JOURNAL OF NEUROBIOLOCY, VOL. 7, NO. 4, PP. 377-381

BRIEF COMMUNICATIONS

A Model for the Localization of Acetylcholine Receptors at the Muscle Endplate

In the normally innervated neuromuscular junction of the frog and rat, the area of greatest sensitivity to the transmitter, acetylcholine, is the region of the muscle membrane immediately adjacent to the nerve terminal; however, there is a low, but measurable sensitivity in the adjacent region (Miledi, 1960b). The sensitivity to acetylcholine is presumably due to the presence of specific protein molecules, or receptors, which lie in the muscle membrane; these have been shown to be highly concentrated in the region of the muscle membrane closely apposed to the nerve terminal (Miledi and Potter, 1971; Barnard, Wieckowski, and Chiu, 1971; Fambrough and Hartzell, 1972; Chang, Chen, and Chuang, 1973; Fertuck and Salpeter, 1974).

The local concentration of the receptor protein molecule raises the question of the underlying mechanism for this localization. The two obvious extreme cases are:

( 1 ) Receptor molecules are Synthesized throughout the cell, are inserted at random into the membrane, where they are free to diffuse; in the region of the nerve terminal, there is a “sticky zone” so that all receptors that approach this zone are stuck, or confined, to this area. The receptors, both in the membrane and in the sticky zone, have a finite half-life.

(2) The receptor molecules are placed in the membrane only in the region of the nerve terminal, and remain confined there.

We propose to demonstrate mathematically the feasibility of mechanism (1 1. The published solution of an electrostatic potential problem will be shown to be essentially applicable to the proposed model. Published values for the muscle fiber and endplate size, and extrajunctional receptor density and possible values for the diffusion constant will be used to calculate the flux into the endplate re- gion. The steady-state concentration in the junctional region will be calculated from the flux and the turnover time of the receptor molecules as estimated from labeled a-bungarotoxin studies.

To estimate the steady-state flux, F, of the diffusing particles into the “sticky zone” we must make two approximations. First the surface of the muscle fiber cell (whose length is 10-100 times its mean circumference) will be replaced by the surface of an infinitely long, right circular cylinder of circumference, 2c (Fig. 1). Secondly, we replace the roughly oval-shaped “sticky zone” of the endplate by a circular disc of comparable area of radius, a, centered at the origin of the coordinate system.

Finally, we obtain a flat surface region (between x = f c , --a, < y < -a,) by slitting the cylinder lengthwise through a point diametrically opposed to the origin and flattening the resultant curved surface. A solution of V2+ = 0 is re- quired satisfying the conditions: ( 1 ) @ = @,,, a constant, when x = f c , c > a; and

377

0 1976 by John Wiley & Sons, Inc.

378 EDWARDS AND FRISCH

/---Slit here a n d flatten

I 1 I

.,J L ++-I -2c-1

Fig. 1. Sketch, to scale, of model of endplate and muscle membrane used; the procedure for slitting and flattening the membrane is illustrated.

(2) @ = 0 on the circle x + y2 = a 2. The solution of this steady-state diffusion problem can be derived directly from the solution to an electrostatic problem published by Knight (1935). Knight found the (dimensionless) potential V of a circular cylinder between two infinite plates (and specifically @/@o = 1 - V). The desired reduced flux Fl@., is numerically equal to the capacity of the cylinder, per unit length, computed by Knight if the constant used in the potential prob- lem, the specific inductive capacity, is replaced by the diffusion coefficient, D, of the particles. The steady-state flux F into the circle of radius, a, can thus be obtained from

where @., is the concentration remote from the endplate. The function ~ ( a / 2 c ) of the dimensionless ratio aI2c is tabulated by Knight in his Table 11, and is plotted in our Figure 2 as ~ ( a / 2 c ) versus aI2c. We find that in the region 0.1 5 a/2c 5 0.25 of biological interest ~ ( a I 2 c ) can be represented to a few percent error by the approximation

5a 3c

K(U12C) 3 0.2 + - For a/2c 5 0.05,

~ ( a / 2 c ) 3 l/ln ( 4 c l ~ a ) , (3 )

otherwise ~ ( a 1 2 c ) is given by a mathematically complicated series. For the endplate, we take the following values for the parameters: D = 2 X

cm2/sec, as measured by Edidin and Fambrough (1973) for the movements of protein antigens on the membrane of cultured rat embryo muscle; @., = 5 re-

LOCALIZATION OF ACETYLCHOLINE RECEPTORS 379

V I I I I 0 0. I 0.2 0.3 0.4

0 / 2c Fig. 2. Plot of the reduced flux (dimensionless), K , as a function of the (dimensionless) ratio, ra-

dius/circumference.

ceptors/p2 (Hartzell and Fambrough, 1972); a = 17p and 2c = 118p (rat dia- phragm, Salpeter and Eldefrawi, 1973). Then a/2c = -14 and K = 0.68 from Figure 2. The steady-state flux into the endplate region may be calculated to be 4.3 receptors/sec.

