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Acid–base equilibria of polyvalent electrolytes and theoretical description of polyelectrolyte behavior in electrokinetic separations
J. Biochem. Biophys. Methods 46 (2000) 21–30 www.elsevier.com / locate / jbbm
Acid–base equilibria of polyvalent electrolytes and theoretical description of polyelectrolyte behavior in electrokinetic separations a b c, Alexander V. Stoyanov , Irina N. Stozhkova , Pier Giorgio Righetti * a
b
University of Milan, LITA, Segrate, Via Fratelli Cervi 93, 20090 Milan, Italy Frumkin Institute of Electrochemistry, Russian Academy of Sciences, Leninsky Av. 31, 117071 Moscow, Russia c University of Verona, Department of Agricultural & Industrial Biotechnologies, Strada Le Grazie, Ca’ Vignal, 37134 Verona, Italy Received 3 July 2000; accepted 3 July 2000
Abstract The problem of the theoretical description of the dissociation of polyvalent electrolytes is considered. It is shown that the traditional model which assumes some constant microstate mobility values may have limited applications, since for each microstate the rate of conversion into other states is strongly dependent on pH and thus it may often become comparable with the characteristic times of other important processes (ionic atmosphere relaxation, biopolymer acceleration by electric field). The question of correct modelling of the electrophoretic flux, electrophoretic mobility and conductivity is discussed. 2000 Elsevier Science B.V. All rights reserved. Keywords: Acid–base equilibria; Polyelectrolytes; Amphoteric electrolytes; Electrophoretic mobility; Conductivity
1. Introduction It does not seem to be possible to speak about a complete description of any rather complicated multi-interacting system neither from a kinetic, nor from a thermodynamic *Corresponding author. Tel. / fax: 139-45-802-7901. 0165-022X / 00 / $ – see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S0165-022X( 00 )00120-2
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point of view. For a correct description of any system containing polyvalent electrolytes and of their behavior in electrophoresis, we may not need a complete information about such a system, but at least we need some ideas about the kinetic constants of the processes taking place in our system. When acid–base equilibria are treated with a thermodynamic approach, the equilibrium constant values allow us to calculate the buffer properties, such as pH and buffering capacity. For this purpose it is sufficient to know only the set of so-called macroscopic or ‘‘apparent’’ constants, which can be measured experimentally. Generally, for any system (also of the hybrid type) the proton binding curve may be described with the number of constants equal to the number of dissociation steps [1]. In practice it is more suitable to treat this problem in terms of ‘‘reduced’’ intrinsic constants, formally calculated by an assumption of independent dissociation [1]. The advantage of this approach is that the latter values can be directly applied to any other substances containing the appropriate ionogenic groups, and this approximation is in general in reasonable agreement with experimental results. Although ‘‘N’’ equilibrium dissociation constants are adequate to obtain a charge-pH relationship for a system with ‘‘N’’ dissociation steps, in reality we have only the effective concentrations of ‘‘macrostates’’, i.e. the total sums of microstates with the same electric charge value, thus the information about the concentration of each microstate within a macrostate is lost, and this not only can lead to wrong results in calculating some other equilibrium properties of such a system, but also negates an opportunity for writing a correct kinetic scheme. Without any information about proton-binding reaction kinetics, one is completely unable to propose some expressions for a polyelectrolyte electophoretic mobility or for the conductivity of its solution. Any useful theoretical model (a system of equations, that can be solved) may be created on the basis of some simplifications, and in order to make these assumptions the life-time of an appropriate microstate should be compared with the time needed to a species acquiring a charge to be accelerated to the friction-limited velocity, and also with the ionic atmosphere creation time. These questions will be analyzed below.
