the rate of glucose consumption, within reasonable limits (0-200 flmol hg-I min-I). However, the mag­ nitude of the effect depends on the relationship be­ tween K� and K�, and the greater the difference between these values, the greater the effect. Discrepancy between tracer and net transport

The Michaelis-Menten constants that describe the tracer flux need not describe the net flux of the parent substance or native substrate. In other words, the Michaelis-Menten constants determined in a tracer experiment need not equal the apparent constants determined in a steady-state experiment with native substrate, unless membranes a and b are equal. We originally defined the net transport by Eqs. 10 and 11 as a function of the properties of the individual membranes. In the steady state, �J is defined by a relationship used by Buschiazzo et al. (1970) and Pappenheimer and Setchell (1973):

�J= Tmax

( K(

Ca

+ Ca

-

Ce

K( + Ce

)

(14)

This equation, however, does not yield the "cor­ rect" value of Ce if the membranes are unequal. In case D in Table 1, the correct interstitial glucose concentration calculated from Eqs. 10 and 11 is 3.18 mM. The interstitial glucose concentration calcu­ lated by Eq. 14 would be 6.71 mM, using the values of Tmax and K( for case D in Table 1. As a result, the endothelial Michaelis-Menten constants cannot be used to estimate the interstitial substrate con­ centration if the membranes are unequal. Our analysis demonstrates that the discrepancy between the brain glucose concentration calculated

J

Cereb Blood Flow Metabol, Vol. 4, No.2, 1984

A. GJEDDE AND O. CHRISTENSEN

246

with T max and Kt for the single-membrane model and those calculated with the correct constants for the double-membrane model increases with in­ creasing plasma glucose concentration. Figure 3 shows a whole range of glucose concentration in the brain calculated for case D in Table 1. The lower curve (open circles) represents the brain glucose concentrations calculated by Eqs. 10 and 11 with the Michaelis-Menten constants for the luminal and abluminal membranes; the upper curve (filled cir­ cles) represents the brain glucose concentrations calculated by Eq. 14 with T max and Kt for the single­ membrane model. The logic of the discrepancy illustrated above can be used experimentally to address the question of the relationship between the membranes. For ex­ ample, if actual experiments yielded the values of T max and Kt shown in Table 1 for case D, while the brain concentrations varied with the arterial plasma concentrations as shown in the lower curve in Fig. 3, then the two membranes would be judged to be unequal. If, on the other hand, Eq. 14 could be fitted to the concentrations of the upper curve in Fig. 3 and yield Michaelis-Menten constants iden­ tical to the constants measured in tracer experi­ ments, then the two membranes would be judged to be equal. The sensitivity of this comparison must necessarily depend on the accuracy of the mea­ surements. Therefore, it is not possible generally to state how different the constants would have to be to exclude the null hypothesis that the membranes are equal.

o E

=loo ()

OJ

ui

-

Reference

10.

20.

30.

plasma glc

40.

[mMJ

Buschiazzo et al. (1970) Lewis et al. (1974)

50.

Betz et al. (1975)

FIG. 3. Relationship between plasma and brain glucose con­

centrations in case D in Table 1. Abscissa: plasma glucose concentration (mM ); ordinate: brain glucose content (IJ..mol/g). The upper graph (filled circles) represents values calculated by Eq. 14 using estimates of Tmax and Kt. The lower graph (open circles) represents values calculated by Eqs. 10 and 11 using the Michaelis-Menten constants of the indi­ vidual membranes. In both graphs, the glucose consumption (net glucose transport) rate was assumed to be 50 IJ..mol hg t min-1. -

J

-

TABLE 2. Michaelis-Menten constants estimated from four studies on the relationship between plasma and brain glucose concentrationsa

.

N

C -

co L .n

Tracer studies on the transport of glucose from blood to brain have yielded T max values of 200-400 j..Lmol hg 1 min 1 and Kt values of 6-7 mM for dogs and rats, respectively (Christensen et al., 1982; Gjedde, 1983). If the analysis of steady-state plasma and brain glucose concentrations over a sufficiently wide range were to yield similar values, the two membranes would not have grossly different glu­ cose transport properties. In no case, however, have both analyses (tracer and steady-state esti­ mates) been performed and subsequently com­ pared. Measurements of corresponding values for brain and plasma glucose do, of course, exist in the lit­ erature, but only Buschiazzo et al. (1970) attempted to analyze steady-state brain glucose concentra­ tions as a function of the plasma glucose concen­ tration as expressed in Eq. 14. The result of their analysis is shown in Table 2. We chose, in addition, to analyze the measurements of Lewis et al. (1974), Betz et al. (1975), and Gjedde and Diemer (1983). The first two studies were discussed previously by Lund-Andersen (1979), and the study by Betz et al. (1975) was also analyzed by Christensen et al. (1982). In the study by Betz et al. (1975), both steady-state and tracer measurements were carried out, but kinetic analyses of the two types of mea­ surements were not compared. For the three studies, we fitted Eq. 14 to the pub­ lished plasma and brain glucose values, as shown in Fig. 4. The fitting used a nonlinear, least-squares normalized regression analysis that yielded the most likely estimates of the T maxi6.J ratio and Kt. The estimates are given in Table 2. We also estab­ lished theoretical relationships between brain and plasma glucose based on the estimates. The estimated values all lie within the range mea­ sured in tracer experiments. In the study by Betz

