ZntJBiomedComput, 24(1989) 119-126 Elsevier Scientific Publishers Ireland Ltd.
COMPUTER SIMULATION FUNDAMENTALS
ENRIQUE SOT0 Universidad Authoma de Pueblo, ZCUAP, Departamento de Ciencias Fisiol6gico.q Apartado Postal 4M, Pueblo; Pue. (Mexico) (Received November 17th. 1988) (Revision received March 17th. 1989) (Accepted March 17th, 1989) A computer simulation of membrane potential generation is presented. It allows the user to vary the intra and extracellular concentrations of sodium, potassium and chloride ions (Na+, K’ and Cl-), and de-termine the membrane potential and the equilibrium potential for each ion. The permeability coefficients for the ionic species considered, and the temperature may also be changed at will. Concentration-potential curves may be obtained at any time. The user may also select a voltage clamp option, which allows him to impose a certain membrane potential value, and study the resulting driving force for each ion. The use of the program in general physiology courses has shown it to be a useful aid for teaching the principles of membrane potential generation. Keywords: Membrane potential; Ion; Teaching; Education; Computer; Models
Introduction With the advent of personal computers, simulation has become a very well suited aid for teaching in science. When the variables which determine a process are well known and susceptible to numerical treatment, and the experimental procedures required to analyze it are difficult to perform, simulation seems a particularly useful tool [1,2]. Membrane potential is one of those processes in which experimental manipulation is difficult. Furthermore, since equations describing it may be solved by a simple mathematical treatment, a complete digital simulation of the process is feasible. In general physiology courses, the lack of knowledge of the mechanisms of ionic equilibrium represents a major obstacle for the students to fully comprehend excitability phenomena. However, learning and teaching the basics of membrane potential is troublesome. In addition, recording the membrane potential experimentally is quite difficult and requires particular skills and equipment that is not commonly available. Furthermore, even when intracellular recording is feasible, technical problems limit the experimental analysis of variables involved in the process and usually hinder their quantification . Thus, I decided to develop a computer simulation that would render teaching and learning the mechanisms of membrane potential generation easy and interactive. The software presented is intended to be used as a complement to the 0020-7101/89/$03.50 0 1989 Elsevier Scientific Publishers Ireland Ltd. Printed and Published in Ireland
experimental and theoretical study of the ionic basis of membrane potential. It allows the student to determine the influence of the intra and extracellular ionic concentrations on the equilibrium potentials of Na+, K+ and Cl-, and on the membrane potential. The program also permits the variation of the membrane permeability coefficients for each ion, and the temperature at which the process occurs. Hence, the user may study the influence of certain parameters over the electrical and diffusional forces which determine the tendency of ions to move across cell membranes. Furthermore, the user may arbitrarily impose a membrane potential (voltage clamp), and study the resultant driving forces for the different ions considered. The simulation focuses on the electrochemical forces which determine ionic movements under voltage clamp conditions. This elementary approach seemed desirable because the voltage clamp literature and simulations are usually complex descriptions of the dynamics of membrane channels and currents [4-71. The program is designed for students taking physiology courses at graduate and undergraduate levels. It can be used as a complement of conventional physiology laboratory experiments, or as a lecture aid. Software Description The program was developed in Turbo PascalTM, in an IBM PC compatible computer (Columbia Data Products, MPC-VP) configured with 256 K RAM and one disk drive. Physiological basis The simulation is based on the very well known Nernst and Goldman equations , which describe the electrical and chemical equilibrium conditions to which ionic concentrations and membrane potential will tend. The equilibrium potential for each ion is calculated using the Nernst equation: RT Vcq. =-InZF
where R, T, 2 and F have their usual meaning; [x], and [X&, are the extra- and intracellular ionic concentrations. According to this equation, the equilibrium potential varies linearly with the temperature and logarithmically with the ionic concentration ratio. The equilibrium potential is the value at which the flow of a certain ion, under its chemical gradient in one direction, is counterbalanced by the flow in the opposite direction that is driven by the electrical gradient. Thus, when the membrane potential reaches the value given by the Nernst equation, for a certain ion, its net passive flow across the membrane ceases [7,8]. The membrane potential (Vm) was calculated using the Goldman’s equation: RT V, = ZF
PNa [Na’l, + PK [K+], + PC1 [Cl-], In PNa [Na+li + PK [K+li + PC1 [Cl-],
Membrane potential simulation
where Px is the membrane permeability for a given ionic species. The equation describes an equilibrium condition, which considers the interaction of various ionic flows, as well as the diffusional restrictions due to the membrane permeability to each ionic species; it determines the membrane potential at which the ionic flow in one direction is counterbalanced by that occurring in the opposite one, thus no net membrane current is generated [7,8]. Program operation The program has been designed in a friendly oriented way and the simulation is lively enough to maintain the student’s interest. Figures presented in this paper are the direct output of the program, and were obtained using the printing routine which is included. Menu selected options define the action to be taken. An introductory text tutorial is incorporated for those students not acquainted with the theoretical background of membrane potential generation. From the main menu, the