2. Theory
2.1. Thermodynamic aspects of acid–base equilibria As we already mentioned in the introduction, a reduced number of equilibrium constants (N constants for N dissociation steps) is sufficient for modelling the pH-charge relationship, but in this way all the information about concentration distribution between microstates with the same charge value is lost. As a simplest example consider the glycine molecule. The tabulated values of dissociation constants may be easily found in any handbook of chemistry, e.g. [2], and, obviously, they should be considered as macroscopic (or apparent) ones. The appropriate pK values are pK1 5 2.35 and pK2 5 9.78. Due to a high DpK difference, when describing the proton binding curve in terms of ‘‘reduced’’ intrinsic constants, one obtains practically the same values. The assumption of independent dissociation allowed us to determine all the intrinsic constants, since
A.V. Stoyanov et al. / J. Biochem. Biophys. Methods 46 (2000) 21 – 30
23
it made possible to reduce the number of independent constants (in term of paper [1] we suppose the equivalence of k Pa and k bU , and k Pb and k aU ): k Pa 5 a(H) /P
(1)
k Pb 5 b(H) /P
(2)
k aU 5 U(H) /a
(3)
k bU 5 U(H) /b
(4)
Here symbols P and U are used to denote the completely protonated (01) and unprotonated (02) states, correspondingly; a and b are the zwitterionic (21) and true zero (00) states. Thus, in our case: k Pa 5k bU 5k 1 , k Pb 5k aU 5k 2 and k 1 510 22.35 , k 2 510 29.78 . Such a system is characterized by a predominant concentration of a zwitterionic microstate and by a negligible concentration of a ‘‘true zero’’ state, see Fig. 1. At the same time, another approach, in which the dissociation constant value of glycine ethyl ester (2310 28 ) is used to approximate the appropriate (k Pb ) microconstant value, leads to the following set of microconstants: k Pa 510 22.35 (k bU 510 24.43 ), k Pb 5 10 27.8 , k aU 510 29.78 [3]. In the latter case, despite the same general pattern, the ratio
Fig. 1. Relative concentrations of different microstates for the glycine molecule as a function of pH. Curves I, II, and III correspond to positive, negative, and zwitterionic states, respectively. Microconstant values used are: k Pa 5k bU 5k 1 , k Pb 5k aU 5k 2 and k 1 510 22.35 , k 2 510 29.78 . By these conditions the ’zwitterion / zero concentration ratio is maintained at 2.7310 7 .
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‘‘zero / zwitterion’’ which is pH independent, although being very low, nevertheless increases by a factor of more than two hundred times. As an example of a three dissociation step system let us consider glutamic acid. The treatment of this system gives the pattern shown in Fig. 2 (dotted line), in which only two zwitterionic states are present in appreciable amounts. A more careful analysis of the same system may be performed by using the experimental macroscopic constants of three ethyl esters of glutamic acid and by taking into account some simple reasonable assumptions [4]. The results obtained are represented in Fig. 2 (solid line). Here we used the same experimental dissociation constant values of Neuberger [5] as used in the work [4], but the method of microconstant calculations was slightly different. In contrast with the system assuming independent dissociation, a new microstate appears in considerable concentration (C 021 ), and for each other microstate some visible deviations from their previous values can be observed (Fig. 2). Generally, for the case of many dissociating groups, the evaluation of the concentration distribution between possible microstates is a very complicated task, but the latter is a matter of great importance, since it influences the electrophoretic behavior of such a system, and for a correct description we need also the information about the kinetics of the reactions taking place between microstates.
Fig. 2. Microstates in the process of dissociation of glutamic acid. Zero (0) or minus (–) in subindexes at the first position reflect an appropriate state of a-carboxyl group; the same for g-carboxyl group (second position); the third position is reserved for amino group, here the two possible states are plus (1) and zero (0). Dotted lines correspond to microstates concentration calculated with assumption of independent dissociation. The maximum in the curve C 201 corresponds to the pI value of Glu.
A.V. Stoyanov et al. / J. Biochem. Biophys. Methods 46 (2000) 21 – 30
25
2.2. Life-time of microscopic states The recombination reactions of the hydrogen ion with appropriate anions or of the hydroxyl ion with its cation are extremely fast reactions [the kinetic constants may achieve values of 10 10 –10 11 l /(mol s)]. But, obviously, for this reaction of the second order the rate strongly depends on the pH value of the solution. Coming back to the most simple case of only two dissociation centers, let us consider again a true ampholyte with one acid and one basic group. The characteristic time of proton (hydroxyl) association reaction varies essentially with pH, and for different microstates, their life-times may differ considerably from each other; and also, in our example, the situation completely changes when moving from one pH extreme to the other. Besides this fact, at pH extremes, for rather large molecules, we may expect that the characteristic time of microreaction may approach the time needed for these species to be accelerated to the velocity limit dictated by frictional forces. The latter value is easy evaluated (see e.g. [6,7]) by analyzing the equation of motion
S
dx f ] ¯ 1 2 exp 2 ]t dt m
D
(5)
where dx / dt is the velocity of moving, f is the frictional coefficient, and m is the mass of the particle. Generally, it is supposed that, for a time interval greater than 10 211 s (assuming, also, that the f/m ratio does not exceed 10 212 ), the velocity may be considered as a constant value. But there is, nevertheless, an important circumstance in the behavior of multidissociating systems, namely that each microstate is involved in many reactions simultaneously. More precisely, for a true parallel process, one microstate directly participates in N (N is the number of ionogenic groups) different reactions (both direct and inverse in terms of proton loss), as illustrated in Fig. 3. Thus, in the case of many ionizable groups, the ‘‘effective’’ life-time of one microstate should be considerably smaller in comparison with the characteristic time of any single microreaction.
