MODELLING OF MINIATURE ENDPLATE CURRENT UDC 57.001.57:612.815

N. R. Nigmatullin, V. A. Snetkov, E. E. Nikol'skii, and L. G. Magazanik

From analyzing the mathematical model of miniature endplate current (MEPC) generation described previously an optimal set of parameters was found providing the best match between the results of stimulation and experimentally obtained findings. The time course of MEPC in the controls, after cholinesterase inhibition, and following a bungarotoxin-induced reduction in the density of unoccupied cholinoreceptors were described within the framework of the model. This model also satisfactorily describes voltage-dependent current decay, and the current-voltage relationship of MEPC, as well as the kinetics of "giant" MEPC observed during cholinesterase inhibition. The influence of the parameters of the model on "model" MEPC is also examined. The good match between the results of modelling and experimental findings leads to the conclusion that the model gives a true picture of different processes contributing to the generation of MEPC. INTRODUCTION Combined experimental findings help us to quantify only some of the processes underlying synaptic transmission. Traditional approaches therefore require supplementing with mathematical modelling. Models make it possible to examine stages in the generation of synaptic response not yet (directly) experimentally measured and to verify hypotheses on the mechanism underlying interaction between the transmitter and postsynaptic receptors, as well as postulate values for unknown parameters and predict changes in synaptic response during further experimental procedures. A range of models is now available reproducing, with various degrees of accuracy, the time course of postsynaptic currents at the neuromuscular junction both in the normal and during certain types of intervention [2, 9, i0, 16, 18j. This article set out to produce a detailed analysis of the relationship between model MEPC and a wide range of modeled parameters [2]. Findings from this analysis were used to further optimize the series of parameters and define the limits of applicability Of the MEPC model. DESCRIPTION OF THE MODEL The numerical model of 5~PC generation at the frog neuromuscular junction produced previously [2, i8] describes MEPC during voltage gating at the membrane under differing experimental conditions. It supposed that four main processes underly the rise of MEPC. The first of these processes consisted of release of one quantum of acetylcholine (ACh) by the nerve ending. In the model, ACh appears at the center of the synaptic zone within a circle of radius V almost instantaneously (starting condition). The second process consists of interaction between ACh and the acetylcholine receptor (AChR) of the postsynaptic membrane, described in the following simplified form:

2k+l

k+2

2A -F P ~---" A -}- AP .~-~-- ASP',

where A = ACh and P = AChR. According to the above, time during which the receptor (channel) remains in the open state A2P* equals the period of the receptor complex persising with two molecules of agonist. Transitory open/closed and desensitized states of the channel are not included, since they barely affect the time course of MEPC [5]. S. V. Kurashov Medical Institute, Kazan', Institute of Evolutionary Physiology and Biochemistry, Academy of Sciences of the USSR, Leningrad. Translated from Neirofiziologiya Vol. 20, No. 3, pp. 390-397, May-June, 1988. Original article submitted M a y 5, 1987.

0090-2977/88/2003-0289512.50

9 1988 Plenum Publishing Corporation

289

TABLE i.

Standard Set of Model Parameters .Value

[

parameter

Number of .A_Chmolecules per quantum (AO) Density of AChR (CO) 2k+t k-i k+~

12200~m- z 10000 3,3.107mole- I .liter.sec-l 1,3 msec- • 2,0.107mole -I .liter.sec- 1

0,82 msec - I 2400 IJm-2 20,0.107 tool!e- z . l i t e r - s e c -

Density of active AChE center (EO)

k~

16,0

d o e f f i ~ i e n t of ACh fl~ffusio~ D ~4adius6f p o s t s y n a p t i c zone R[ O r ~ i n a t - i n g radius" of ACA~V~HeiEht of synap_tic_ dleft R" Period of modelling

9

msec-

6,0.10 -6 msec- i 1,3 ~m 0,025 ]~m 0,05 msee more than 6 msec

Source

[121

fl7] 91

.

this stuay thls study

[3] thls study thzs study,

risi Fls]

Note. ACh) acetycholine; ACHE) acetylcholine esterase; AChR) acetylcholine receptor; 2k+,, k-l, k+a, 2k-2, kl, ks) kinetic constants.

