390
J. Sep. Sci. 2014, 37, 390–392
Leonid D. Asnin Yuliya Nikitina
Short Communication
Perm National Research Polytechnic University, Department of Chemistry and Biotechnology, Perm, Russian Federation
Effect of competing binding modes on retention in chromatography and capillary electrophoresis. A theoretical consideration
Received September 30, 2013 Revised December 2, 2013 Accepted December 3, 2013
A mathematical formalism has been developed to describe equilibrium in a system involving a single selector and a selectand molecule capable of binding to this selector by different modes. The generalization of this model to a multiselector system has also been considered. Applications of the developed equations in chromatography and CE are discussed. Keywords: Binding / Capillary electrophoresis / Chromatography / Equilibrium DOI 10.1002/jssc.201301074
1 Introduction Numerous studies of binding processes revealed that if a selectand (SA) molecule has several binding sites (several functionalities or otherwise distinguishable fragments) one may bind to a selector (SO) by any of these sites [1–5], which results in a competition between different binding modes. This binding scenario is illustrated in Fig. 1B. For the sake of comparison, Fig. 1A demonstrates the classic single binding mode model and Fig. 1C shows another well-studied situation in chromatography [6–8] and CE [9] when a single binding mode SA interacts with several SOs (two in the figure, designated S1 and S2). Figure 1D represents a generalization combining a multiple binding mode SA with multiple SOs. The model of competing binding modes shown in Fig. 1B has been developed in biochemistry [10] to solve specific problems of substrate–enzyme interaction kinetics. It has never been adopted for separation problems. Thus separation methods lack a formal mathematical justification for this frequently used concept. The present report is aimed at filling this gap.
2 Results and discussion For the effect of interest to take place, an SA molecule has to contain several binding sites, each of them able to interact with an SO independently from the others. First, let us consider a molecule AB with two binding sites A and B. If either of them is able to bind to an SO S having only one active site, two equilibria are possible with respective equilibrium constants KABS and KBAS :
AB + S ↔ ABS K ABS =
[ABS] [AB] · [S]
(1)
AB + S ↔ BAS K BAS =
[BAS] [AB] · [S]
(2)
Obviously, the apparent equilibrium constant K(AB) is given by K (AB) =
[ABS] + [BAS] = K ABS + K BAS [AB] · [S]
In the general case of a molecule A···N possessing N binding sites, the apparent equilibrium constant K(A···N) is K (A···N) = K A···NS + . . . + K N···AS =
Abbreviations: SA, selectand; SO, selector C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
N···A
K JS
(4)
J =A···N
Now let us assume that there are n SOs: S1 , S2 , . . . ,Sn , each interacting differently with each binding site of the molecule A···N. The total SO concentration is [S] = n [Si ]. The equilibrium constant for a particular SO–SA complex JSi (J symbolizes any of the possible binding modes of the molecule A···N) is by definition JSi (5) K JS,i = [A · · · N] · [Si ] The partial equilibrium constant for a particular SO i that takes into account all possible modes of binding is N···A
K (A···N),i = Correspondence: Dr. Leonid Asnin, Perm National Research Polytechnic University, Department of Chemistry and Biotechnology, 29 Komsomilsky Ave., Perm 614000, Russian Federation E-mail: [email protected] Fax: +7(342)391-511
(3)
J =A···N
JSi
[A · · · N] · [Si ]
(6)
Designating a molar fraction of the ith SO as i , one obtains an expression for the overall equilibrium constant K (A···N){n} K (A···N){n} =
n
K (A···N),i i
(7)
i=1
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Liquid Chromatography
J. Sep. Sci. 2014, 37, 390–392
The additional subscript index {n} symbolizes here summation over all n SO types. Note that Equation (7) reduces to Equation (4) when n = 1.