2.3. Relaxation of the ionic atmosphere It is interesting to evaluate the time of ionic atmosphere relaxation, since solvation usually is supposed to cause a difference in mobility for differently charged forms. We can use the Einstein-Smoluchowsky formula: kx 2 l 5 2 Dt where kx 2 l is the mean square value of the replacement during the time t. D is the diffusion coefficient. Replacing the average distance (kx 2 l)1 / 2 , by the radius of ionic atmosphere, i.e. the Debye-length, and also, changing the time t by the relaxation time t, we thus obtain: 2 e0r 2 r t ¯ ] ¯ ]] 2D 2kTu
(6)
where r is the Debye radius, e 0 is the electron charge, k is the Boltzmann constant, u the
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Fig. 3. The number of possible microreactions for one microscopic state. When describing the parallel dissociation process in terms of proton loss, starting from the unprotonated state we have a progressive decrease in the number of possible ‘‘direct’’ reaction, connected with an appropriate increase of ‘‘inverse’’ reactions, so that the total number of possible reactions for anyone microstate remains the same and equals N. Note that the latter result is valid only in the case of pure parallel dissociation, see, for instance, some examples of ‘‘hybrid’’ dissociation schemes in [1].
relative mobility (divided by the charge) and T is the absolute temperature. That gives, for example, for the concentration of hydrogen ions of 10 23 (in the absence of other mobile ions) t -values of 10 26 –10 27 s. And, obviously, the results obtained should be every time compared with the appropriate microstate life-time. Although this kind of estimation is a rather crude one, it illustrates that the assumption of ionic atmosphere relaxation is mostly due to diffusion mechanism, thus leading to a rather high time values. In frames of this approximation we have a strong dependence on ionic strength: with a decrease of which, the r 2 increases causing an appropriate growth of the relaxation time.
2.4. Modeling of the electrophoretic flux, electrophoretic mobility and conductivity An important question is how to model an electrophoretic flux in such a system with many reactions. While the total flux is a sum of ones for each component:
A.V. Stoyanov et al. / J. Biochem. Biophys. Methods 46 (2000) 21 – 30
Om (t)q C E
f5
i
i
i
27
(7)
i
where E is the electric field strength, Ci are the concentrations of each microform, mi (t) is the relative mobility (should be treated as time depending value) and qi is electric charge value (electron charge units). In the theory of electrophoresis the values of mi are generally supposed to be constant, with respect to time (for example [8,9]), although there is still a possibility to attribute some different values to them and some kind of relationship from other parameters (from pH for example). Such an approximation may be treated as an assumption of instant acquiring the appropriate velocity during conversion of one microstate to another, or in other words, the time needed for acceleration, after the conversion, is negligible in comparison with microstates life-time. When it is not true (see Section 2.2), the effective mobility can not be expressed as an arithmetic sum of the ones corresponding to each microstate (multiplied by an appropriate concentration factor) and it’s final value should be smaller. For bivalent ampholyte with further simplifications of equal relative mobilities the expression for an electophoretic flux rewrites as:
f 5 ( m 1 C 1 2 m 2 C 2)E
(8)
where C 1 and C 2 are the concentrations of the positively and negatively charged forms. Generally speaking the mobility of an arbitrary microstate hardly can be measured, at least by direct measurements. The only thing that can be measured is the effective mobility of the whole complex, although, for some microscopic states, there exists a direct opportunity of mobility measurements. For example, at pH extremes, the concentration of protonated or unprotonated state is a predominant value; in addition, for some simple systems, the concentration of one intermediate microstate may become practically equal to unity at some pH conditions (see C 221 around pH 7, Fig. 2). The comparability of the following times microstate: life-time and ionic atmosphere creation, in the cases when it take place, is an additional strong argument against the approach in which, for example for simple ampholyte, the mobilities of the anionic and cationic forms being determined initially at pH extremes then approximates to all pH scale. It should be noted, also, that not only the mobility of microstate can be treated as pH-dependent, but the limiting mobility value cannot be achieved due to very high conversion rate.
