STERIDES BEHAVIOUR IN ELECTRICAL FIELD

Transcription

1 STERIDES BEHAVIOUR IN ELECTRICAL FIELD B. OPRESCU, C. TOPALĂ University of Piteºti, 1, Târgul din Vale street, Piteºti , Romania, Received December 1, 004 By means of a mathematical model of the non-facilitated passive transmembrane microtransport, the influence of the relaxation time of the various ions on the dynamics of cell parameters is demonstrated. Through the equating of the non-facilitated trans-membrane microtransport across the cell membrane with the relaxation processes in a polarized electrolytic cell, the experimental determination is sought of the relaxation time for the solutions of two cholesterol esters in glycerol. The influence of the nature of the compound on its concentration on the relaxation time is found. Key words: trans-membrane potential, relaxation time, cholesterol esters, glycerol. 1. INTRODUCTION A special place is held, within the diversity of the biological membranes, by the cell membranes, seen as natural edges of the building blocks that any living organism is made up of. This part of the biological cell does not only represent a mere geometric limit partly separating its inside from the rest of the universe, but it is a structural part, which discharges functions that are, in all probability, only partially known. Among these functions one should mention: the selective ione, as well as micro- and macromolecules, exchange, through both passive and active processes; cellular recognition, interactions with viruses, with toxins, medicines, and with other cells; bio-electrogenesis (in the case of excitable cells); the extracellular information transfer; morphogenesis. In order to be able to achieve all these functions, the cellular membrane cannot be a homogeneous and isotropic medium, even for the simplest cells. As mentioned earlier, one of the functions of the cell membrane is to transport the various physico-chemical parameters. Out of the diversity of the transport processes, we are going to approach, in the present paper, only the restricted type of the unfacilitated passive trans-membrane microtransport (which means that they lack Paper presented at the 5th International Balkan Workshop on Applied Physics, 5 7 July 004, Constanþa, Romania. Rom. Journ. Phys., Vol. 50, Nos. 9 10, P , Bucharest, 005

2 1180 B. Oprescu, C. Topalã the assistance of an intra-membrane transporter ). Such processes are simple diffusion, osmosis and the drift produced by the gradient of the electric potential. The former two processes of transport are based on the random movement of the molecules or ions as a result of thermic agitation, while the second process appears as a result of the orientation of the ion movement in the membrane electric field. In order for such transport processes to occur it is necessary that the substances to be transported should be dissolved either in the lipid matter (alcohols, fatty acids, a number of medicines), or in the aqueous matrix of the ion canals, as is the case of the ions Na +, K +, Cl, etc. It has been found that sterides (the esters of cholesterol with fatty acids) exhibit mesogeneous properties [1, ]. Sterides that originate in the polynonsaturated fatty acids, such as: cholesteril linoleate (18:), cholesteril linolenate (18:3), which contain two, and, respectively, three double links and are liquid crystals having a melting temperature, or t.t., (T C ) situated below the body temperature (0 35 C). When the sterides originate in the monononsaturaed acids, as is the case of cholesteril oleate (18.1), cholesteril nervonate (4:1), the droplets (or the Malta crosses ) appear at higher temperatures, ranging between 4 C and 5 C. The esters of cholesterol with the saturated fatty acids, like the cholesteril stearate (18:0) or the cholesteril palmitate (16:0) melt at temperatures higher than 70 C [3]. Most biological tissues contain mixtures of the above three types of sterides. As the cholesteric mesogenes which are to be found in the human body have low melting temperatures, the prevalent type of sterides is the one derived from the poly- or non-saturated fatty acids. So far, no single biological tissue was found as containing only cholesteric esters of the saturated fatty acids [4, 5]. Thus, poly- and mono-nonsaturated fatty acids are essential as far as their presence in the membrane structure is concerned. Studies have been carried out that have demonstrated that the major changes in the membrane structure occur as a result of a weak modification of the non-saturation state, which can elucidate the crucial role of the double link in the membrane functions. In the present study, cholesteric esters of a non-saturated acid are used namely, the ricinoleic acid (the I and II compounds). The I and II mesogeneous O CH 3 (CH ) 5 CH CH CH CH (CH ) 7 C O OH I HOOC (CH ) 7 CH CH CH CH O O C O II (CH ) 5 CH 3