2.1 Chromatography Chromatographic retention is characterized by the value of retention time tR , which relates to an equilibrium isotherm through the well-known Equation (8) [11] tR = t0 [1 + F (dq (c)/dc)]
(8)
where t0 is the hold-up time, F the phase ratio, q and c the equilibrium concentrations of an analyte (SA in our terminology) in the stationary and mobile phases, respectively. Obviously, q = [A···NS] + ··· + [N···AS], c = [A···N] if we consider a molecule A···N. According to this equation, the retention depends on the nature of SO–SA interactions only through the equilibrium isotherm. Designating the concentration of the adsorption sites on the surface of the stationary phase as q*, we arrive at the adsorption isotherm q ∗ K (A···N) c q = 1 + K (A···N) c
q =
max min
f K (A···N) K (A···N) c d K (A···N) 1 + K (A···N) c
C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
An addition of specific SOs to the running buffer is commonly used in chiral CE for the separation of enantiomers. A theory describing the migration of an analyte in the presence of a chiral SO was developed by Wren and Rowe [13]. According to this theory, the effective mobility of a single binding mode analyte A is given by A eff =
0 + AS K A [S] 1 + K A [S]
(12)
Here 0 is the mobility of the free analyte, AS is the mobility of the analyte–SO complex, and KA is the equilibrium binding constant. We shall extend this theory to the case of a multiplebinding mode equilibrium. The effective mobility is defined as the sum of the electrophoretic mobilities of all different species in which the analyte may exist weighted by the mole fraction ϕ of the respective species [14]. Consequently,
A···N eff
= 0 1 −
N···A
N···A
JS +
JS JS
(13)
J =A···N
(9)
(10)
where q i∗ is the concentration of the ith SO. Provided that the SA has a single binding mode, Equation (10) reduces to the multisite Langmuir isotherm well known in the theory of adsorption on a heterogeneous surface [6], describing the binding equilibrium depicted in Fig. 1C. Equation (10) implies a discrete distribution of SOs with respect to the equilibrium constant. In chromatography, the concept of a continuous equilibrium constant distribution is frequently used [11, 12]. Let us assume such a distribution f(K(A···N) ) defined between the minimum (min) and the maximum (max) possible values of the apparent equilibrium constant K(A···N) (see Eq. (4)) so max that min f (K (A···N) )d K (A···N) = q ∗ , the total concentration of SOs. Then Equation (10) should be rewritten replacing summation over a finite number of SO types by integration over a continuous range of K(A···N ) :
2.2 Capillary electrophoresis
J =A···N
This equation is similar to the Langmuir adsorption isotherm for a single binding mode adsorption. Nevertheless, it differs from the latter in the meaning of the equilibrium constant defined by Equation (4). In the general case of a multiple binding mode SA and multiple SOs (Fig. 1D), the adsorption isotherm reads n q i∗ K (A···N),i c q = 1 + K (A···N),i c i=1
391
(11)
After rearrangement and taking into account Equation (4) we derive 0 + = A···N eff
N···A J =A···N
JS K JS · [S] (14)
1 + K (A···N) [S]
As seen, the effective mobility of a multiple binding mode SA is defined by an equation similar to Equation (12). An essential difference lies in the definitions of the coefficients at the equilibrium SO concentration. Earlier, Dubsk´y et al. [9,15] developed a model involving a single binding mode SA and multiple SOs, with the expression for the effective mobility given by 0 + A···N eff
=
n
JS,i K JS,i i
i=1
1 + K (A···N){n} [S]
· [S] (15)
where summation is carried out over all SOs and the index J, denoting an SA’s binding mode, takes a certain single value. The above expression corresponds to the model depicted in Fig. 1C (with reservation that there is no stationary phase in CE), while Equation (14) reflects the model shown in Fig. 1B. From a formal mathematical point of view, the difference between Equations (14) and (15) concerns the molar fraction of the ith SO i , which is present in Equation (15) and lacking in Equation (14). Consider the situation with competing binding modes and multiple SOs. Omitting routine mathematical
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392
J. Sep. Sci. 2014, 37, 390–392
L. D Asnin and Y. Nikitina
Figure 1. Illustration of four scenarios of binding equilibria: (A) a single-binding mode SA—a single SO; (B) a multiple-binding mode SA—a single SO; (C) a single-binding mode SA—multiple SOs; (D) a multiple-binding mode SA— multiple SOs. The molecule AB is an SA, Symbols S, S1, and S2 denote SOs. S1 and S2 designate different SOs. The figure illustrates equilibria in chromatography. It will be in principle the same in CE except there will be no stationary phase.
