J. theor. Biof. (1979) 81, 123-127
The Effect of Multivalency on the Specificity of Protein and Cell Interactions PAUL
H.
EHRLICH
Cellular and Molecular Research Laboratory, Massachusetts General Hospital, Boston, Massachusetts (Received 8 January
1979, and in revised-form
02114
3 1 May 1979)
The specificity of interaction between proteins and ligands is shown to depend on the multivalency of the interaction. Increases in specificity are possible by increasing the number of protein subunit-ligand interactions. Since there are many receptors on each cell membrane, cells can be considered multivalent and cell-cell interactions can be very specific. 1. Introduction
Proteins can be very specific in their interactions with other proteins or ligands, even discriminating between molecules which are quite similar. However, there are certain circumstances in which biological specificity is so important that enhanced selectivity of binding is desirable. One such example is in protein synthesis where few errors are tolerable. Enzymes involved in the biosynthesis of proteins may have devised a “proofreading” kinetic mechanism (Hopfield, 1974; Ninio, 1975; Hopfield, Yamane, Yue & Coutts, 1976) to decrease errors. This is an enzymatic pathway requiring energy which enables the enzyme to re-select a substrate after a preliminary selection has been made, thus greatly enhancing the specificity. In this paper, the possibility of enhancement of specificity in immunology by multivalency is examined. It is shown that the multivalency of many proteins of the immune system allows the protein to re-select, in a sense, the substrate, and therefore increase the specificity of the interaction. This work has also been extended to the case of cell-cell interactions and it is shown that the properties of isolated receptors may differ from the receptors on the cell surface. Isolated receptors may bind different ligands with little specificity but the possibility of binding many ligands in cell-cell interactions can result in very specific interactions. 2. Antibodies
Enhancement of specificity of antibodies would mean an improvement in discrimination between two similar antigens above that selectivity possible 123 0022-5193/79/210123+05
%02.00/O
‘(I”: 1979 Academic
Press Inc. (London)
Ltd.
124
P. H. EHRLICH
with monovalent antibody-hapten interaction. If multivalent antibodies were more specific than monovalent antibodies, this specificity would be derived at little cost in terms of genetic input. The increased avidity of some multivalent antibodies has been well documented (Hornick & Karush, 1972; Gopalakrishnan & Karush, 1974). The following derivation shows that specificity could be increased as well. Specificity can be defined as the ratio of substrate (or antigen) which would be bound to the protein to that of a different antigen when both are present at the same concentration (Hopfield, 1974; Ninio, 1975). Therefore, s+
(1) B
where S is the selectivity, K, is the equilibrium binding constant of antigen A to the antibody and KB is the binding constant of a similar, but different, antigen B to the same antibody. If the antibody is divalent and the antigens are multivalent, then, using the terminology of Crothers & Metzger (1972) K;& = 2K\(l +K;)
(2)
where K; is the binding constant of the F(ab) (antigen binding fragment of an immunoglobulin, which is monovalent), K; is the equilibrium constant for the second step of binding to the particle, K& is the observed equilibrium constant. For K; > 1 (meaning for any antibody for which the second binding step is favorable)
K& = 2K;K;.
(3)
K& = 2K;,K;,
(4)
Kbbs= 2K;BK;B.