3 3 Sterides behaviour in electrical field 1181 sterids have t.t. 41 C, and, respectively, 44 C. This is a principle study, as the ricinoleic acid can only be found in vegetable lipids. The relaxation time of these sterides derived from the ricinoleic acid will be focused on, through modelling the cellular membrane by means of an electrolytic cell.. THE EXPERIMENTAL PART.1. THE PRINCIPLE OF THE METHOD In order to determine the relaxation time of the electrolytic solutions, an electrolytic cell is used, which is polarised through the agency of a direct current power source. After the polarisation, the voltage source is removed, and its electrodes are linked up to a metal conductor, by means of a high resistance. The electric scheme corresponding to such an electric circuit is that presented by Fig. 1. In this figure, we have marked by R 0 and, respectively, L 0 the equivalent electric resistance and, respectively, the equivalent inductance of the portions of the circuit other than the electrolyte (electrodes, linking wires, charge resistors, measuring devices); we have marked by R and L, respectively, the resistance and the inductance of the circuit portion made up of the electrolytic conductor; C K and C A represent the electric capacities of the electrolyte-electrode interfaces for the cathodic region, and, respectively, for the anode one. The two capacities can be replaced by one equivalent capacity, C. Fig. 1. The equivalent electric scheme of a depolarising electrolysor. Let us assume that, at the initial moment, the charge q 0 is situated on the equivalent capacity of the electrolysor. This electric charge tends to be discharged through the external metallic conductor, as well as the internal electrolytic conductor. As a result of the motion and displacement of the electric charges, a time-dependent electric current appears. The electric current that goes through the equivalent inductancies generates, according to the law of self-induction,

4 118 B. Oprescu, C. Topalã 4 electromotive voltages. Taking into account the form of the self-induced electromotive voltages, Kirchhoff s second law, as applied to this circuit, has the following expression [7]: d q ( L + L0) I ( R R0) I dt = + + C (1) Because the following relation holds between the conduction electric current and the electric charge on the equivalent capacity: d q I =, dt the equation (1) is equivalent to the differential linear equation of the second order R+ R + + q = 0 L L d t ( L L ) C dq 0 dq 1 dt We shall make the following notations: R+ R λ= L+ L 0 0 ω = 1 ( L L ) C With the (3) and (4) notations, equation () becomes: d q dq + λ +ω 0 q = 0 (5) dt dt The equation characteristic of this differential linear equation of the second order is: x + λ x+ω = 0 0 The roots of the characteristic equation are: x 1, = λ± λ ω 0 (6) Making use of the roots of the characteristic equation, the form of the general solution of the (5) equation is: xt q A B xt () (3) (4) = e 1 + e (7) where A and B are two arbitrary constants. To be able to continue the analysis of the (7) solution, it is important to know the nature of the roots (6), i.e. whether they are real or complex. In order for these numbers to be real, the following condition must be met:

5 5 Sterides behaviour in electrical field 1183 λ ω 0 Using the (3) and (4) notations, this condition is equivalent to: R R 0 L + L 0 + (8) C As far as the circuit under study is concerned, the R and R 0 resistances are high (of the order of the MΩ s), whereas the inductances are very low, so that condition (8) is abundantly met. The roots (6) can also be written in the following form: ω0 x 1, = λ 1± 1 λ Because, as we have mentioned above, λ>>ω 0 the radical can be written in a series of powers around the 1 value. Focusing only on the first degree terms of the development, the roots of the characteristic equation assume the approximate form: 1 ω0 x1, λ 1± 1 λ Substituting this form of the roots in the (7) solution, we get: 1 ω0 1 ω0 q= Aexp λ t + Bexp λ t λ λ The argument of the first exponential can be approximated by - λ t, so that the solution can be brought to the final form: exp[ ] exp 1 ω q A t B 0 = λ + t λ (9) The dependency of the intensity of the electric current upon the time factor is determined by deriving the (9) relation as to time. Thus we shall obtain: exp[ ] exp 1 ω I C t D 0 = λ + t λ where C and D are two arbitrary constants. From the condition (10)