manipulations, we arrive at the following expression for the effective mobility
N···A n 0 + JS,i K JS,i i · [S] J =A···N i=1 (16) A···N = eff 1 + K (A···N){n} [S] Assuming a single binding mode SA, one can obtain Equation (15). The assumption of a single SO results in Equation (14).
3 Concluding remarks
4 References [1] Lipkowitz, K. B., Raghothama, S., Yang, J., J. Am. Chem. Soc. 1992, 114, 1554–1562. [2] Maier, N. M., Schefzick, S., Lombardo, G. M., Feliz, M., Rissanen, K., Lindner, W., Lipkowitz, K. B., J. Am. Chem. Soc. 2002, 124, 8611–8629. [3] Zhao, C. F., Cann, N. M., Anal. Chem. 2008, 80, 2426–2438. [4] Angulo, J., D´ıaz, I., Reina, J. J., Tabarani, G., Fieschi, F., Rojo, J., Nieto, P. M., ChemBioChem 2008, 9, 2225–2227. [5] Pagnozzi, D., Birolo, L., Leo, G., Contessi, S., Lippe, G., Pucci, P., Mavelli, I., Biochemistry 2010, 49, 7542–7552.
Equation (4) in chromatography and Equation (14) in CE provide a mathematical basis for the use of the model in which an analyte binds to an SO by different (competing) modes. Supposedly, this model plays a role in the retention of analytes having several functionalities, each capable of interacting with an SO. Chiral systems involving an SA compound with two distant stereogenic centres and a single binding site chiral SO can be useful to investigate this phenomenon. For this purpose, chromatographic (electrophoresis) results should be supplemented by the data from molecular dynamic simulation or sophisticated spectroscopic techniques, which are able to distinguish contributions of particular binding modes. This sort of information is unattainable using chromatographic or CE experiment alone. Indeed, what is measured is the lumped equilibrium constant (Eq. (4)) in chromatography and the lumped constant and the sum JS K JS in CE.
[10] Hein, G. E., Niemann, C., J. Am. Chem. Soc. 1962, 84, 4495–4503.
This work was supported by a grant of the Russian Foundation for Basic Research (No. 13-03-92692).
[14] Scriba, G. K. E., Topics in Current Chemistry, Springer-Verlag, Berlin, Heidelberg 2013, pp. 1–67. DOI: 10.1007/128_2013_438.
The authors have declared no conflict of interest.
C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
[6] Asnin, L., J. Chromatogr. A 2012, 1269, 3–25. [7] Levkin, P. A., Schurig, V., J. Chromatogr. A 2008, 1184, 309–322. [8] Fornstedt, T., J. Chromatogr. A 2010, 1217, 792–812. ´ P., Svobodova, ´ J., Gas, ˇ B., J. Chromatogr. B [9] Dubsky, 2008, 875, 30–34.
[11] Guiochon, G., Felinger, A., Shirazi, D. G., Katti, A. M., Fundamentals of Preparative and Non-Linear Chromatography, Academic Press, Boston 2006. [12] Samuelsson, J., Arnell, R., Fornstedt, T., J. Sep. Sci. 2009, 32, 1491–1506. [13] Wren, S. A. C., Rowe, R. C., J. Chromatogr. 1992, 603, 235–241.