Cereh Blood Flow Melahol, Vol. 4. No.2, 1984

Gjedde and Diemer (1983)

Tma/Il.! ratio

Kt

(mM)

7

3.1 3.7 5.5 7.9

±

±

±

n

0.1 1.1 1.3

7.2 6.8 5.0

±

±

±

0.6 2.4 1.8

42 8 18

a The values for Buschiazzo et al. (1970) are estimates re­ ported by the authors. In the other cases, the values are non· linear least-squares computer estimates ± SO, obtained by fit· ting Eq. 14 to the original numbers reported by the authors. In the case of Lewis et al. (1974), we converted arterial blood glu­ cose to plasma glucose values by assuming 80% of whole-blood glucose to be in the plasma fraction. In the case of Gjedde and Diemer (1983), only the average values have been published pre­ viously.

TWO MEMBRANES OF BRAIN ENDOTHELIUM

247

Equation 15 now reduces to the simple expression

Kt o E . =1'"

=

x� +

(1 -

x)Kf

( 16)

The quantity x is zero when J�ax 0 and ap­ proaches unity when J�ax becomes much larger than J�ax' The value of unity can, however, never be reached unless !1J O. Equation 16 shows that Kt varies linearly between Kf and � as x varies from zero to unity. We demonstrate that the ap­ parent Kt for a single barrier must always assume a value between Kf and K� for a cell with two mem­ branes in series. In addition, as J �ax increases rel­ ative to J�ax' Kt approaches �; similarly, as J�ax becomes large relative to J �ax' Kt approaches Kf. Thus, Kt approaches the value of the transport-lim­ iting membrane, i.e., the cell membrane with the lowest J max value. More important is the demonstration that changes in the transport kinetics of only one of the two membranes in series may change the measured constants T ma x and K t. This explains why some studies on kinetic changes in the blood-brain bar­ rier may give evidence of a change in both T max and Kt. =

32.

Betz

64#

8.

Lewis

ImMJ

16.

ImMJ

10.

GJedde

20.

ImMI

FIG. 4. Three graphs showing the relationship between

plasma and brain glucose concentrations, redrawn from values published by Betz et al. (1975), Lewis et al. (1974), and Gjedde and Diemer (1983). Abscissae: plasma glucose con­ centrations (mM); ordinates: brain glucose contents (fLmol/ g). Equation 14 was fitted to all values and the resulting es­ timates of Tmax and Kt listed in Table 2. The curves were drawn on the basis of the estimates, also using Eq. 14. Only in the case of the isolated, perfused dog head was it possible to reach plasma (perfusate) concentrations close to 80 mM.

et ai. (1975), the tracer part of the study yielded the values T max 165 ± 6 f.lmol hg-I min-I and Kt 6.2 ± 0.4 mM (Christensen et ai., 1982). When a net glucose transport of 30 f.lmol hg-I min - I in the dog (Lund-Andersen, 1979) is assumed, the tracer esti­ mate corresponds precisely to the T maxf!1 J ratio es­ timated from the steady-state measurements. It is true that T max/!1 J ratios have shown a steady in­ crease since 1970. The main reason for this increase is the more accurate determination of brain glucose contents with rapid inactivation of glycolytic en­ zymes. The similarity of the Kt and T max values for glucose transport across brain capillaries estimated from tracer and glucose measurements and our theoretical analysis of transport across two mem­ branes in series suggests that transport across the luminal and abluminal membranes is the same. Adaptive changes

If the transport of one of the two membranes changes, both T max and Kt may change. Assume, for example, that the J �ax of membrane a doubles as a result of an inductive process (a change from case D to case E in Table 1). Although Kf and K� do not change in this example, T max must increase from 507 to 867 f.lmol hg-I min-I and Kt must de­ crease from 8.3 to 7.1 mM. This is an example of a general relationship be­ tween Kl' and Kf and K�. In Eq. 13 we introduce the expression Q !1 J( K� - Kf). Multiplying the numerator and denominator in this relation by Kf, we obtain =

J �axKfK� J �axKf For simplicity, let

+ ( J�axKf + + J�axKf +

Q)Kf Q

(15)

=

Biaffinity transport

Transport across the blood-brain barrier has oc­ casionally been described in terms of more than one transport system (Gjedde, 1981). One question that naturally arises when two membranes are involved in the transport is whether different transport prop­ erties of the two membranes can be manifested as biaffinity transport and hence spuriously inter­ preted as evidence of more than a single transport system in a single membrane. In other words, do unequal membranes per se yield nonlinear Eadie­ Hofstee plots (in the same manner that heteroge­ neous receptor populations yield nonlinear Scat­ chard plots)? The answer, based on the plots in Fig. 2, is negative; the two membranes, when unequal, do not cause the plot of P SCa versus P S to depart I I from that of a straight line at either extreme. CONCLUSIONS