user may vary the intra- and extracellular concentrations of Cl-, Na+ and K+, the membrane permeability coefficients and the temperature. Default values for ionic concentrations correspond to those found in human red blood cells . The results are graphically displayed (Fig. 1). A schematic cell, net driving forces (represented as vectors), and the corresponding membrane potential are shown on the left portion of the screen. The equilibrium potential value of each ion, the resting membrane potential level, and the zero value are numerically and graphically illustrated on the right. The user may vary the concentration of only one ion at a time, in order to plot the membrane potential against the concentration of the ionic species being studied (Fig. 3). In this case, the concentration is varied by means of the arrow keys, maintaining all the other parameters constant, in the way that is usually performed experimentally. The program allows the user to introduce the data obtained from experimental recordings of membrane potential to compare them with those predicted by the Nernst and Goldman equations. Finally, a table of results obtained from the ionic conditions selected during the whole session may be printed. Once the resultant data are shown, in any of the above described conditions, the user may choose to impose a membrane potential upon the cell (voltage clamp). Hence, the net driving force for each ion varies, depending on the relationship between the ion’s equilibrium potential, and the value at which the membrane potential was clamped [5,9]. The simulation also considers the osmotic equilibria of the cell, calculated by using a simple algebraic analysis of the intra and extracellular concentrations. A maximal difference of 100 mOsm between the intra- and extracellular medium is permitted. Beyond these extreme values, the program shows a cell shrinkage or burst, indicating that these particular ionic concentrations are not likely to occur in reality. Results By using the program, the students may stimulate a variety of experimental conditions. As an example, Fig. 1 illustrates the effect of a permeability change
i .___ EJi + -83 i ____.-__
E-Cl - -74
cl- Y _----- ---_
( J3 35
i ____I_ EJI + -83 : ___._.-__
E-Cl - -74
Fig. 1. Once certain ionic conditions and permeability values have been specified, the main display is shown. The right portion of the screen illustrates the equilibrium potential for each ion. The darkest line represents the membrane potential, and the dotted line the value of zero potential. At the left, a schematic cell is shown with its corresponding membrane potential, and three vectors that represent the net driving forces for Na+, K* and Cl-. A: results obtained using the default values. B: lO&fold permeability change for Na+ was simulated. Comparison of the results in A and B, illustrates the influence of permeability changes upon the membrane potential, and on the tendency of other ions to move. In this particular case, the membrane potential varied from - 76 to + 35 mV, and Cl- and K’, which were near equilibrium in A, showed a clear tendency to enter and leave the cell, respectively. A similar change of Na+ permeability occurs during the generation of the action potential and is responsible for its depolarizing phase.
Membrane potential simulation
upon the membrane potential. Using the default ionic concentrations, and membrane permeability values, a membrane potential of - 76 mV is obtained. Since this value is very close to the equilibrium potential of K+ and Cl-, these ions are almost in equilibrium. In contrast, Na+ tends to flow into the cell, due to the difference between its equilibrium potential and the prevailing membrane potential. If the permeability coefficient for Na’ is increased lOO-fold (Fig. lB), the membrane
U: Uoltaee Clamp E_Ha+ 53 I .._ ____.___.___ .._ _.
EJ + -83 ; ________I
E-Cl - -74
I _-__ E-Cl - -74 E__lt+ -83 i ____.__..____ .._......... .._... Fig. 2. A: The simulation was performed using the default parameters. B: the effect of voltage clamping the membrane potential at - 10 mV is illustrated (V.clamp). As a result of the membrane potential change, Cl- tends to enter, and K* to leave the cell. The net driving force for Na’ is reduced. A similar condition, occurs naturally during the generation of an action potential, in which, the membrane permeability to Na’ increases, then the cell becomes depolarized (as shown in Fig. 1B). Subsequently, K’ tends to move out of the cell, generating the repolarizing phase of the action potential.
potential changes from - 76 to 35 mV. This potential is far from the equilibrium potential for K+ and Cl-. Thus, these ions tend to flow across the cellular membrane as shown by the vectors in Fig. 1B. Since the membrane potential is near the equilibrium potential for Na+, this ion shows a very small tendency to flow into the cell. Note that the equilibrium potentials are not modified by the changes in the permeability coefficients. Figure 2 illustrates a printout of a voltage clamp experiment. The initial condition (Fig. 2A) is the same as that in Fig. 1A. By choosing to clamp the membrane potential, the user may study the driving forces to which ions are subject when current is injected into the cell. In this particular case, the membrane potential was clamped at - 10 mV. Due to the change in membrane voltage, K’ and Cl- which were almost in equilibrium, are driven away from their equilibrium condition and, therefore subject to strong driving forces which tend to move Cl- and K’ in and out of the cell, respectively. In contrast, the net driving force for Na+ ions is reduced (Fig. 2B). In reality, these experimental procedures generate a flow of ions through voltage sensitive ionic channels; the flow of each ionic species depends on its driving force and the specific permeabilities of the corresponding membrane channels
[%7,91. The simulation also allows the user to study the dependence of resting membrane potential on the extra and intracellular ionic concentrations. In Fig. 3, the extracellular K+ concentration is varied from 1 to 500 mEq/l, the membrane potential shows a logarithmic dependence on the external potassium concentration.