3. Discussion The best correlation between theoretical prediction and experimental results when doing electrokinetic separations can be achieved by using maximum of experimental information about the object of separation (or in other words in order to obtain an opportunity to describe (or to optimize) a separation process we can use some experimentally obtained relationships — for example pH-mobility relationship), while someone could ask for a theory that needs some minimal number of universal parameters. Since the precise and exhaustive information is often not available and
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moreover it is impossible or unreasonable to obtain such an information, we need to take advantage of some simple general model, which can provide rather good qualitative agreement, but for analyzing some complicated phenomena it may be unuseful. In particular, the question exists to what extent it is justified that the kinetic aspects of the acid base equilibria are essentially ignored? The appropriate evaluations are easily obtained for a concrete situation, but to create a model taking into account the effect of appropriate mobility changes (acceleration or retardation to a new value, after conversion of one microstate to another) is not easy. For this purpose we need, also the precise distribution of substance between all possible microscopic states, and this task, probably, can be solved correctly in some limited cases only. For biopolymers, when interpreting the results of direct conductivity measurements also some problems may arise, since there should exist an effect of conductivity decrease due to ion–ion (ion–dipole) interaction. Nevertheless, the conductivity level of some commercially available proteins (with high degree of purity) exceeds considerably the level calculated theoretically even with using the formula (9), given below:
l 5 Fm Ckq 2 l
(9)
where kq 2 l is the mean square value of electric charge, F is the Faraday of electricity, the relative mobility m is supposed to be the same for all microstates, and C is the protein concentration (see Fig. 4), although this evaluation may be considered only as an
Fig. 4. Experimentally measured conductivity with respect to concentration: I-albumin, II-pepsin. The theoretically calculated value for albumin at 1 mM concentration, in accordance with formula (9), is approximately 0.2310 22 S / m, which is ten times lower than that found experimentally. In the case of pepsin, the theoretical protein conductivity contribution is about of the same order as that measured experimentally, although here there is a very high contribution of bulk water, due to very low pH value of the protein solution.
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29
Fig. 5. Theoretically calculated values of mean electric charge and 1 / 2 power of mean square charge for albumin, with assumption of independent dissociation. Curve I: absolute value of mean charge; curve II: square root out of mean square charge (curve III represents 1 / 2 power of b / ln 10). These two curves (I and II) illustrate the difference between the two extreme approaches: very fast transition between the microstates results in the microspecies being unable to get the appropriate velocity, and one can attribute to each molecule the same average charge (curve I) and the same mobility value; in the opposite case, one should operate with mean square charge value (curve II), as stated by (9). Amino acid composition data are taken from [10].
upper limit for conductivity, see Fig. 5. One of the possible explanations is the microheterogeneity phenomenon, which leads to the presence of a number of forms with mean charge value considerably different from zero.
Acknowledgements This work was supported by grants RFFI and from MURST (Coordinated Project Protein Folding and Misfolding, 40%) and from ASI (Agenzia Spaziale Italiana, Roma), grant I / R / 28 / 00.
References [1] [2] [3] [4]
Stoyanov AV, Righetti PG. Dissociation of polyvalent electrolytes. J Chromatogr A 1999;853:35–44. Weast RC, editor, CRC handbook of chemistry and physics, Boca Raton, Florida: CRC Press, 1987. Mahler HR, Cordes EH. Biological chemistry (2nd ed.), New York: Harper & Row Publishers, 1971. Saroff H. From glycine to glutamic acid: analysis of the proton binding isoterm of glutamic acid. Analyt Biochem 1998;257:71–9. [5] Neuberger A. Biochem J 1936;30:2085–94. [6] Bier M, editor, Electrophoresis, theory, methods, and applications, New York: Academic Press, 1959. [7] Shaw DJ. Electrophoresis, New York: Academic Press, 1969.