´ P., Svobodova, ´ J., Tesaˇrova, ´ E. J., Gas, ˇ B., Elec[15] Dubsky, trophoresis 2010, 31, 1435–1441.
www.jss-journal.com
J. Sep. Sci. 2014, 37, 390–392
Leonid D. Asnin Yuliya Nikitina
Short Communication
Perm National Research Polytechnic University, Department of Chemistry and Biotechnology, Perm, Russian Federation
Effect of competing binding modes on retention in chromatography and capillary electrophoresis. A theoretical consideration
Received September 30, 2013 Revised December 2, 2013 Accepted December 3, 2013
A mathematical formalism has been developed to describe equilibrium in a system involving a single selector and a selectand molecule capable of binding to this selector by different modes. The generalization of this model to a multiselector system has also been considered. Applications of the developed equations in chromatography and CE are discussed. Keywords: Binding / Capillary electrophoresis / Chromatography / Equilibrium DOI 10.1002/jssc.201301074
1 Introduction Numerous studies of binding processes revealed that if a selectand (SA) molecule has several binding sites (several functionalities or otherwise distinguishable fragments) one may bind to a selector (SO) by any of these sites [1–5], which results in a competition between different binding modes. This binding scenario is illustrated in Fig. 1B. For the sake of comparison, Fig. 1A demonstrates the classic single binding mode model and Fig. 1C shows another well-studied situation in chromatography [6–8] and CE [9] when a single binding mode SA interacts with several SOs (two in the figure, designated S1 and S2). Figure 1D represents a generalization combining a multiple binding mode SA with multiple SOs. The model of competing binding modes shown in Fig. 1B has been developed in biochemistry [10] to solve specific problems of substrate–enzyme interaction kinetics. It has never been adopted for separation problems. Thus separation methods lack a formal mathematical justification for this frequently used concept. The present report is aimed at filling this gap.
2 Results and discussion For the effect of interest to take place, an SA molecule has to contain several binding sites, each of them able to interact with an SO independently from the others. First, let us consider a molecule AB with two binding sites A and B. If either of them is able to bind to an SO S having only one active site, two equilibria are possible with respective equilibrium constants KABS and KBAS :
AB + S ↔ ABS K ABS =
[ABS] [AB] · [S]
(1)
AB + S ↔ BAS K BAS =
[BAS] [AB] · [S]
(2)
Obviously, the apparent equilibrium constant K(AB) is given by K (AB) =
[ABS] + [BAS] = K ABS + K BAS [AB] · [S]
In the general case of a molecule A···N possessing N binding sites, the apparent equilibrium constant K(A···N) is K (A···N) = K A···NS + . . . + K N···AS =
Abbreviations: SA, selectand; SO, selector C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
N···A
K JS
(4)
J =A···N
Now let us assume that there are n SOs: S1 , S2 , . . . ,Sn , each interacting differently with each binding site of the molecule A···N. The total SO concentration is [S] = n [Si ]. The equilibrium constant for a particular SO–SA complex JSi (J symbolizes any of the possible binding modes of the molecule A···N) is by definition JSi (5) K JS,i = [A · · · N] · [Si ] The partial equilibrium constant for a particular SO i that takes into account all possible modes of binding is N···A
K (A···N),i = Correspondence: Dr. Leonid Asnin, Perm National Research Polytechnic University, Department of Chemistry and Biotechnology, 29 Komsomilsky Ave., Perm 614000, Russian Federation E-mail: [email protected] Fax: +7(342)391-511
(3)
J =A···N
JSi
[A · · · N] · [Si ]
(6)
Designating a molar fraction of the ith SO as i , one obtains an expression for the overall equilibrium constant K (A···N){n} K (A···N){n} =
n
K (A···N),i i
(7)
i=1
www.jss-journal.com
Liquid Chromatography
J. Sep. Sci. 2014, 37, 390–392
The additional subscript index {n} symbolizes here summation over all n SO types. Note that Equation (7) reduces to Equation (4) when n = 1.