P: to printout
any other to continue
7 i ! rru
#---_.i,--_a+__/.-,,.... ,..’ ,. .’
Fig. 3. Plots of the concentration vs. membrane potential may be obtained at any time. The program allows the user to select standard or semilogarithmic plots. The depolarizing effect of increasing extracellular K+ concentrations, from 1 to 500 mM is shown in a semilogarithmic plot. The user may also introduce values obtained experimentally, to compare the results with those predicted by the Goldman equation.
Membrane potential simulation
Due to the membrane potential variation produced by K’ concentration net driving forces for the remaining ions are also modified.
The program developed provides the student with a tool to achieve a better understanding of the mechanisms responsible for the generation of membrane potential. It allows the user to study the influence of different ionic concentrations and membrane permeabilities upon membrane and equilibrium potentials. Border conditions and a variety of situations difficult to achieve experimentally may be simulated. Briefly, the program has been designed as a graphical window to the Nernst and Goldman equations, by which any of their parameters may be dynamically modified. No consideration is given to other variables which influence the membrane potential, such as: the activity of pumping mechanisms (Na’ - K’ pump being particularly relevant), the activity coefficient of ions in the intra- and extracellular medium, fixed charge layers associated with the membrane, and the influence of other permanent ions such as Ca2+ which may become important under certain circumstances [6,7]. The simulation has been used in several graduate general physiology courses taught at our university. Students were encouraged to use the simulation to test their ideas concerning the influence of different ionic conditions and permeabilities on the electrical behavior of a cell. After using the simulation students acquired a deeper comprehension of the functional significance of ionic concentration variations. Hence, users were able to quickly grasp the electrochemical behavior of a cell which is subjected to ionic concentration or membrane permeability changes, and clearly understand the tendency of ions to move, under voltage clamp conditions. The simulation was also helpful in designing an experimental protocol, when students used it previously to laboratory experiments. It provides acquaintance with the experimental data, and may additionally be used to analyze the experimental results HOI. Instructions, texts and related materials have been written in Spanish and English. The program has been fully debugged and is self-documented. However, an additional operation guide facilitates its use. Computational expertise is not necessary. Procedures for keyboard and printer error-trapping are employed to avoid invalid data entries and protect from keyboard mistakes. The main features of the program are: (i) it is easy to use, (ii) it graphically presents the data and (iii) it employs only parameters which have a strict physiological meaning. Acknowledgements
I wish to thank Dr. Oscar Diez-Martinez and Miguel Holmgren for critically reading the manuscript. This research was partially supported by grants from the Consejo Estatal de Ciencia y Tecnologia, Puebla (CECyT, 1987), and by the Secretarla de Educacibn Pdblica (SEP, DGICSA C87-01-0239).
References Ellington HI, Addinall E and Percival F: Games & Simulations in Science Education. Kogan Page Limited/ Nichols Publishing, Co., New York, 1981. Peterson NS and Campbell KB: Simulated laboratory for teaching cardiac mechanics, The Physiologist, 27 (1984) 165-169. Holmgren M, Budelli R and Diez-Martinez 0: Computer simulation of diffusion processes as a teaching aid, Comput Methods Prog Biomed, 27 (1988) l-5. Bezanilla F, Vergara J and Taylor RE: Voltage clamping of excitable membranes, Methods Exp Phys, 20 (1982) 445-511. Moran 0, Smith JE and Requena J: Generaci6n de1 impulso nervioso: un modelo de simulaci6n para microcomputadoras personales coma herramienta auxiliar en la docencia universitaria, Interciencia, 11 (1986) 69-76. Barach JP: A computer simulation routine for cardiac action potentials, J Electrophysiol Tech, 14 (1987) 91-102. MacGregor RJ and Lewis ER: NeuralModeling, Plenum Press, New York, 1977. DeVoe RD and Maloney PC: Principles of cell homeostasis. In Medical Physiology (Ed: V.B. Mountcastle), C.V. Mosby Co., St. Louis, 1980, pp. 3-43. Hille B: Ionic ChanneIs of Excitable Membranes, Sinauer Associates Inc. Sunderland Massachusetts, 1984. Smith ICH: Microcomputers and biologists, Arch BiolMedExp, 19 (1986) 313-321.