The third of the processes mentioned consists of a c e t y l c h o l i n e s t e r a s e - i n d u c e d hydrolysis at the synaptic cleft, expressed thus: kI

ACh

~za

A+E.-~AE--~E+II, where E = acetylcholinesterase (ACHE), P = product of ACh hydrolysis, and AE = enzyme-substrate complex. The above does not describe the phenomenon of substrate inhibition, which is thoughtto make only a small contribution to the kinetics of MEPC [16]. The acetylation stage has also been omitted, since deacetylation was known to be the rate-limiting step [7]. The fourth process is that of ACh diffusing away from the center of the synaptic zone. A disk stands for radius R and H (height) in the model of the ACh action zone. Processes occurring within the space of this disk are not examined in the model (limiting conditions). The MEPC obtained in the model, expressed as l(t), is a change in time in the total number of open ionic channels n(t) multiplied by conductance of a single channel y and the difference between membrane potential (MP = U) and reversal potential Uo: 1(0---- V " n (0 9 ( U - - U0).

Values of 25 pCm and --i0 mV for y and Uo respectively were used during this study. All the results from modelling described previously [18] were reproduced in our earlier article [2]. It was found from analyzing the latter that adopting the set of parameters recommended by the writers [18] only produced a very rough a p p r o x i m a t i o n of the effects of AChE inhibition and ensuing reduction in the density of AChR. Accordingly, decay rate of model t~PC during AChE inhibition was 1 89 times less than found during experimentation [ii]. Reduced density of unoccupied A C h R induced by ~-bungarotoxin during AChE inhibition led to accelerated decay of model MEPC to a level exceeding control MEPC decay rate, which similarly conflicts with experimental findings [3, 8, ii]. These discrepancies could be eliminated by refining the set of parameters used and the contribution of factors determining the time course of ~ P C in phasic and tonic frog fibers [2]. Successful modeling of MEPC could thus be achieved during intact AChE and normal AChR distribution with fairly divergent sets of parameters [2, 18] -- a fact worth further study. Model MEPC depend, moreover, on large numbers of parameters, many of w h i c h are only roughly estimated during experiments and subsequently further refined. Parameters of the model were therefore varied systematically in order to pinpoint their effects on the pattern of MEPC. The main processes listed above may be described by a set of nonlinear differential equations in quotients of derivatives within a two dimensional system of coordinates with initial and limiting conditions [18]. Velocity constants 2k+i, k+2, and kl of binding between ACh on the one hand and AChR and AChE on the other were counted into a two-dimensional 290

TABLE 2. Model

Findings Showing Variations

in Parameters of the

Parameters of end late currents Irisfng [ number of decay t i m e I time to I channels constant, Ipeak Isvel,Jopen at peak ~ec to 0.8 of ~ ~/sec 'J amplitude~~ | rising time from 0.2

Measurements

C o n t r o l l e v e l (see Table 1) AO 6100 2444)0 24400(EO=0) CO 3500 12000 I000 ( E O = 0 ) 1,0 2k+l 2,0 20,0 k_~ 0,5 5,0 1,0 k+2 3,0 0,25 2k- 2 0,5 1,0 2,0 EO 0 500 3000 k i 1,0 l 0,0 30,0 k3 1,0 5,0 32,0 D 3,0 4,0 10,0 R 0,5 0,7 2.3 2,3 (EO=O) 2,3 lEO = 0, AO = 24400) V 0,1 0,5 H 0,04 0,06

71 57 93 165 95 66 314 87 78 57 71 73 74 70 80 76 69 62 129 105 66 116 81 67 93 83 64 93 83 6! 57 69 72 130 168 71 67 66 77

211 174 256 531 250 203 778 246 217 182 203 219 209 202 293 235 194 162 450 344 188 385 244 188 299 250 182 262 225 l&7 167 197 219 442 563 203 213 189 226

1432 496 3603 5614 731 1563 754 934 1244 1776 1435 1399 9S4 1738 1546 1489 1404 1283 2263 1966 1334 2065 1568 1369 1682 1580 1319 1804 1667 If08 1242 1406 1434 2272 5658 1430 1085 1638 1270