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[8] Almgren M. Isoelectric fractionation, analysis, and characterization of ampholytes in natural pH gradients. Chemica Scripta 1971;1:69–75. [9] Mosher R, Saville DA, Thorman W. The dynamics of electrophoresis, Weincheim: VCH, 1992. [10] Brown JR. Structural origins of mammalian albumin. Fed Proc 1976;35:2141–4.
Acid–base equilibria of polyvalent electrolytes and theoretical description of polyelectrolyte behavior in electrokinetic separations a b c, Alexander V. Stoyanov , Irina N. Stozhkova , Pier Giorgio Righetti * a
b
University of Milan, LITA, Segrate, Via Fratelli Cervi 93, 20090 Milan, Italy Frumkin Institute of Electrochemistry, Russian Academy of Sciences, Leninsky Av. 31, 117071 Moscow, Russia c University of Verona, Department of Agricultural & Industrial Biotechnologies, Strada Le Grazie, Ca’ Vignal, 37134 Verona, Italy Received 3 July 2000; accepted 3 July 2000
Abstract The problem of the theoretical description of the dissociation of polyvalent electrolytes is considered. It is shown that the traditional model which assumes some constant microstate mobility values may have limited applications, since for each microstate the rate of conversion into other states is strongly dependent on pH and thus it may often become comparable with the characteristic times of other important processes (ionic atmosphere relaxation, biopolymer acceleration by electric field). The question of correct modelling of the electrophoretic flux, electrophoretic mobility and conductivity is discussed. 2000 Elsevier Science B.V. All rights reserved. Keywords: Acid–base equilibria; Polyelectrolytes; Amphoteric electrolytes; Electrophoretic mobility; Conductivity
1. Introduction It does not seem to be possible to speak about a complete description of any rather complicated multi-interacting system neither from a kinetic, nor from a thermodynamic *Corresponding author. Tel. / fax: 139-45-802-7901. 0165-022X / 00 / $ – see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S0165-022X( 00 )00120-2
22
A.V. Stoyanov et al. / J. Biochem. Biophys. Methods 46 (2000) 21 – 30
point of view. For a correct description of any system containing polyvalent electrolytes and of their behavior in electrophoresis, we may not need a complete information about such a system, but at least we need some ideas about the kinetic constants of the processes taking place in our system. When acid–base equilibria are treated with a thermodynamic approach, the equilibrium constant values allow us to calculate the buffer properties, such as pH and buffering capacity. For this purpose it is sufficient to know only the set of so-called macroscopic or ‘‘apparent’’ constants, which can be measured experimentally. Generally, for any system (also of the hybrid type) the proton binding curve may be described with the number of constants equal to the number of dissociation steps [1]. In practice it is more suitable to treat this problem in terms of ‘‘reduced’’ intrinsic constants, formally calculated by an assumption of independent dissociation [1]. The advantage of this approach is that the latter values can be directly applied to any other substances containing the appropriate ionogenic groups, and this approximation is in general in reasonable agreement with experimental results. Although ‘‘N’’ equilibrium dissociation constants are adequate to obtain a charge-pH relationship for a system with ‘‘N’’ dissociation steps, in reality we have only the effective concentrations of ‘‘macrostates’’, i.e. the total sums of microstates with the same electric charge value, thus the information about the concentration of each microstate within a macrostate is lost, and this not only can lead to wrong results in calculating some other equilibrium properties of such a system, but also negates an opportunity for writing a correct kinetic scheme. Without any information about proton-binding reaction kinetics, one is completely unable to propose some expressions for a polyelectrolyte electophoretic mobility or for the conductivity of its solution. Any useful theoretical model (a system of equations, that can be solved) may be created on the basis of some simplifications, and in order to make these assumptions the life-time of an appropriate microstate should be compared with the time needed to a species acquiring a charge to be accelerated to the friction-limited velocity, and also with the ionic atmosphere creation time. These questions will be analyzed below.