6 1184 B. Oprescu, C. Topalã 6 results: t = 0 I = I C+ D = I 0 By carefully watching the form of the (10) solution, we find that the first term is a much more rapidly decreasing function than the second term. This makes it possible that, at moments sufficiently far from the initial moment, the evolution of the electric current should be described as an exponential function. When the initial moment is near, its behavior is rather more complex. The semilogarithmic representation of the theoretical dependency of the relaxation current upon the time factor is given in Fig.. 0 Fig.. The theoretical dependency of the intensity of the depolarizing electric current upon the time factor. From the (10) relation, and by considering the expressions of the λ and ω 0 parameters, the expression of the relaxation time results as: τ= λ = ( R+ R 0 ) C (11) ω 0 Let us admit that the electric parameters of the metallic circuit portions remain unaltered. Let us further admit that the geometry of the electrolysor stays the same, as well. In these conditions, for a certain type of electrolytic solution, the relaxation time is dependent only upon the concentration of the electrolyte solution. As the geometry of the system is constant, the variation of the electric resistance of the electrolyte column is determined only by the modification of the electric conductibility of the solution, at the same time as the concentration of the electrolytic solution rises. As is known [8], the electric conductibility of a monovalent electrolytic solution is given by the relation: σ= en ( μ + +μ ) (1) where: e represents the electric charge of an ion, n the ion concentration in the electrolytic solution, and μ + and μ represent the mobilities of the anions and

7 7 Sterides behaviour in electrical field 1185 those of the cations. It is found that the electric resistance of the electrolytic solution column must decrease once with the increase in its concentration. Supposing the capacity of the electrolytic solution-electrode interfaces as being constant, it follows from (11) and from what we have stated above that the relaxation time decreases as the concentration of the electrolytic solution decreases. By experimentally determining the time of relaxation for various concentrations of the electrolytic solution, the magnitude of μ + +μ along the slope of the straight line can be determined: τ= an 1 + b (13) where a and b are two constants... PROCEDURE The I compound, i.e. the 1-hydroxy-9-octadecenoate of cholesteril, was synthesized from cholesterol and ricinoleic acid, in the presence of the p-toluensulphonic acid, according to the indications in the literature. The II compound, i.e. the 1-(cholesteriloxycarboniloxy)-9-octadecenoic acid, was obtained through the reaction of the cholesteril chlorophormate with the ricinoleic acid [6]. For these compounds, determinations of the relaxation time, at concentrations smaller than their concentrations in glycerine, were done. In these conditions, we can consider that all their molecules are dissociated. As the I and II compound hardly dissolve in glycerine at room temperature, all the measurements were made at 90 C. In the final part, some results were presented, concerning the relaxation time in the case of the solutions which contain both compounds. The manner in which the experiment was carried out is the classic one: a potential difference of one volt was applied across the two electrodes immersed in the electrolytic solution. After a certain time was allowed in order to reach the stationary regimen for the functioning of the electrolysor, the electrolysor was commuted in another circuit, where there is no external power supply, but there is instead a high resistance resistor (1,5 MΩ), connected serially between the two electrodes. The time variation of the depolarizing electric current intensity was recorded by means of a data-acquisition system, employing an interface of the UT70B type. To obtain the experimental results presented in the last column of the table, the measurements were repeated for all the samples, and after that the relaxation time was determined for each measurement and, finally, its average value was calculated, as well as the average square deviation..3. RESULTS AND DISCUSSIONS To fully understand the way each individual relaxation time was calculated, let us concentrate on one of the experimental graphs (for instance, the graph in

8 1186 B. Oprescu, C. Topalã 8 Fig. 3). In the top left corner of the image, the experimental evolution of the intensity of the depolarizing electric current is presented in keeping with the time Fig. 3. The time evolution of the depolarizing current for sample 1. Fig. 4. The time evolution of the depolarizing current for sample. factor. What one can find is that, as naturally expected, the intensity of the electric current keeps decreasing in time (except for certain relatively small fluctuations). In the centre is shown the dependency of the logarithm of the ratio between the initial intensity of the electric current and its instantaneous value in

9 9 Sterides behaviour in electrical field 1187 Fig. 5. The time evolution of the depolarizing current for sample 3. keeping with the time factor, after achieving the necessary corrections, as presented in [6]. In the right bottom corner is shown the dependency of the parameter t/ln(i 0 /I) upon time, which is necessary in determining the relaxation time in the Honciuc-Pãun method [9]. By ignoring the insignificant deviations from linearity within the region of the high time periods, caused by the high measuring errors, corrresponding to the very low depolarizing currents, we find that the shape of experimental curves is identical to the one deduced by means of the equivalent electric scheme. The samples analysed, the chemical composition, both qualitative and quantitative, as well as the values of the relaxation time determined experimentally are presented in Table 1. Table 1 The structure of the samples investigated and the values of the relaxation time Sample Compound (mols) Glycerine (mols) Relaxation time (s) I II 1 1, , ±10 1, , ±15 3 1, , , ±3 3. THEORETICAL CONSIDERATIONS The volume of the biological cell is much smaller than that of the medium it is in contact with. That is why we can consider that the natural medium is a

