Appendix D. Kinetics of glucose transport across the blood–brain barrier
The concentration dependence of the rate of glucose influx, Tinf, has often been described empirically using the simplest form of Michaelis–Menten kinetics [310] 28 Tinf=Tmax∗cplasma/Kt+cplasmawhere Tmax is the maximum transport rate and Kt is the Michaelis constant for influx (t and T stand for “transport”). From this expression for the rate, the decrease in clearance with concentration is predicted to be29 CL=Tmax/Kt+cplasma.
Betz et al. [327] reported Tmax = 1.6 µmol g−1 min−1 and Kt= 8.6 mM (based on blood rather than plasma) in the dog based on tracer uptake rate after 1 min of hypoxia, while Pardridge and Oldendorf [538] investigated transport of five different hexoses into rat brains and found Tmax = 1.6 µmol g−1 min−1 for all, but differing apparent dissociation constants with 9 mM for glucose, which they took to imply that the conformation changes of the carrier rather than binding of the substrates were rate limiting. This is plausible because relatively low affinity binding of small substrates to sites is often diffusion controlled and rapid while conformation changes of large molecules may well be much slower. At a plasma concentration of 6 mM these values correspond to a glucose clearance of 100 µL g−1 min−1. Mason et al. [334] lists many values of Tmax and Kt determined from flux studies prior to 1992. These range from 0.5 to 6.7 µmol g−1 min−1 for Tmax and 4.9–11 mM for Kt.
Efflux has not been measured directly, but it can be calculated by difference from the net flux and the influx. The net flux can be calculated from the rates of extraction of glucose from the blood as indicated in Sect. 5.3 (assuming negligible metabolism within the endothelial cells), or from the rates of glucose metabolism, perivascular loss and accumulation within the brain. Perivascular loss is likely to occur, but at a much lower rate than metabolism. At steady-state the rate of accumulation is zero and thus the net flux equals the rate of glucose metabolism,30 Tnet=CMRglc.
The net flux and efflux have often been interpreted using the simplest extension of the Michaelis–Menten description used for influx, i.e. (see e.g. [310, 323, 539]) leading to31 Teff=Tmax∗cisf/Kt+cisfand32 Tnet=Tinf-Teff=Tmaxcplasma/Kt+cplasma-cisf/Kt+cisf.
These equations have been called irreversible Michaelis–Menten kinetics [540] because in Eqs. 31 and 32 the product of the “reaction”, which is the substrate on the far side of membrane after transport, has no effect on the rate of the reaction. This has been described as implying that influx and efflux occur by completely separate mechanisms, which was regarded as being most unlikely (see e.g. [338]). However, it should be noted that Eqs. 31 and 32 and even the extensions of these when two species are present are the same as equations that can be derived from the simple carrier model with the additional assumptions that association and dissociation are rapid, the rate constants for the conformation changes of the carrier are the same with or without a bound substrate and the mechanism is symmetrical, i.e the same viewed from either side [324].
The carrier model for kinetics was introduced to account for the transport of sugars across sheep placenta [323] and human red blood cells [324]. The name “carrier” arose because the model should apply when the transporter collects the substrate on one side of the membrane and “carries” it across the membrane to deposit it on the other. This appears to be the actual physical mechanism for ion transport by low molecular weight ion carriers like nonactin and trinactin ([530, 541, 542] and probably valinomycin [530, 543–545]. However for much larger transporters such as GLUT1, it has always been much more likely that the physical mechanism is somewhat different. The essential feature of carrier kinetics is not transfer of the carrier molecule across the membrane, but rather the change in exposure of the binding site for the substrate. It must be possible for this site to be exposed on each side of the membrane, but not both at the same time. The structures of GLUT1 (see Fig. 12) and related transporters all indicate that there is a transport pathway through the molecule which is occluded or gated at one end or the other and furthermore suggest conformation changes that could close the gate at one end while opening the gate at the other.