(5)
Therefore, for antigen A and for antigen B Then, (6)
Therefore, the difference in specificity of the bivalent antibody-antigen interaction and the monovalent antibody-antigen interaction is the factor K;JK& If this factor is 1, then there is no change in selectivity. If the ratio is more than one, specificity is increased and if less than one, specificity is decreased. It is very likely that the selectivity is increased (K&/Y& > 1). This is due to the fact that the monovalent antibody was assumed to interact more strongly with antigen A than antigen B (K;A/K;B > 1). This implies that, everything else being equal, any other F(ab) fragment of a multivalent
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AND
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antibody will also interact more strongly with A than B. The only factor that can change this is if the three dimensional geometry of determinant B on a multivalent particle is much more favorable than determinant A. This is certainly possible. However, it is more probable that any natural particles containing antigens A or B which are close enough in structure for determinants of A and B to cross-react, will also have a similar three dimensional array of A and B. One example could be an anti-idiotypic antibody which reacts with the idiotype (an antigenic determinant present on an individual immunoglobulin molecule ; for a discussion of idiotypy see Nisonoff, Hopper & Spring, 1975) on both F(ab) fragments of an IgG molecule. Its specificity will be enhanced over a cross-reacting idiotype with which it interacts since the relative orientation of the F(ab) arms of the two idiotypic antibodies will be about the same. Another example would be an antibody which reacts with the histocompatibility antigens on a cell surface (Klein, 1979). Different allelic products are probably on the cell surface in a similar conformation and in similar concentration. In addition, many cell surface antigens are mobile on the membrane and movement of the antigen could lower many geometric constraints. In summary, there should be many examples of antigens of similar tertiary structure on particles of similar quaternary structure with which multivalent antibodies would interact more specifically than monovalent antibodies. By a similar argument, for a multivalent antibody and antigen,
where ?i is the valency of the antibody (this formula assumes that a degeneracy factor needed because of the different number of ways of binding multivalent antibody cancels out). The difference in selectivity between the monovalent and multivalent antibody, S,, is then
Therefore, if antigens A and B have similar geometric distributions on a particle but the monovalent affinity of the antibody for determinant A is higher than for determinant B, the selectivity of the antibody for A can be greatly increased. For IgG, an antibody whose monovalent form binds determinant A 100 times stronger than determinant B could bind determinant A with a selectivity of 10,000 or more. For IgM antibodies the selectivity could be even higher.
126
P. H.
EHRLICH
3. Cell-Cell
Interactions
The receptor-ligand interaction in the case where the ligand is present on a cell membrane is formally identical to the multivalent antibody-antigen interaction. Indeed, the increased avidity of cell receptor-multivalent antigen interactions has been shown (Bystryn, Siskind & Uhr, 1973). The diffusion on the membrane and of the cells should be much slower than free molecule interactions, but assuming the state of the membrane does not affect the receptor or ligand, the specificity of each receptor should remain unchanged. The loss of translational freedom should not affect the relative strengths of monovalent receptor-ligand interaction. This correction for loss of translational freedom can be validated by a statistical mechanical argument. Since the selectivity is related to equilibrium constants, it can be expressed in terms of partition functions. A separation of the translational variables of the particle from the intramembrane translational, rotational and vibrational variables of the individual receptors is possible because these two sets of variables should be uncoupled according to the fluid membrane model (Singer, 1974). If the particles containing antigens A and B are similar, the translational partition functions cancel out, Bell (1978) has also assumed that the equilibrium constant can be approximated by an intramembrane two dimensional equilibrium constant. Therefore, G, = n,g, and G, = nsg, (9) where GA is the free energy of interaction of receptors on the surface of a cell with ligand A on another cell, G, is the free energy of interaction with ligand B, nA is the number of receptors for ligand A which can participate simultaneously in a cell-cell interaction, nB is the number of receptors for ligand B, and gA and ge are the interaction free energy of the isolated receptor with the monovalent ligand A and B, respectively, corrected for the loss of translational freedom. Then, e-G” = e-n”&+%g. s= __ (10) e-G, Using the same arguments as in the antibody case that A and B are probably on similar particles (or identically, we can consider the result valid only for such particles), then nA equals n, and: s = e-&,-g.) (11) Since ga and ge are negative for favorable interactions, and gA is larger than gB (interaction of receptor with ligand A is assumed to be stronger), S is