10 1188 B. Oprescu, C. Topalã 10 thermodynamic reservoir for the cell. That means that, irrespective of the values assumed by the thermodynamic flows that cross the cellular membrane, the thermodynamic parameters of the medium remain unaltered. Obviously, in the immediate vicinity of the cell, the concentration of the metabolic products, for example, will be higher than that occurring at a greater distance, but, taking into account the fact that membranes exhibit much smaller diffusion coefficients and mobility than the media within and without the cell, the gradients within and without the cell can be ignored if compared with the gradients inside the cellular membrane. Given the equilibrium trans-membrane potential V 0, if we select, by convention, the medium potential equal to zero. We suppose that in the medium around the cell the substances A, B,. are introduced. These substances penetrate througn the cellular membrane and give rise to a chain of reactions that we can represent symbolically as follows: A + B + X + Y + C + D + where X + Y is an ionic substance that can be dissociated in the intracellular aqueous medium. As a result of these reactions, a continuous transfer of substances A, B, from the medium towards the inside of the cell, and, from the inside of the cell, there is a transfer of X +, Y ions, as well as a transfer of C, D,. molecules. Since we have presupposed that the ambient medium is a thermodynamic reservoir for the cell, we can consider that the density of the A, B,, C, D, molecules, and that of the X +, Y ions is constant with regard with the time factor in the respective medium. Because both the X +, Y ions and the C, D, substances are produced by the cell and they did not exist in the medium before the penetration of the A, B, substances into the cell, it follows that we can consider their concentrations in the ambient enviroment as always null. Within the cell, the thermodynamic parametres are not necessarily constant with regard to the time factor, because of the finite dimensions of the cell volume. In a special manner, the V, [X + ], [Y ] parameters can be altered in time. We will presuppose that, irrespective of the values assumed by the other thermodynamic parametres, the concentrations of the molecules A, B, that penetrate from the medium into the cell are always null inside the cell (because cellular metabolism transforms them into ions and other molecules much faster than the flows through which these substances penetrate). In order to simplify calculation, we shall reduce the problem to the unidimensional case. To do that, we shall consider the cell in Fig. 6. As can be noticed in this figure, the substance transfer is only done through a plane wall, perpendicular to the OZ axis. To simplify writing, we shall further on note the concentrations of the X + and Y ions by X and Y. Let us consider a plane domain having the thickness dz inside the cellular membrane, placed at the z distance along the OZ axis, as in Fig. 7. For the X + ion, the continuity equation is:

11 11 Sterides behaviour in electrical field 1189 Fig. 6. The schematic representation of the problem under study, in the unidimensional situation. Fig. 7. The transport of ions through the infinitesimal element in the cell membrane. Xz d =Φ ( z) Φ ( z+ d z) X because in the volume element there are no processes of generation or recombining of the ion. Admitting the flow as being a continuous, indefinitely derivable variable, we can develop this variable in the Taylor series. Thus, it follows that: Φ X Φ X( z+ d z) =Φ X( z) + d z+... Focusing only on the first two terms of the development, it follows: Φ X = z A similar equation is also obtained for the Y ion: X X (14) Φ = z Y Y (15)

12 1190 B. Oprescu, C. Topalã 1 In order to find the flows of the two ions in position z, we have to know the physical factors which determine them. From the data of the problem can be noticed that these physical factors are: the free diffusion (determined by the concentration gradient between the interior and the exterior of the cell) and the electric drift (determined by the potential gradient between the interior and the exterior of the cell). Considering these two factors, the expressions of the flows will write as: Φ X() z = D X V X μxx (16) and Φ Y() z = D Y V Y +μyy (17) Substituting the relations (16) and (17) in the equations (14) and (15), the result is: X = D X X V V X +μ X +μxx (18) and Y = D Y Y V V Y μ Y μyy (19) From Poisson s equation it follows that: V = e ( X Y ) ε In this relation, we have presupposed that the ions have the electrovalence equal to one. Substituting in relations (18) and (19) the result is: e X X X X X V μ = D X +μ ( ) X X Y (0) ε e YY Y Y Y V μ = D Y μ ( ) Y + X Y (1) ε Because we are interested in the evolution of the physical parameters inside the cell, we will rewrite the above equations for the situation in which z = 0. In this plane, the derivate of the electric potential compared with z is null (because the potential inside the cell is virtually constant). Moreover, in the plane z = 0 a process of generation of the X + and Y ions takes place, as a result of the continuous flow of A, B, molecules, and also of the ensuing chemical reactions inside the cell. In these conditions, it follows that: X(0) e X X(0) X μ (0) = g+ DX [ X(0) Y(0)] () ε