Experimental determinations of clearance across the cerebral capillary endothelium and the subse­ quent calculation of an apparent permeability for facilitated diffusion across the endothelium permit estimates to be made of the Michaelis-Menten con­ stants T max and Kt· For apparent permeabilities well above 10-6 cm s-I, the volume of the endothelial cell is too small to delay tracer transport measurably. In that case, simple mathematical expressions describe the re­ lationship between the calculated MichaelisJ

Cereb Blood Flow Metabol, Vol. 4, No.2,

1984

A. GJEDDE AND O. CHRISTENSEN

248

Menten constants and the properties of the indi­ vidual membranes. If the membranes are identical, Tmax will be half the maximal transport rate of the individual membranes. If the membranes are not identical, a discrepancy will exist between the values of Tmax and Kt calcu­ lated in tracer experiments and those of Tmax and Kt estimated from glucose concentrations in plasma and brain. In that case, the steady-state formulation of the Michaelis-Menten equation (Eq. 14) will yield tissue concentrations that are too high. The dis­ crepancy in turn is experimental proof of unequal membranes. However, an analysis of three pub­ lished sets of values of brain and plasma glucose concentrations gave no evidence of unequal mem­ branes. Adaptive changes in the transport (1maJ of only one membrane lead to changes in both Tmax and Kt when Kf and K� differ.

6.J PIS

K�

PjS

+

J�ax

J�ax

K�

+

K�

CI Cj

+

+

Cj

J�ax

K�

+

(9a)

Cj

The cell concentration Cj in this equation is implic­ itly given by Eq. 10:

6.J

=

J�ax

(

CI

Kf

+

CI

(10)

To facilitate algebraic manipulations, we introduce dimensionless units. Fluxes are measured in units of 6.J:

, Ja

=

J�ax 6.J

Concentrations are measured in units of Kf: _

CI Kat

C1

(A2)

+

+

C;

ex

+

C;

+

C;

' Jb

(9b)

PIS

=

--- --- -- /( l + C;)] C;) '-------'-.:.... CD + J;/ex 1)

+

[(1 6.J J;J;' --"---"--" K�' (J� + J;')(I

+

We need expressions for 1 + C; and (I C;). Equation 10 is rewritten as

C;

C1

+

+

C;

+

C;

(1 + C;> C;J� J� + (1 + C;>

and therefore +

C ;)/( I

-

which yields

1

+

(9c)

=

C;

J�

' Ja

+

+

C;

(1

+

C;)

This expression for the endothelial cell concentra­ tion is inserted into Eq. 9c. Multiplying the numer­ ator and denominator by J� + (I + C;) and rear­ ranging terms, we obtain

J�J;'

6.J

-------

(J�

Kat C'I

+

+

Jt,)

+

Jt,

+

exJ�

+

(ex - 1)

J()

+

(ex - I)

(1;,

+

(ex - 1)

(9d)

Equation 9d is put into the form of a usual Mi­ chaelis-Menten relation. When dimensionless vari­ ables are replaced by original variables from Eqs. A 1- A3, one obtains

ex

=

Kbt Kat

-

Now Eq. 9a can be written as

Cereh Blood Flo\\' Me/abol, Vol. 4, No.2, 1984

where the apparent single-membrane transport con­ stants Tmax and Kt are given by Eqs. 12 and 13. GLOSSARY

and we introduce the ratio

J

J�

ex

(AI)

-

' C 1-

C;

Numerator and denominator are multiplied by the common factor (I + C;)(ex + C i):

APPENDIX

J�ax

+

1

Kat

C1

Derivation of the transport constants Tmax and Kt for the two-membrane model Equations 5 are substituted into Eq. 9:

Jt,

J;,

(A3)

The letter "x" in the terms below stands for the symbols a, - a, or b, for luminal membrane in the direction blood-to-cell or cell-to-blood, and the abluminal membrane, respectively.

TWO MEMBRANES OF BRAIN ENDOTHELIUM

C i( 1) C.I q(t) or C'i'I C, q(t) or Cj F ilg

J�x a

Arterial concentration of native sub­ strate in source fluid (plasma) Tracer concentration in arterial plasma at time T Concentration of native substrate in endothelial cells Local concentration of tracer in endo­ thelial cells Local plasma concentration of native substrate Local plasma concentration of tracer Source fluid (plasma) flow to brain or unit weight of brain Weight of infinitesimally small region of brain Maximal transport capacity of single endothelial membrane Net steady-state flux of native sub­ strate across individual endothelial membranes Unidirectional clearance of tracer from blood to unit weight of brain Half-saturation concentration of trans­ endothelial transport

M* (1)

Half-saturation concentration of single endothelial membrane transport Tracer amount per unit weight of brain at time T Apparent permeability of endothelium during transendothelial transport, per unit weight of brain

S

Apparent permeability of individual en­ dothelial membrane during transport of native substrate, per unit weight of brain Endothelial surface area, per unit weight of brain Maximal transport capacity per unit weight of brain of trans endothelial transport of native substrate

249

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J

Ce/'eb Blood Flow Metabol, Vol.

4,

No.2,

1984