2.1 Chromatography Chromatographic retention is characterized by the value of retention time tR , which relates to an equilibrium isotherm through the well-known Equation (8) [11] tR = t0 [1 + F (dq (c)/dc)]
(8)
where t0 is the hold-up time, F the phase ratio, q and c the equilibrium concentrations of an analyte (SA in our terminology) in the stationary and mobile phases, respectively. Obviously, q = [A···NS] + ··· + [N···AS], c = [A···N] if we consider a molecule A···N. According to this equation, the retention depends on the nature of SO–SA interactions only through the equilibrium isotherm. Designating the concentration of the adsorption sites on the surface of the stationary phase as q*, we arrive at the adsorption isotherm q ∗ K (A···N) c q = 1 + K (A···N) c
q =
max min
f K (A···N) K (A···N) c d K (A···N) 1 + K (A···N) c
C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
An addition of specific SOs to the running buffer is commonly used in chiral CE for the separation of enantiomers. A theory describing the migration of an analyte in the presence of a chiral SO was developed by Wren and Rowe [13]. According to this theory, the effective mobility of a single binding mode analyte A is given by A eff =
0 + AS K A [S] 1 + K A [S]
(12)
Here 0 is the mobility of the free analyte, AS is the mobility of the analyte–SO complex, and KA is the equilibrium binding constant. We shall extend this theory to the case of a multiplebinding mode equilibrium. The effective mobility is defined as the sum of the electrophoretic mobilities of all different species in which the analyte may exist weighted by the mole fraction ϕ of the respective species [14]. Consequently,
A···N eff
= 0 1 −
N···A
N···A
JS +
JS JS
(13)
J =A···N
(9)
(10)
where q i∗ is the concentration of the ith SO. Provided that the SA has a single binding mode, Equation (10) reduces to the multisite Langmuir isotherm well known in the theory of adsorption on a heterogeneous surface [6], describing the binding equilibrium depicted in Fig. 1C. Equation (10) implies a discrete distribution of SOs with respect to the equilibrium constant. In chromatography, the concept of a continuous equilibrium constant distribution is frequently used [11, 12]. Let us assume such a distribution f(K(A···N) ) defined between the minimum (min) and the maximum (max) possible values of the apparent equilibrium constant K(A···N) (see Eq. (4)) so max that min f (K (A···N) )d K (A···N) = q ∗ , the total concentration of SOs. Then Equation (10) should be rewritten replacing summation over a finite number of SO types by integration over a continuous range of K(A···N ) :
2.2 Capillary electrophoresis
J =A···N
This equation is similar to the Langmuir adsorption isotherm for a single binding mode adsorption. Nevertheless, it differs from the latter in the meaning of the equilibrium constant defined by Equation (4). In the general case of a multiple binding mode SA and multiple SOs (Fig. 1D), the adsorption isotherm reads n q i∗ K (A···N),i c q = 1 + K (A···N),i c i=1
391
(11)
After rearrangement and taking into account Equation (4) we derive 0 + = A···N eff
N···A J =A···N
JS K JS · [S] (14)
1 + K (A···N) [S]
As seen, the effective mobility of a multiple binding mode SA is defined by an equation similar to Equation (12). An essential difference lies in the definitions of the coefficients at the equilibrium SO concentration. Earlier, Dubsk´y et al. [9,15] developed a model involving a single binding mode SA and multiple SOs, with the expression for the effective mobility given by 0 + A···N eff
=
n
JS,i K JS,i i
i=1
1 + K (A···N){n} [S]
· [S] (15)
where summation is carried out over all SOs and the index J, denoting an SA’s binding mode, takes a certain single value. The above expression corresponds to the model depicted in Fig. 1C (with reservation that there is no stationary phase in CE), while Equation (14) reflects the model shown in Fig. 1B. From a formal mathematical point of view, the difference between Equations (14) and (15) concerns the molar fraction of the ith SO i , which is present in Equation (15) and lacking in Equation (14). Consider the situation with competing binding modes and multiple SOs. Omitting routine mathematical
www.jss-journal.com
392
J. Sep. Sci. 2014, 37, 390–392
L. D Asnin and Y. Nikitina
Figure 1. Illustration of four scenarios of binding equilibria: (A) a single-binding mode SA—a single SO; (B) a multiple-binding mode SA—a single SO; (C) a single-binding mode SA—multiple SOs; (D) a multiple-binding mode SA— multiple SOs. The molecule AB is an SA, Symbols S, S1, and S2 denote SOs. S1 and S2 designate different SOs. The figure illustrates equilibria in chromatography. It will be in principle the same in CE except there will be no stationary phase.