13 13 Sterides behaviour in electrical field 1191 Y(0) e XY(0) Y μ (0) = g+ DY + [ X(0) Y(0)] (3) ε where g is a positive parameter, which characterizes the rate of ion-generating. It is obvious that the derivate of the second order of the ion concentration as to the position must be negative, in the z = 0 plane. We produce the physically reasonable hypothesis that this derivate is a linear function of the ion concentration. With this hypothesis, the previous equations become: and X = g cx dx( X Y) (4) Y = g ey + fy( X Y) (5) where c, d, e and f are constants of the system, which are also positive (in these equations, we have no longer explicitly mentioned the fact that z = 0, as this goes without saying). There is a last question, concerning the variation rate of the cellular potential. To settle it, let us analyse the factors that can lead to the variation of that parameters. As the cellular potential is proportional to the net charge brought about by the two ions (that is to say, to X-Y), it follows that the variation rate of the potential is directly proportional to the difference of the the variation rates of the concentrations of the two ions. The deviation of the cellular potential from the equilibrium value implies supplementary flows of the ions that are essential in determining the membrane potential (i.e. the ions Na +, K + and Cl ). The aim of the migration of these ions (passive or active) is the restoration of the equilibrium potential. We can suppose that the variation rate of the membrane potential determined by this factor is proportional to the value of the difference between the instantaneous potential of the cell and the equilibrium potential. Synthetically summing up all the above remarks, we can make up the following differential linear equation system of the first order: V = a( V V0 ) + b( X Y ) X = g cx dx( X Y) Y = g ey + fy( X Y) (6) Intuitively, we can notice the fact that, as the concentration of the two ions, X and Y, rises inside the cell, as a result of the generation processes g, the flow of their expulsion out of the cell increases (through the processes of diffusion and drift). This leads to the situation where, at a certain moment in time, the number of the ions produced in a time unit is equal to the number of the ions expelled; which means a situation of stationarity. In this stationary regimen, the number of

14 119 B. Oprescu, C. Topalã 14 the positive ions going out of the cell in a time unit is equal to the number of negative ions going out of it in the same time interval. In ionized media physics, such a collective displacement of the two types of ions from the region where they exhibit a higher concentration towards a zone having a lower ion concentration is called dipolar diffusion. Although, in the stationary regimen, the number of the positive ions going out of the cell in a time unit is equal to the number of negative ions going out of the cell in the same time unit, the concentrations of stationarity of the two species are not, however, equal. This can be noticed, for instance, in Fig. 8. We find that, in a stationary regimen, the concentration of the negative ion is higher than that of the positive ion. This Fig. 8. The evolutions of the variables X, Y and V, for g = 10, c = d = 10 and e = f =3, V 0 = 1. Fig. 9. The evolutions of the variables X, Y and V, for g = 100, c = d = 10 and e = f = 3, V 0 = 1.

15 15 Sterides behaviour in electrical field 1193 aspect is generated by the fact that the action of the membrane electric field is exerted on the two types of ions in opposite senses, and also due to the fact that the mobilities of the positive ions are, in this case, higher than those of the negative ions. There is an interesting aspect concerning the manner in which the cellular potential evolves. When the variation rate of the ion concentration is very high, the cell metabolism cannot compensate for the modification of the trans-membrane potential, which is caused by the difference of concentration of the two ions. If the variation rate of the ion concentration is low, the mechanisms specific to the biological system allow for the compensation of the tendency to variation of the potential, which is caused by the concentration differences between the two ions, so that the cell is restored to its equilibrium potential. As the rate of ion generation is increased, the values of the stationarity concentrations of the two ions increase, too, no less than the maximal deviation of the trans-membrane potential. If the mobilities of the positive ions are lower than those of the negative ions, the stationarity concentrations of the positive ions are higher than those of the negative ion. The evolutions of the parameters X, Y and V with respect to the ion generation rate, in this case, are similar to those in the previous case. The only difference is, obviously, provided by the fact that the trans-membrane potential exhibits a maximum, not a minimum. In Figs. 11 and 1 are presented two situations where the mobility of the positive ions is lower than that of the negative ions. From the above theoretical analysis one can derive the conclusion that the evolutions of the cellular parameters, as a result of the generation of a dissociable ionic substance in an aqueous intracell medium essentially depend on the mobilities of the ions produced as a result of the dissociation across the wall Fig. 10. The evolutions of the variables X, Y and V, for g = 10, c = d = 3 and e = f = 10, V 0 = 1.