1,44 1,4I 1,47 4,25 1,30 1,48 1,55 1,33 1,39 1,74 1,67 1,26 1.37 1,51 4,21 2,20 1,22 0,69 3,87 2,13 1,40 3,01 1,64 1,37 1,65 1,46 1,43 1,47 1,46 1,42 1,38 1,43 1,44 8,16 9,58 1,44 1,44 1,46 1,43

Note. Measurements given according to the following system: umole-msec--~m. Notations as for Table i. system of coordinates according to the equation k' = k/NAH so as to conserve the balance of the equation, where k = the corresponding velocity constant, N A = Avogadro number, and H = width of synaptic cleft. RESULTS AND DISCUSSION An initial set of parameters was selected (see Table i) for use in modeling different experimentally-produced effects at the neuro-muscular junction and for studying the effects of different parameters of the model on pattern of ~ P C . Modelling Action of AChE Inhibitors. Initial parameters of the model were selected primarily so as to match amplitude and decay time constant of MEPC (TMEPC) and tlle pattern of naturally-occurring MEPC both in controls and with total AChE inhibition. The dependence o f TMEPC and the density of AChE in the given situation resembled what was described earlier [2] (Fig. i). Deceleration in MEPC decay only became noticeable with a considerable decline in the number of active AChE cunters. Accordingly, TMEPC increased by only 150% when density of AChE declined five-fold (see Table 2). Complete AChE inhibition changed the entire pattern of MEPC: rMEPC increased 2.8-fold, rising time (20-80%) two-fold, and amplitude rose 1.7fold (see Fig. i). Parameters of MEPC also became far more sensitive to AChR distribution, the geometry of the synaptic cleft, and other factors following AChE inhibition. Thus, the function AChE does nQt just consist of limiting the "life" of free ACh at the svnaptic c l e f t a fact of great importance for preventing residual response at rates of transmission through

291

% 250-

I

meC

2

200-

/5O -

loft.

I

0

25

50

75

Fig.

lflO

1

%

Fig. 2

Fig. i. Parameters of miniature endplate currents (MEPC) plotted as a function of density of active acetylcholine esterase enters (EO). Abscissa) EO expressed as a % of control EO level (see Table i). Ordinate) changes in MEPC parameters expressed as a % of control values (see Table 2); plot I) decay time constant of MEPC; 2) rising time (20-80%); 3) amplitude. Fig. 2. Ratio between amplitude and decay time constant of miniature endplate current (TMEPC) with decline in the density of unoccupied acetylcholine receptors with intact (curve i) and inhibited (2) ACHE. the cholinergic neuromuscular synapse typical of naturally-occurring physiological conditions. A high level of AChE activity is likewise essential for generating highest degree ~f uniform response to quanta of ACh when "natural" variations in parameters (such as number of ACh molecules per quantum, distribution of AChR, and morphology) apply at different synaptic zones. In fact, MEPC does not manifest as a completely uniform population of signals, as emerges particularly clearly following AChE inhibition [3]. It might be enquired which criterion should be adopted to assess the functional activity of ACHE. Various possibilities include MEPC amplitude, rising time, decay time, and ratio between amplitude and ~MEPC" Both experimentation and results from modelling would favor MEPC decay rate, or more precisely, the rise of THEPC above average "life" of channel opening (see Fig. 1 and Table 2) [2]. Amplitude of MEPC is less sensitive to AChE activity. Parameters of the lead front of MEPC could serve as a suitable criterion of AChE activity with this model, but are unsuitable for accurate experimental evaluation [9, 14]. The correlation found between amplitude and TMEPC [i, 14] gives a picture of the whole and thus, while reflecting level of AChE activity, can also affected by the divergence between other parameters at different synaptic zones. The ratio between ~MEPC and average channel opening time is apparently always in excess of unity, since these values may only be matched by virtue of very large numbers of active AChE centers and a decline in MEPC amplitude, which would reduce the efficacy of synaptic transmission. The level of AChE activity existing in reality, whereby the aforementioned ratio equals 1.1-1.4, may thus, in a way, be considered optimal. Modelling the Action of a-Bungarotoxin. Declining numbers of unoccupied receptors (those free to interact with ACh) led to a reduction in MEPC amplitude, accompanied by a slight (just significant) acceleration of decay with intact AChE (see Fig. 2, plot i). This completely fits in with most experimental findings [3, 8, Ii] obtained when investigating the effects of a-bungarotoxin on frog phasic muscle fiber. The picture changed substantially following AChE inhibition, however (see Fig. 2, plot 2); this also fits in with experimentally obtained findings [II]. The initial phase of decline in HEPC amplitude was accompanied by a far more intensive accleration of decay. The angle of slope of the lead front of MEPC (ratio between 20-80% increase in amplitude and duration of this increase) also altered substantially with a decline in the density of AChR. A two-fold reduction in amplitude was accompanied by a 2.5-fold decline in this angle with intact as against 12-fold with inhibited ACHE. Few data are available regarding the parameters of the lead front ef MEPC in view of the difficulties over measuring these under experimental conditions, although it was established that the angle of slope of this front did become less steep during AChE inhibition under the effects of a-bungarotoxin. 292