SPEClFlCITY
AND
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127
larger than 1. Since nA can be large (but may be limited by steric factors), the selectivity of cell-cell interactions can be significantly greater than the specificity of individual receptors. This may be important in explaining some properties of receptors. The T cell receptor may have only V, variable regions or about half the possible interactions with antigen as immunoglobulins (Krawinkel et al., 1976). However, the T cell receptor is, at least in some cases, more selective in recognizing some antigens than immunoglobulins (Paul & Benacerraf, 1977). The scheme mentioned above could greatly increase the specificity of an immunoglobulin domain if many domains were attached to one cell and they could interact only with antigen on a cell containing many antigens. A kinetic analysis of cell-cell interactions also reveals that small differences in the strength of binding of the receptor and ligand can lead to large differences in the time two cells are bound. Bell (1978) has shown that under an applied force a, relatively small difference in T (lifetime of an individual receptor-ligand bond, which is related to applied force and the equilibrium constant) can lead to very large differences in t (time to separate cells which are bound together by individual bonds of lifetime T). Therefore, while a receptor may not bind two different ligands for a significantly different length of time, a cell containing large numbers of receptors may bind cells containing the different ligands with an appreciably different lifetime. The author would like to thank Professor E. Haber, Mr J. Newell, Dr C. Homey and Dr V. Zurawski for a critical review of the manuscript. REFERENCES BELL. G. I. (1978). Science 200, 618. BYSTRYN, J.-C., &KIND, G. W. & UHR, J. W. (1973). J. Exp. Med. 137, 301. CROTHERS,P. M. & METZGER, H. (1972). 1mmunochemistr.v 9, 341. G~PALAKRISHNAN, P. V. & KARUSH, F. (1974). J. Immunol. 113, 769. HOPFIELD. J. J., YAMANE, T., YLJE, V. & Courts, S. M. (1976). Proc. nom. Acad. Sci. U.S.A. 73, 1164. HOPFIELD, J. J. (1974). Proc. nutn. Acad. Sci. U.S.A. 71, 4135. HORNICK, C. L. & KARUSH, F. (1972). fmmunochemistr.v 9, 325. KLEIN, J. (1979). Science 203, 516. KRAWINKEL, U., CRAMER, M., BEREK, C., HAMMERLING, G., BLACK, S. J., RAJEWSKY. K. & EICHMANN, K. (1976). Cold Spring Harb. Symposium Quant. Biol. XLl, 285. NINIO, J. (1975). Biochimie. 57, 587. NISONOFF, A., HOPPER, J. E. & SPRING, S. B. (1975). The Antibody Molecule. New York:
Academic Press. PAUL, W. E. & BENACERRAF, B. (1977). Science 195, 1293. SINGER, S. J. (1974). Advances in Immunology 19, I.
The Effect of Multivalency on the Specificity of Protein and Cell Interactions PAUL
H.
EHRLICH
Cellular and Molecular Research Laboratory, Massachusetts General Hospital, Boston, Massachusetts (Received 8 January
1979, and in revised-form
02114
3 1 May 1979)
The specificity of interaction between proteins and ligands is shown to depend on the multivalency of the interaction. Increases in specificity are possible by increasing the number of protein subunit-ligand interactions. Since there are many receptors on each cell membrane, cells can be considered multivalent and cell-cell interactions can be very specific. 1. Introduction
Proteins can be very specific in their interactions with other proteins or ligands, even discriminating between molecules which are quite similar. However, there are certain circumstances in which biological specificity is so important that enhanced selectivity of binding is desirable. One such example is in protein synthesis where few errors are tolerable. Enzymes involved in the biosynthesis of proteins may have devised a “proofreading” kinetic mechanism (Hopfield, 1974; Ninio, 1975; Hopfield, Yamane, Yue & Coutts, 1976) to decrease errors. This is an enzymatic pathway requiring energy which enables the enzyme to re-select a substrate after a preliminary selection has been made, thus greatly enhancing the specificity. In this paper, the possibility of enhancement of specificity in immunology by multivalency is examined. It is shown that the multivalency of many proteins of the immune system allows the protein to re-select, in a sense, the substrate, and therefore increase the specificity of the interaction. This work has also been extended to the case of cell-cell interactions and it is shown that the properties of isolated receptors may differ from the receptors on the cell surface. Isolated receptors may bind different ligands with little specificity but the possibility of binding many ligands in cell-cell interactions can result in very specific interactions. 2. Antibodies
Enhancement of specificity of antibodies would mean an improvement in discrimination between two similar antigens above that selectivity possible 123 0022-5193/79/210123+05
%02.00/O
‘(I”: 1979 Academic
Press Inc. (London)
Ltd.