2. Theory
2.1. Thermodynamic aspects of acid–base equilibria As we already mentioned in the introduction, a reduced number of equilibrium constants (N constants for N dissociation steps) is sufficient for modelling the pH-charge relationship, but in this way all the information about concentration distribution between microstates with the same charge value is lost. As a simplest example consider the glycine molecule. The tabulated values of dissociation constants may be easily found in any handbook of chemistry, e.g. [2], and, obviously, they should be considered as macroscopic (or apparent) ones. The appropriate pK values are pK1 5 2.35 and pK2 5 9.78. Due to a high DpK difference, when describing the proton binding curve in terms of ‘‘reduced’’ intrinsic constants, one obtains practically the same values. The assumption of independent dissociation allowed us to determine all the intrinsic constants, since
A.V. Stoyanov et al. / J. Biochem. Biophys. Methods 46 (2000) 21 – 30
23
it made possible to reduce the number of independent constants (in term of paper [1] we suppose the equivalence of k Pa and k bU , and k Pb and k aU ): k Pa 5 a(H) /P
(1)
k Pb 5 b(H) /P
(2)
k aU 5 U(H) /a
(3)
k bU 5 U(H) /b
(4)
Here symbols P and U are used to denote the completely protonated (01) and unprotonated (02) states, correspondingly; a and b are the zwitterionic (21) and true zero (00) states. Thus, in our case: k Pa 5k bU 5k 1 , k Pb 5k aU 5k 2 and k 1 510 22.35 , k 2 510 29.78 . Such a system is characterized by a predominant concentration of a zwitterionic microstate and by a negligible concentration of a ‘‘true zero’’ state, see Fig. 1. At the same time, another approach, in which the dissociation constant value of glycine ethyl ester (2310 28 ) is used to approximate the appropriate (k Pb ) microconstant value, leads to the following set of microconstants: k Pa 510 22.35 (k bU 510 24.43 ), k Pb 5 10 27.8 , k aU 510 29.78 [3]. In the latter case, despite the same general pattern, the ratio
Fig. 1. Relative concentrations of different microstates for the glycine molecule as a function of pH. Curves I, II, and III correspond to positive, negative, and zwitterionic states, respectively. Microconstant values used are: k Pa 5k bU 5k 1 , k Pb 5k aU 5k 2 and k 1 510 22.35 , k 2 510 29.78 . By these conditions the ’zwitterion / zero concentration ratio is maintained at 2.7310 7 .
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‘‘zero / zwitterion’’ which is pH independent, although being very low, nevertheless increases by a factor of more than two hundred times. As an example of a three dissociation step system let us consider glutamic acid. The treatment of this system gives the pattern shown in Fig. 2 (dotted line), in which only two zwitterionic states are present in appreciable amounts. A more careful analysis of the same system may be performed by using the experimental macroscopic constants of three ethyl esters of glutamic acid and by taking into account some simple reasonable assumptions [4]. The results obtained are represented in Fig. 2 (solid line). Here we used the same experimental dissociation constant values of Neuberger [5] as used in the work [4], but the method of microconstant calculations was slightly different. In contrast with the system assuming independent dissociation, a new microstate appears in considerable concentration (C 021 ), and for each other microstate some visible deviations from their previous values can be observed (Fig. 2). Generally, for the case of many dissociating groups, the evaluation of the concentration distribution between possible microstates is a very complicated task, but the latter is a matter of great importance, since it influences the electrophoretic behavior of such a system, and for a correct description we need also the information about the kinetics of the reactions taking place between microstates.
Fig. 2. Microstates in the process of dissociation of glutamic acid. Zero (0) or minus (–) in subindexes at the first position reflect an appropriate state of a-carboxyl group; the same for g-carboxyl group (second position); the third position is reserved for amino group, here the two possible states are plus (1) and zero (0). Dotted lines correspond to microstates concentration calculated with assumption of independent dissociation. The maximum in the curve C 201 corresponds to the pI value of Glu.