2,5. a

HA

------u_

2o.

Z~

oJ

o

-/0

?--.

-25G

.

-/~o

4o

.~o my

~250

-/5o

-~o

-.4a-my

Fig. 3. Relationship between amplitude (a) and decay time constant (b) of miniature endplate currents (MEPC) plotted as a function of membrane potential; a) voltage dependence calculated of (only) kinetic constants 2k-= (plot i) and 2k-2 and k+= (2); b, plot i) with intact and 2) inhibited acetylcholinesterase; 3) "giant" MEPC. It should also be mentioned that the wide scatter observed in measurements of duration of MEPC decay declines substantially with reduced density of AChR [3, 8]. In fact, we demonstrated in a previous publication [2] that rMEPC tends towards average channel opening time under these circumstances and that part played by naturally-occurring discrepancies in the conditions of MEPC generation at differen~ synaptic zones declines. Modelling Voltage Dependence. A simple change in the constant 2k_2 [18] enables the relationship between TMEPC and MP to be described satisfactorily. The current--voltage relationship of MEPC does not then fit in with experimental findings [6], however, which pointed to a nonlinear relationship. We therefore postulated that two kinetic constants, k+2 and 2k_2 are at once dependent on MP, the values of which could be determined by the equation

2k_2=Bzexp(AzU), k + 2 = B 2 exp(AzU).

where A~ ~ 0.00795 m V -I, Bz = 1.67 msec -z, A2 = 0.0032 mV -I, B2 = 0.0267 uM -I x L x msec -I (at 20-13~ and U is the ~evel of MP [6]. This made a correct description of the naturally occurring current-voltage relationship possible while maintaining the voltage dependence of z ~ p c , which declined somewhat following AChE inhibition (see Fig. 3). Modellin$ "Giant" M~PC. Findings have already been published on the pattern of "giant" MEPC (GHEPC). It was found that TMEPC was twice (or more) above control level. This discrepancy increased even further following AChE inhibition. Since there are reasons to believe that G ~ P C result from the simultaneous release of two or more quanta within a single quantal zone, attempts to reproduce GMEPC in the model used ended in an increase in numbers of ACh molecules per quantum. This increased T ~ p c only marginally, however, whether AChE was inhibited or remained intact. We would note that increased numbers of ACh molecules per quantum actually led to a reduction in rMEPC in comparison with the controls using the set of parameters recommended previously [18]. Attempts to model GMEPC by simply doubling the number of ACh molecules per quantum thus produced results diverging considerably from those of experimentation. Nor did this approach take account of the possibility of AChR activation at neighboring postsynaptic zones due to surplus ACh "seeping" into them and thereby contributing an additional input to GMEPC. The area of transmitter activation tripled (one transmitter release site and two adjoining it) for GMEPC modelling in this context, giving a zone R radius measuring 2.3 ~m in the model. Tripling the action area of the double-size quantum of ACh did not alter GMEPC with AChE remaining intact, but TGMEP C rose to double the level of single quantum response of one zone when AChE was inhibited, which matches experimental findings [4]. A change in MP from --150 to +50 mV was modelled for GMEPC by the above method during AChE inhibition. The results obtained (see Fig. 3b) indicate some reduction in the voltage dependence of ~GMEPC compared with that of ~MEPC -- another characteristic of naturally-occurring GMEPC [4]. Discrepancies between findings from model and experimental

293

20-

20

I0-

-10

9

/_

-

3-20-frO--.~ O-

.