124
P. H. EHRLICH
with monovalent antibody-hapten interaction. If multivalent antibodies were more specific than monovalent antibodies, this specificity would be derived at little cost in terms of genetic input. The increased avidity of some multivalent antibodies has been well documented (Hornick & Karush, 1972; Gopalakrishnan & Karush, 1974). The following derivation shows that specificity could be increased as well. Specificity can be defined as the ratio of substrate (or antigen) which would be bound to the protein to that of a different antigen when both are present at the same concentration (Hopfield, 1974; Ninio, 1975). Therefore, s+
(1) B
where S is the selectivity, K, is the equilibrium binding constant of antigen A to the antibody and KB is the binding constant of a similar, but different, antigen B to the same antibody. If the antibody is divalent and the antigens are multivalent, then, using the terminology of Crothers & Metzger (1972) K;& = 2K\(l +K;)
(2)
where K; is the binding constant of the F(ab) (antigen binding fragment of an immunoglobulin, which is monovalent), K; is the equilibrium constant for the second step of binding to the particle, K& is the observed equilibrium constant. For K; > 1 (meaning for any antibody for which the second binding step is favorable)
K& = 2K;K;.
(3)
K& = 2K;,K;,
(4)
Kbbs= 2K;BK;B.
(5)
Therefore, for antigen A and for antigen B Then, (6)
Therefore, the difference in specificity of the bivalent antibody-antigen interaction and the monovalent antibody-antigen interaction is the factor K;JK& If this factor is 1, then there is no change in selectivity. If the ratio is more than one, specificity is increased and if less than one, specificity is decreased. It is very likely that the selectivity is increased (K&/Y& > 1). This is due to the fact that the monovalent antibody was assumed to interact more strongly with antigen A than antigen B (K;A/K;B > 1). This implies that, everything else being equal, any other F(ab) fragment of a multivalent
SPECIFICITY
AND
MULTIVALENCY
125
antibody will also interact more strongly with A than B. The only factor that can change this is if the three dimensional geometry of determinant B on a multivalent particle is much more favorable than determinant A. This is certainly possible. However, it is more probable that any natural particles containing antigens A or B which are close enough in structure for determinants of A and B to cross-react, will also have a similar three dimensional array of A and B. One example could be an anti-idiotypic antibody which reacts with the idiotype (an antigenic determinant present on an individual immunoglobulin molecule ; for a discussion of idiotypy see Nisonoff, Hopper & Spring, 1975) on both F(ab) fragments of an IgG molecule. Its specificity will be enhanced over a cross-reacting idiotype with which it interacts since the relative orientation of the F(ab) arms of the two idiotypic antibodies will be about the same. Another example would be an antibody which reacts with the histocompatibility antigens on a cell surface (Klein, 1979). Different allelic products are probably on the cell surface in a similar conformation and in similar concentration. In addition, many cell surface antigens are mobile on the membrane and movement of the antigen could lower many geometric constraints. In summary, there should be many examples of antigens of similar tertiary structure on particles of similar quaternary structure with which multivalent antibodies would interact more specifically than monovalent antibodies. By a similar argument, for a multivalent antibody and antigen,
where ?i is the valency of the antibody (this formula assumes that a degeneracy factor needed because of the different number of ways of binding multivalent antibody cancels out). The difference in selectivity between the monovalent and multivalent antibody, S,, is then
Therefore, if antigens A and B have similar geometric distributions on a particle but the monovalent affinity of the antibody for determinant A is higher than for determinant B, the selectivity of the antibody for A can be greatly increased. For IgG, an antibody whose monovalent form binds determinant A 100 times stronger than determinant B could bind determinant A with a selectivity of 10,000 or more. For IgM antibodies the selectivity could be even higher.