A.V. Stoyanov et al. / J. Biochem. Biophys. Methods 46 (2000) 21 – 30
25
2.2. Life-time of microscopic states The recombination reactions of the hydrogen ion with appropriate anions or of the hydroxyl ion with its cation are extremely fast reactions [the kinetic constants may achieve values of 10 10 –10 11 l /(mol s)]. But, obviously, for this reaction of the second order the rate strongly depends on the pH value of the solution. Coming back to the most simple case of only two dissociation centers, let us consider again a true ampholyte with one acid and one basic group. The characteristic time of proton (hydroxyl) association reaction varies essentially with pH, and for different microstates, their life-times may differ considerably from each other; and also, in our example, the situation completely changes when moving from one pH extreme to the other. Besides this fact, at pH extremes, for rather large molecules, we may expect that the characteristic time of microreaction may approach the time needed for these species to be accelerated to the velocity limit dictated by frictional forces. The latter value is easy evaluated (see e.g. [6,7]) by analyzing the equation of motion
S
dx f ] ¯ 1 2 exp 2 ]t dt m
D
(5)
where dx / dt is the velocity of moving, f is the frictional coefficient, and m is the mass of the particle. Generally, it is supposed that, for a time interval greater than 10 211 s (assuming, also, that the f/m ratio does not exceed 10 212 ), the velocity may be considered as a constant value. But there is, nevertheless, an important circumstance in the behavior of multidissociating systems, namely that each microstate is involved in many reactions simultaneously. More precisely, for a true parallel process, one microstate directly participates in N (N is the number of ionogenic groups) different reactions (both direct and inverse in terms of proton loss), as illustrated in Fig. 3. Thus, in the case of many ionizable groups, the ‘‘effective’’ life-time of one microstate should be considerably smaller in comparison with the characteristic time of any single microreaction.
2.3. Relaxation of the ionic atmosphere It is interesting to evaluate the time of ionic atmosphere relaxation, since solvation usually is supposed to cause a difference in mobility for differently charged forms. We can use the Einstein-Smoluchowsky formula: kx 2 l 5 2 Dt where kx 2 l is the mean square value of the replacement during the time t. D is the diffusion coefficient. Replacing the average distance (kx 2 l)1 / 2 , by the radius of ionic atmosphere, i.e. the Debye-length, and also, changing the time t by the relaxation time t, we thus obtain: 2 e0r 2 r t ¯ ] ¯ ]] 2D 2kTu
(6)
where r is the Debye radius, e 0 is the electron charge, k is the Boltzmann constant, u the
26
A.V. Stoyanov et al. / J. Biochem. Biophys. Methods 46 (2000) 21 – 30
Fig. 3. The number of possible microreactions for one microscopic state. When describing the parallel dissociation process in terms of proton loss, starting from the unprotonated state we have a progressive decrease in the number of possible ‘‘direct’’ reaction, connected with an appropriate increase of ‘‘inverse’’ reactions, so that the total number of possible reactions for anyone microstate remains the same and equals N. Note that the latter result is valid only in the case of pure parallel dissociation, see, for instance, some examples of ‘‘hybrid’’ dissociation schemes in [1].
relative mobility (divided by the charge) and T is the absolute temperature. That gives, for example, for the concentration of hydrogen ions of 10 23 (in the absence of other mobile ions) t -values of 10 26 –10 27 s. And, obviously, the results obtained should be every time compared with the appropriate microstate life-time. Although this kind of estimation is a rather crude one, it illustrates that the assumption of ionic atmosphere relaxation is mostly due to diffusion mechanism, thus leading to a rather high time values. In frames of this approximation we have a strong dependence on ionic strength: with a decrease of which, the r 2 increases causing an appropriate growth of the relaxation time.