Let us assume further, that the receptors that enter the endplate region remain there until they are destroyed by some reaction that takes place in the membrane. Chang and Huang (1975) have shown that the half-time of the turnover of [3H]-a-bungaratoxin labeled receptors is about 7.5 days (or 6.5 X lo5 sec). It shall be assumed that this reflects the turnover of the actual receptor molecules. If receptors enter the endplate region at a rate F and turnover with a rate con- stant, 6 (which can be calculated to be 1.1 X lou6 sec-l from the half-life given above assuming a first order reaction), the number of receptors in the endplate in the steady state is the ratio F/S or 3.9 X 106. The values given in the literature for the number of a-bungarotoxin binding sites in rat and mouse diaphragm muscle are somewhat higher, about 1.6 - 4.7 X lo7 (Barnard, Wieckowski and Chiu, 1971; Miledi and Potter, 1971; Fambrough and Hartzell, 1972; Chang, Chen and Chuang, 1973).

Thus, the calculated receptor content a t the endplate agrees with that mea- sured experimentally to within an order of magnitude. There are a number of possibilities which could make this agreement closer or less close. The value for D was taken from measurements of the movement of plasma membrane proteins of molecular weight approximately 200,000 in muscle membrane. Similar measurements in other cells have given a range of values between 10-8 and cm2/sec (Edidin, 1974). In the measurement of extrajunctional re- ceptor density the number is so small that statistical factors introduce a signif- icant error. Finally, in the experiments measuring the half-life of receptors, the time course of the loss of labeled a-bungarotoxin was examined. Therefore, the estimate assumes that the loss of the label reflects the turnover of receptor molecules. There is no evidence to support this assumption, however, and the loss of label could also be brought about by degradation or slow dissociation of

380 EDWARDS AND FRISCH

the toxin. In either case the half-life would be longer, and the number of re- ceptors would be greater.

There is much other evidence to support the proposed model for the local- ization of the acetylcholine receptor. In the course of development, the receptors are apparently distributed uniformly over the surface of muscle fibers, both in uiuo (Diamond and Miledi, 1962; Hartzell and Fambrough, 1972) and in uitro (Hartzell and Fambrough, 1973). Indeed Fambrough and Rash (1971) have noted that while myoblasts lack sensitivity to acetylcholine, fusion to form my- otubes seems to be followed by synthesis of both acetylcholine receptors and myosin; they suggest that this may be due to the activation of a single set of genes. Further the movement of protein molecules in lipid membranes has been dem- onstrated in a number of cells (reviewed by Edidin, 1974).

Following innervation, receptors become highly concentrated in the region of the muscle closely adjacent to the nerve terminal. This could well be a con- sequence of a change in the properties of the muscle membrane of an unknown nature. A gradient in these properties in the region of the membrane closely adjacent to the endplate could possibly be responsible for the gradient of receptor density that has been reported (Miledi, 1960a; Dreyer and Peper, 1974). Den- ervation is followed by an increase in the area of the membrane sensitive to acetylcholine (Axelsson and Thesleff, 1959; Miledi, 1960a). This is due appar- ently to a large increase in the content of receptors in the membrane (Hartzell and Fambrough, 1972). The newly formed receptor molecules seem to be dis- tributed uniformly on the membrane except that a high concentration of receptor molecules is still present at the endplate (Albuquerque and McIssac, 1970; Fambrough, 1970; Hartzell and Fambrough, 1972). The increase in receptor number following denervation may be due to inactivity, since stimulation of a denervated muscle blocks the increase in sensitivity to acetylcholine, and thus presumably the increase in receptor number (Drachman and Witzke, 1972; L@mo and Rosenthal, 1972; Jones and Vrbova, 1974).

Another calculation of interest is to determine how fast receptors would leave the endplate region should they become “unstuck.” The solution for this problem is given and graphed in Figure 4b of Carslaw and Jaeger (1959). The mean concentration of receptors in the sticky zone will be reduced to about 12% when Dtla2 = 2, or for the values assumed, t = 48 min.

The authors are indebted to Drs. Douglas Fambrough and Gary Banker for reading and criticizing the manuscript. This work was supported by grants from the USPHS (NS 07681), NSF (CHE 7404171A02) and ACS (PRF 3519C5,6).

REFERENCES

ALBUQUERQUE, E. X. and MCISSAC, R. J. (1970). Fast and slow mammalian muscles after dener- vation. E x p . Neurol. 2 6 183-202.