A second substrate can inhibit transport of the first by binding to the carriers thus reducing the number of carriers free to complex with the first. However, it is also possible for a second substrate to increase transfer of the first. In the extreme case if the carrier can only change conformation while a substrate is bound, the carrier is an obligatory exchanger and net transfer of one solute can only occur in the presence of another. More generally efflux of a substrate can, by increasing the rate of changes from inward to outward facing conformations increase the availability of carrier to collect a different substrate on the outside and hence its influx. Exchange whether or not obligatory can result in secondary active transport in which uphill transport of one solute is driven by downhill flux of another [322–324, 326]. Manifestations of this coupling are sometimes called variously counter-transport, counter-flow or trans-stimulation (see Fig. 12).
There are, of course, many extensions that can be made to the carrier model, examples include invoking more than one binding site, allowing co-transport and accounting for diffusion limited access in unstirred layers. Some form of extension has been found necessary for GLUT1 transport in red blood cells (see e.g. [546]) and almost certainly, given its greater complexity, will be necessary for glucose transport at the blood–brain barrier. The following is more an empirical description of results than a mechanistic model.
The general solution of the simple carrier model for the steady-state fluxes in terms of the concentrations and the constants of the model can be derived using standard methods for reaction kinetics [547–549]. This solution has been reported and discussed a number of times (see e.g. [326, 327, 339, 550] together with some possible extensions [329, 546].
The “simple” pore is another type of mechanism that has been considered for glucose transport across the blood–brain barrier. Pores can be gated but when the gates are open the transport pathway allows solute movement from one side of the membrane to the other with no further movement of the gates. Lieb and Stein [551] have described the kinetics for movements through “simple” pores that are defined as pores that can be occupied by only one substrate at a time. Generally, because the substrates are small and can move rapidly and no conformation changes of the pore are required, transport rates through an open pore can be large. By contrast for a simple carrier movement of each substrate molecule requires conformation changes, which are likely to be slow.
The simple pore mechanism does not predict or explain counter-transport or trans-stimulation. Reversible Michaelis–Menten kinetics (see [337, 338, 540, 552], which have been used in some descriptions of glucose transport at the blood–brain barrier, are the same as the kinetics of the simplest single occupancy pore [551].
As counter-transport is well established for GLUT1 in red blood cells and both counter-transport and trans-stimulation have been demonstrated for glucose transport at the blood–brain barrier, the simple pore model and reversible Michaelis–Menten kinetics are not considered further here (compare [339]).
Caution is advisable in the interpretation of flux data using the carrier model for at least three reasons: steady-state data cannot determine all of the rate constants even in the “simple” carrier model (see e.g. [326]); there must be transport across two membranes in series; and there is likely to be interaction between the GLUT1 monomers in the tetramers thought to be present in membranes. Thus it would be unwise to attach mechanistic significance to the values of constants determined by fitting the model to data. However, the forms of the equations relating the fluxes to the concentrations [325–327, 550] remain the simplest available framework capable of describing the transport. Using the constants that are defined below,33 Tinf=T1,maxK2cplasma1+cisfRK1K2+K2cplasma+K1cisf+cplasmacisfKinh34 Teff=T2,maxK1cisf1+cplasmaRK1K2+K2cplasma+K1cisf+cplasmacisfKinhand35 Tnet=T1,maxK2cplasma-cisfK1K2+K2cplasma+K1cisf+cplasmacisfKinh36 withT1,max/K1=T2,max/K2.
In these equations cplasma and cisf are concentrations, the Ks are apparent dissociation constants, and the Tmax values are transport maxima for transport in the two directions. The inhibition constant, Kinh, which is dimensionless, and the trans-stimulation constant, R, are related to the constants used by Betz et al. [327] by37 Kinh=K1i/K1=K2i/K2and38 R=R1=R2.1/Kinh indicates a combined effect of glucose in plasma and ISF to compete for transport and 1/R indicates the strength of trans-stimulation. An equation of the same form as Eq. 35 was used by Simpson et al. [315] to describe glucose transport in their modelling of cerebral energy metabolism. A symmetrical version was used by Duarte et al. [341] in their interpretation of data for the amount of glucose in the brain versus plasma concentration.