16 1194 B. Oprescu, C. Topalã 16 Fig. 11. The evolutions of the variables X, Y and V, for g = 100, c = = d = 3 and e = f = 10, V 0 = 1. of the cell. That is why it is absolutely essential to know the values of these parameters for the various substances of a biological interest. 4. CONCLUSIONS Biological cells exhibit stable stationary states. These states are the result of the constant substance and energy transfer between the cells and the medium in which they are placed. Among the physical parameters of a major importance in characterizing the stationary state of the cell count the difference of membrane potential (V 0 ). If the cell system is perturbed, the physiological mechanisms tend to bring the cell back to the initial state, which includes the restoring of the V 0 value for the membrane potential. In the present paper, we have investigated, by means of a theoretical model, the behaviour of the cells when their equilibrium state is perturbed by injecting into them a substance capable of dissociating in the ions. The ions thus appeared tend to leave the interior of the cell, through processes of diffusion and drift. Owing to the mobilities, the differing diffusion coefficients, as well as the contrary effect the membrane field has on the motion of the ions, there appear differences as far as the ion concentrations within the cell are concerned. The existence of a different number of opposite sign ions inside the cell causes the modification of the potential difference between the interior and the exterior of the cell, that is to say to the perturbation of the state of physiological equilibrium. The perturbation of the membrane potential tends to be annihilated through the processes of active transport of the essential Na +, K + and Cl ions. As a result of the interplay among these factors, the cell can evolve to new stable stationary states or reach unstable regimens, where various

17 17 Sterides behaviour in electrical field 1195 oscillations can develop (be they periodical or random). A determining factor in the evolution of the cell behaviour is represented by the coefficients of diffusion and the mobility of the ions that are being transported. By studying the cell behaviour, we have reached the following conclusions: 1. The behaviour of the cell depends on the rate of ion generation;. At generation rates which are not exceedingly high, the cell reaches, after a certain time interval, a new stationary state; 3. In the new stationary state, the concentrations of the opposite signs ions are different; 4. The ion concentrations corresponding to the stationarity states increase as their generation rate is increased; 5. The trans-membrane potential is deviated from the state of physiological equilibrium when the variation rates of the ion concentration are too high; 6. The maximal deviation of the potential from the equilibrium state increases as the generation rate is increased; 7. The concentrations of equilibrium of the ions, as well as the maximal deviation of the potential from the state of stationarity depend on the mobilities of the ions; 8. The experiment has proved that the relaxation time of the ions depends on the nature of the chemical compounds they originate in, and also on their concentration in the respective solutions. REFERENCES 1. G. H. Brown and W. G. Show, Chemistry Review, 57, 1049 (1957).. M. J. Vogel and E. M. Barall II, C. P. Mignosa, IBM J. Res. Develop., 5 (1971). 3. D. M. Small, The Physical Chemistry of Lipids, Plenum Press New York, 395 (1986). 4. D. A. Waugh and D. M. Small, Lab Invest, 51, 70 (1984). 5. D. M. Small, Arteriosclerosis, 8, 103 (1988). 6. B. Oprescu and C. Topalã, Revista de chimie (Bucureºti), 55(5), 341 (004). 7. B. Oprescu and G. Chirleºan, Bazele fizice ale electromagnetismului, Editura Universitãþii din Piteºti, Piteºti, 317 (003). 8. C. D. Neniþescu, Chimie generalã, Editura Didacticã ºi Pedagogicã, Bucureºti, 44 (1980). 9. M. Honciuc and V. Puiu-Pãun, Revista de chimie (Bucureºti), 54(), 117 (003).