AXELSSON, J. and THESLEFF, S. (1959). A study of the super-sensitivity in denervated mammalian skeletal muscle. J . Physiol. 147: 178-193.

BARNARD, E. A., WIECKOWSKI, J., and CHIU, T. H. (1971). Cholinergic receptor molecules and cholinesterase molecules a t mouse skeletal muscle junctions. Nature 234: 207-209.

CARBLAW, H. S. and JAEGER, J. C. (1959). Conduction of Heat in Solids, 2nd ed., Oxford University Press, Oxford, 510 pp.

LOCALIZATION OF ACETYLCHOLINE RECEPTORS 381

CHANG, C. C., CHICN, T. F., and CHUANG, S.-T. (1973). N,O-Di and N,N,0-Tri[3H]acetyl a-bun- garotoxins as specific labelling agents of cholinergic receptors. Brit. J . Pharmacol. 47: 147- 160.

CHANG, C. C. and HUANG, M. C. (1975). Turnover of junctional and extrajunctional acetylcholine receptors of the rat diaphragm. Nature 253: 643-644.

CHANG, C. C. and LEE, C. Y. (1963). Isolation of nerotoxins from the venom of Bungarus multi- cinctus and their modes of neuromuscular blocking action. Archs. int. Pharmocodyn Ther. 144: 241-257.

DIAMOND, J. and MILEDI, R. (1962). A study of foetal and newborn rat muscle fibers. J. Physiol.

DRACHMAN, D. B. and WITZKE, F. (1972). Trophic regulation of acetylcholine sensitivity of muscle: effect of electrical stimulation. Science 176: 514-516.

DREYER, F. and PEPER, K. (1974). The acetylcholine sensitivity in the vicinity of the neuromuscular junction of the frog. Pfliigers Arch. 348: 273-286.

EDIDIN, M. (1974). Rotational and translational diffusion in membranes. Anna. Reu. of Biophys. Bioengin. 3: 179-201.

EDIDIN, M. and FAMBROUGH, D. (1973). Fluidity of the surface of cultured muscle fibers. J. Cell Biol. 57: 27-37.

FAMBROUGH, D. M. (1970). Acetylcholine sensitivity of muscle fiber membranes: Mechanism of regulation by motoneurons. Science 1'68: 372-373.

FAMBROUGH, D. M. and HARTZELL, H. C. (1972). Acetylcholine receptors: Number and distri- bution at neuromuscular junctions in rat diaphragm. Science 176: 189-191.

FAMBROUGH, D. M. and RASH, J . E. (1971). Development of acetylcholine sensitivity during my- ogenesis. Deoelop. Biol. 26: 55-68.

FERTUCK, H. C. and SALPETER, M. M. (1974). Localization of acetylcholine receptor by 1251-labeled a-bungarotoxin binding at mouse motor endplates. Proc. Nat. Acad. Sci. U.S.A. 71: 1376-

HARTZELL, H. C. and FAMBROUGH, D. M. (1972). Acetylcholine receptors: distribution and ex- trajunctional density in rat diaphragm after denervation correlated with acetylcholine sensitivity. J. Gen. Physiol. 6 0 248-262.

HARTZELL, H. C. and FAMBROUGH, D. M. (1973). Acetylcholine receptor production and incor- poration into membranes of developing muscle fibers. Deuel. Biol. 30: 153-165.

JONES, R. and VRBOVA, G. (1974). Two factors responsible for the development of denervation hypersensitivity. J. Physiol. 236: 517-538.

KNIGHT, R. C. (1935). The potential of a circular cylinder between two infinite planes. Proc. London Math. SOC. 39: 272-281.

L ~ M o , T. and ROSENTHAL, J. (1972). Control of ACh sensitivity by muscle activity in the rat. J . Physiol. 221: 493-513.

MILEDI, R. (1960a). The acetylcholine sensitivity of frog muscle fibers after complete or partial denervation. J . Physiol. 151: 1-23.

MILEDI, R. (1960b). Junctional and extrajunctional acetylcholine receptors in skeletal muscle fibers. J . Physiol. 151: 24-30.

MILEDI, R. and POTTER, L. T. (1971). Acetylcholine receptors in muscle fibers. Nature 233: 599-603.

SALPETER, M. M. and ELDEFRAWI, M. E. (1973). Sizes of endplate compartments, densities of acetylcholine receptor and other quantitative aspects of neuromuscular transmission. J. Histo- chem. Cytol. 21: 769-778.

162: 393-408.

1378.

CHARLES EDWARDS H. L. FRISCH

Neurobiology Research Center and Departments of Biology and Chemistry State University of New York a t Albany Albany, New York 12222

Accepted for publication January 7,1976