In the simple carrier model trans-stimulation can increase the unidirectional flux but not the net flux. For the unidirectional flux if conformation changes of the carrier are faster when substrate is bound, trans-stimulation is expected to be more important than the combined inhibition. Alternatively if the conformation changes are faster when substrate is not bound, trans-inhibition is expected to be more important.
If the transport were by the simple carrier model across a single membrane, the empirical constants could be calculated from the number of carriers and the rate constants of the model [325–327, 339, 550].
The investigation of glucose fluxes in the isolated, perfused dog brain by Betz et al. [327] appears to be the only study that allows calculation of rates of efflux for a range of values of both cplasma and cisf. (They measured concentrations of glucose in whole blood, which are expected to be about 11% smaller than those in plasma. In the following the distinction has been ignored). In their study the steady-state values of cisf were calculated as amount accumulated in the brain divided by the volume of distribution, VD = milliliters of brain water per gram of brain; and the dependence of the rate of influx, Tinf, on cplasma was determined for a range of values of cisf preset by perfusing the brains with different concentrations of glucose.
As shown in Fig. 14 over a wide range of glucose concentrations, from roughly 2 mM to 40 mM, at steady-state the amount in the brain and hence cisf increases proportional to (cplasma − offset) where the offset is about 2 mM.
Betz et al. [327] fitted their influx data using39 Tinf=Tapp/Kapp+cplasmawith apparent values of the Michaelis–Menten constants shown in Table 8 for brains with cisf set by pre-exposure to different cplasma. As cisf is increased both the apparent transport maximum and the apparent Michaelis constants for influx increase which is evidence for both competition and trans-stimulation. Betz et al. interpreted the variation of the apparent constants with cisf in terms of Eq. 33. As shown in the Additional file 1, all of the data for influx can be described empirically using Eq. 35. It is possible to calculate the rate of efflux, under steady-state conditions using Teff = Tinf − Tnet = Tinf − CMRglc for the combinations of cplasma and cisf seen at steady-state. Furthermore, if influx is described empirically by Eq. 33 then Eq. 34 is expected to be a reasonable description of efflux over the same range of concentrations and thus can be used to calculate the efflux for all combinations of concentrations using the fitted empirical constants.
Table 8 Parameters obtained by Betz et al. [327] from fits of the simple Michaelis–Menten expression Tinf = Tapp/(Kapp + cplasma) to glucose influx versus plasma concentration, cplasma, for preset concentrations of glucose in the brain, cisf
cisf/mM 6.11 16.8 26.3 43.9 56
Kapp/mM 8.46 11.2 17.7 28.2 37.7
Tapp/µmol g−1 min−1 1.61 1.84 2.21 2.68 3.83
The fits predict that for all cplasma, the net flux, Tnet when cisf is at the corresponding steady-state value is between 0.6 µmol g−1 min−1 and 0.65 µmol g−1 min−1. If it is demanded that the constants used to fit Tinf produce the same Tnet for all steady-state conditions, i.e. that CMRglc, is constant, then the estimated value is 0.65 µmol g−1 min−1. This value is very close to the value expected for rats but somewhat greater than that expected for humans. The fits indicate that an adequate net flux can be maintained for cplasma as low as about 3 mM. Increases in cplasma produce relatively modest increases in influx with matching increases in efflux at steady-state such that the net flux remains constant. The corresponding increase in cisf is shown in Fig. 14. A notable feature of the fits is that for cplasma = 6 mM, if there were no change in transport capacity, glucose consumption, CMRglc, could increase only to about 0.9 µmol g−1 min−1. At that rate, cisf would be close to 0. This limit on CMRglc is substantially below the Tmax value, which can be approached only if cplasma is increased. As discussed in Sect. 6.2, how transport capacity is increased to support nervous activity is not fully understood.
Betz et al’s data show a pronounced trans-stimulation effect, but only for cplasma > ~ 20 mM. For cplasma < ~ 10 mM increasing cisf decreases influx. In terms of the model this is expected because higher cisf reduces the concentration of free carrier available to complex glucose from plasma.