manipulations, we arrive at the following expression for the effective mobility
N···A n 0 + JS,i K JS,i i · [S] J =A···N i=1 (16) A···N = eff 1 + K (A···N){n} [S] Assuming a single binding mode SA, one can obtain Equation (15). The assumption of a single SO results in Equation (14).
3 Concluding remarks
4 References [1] Lipkowitz, K. B., Raghothama, S., Yang, J., J. Am. Chem. Soc. 1992, 114, 1554–1562. [2] Maier, N. M., Schefzick, S., Lombardo, G. M., Feliz, M., Rissanen, K., Lindner, W., Lipkowitz, K. B., J. Am. Chem. Soc. 2002, 124, 8611–8629. [3] Zhao, C. F., Cann, N. M., Anal. Chem. 2008, 80, 2426–2438. [4] Angulo, J., D´ıaz, I., Reina, J. J., Tabarani, G., Fieschi, F., Rojo, J., Nieto, P. M., ChemBioChem 2008, 9, 2225–2227. [5] Pagnozzi, D., Birolo, L., Leo, G., Contessi, S., Lippe, G., Pucci, P., Mavelli, I., Biochemistry 2010, 49, 7542–7552.
Equation (4) in chromatography and Equation (14) in CE provide a mathematical basis for the use of the model in which an analyte binds to an SO by different (competing) modes. Supposedly, this model plays a role in the retention of analytes having several functionalities, each capable of interacting with an SO. Chiral systems involving an SA compound with two distant stereogenic centres and a single binding site chiral SO can be useful to investigate this phenomenon. For this purpose, chromatographic (electrophoresis) results should be supplemented by the data from molecular dynamic simulation or sophisticated spectroscopic techniques, which are able to distinguish contributions of particular binding modes. This sort of information is unattainable using chromatographic or CE experiment alone. Indeed, what is measured is the lumped equilibrium constant (Eq. (4)) in chromatography and the lumped constant and the sum JS K JS in CE.
[10] Hein, G. E., Niemann, C., J. Am. Chem. Soc. 1962, 84, 4495–4503.
This work was supported by a grant of the Russian Foundation for Basic Research (No. 13-03-92692).
[14] Scriba, G. K. E., Topics in Current Chemistry, Springer-Verlag, Berlin, Heidelberg 2013, pp. 1–67. DOI: 10.1007/128_2013_438.
The authors have declared no conflict of interest.
C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
[6] Asnin, L., J. Chromatogr. A 2012, 1269, 3–25. [7] Levkin, P. A., Schurig, V., J. Chromatogr. A 2008, 1184, 309–322. [8] Fornstedt, T., J. Chromatogr. A 2010, 1217, 792–812. ´ P., Svobodova, ´ J., Gas, ˇ B., J. Chromatogr. B [9] Dubsky, 2008, 875, 30–34.
[11] Guiochon, G., Felinger, A., Shirazi, D. G., Katti, A. M., Fundamentals of Preparative and Non-Linear Chromatography, Academic Press, Boston 2006. [12] Samuelsson, J., Arnell, R., Fornstedt, T., J. Sep. Sci. 2009, 32, 1491–1506. [13] Wren, S. A. C., Rowe, R. C., J. Chromatogr. 1992, 603, 235–241.
´ P., Svobodova, ´ J., Tesaˇrova, ´ E. J., Gas, ˇ B., Elec[15] Dubsky, trophoresis 2010, 31, 1435–1441.
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