'

lb

'

N

~3b

"107 "msec

o

.

.

/

.

2

.~

-

.~

m~8C

Fig. 4. Dependence of the relative changes of MEPC characteristics on values of formation (a) and decay (b) rates of the ACh-ACh receptor complexes; i) time of increase (20-80%), 2) amplitude, 3) constant time of ~MEPC" (Abscissa should read: 107 mole- :.liter.see-:).

findings with intact AChE may indicate that AChE substrate inhibition'could be involved when raised ACh levels are present at the synaptic cleft -- a factor not allowed for in our model. The complexity of modelling G~fEPC, when selecting the kinetics of interaction between ACh and AChR, may indicate that delayed ACh resulting from repeated binding is not the sole reason for the slow decay of GMEPC. The possibility should not be ruled out that a high concentration of ACh in the zone, in the case of naturally occurring GMEPC, leads to altered kinetics of ionic channels, due, for example, to blockade of open channels by ACh molecules [13]. It is also unlikely that the presynaptic mechanism underlying generation of GMEPC consists solely of an increase in ACh concentration at the synaptic cleft. Influence of Individual Parameters of the Model on Pattern of MEPC. The following conclusions were drawn from analyzing findings from variations in parameters (see Table 2). Increasing the binding rate constant of the first ACh molecule 2k+i leads to a rise in TMEPC and a decline in its rising time (see Fig. 4a). The binding rate constant of the second ACh molecule, k+2, has little effect on the time course of M~PC but does affect amplitude substantially. A change in the dissociation rate constant of the first ACh molecule k-~ influences GMEPC only, and according to a complex pattern (see Fig. 4b). An increase in the ACh--AChE kl interaction rate constant leads to some reduction in the numerical values of all ~ P C parameters, as does a rise in the k~ deacetylation rate constant. We calculated the value of k~ as between 5 and 16 msec -I. Variations in diffusion coefficient D with AChE intact produced hardly any change in T ~ p c , but noticeably affected rising time and amplitude, which increased by 31 and 26% respectively when D was halved. Varying D during AChE inhibition exerted the most pronounced effect on TMEPC with the same ~relative) change in amplitude and rising time applying. It was not possible to analyze the results from all conceivable variations in parameters. Problems associated with thermal influences, postsynaptic potenti-ation, the ionic composition of the environment, etc., requiring separate study, have not been dealt with. It should be pointed out that the original set of parameters selected (see Table i) differ from those chosen in our previous study [2]. Accordingly, density of AChE is reduced and the coefficient of ACh diffusion at the synaptic cleft is higher and approximates to the characteristic level for free diffusion. The value of k_~ is also considerably greater. These changes have hardly any compensatory effect, however, since reduced distribution of AChE leads to a greater likelihood of repeated AChR activation, whereas a rise in k_~ and D reduces this likelihood with either inhibited or intact ACHE. Nonetheless, the relationship between rate constants of the first and second stages in ACh--AChR interaction was such that neither could be considered rate-limiting steps. Thus, "absolute" quantification of a proportion of kinetic constant thus lends a rather arbitrary quality to the model, although these are fairly well intercorrelated and this prevents variations between parameters being too-arbitrary. Findings from this study would confirm that the set of parameters suggested (see Table i) satisfactorily described the effects of AChE inhibition on the amplitude and decay of MEPC, thinning out of AChR with

294

intact AChE and AChE inhibition, and the voltage-dependent decay and current-voltage relationship of MEPC in the --250 to +50 mV range. In addition, the model can partially reproduce GMEPC. Successful application of the model justifies the claim that the prefered combination of the basic parameters gives the true picture of the contribution of the represented processes to the generation of postsynaptic current. LITERATURE CITED I.

2.

3.

4.

5.

6.

7.

8. 9.

i0. ii.

12.

13.

14.

15. 16.

17. 18.

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295