2.4. Modeling of the electrophoretic flux, electrophoretic mobility and conductivity An important question is how to model an electrophoretic flux in such a system with many reactions. While the total flux is a sum of ones for each component:
A.V. Stoyanov et al. / J. Biochem. Biophys. Methods 46 (2000) 21 – 30
Om (t)q C E
f5
i
i
i
27
(7)
i
where E is the electric field strength, Ci are the concentrations of each microform, mi (t) is the relative mobility (should be treated as time depending value) and qi is electric charge value (electron charge units). In the theory of electrophoresis the values of mi are generally supposed to be constant, with respect to time (for example [8,9]), although there is still a possibility to attribute some different values to them and some kind of relationship from other parameters (from pH for example). Such an approximation may be treated as an assumption of instant acquiring the appropriate velocity during conversion of one microstate to another, or in other words, the time needed for acceleration, after the conversion, is negligible in comparison with microstates life-time. When it is not true (see Section 2.2), the effective mobility can not be expressed as an arithmetic sum of the ones corresponding to each microstate (multiplied by an appropriate concentration factor) and it’s final value should be smaller. For bivalent ampholyte with further simplifications of equal relative mobilities the expression for an electophoretic flux rewrites as:
f 5 ( m 1 C 1 2 m 2 C 2)E
(8)
where C 1 and C 2 are the concentrations of the positively and negatively charged forms. Generally speaking the mobility of an arbitrary microstate hardly can be measured, at least by direct measurements. The only thing that can be measured is the effective mobility of the whole complex, although, for some microscopic states, there exists a direct opportunity of mobility measurements. For example, at pH extremes, the concentration of protonated or unprotonated state is a predominant value; in addition, for some simple systems, the concentration of one intermediate microstate may become practically equal to unity at some pH conditions (see C 221 around pH 7, Fig. 2). The comparability of the following times microstate: life-time and ionic atmosphere creation, in the cases when it take place, is an additional strong argument against the approach in which, for example for simple ampholyte, the mobilities of the anionic and cationic forms being determined initially at pH extremes then approximates to all pH scale. It should be noted, also, that not only the mobility of microstate can be treated as pH-dependent, but the limiting mobility value cannot be achieved due to very high conversion rate.
3. Discussion The best correlation between theoretical prediction and experimental results when doing electrokinetic separations can be achieved by using maximum of experimental information about the object of separation (or in other words in order to obtain an opportunity to describe (or to optimize) a separation process we can use some experimentally obtained relationships — for example pH-mobility relationship), while someone could ask for a theory that needs some minimal number of universal parameters. Since the precise and exhaustive information is often not available and
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moreover it is impossible or unreasonable to obtain such an information, we need to take advantage of some simple general model, which can provide rather good qualitative agreement, but for analyzing some complicated phenomena it may be unuseful. In particular, the question exists to what extent it is justified that the kinetic aspects of the acid base equilibria are essentially ignored? The appropriate evaluations are easily obtained for a concrete situation, but to create a model taking into account the effect of appropriate mobility changes (acceleration or retardation to a new value, after conversion of one microstate to another) is not easy. For this purpose we need, also the precise distribution of substance between all possible microscopic states, and this task, probably, can be solved correctly in some limited cases only. For biopolymers, when interpreting the results of direct conductivity measurements also some problems may arise, since there should exist an effect of conductivity decrease due to ion–ion (ion–dipole) interaction. Nevertheless, the conductivity level of some commercially available proteins (with high degree of purity) exceeds considerably the level calculated theoretically even with using the formula (9), given below:
l 5 Fm Ckq 2 l
(9)
where kq 2 l is the mean square value of electric charge, F is the Faraday of electricity, the relative mobility m is supposed to be the same for all microstates, and C is the protein concentration (see Fig. 4), although this evaluation may be considered only as an
Fig. 4. Experimentally measured conductivity with respect to concentration: I-albumin, II-pepsin. The theoretically calculated value for albumin at 1 mM concentration, in accordance with formula (9), is approximately 0.2310 22 S / m, which is ten times lower than that found experimentally. In the case of pepsin, the theoretical protein conductivity contribution is about of the same order as that measured experimentally, although here there is a very high contribution of bulk water, due to very low pH value of the protein solution.
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Fig. 5. Theoretically calculated values of mean electric charge and 1 / 2 power of mean square charge for albumin, with assumption of independent dissociation. Curve I: absolute value of mean charge; curve II: square root out of mean square charge (curve III represents 1 / 2 power of b / ln 10). These two curves (I and II) illustrate the difference between the two extreme approaches: very fast transition between the microstates results in the microspecies being unable to get the appropriate velocity, and one can attribute to each molecule the same average charge (curve I) and the same mobility value; in the opposite case, one should operate with mean square charge value (curve II), as stated by (9). Amino acid composition data are taken from [10].
upper limit for conductivity, see Fig. 5. One of the possible explanations is the microheterogeneity phenomenon, which leads to the presence of a number of forms with mean charge value considerably different from zero.
Acknowledgements This work was supported by grants RFFI and from MURST (Coordinated Project Protein Folding and Misfolding, 40%) and from ASI (Agenzia Spaziale Italiana, Roma), grant I / R / 28 / 00.
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