126
P. H.
EHRLICH
3. Cell-Cell
Interactions
The receptor-ligand interaction in the case where the ligand is present on a cell membrane is formally identical to the multivalent antibody-antigen interaction. Indeed, the increased avidity of cell receptor-multivalent antigen interactions has been shown (Bystryn, Siskind & Uhr, 1973). The diffusion on the membrane and of the cells should be much slower than free molecule interactions, but assuming the state of the membrane does not affect the receptor or ligand, the specificity of each receptor should remain unchanged. The loss of translational freedom should not affect the relative strengths of monovalent receptor-ligand interaction. This correction for loss of translational freedom can be validated by a statistical mechanical argument. Since the selectivity is related to equilibrium constants, it can be expressed in terms of partition functions. A separation of the translational variables of the particle from the intramembrane translational, rotational and vibrational variables of the individual receptors is possible because these two sets of variables should be uncoupled according to the fluid membrane model (Singer, 1974). If the particles containing antigens A and B are similar, the translational partition functions cancel out, Bell (1978) has also assumed that the equilibrium constant can be approximated by an intramembrane two dimensional equilibrium constant. Therefore, G, = n,g, and G, = nsg, (9) where GA is the free energy of interaction of receptors on the surface of a cell with ligand A on another cell, G, is the free energy of interaction with ligand B, nA is the number of receptors for ligand A which can participate simultaneously in a cell-cell interaction, nB is the number of receptors for ligand B, and gA and ge are the interaction free energy of the isolated receptor with the monovalent ligand A and B, respectively, corrected for the loss of translational freedom. Then, e-G” = e-n”&+%g. s= __ (10) e-G, Using the same arguments as in the antibody case that A and B are probably on similar particles (or identically, we can consider the result valid only for such particles), then nA equals n, and: s = e-&,-g.) (11) Since ga and ge are negative for favorable interactions, and gA is larger than gB (interaction of receptor with ligand A is assumed to be stronger), S is
SPEClFlCITY
AND
MULTIVALENCY
127
larger than 1. Since nA can be large (but may be limited by steric factors), the selectivity of cell-cell interactions can be significantly greater than the specificity of individual receptors. This may be important in explaining some properties of receptors. The T cell receptor may have only V, variable regions or about half the possible interactions with antigen as immunoglobulins (Krawinkel et al., 1976). However, the T cell receptor is, at least in some cases, more selective in recognizing some antigens than immunoglobulins (Paul & Benacerraf, 1977). The scheme mentioned above could greatly increase the specificity of an immunoglobulin domain if many domains were attached to one cell and they could interact only with antigen on a cell containing many antigens. A kinetic analysis of cell-cell interactions also reveals that small differences in the strength of binding of the receptor and ligand can lead to large differences in the time two cells are bound. Bell (1978) has shown that under an applied force a, relatively small difference in T (lifetime of an individual receptor-ligand bond, which is related to applied force and the equilibrium constant) can lead to very large differences in t (time to separate cells which are bound together by individual bonds of lifetime T). Therefore, while a receptor may not bind two different ligands for a significantly different length of time, a cell containing large numbers of receptors may bind cells containing the different ligands with an appreciably different lifetime. The author would like to thank Professor E. Haber, Mr J. Newell, Dr C. Homey and Dr V. Zurawski for a critical review of the manuscript. REFERENCES BELL. G. I. (1978). Science 200, 618. BYSTRYN, J.-C., &KIND, G. W. & UHR, J. W. (1973). J. Exp. Med. 137, 301. CROTHERS,P. M. & METZGER, H. (1972). 1mmunochemistr.v 9, 341. G~PALAKRISHNAN, P. V. & KARUSH, F. (1974). J. Immunol. 113, 769. HOPFIELD. J. J., YAMANE, T., YLJE, V. & Courts, S. M. (1976). Proc. nom. Acad. Sci. U.S.A. 73, 1164. HOPFIELD, J. J. (1974). Proc. nutn. Acad. Sci. U.S.A. 71, 4135. HORNICK, C. L. & KARUSH, F. (1972). fmmunochemistr.v 9, 325. KLEIN, J. (1979). Science 203, 516. KRAWINKEL, U., CRAMER, M., BEREK, C., HAMMERLING, G., BLACK, S. J., RAJEWSKY. K. & EICHMANN, K. (1976). Cold Spring Harb. Symposium Quant. Biol. XLl, 285. NINIO, J. (1975). Biochimie. 57, 587. NISONOFF, A., HOPPER, J. E. & SPRING, S. B. (1975). The Antibody Molecule. New York:
Academic Press. PAUL, W. E. & BENACERRAF, B. (1977). Science 195, 1293. SINGER, S. J. (1974). Advances in Immunology 19, I.
