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Divalent binding of monoclonal antibody to a cell surface antigen. Modelling of equilibrium data
MolecularImmunology,Vol. 26, No. 8, pp. 735-139, 1989
0161-5890/89 $3.00 + 0.00 Maxwell Pergamon Macmillan plc
Printed in Great Britain.
DIVALENT BINDING OF MONOCLONAL ANTIBODY A CELL SURFACE ANTIGEN. MODELLING OF EQUILIBRIUM DATA
Department
of Immunology,
University
of Stockholm,
S-106 91 Stockholm,
TO
Sweden
(First received 9 December 1988; accepted in revised form28 February 1989) Abstract--The
principles of the binding of a monoclonal Ab to a single epitope of a cell surface antigen were deduced. The equilibrium between the Ab and the antigen was regarded as a two-step reaction, involving free Ab and antigen as well as Ab, monovalently or divalently complexed to its antigen. By a few simple equations the concn of bound Ab was calculated from the values of free Ab concn, cell concn, single cell antigen density and the two equilibrium constants. This cell surface binding was compared to the binding of an ordinary one-step equilibrium reaction. It was shown that a bonus effect due to cell surface divalent binding is obtained when the product of the first association constant and the free Ab concn is less than one and, simultaneously, the product of the second equilibrium constant and the single cell antigen density is greater than one. In addition, when these conditions are fulfilled, an increase in the single cell antigen density may cause an increase in Ab binding that is much more than proportional to the change in antigen density. The implications of these results to tumor immunolocalization and immunotherapy are discussed.
MATHEMATICAL
INTRODUCTION Consider
It
has been found repeatedly that an antibody (Ab) attaching divalently to a cell surface antigen binds more avidly than the same Ab, or the corresponding Fab fragment, attaching monovalently (Mason and Williams, 1980; Parham et al., 1982; Roe et al., 1985). A few models have been published that quantitatively deals with Ab binding to cell surfaces (DeLisi and Metzger, 1976; DeLisi and Wiegel, 1981; Dower et al., 1984; Reynolds, 1979; Roe et al., 1985). The work of Roe and coworkers (1985) is highly interesting. In their model the increased Ab binding to a cell surface antigen is attributed to the fact that the equilibrium reaction occurs in two steps. First, an Ab molecule in solution binds monovalently to a free antigen molecule on the cell surface. Then, in a second equilibrium reaction the monovalent Abantigen complex binds to another free antigen molecule by diffusion in the plasma membrane. In this model, the diffusibility of the antigen molecule, not a high local concn, is the essential point. The model gave an excellent fit to experimental data obtained by using a human osteogenic sarcoma cell line and a monoclonal Ab binding to it. However, the implicit equation describing the state of equilibrium given by Roe is quite tedious, which may explain why this model has not yet obtained the attention it deserves. In the present work an explicit equation was deduced and used for modelling of equilibrium data on Ab binding to a cell surface antigen. The possibility of exploiting the bonus effect of cell surface divalent Ab binding in immunolocalization and immunotherapy of tumours is discussed. MlMM
26,?-D
uniform
a suspension surface
THEORY
of cells
of uniform
size and
composition.
To the suspension is added a monoclonal divalent antibody (IgG) with ability to bind to a single epitope of a surface antigen. The cell surface antigen molecules are assumed to diffuse freely and randomly in the plane of the plasma membrane but are unable to diffuse out of this plane. It is further assumed that the free Ab molecules in the solution in a first step bind single antigen molecules (monovalent binding) and the association (affinity) constant of this reaction is K (first equilibrium). Divalent binding by Ab only results from diffusion of the monovalent Ab-antigen complexes and free antigen molecules on the plasma membrane surface. The equilibrium constant of this second reaction is L. The following abbreviations are used: n = Avogadro’s number = 6.023 x 102’; C = cell concn (cells/liter); B = average concn of bound Ab (M); b = bound Ab (molecules/cell); F = free Ab concn (M); N = single cell antigen density (mol/cell); N’ = single cell antigen density (molecules/cell)
=
Nn; A = NC = average
(M); X = average
antigen concn in the suspension
free antigen
concn
in the suspension
(M); Y = average concn in the suspension Ab complexes (M); Z = average concn in the suspension complexes; 135
of monovalent of divalent
Ab
136
Am
LAI~S~C~N
K = association constant of the first equilibrium reaction, in which a free Ab binds to a free antigen molecule (liter/mol); L = association constant of the second equilibrium reaction, in which a monovalent complex binds another free antigen molecule (liter cells/mol); L’ = L/n (liter cells/molecule); B, = the half-saturation concn of bound Ab, amounting to A 12; F,, = the free Ab concn that accomplishes a B value of l?,= A/2. The law of mass librium gives:
action
applied
to the first equi-
Subtraction
of equation
3 from equation
7 gives:
NC-X-2Z-FXK=Q. Substitution of equation quadratic equation, (2FKLjC)X’
5 into
(8) 8 results
in the
+ (1 + FK)X - NC = 0,
with the solution, X=[-I-FK + J( 1 + FK)* + 8FKLN]/(4FKL/C). Bound
Ab concn
-
FxX
= K.
(11)
(1)
The law of mass action used for the second equilibrium at the cell surface of uniform cells can be written as: Z/C (X/C) x (Y/C)
(2)
= L*
From this equation, Z is first eliminated using equation 6 and then Y using equation 3. This results in B = NC/2 f FXKJ2 - X/2.
(12)
Finally, when X is substituted using equation 10, an equation is obtained that describes B as a function of F in one step:
B=NC+[-l-FK+~(l+FK)2+8FKLN]xCx(FK-1) 2
(13a)
8FKL For calculation of the number molecules per cell the form
b=N’+[-l-FK+j(l+FK)2+8FKL’N’]x(FK-l) 2
where X/C is the single cell density of free antigen, while Y/C and Z/C are the single cell densities of Ab, binding antigen monovalently and divalently, respectively. Equation 1 can be written as Y=FXK
(3)
2 as
Y=Z. Combining gives:
these
two equations
and solving
Z = X*FKLIC. The antigen
concn
can be described
for Z
(5) as:
A=NC=X+Y-+22,
(6)
Y=NC-X-222.
(7)
or
of
bound
Ab
(13b)
8FKL’ is a proper
and equation
(10)
is B=Y+Z.
Y
(9)
one.
For calculation of a state where only monovalent cell surface binding takes place the Langmuir binding isotherm, AKF B=----l+KF’ is used. It is merely a rearrangement of the law of mass action. It corresponds to a situation where the antigen cannot diffuse in the surface and where the sites facilitating divalent binding, as well as the sites where one bound Ab molecule obstructs the binding of another, can be neglected. If instead, the antigen is dissolved in the solution the binding is also approximately described by equation 14, with a similar value of K, provided the mol. wt of the antigen is not much less than that of the Ab (DeLisi and Metzger, 1976). The maximal concn of bound Ab is A. Half of that concn is here denoted B, and the corresponding free Ab concn F,, . With only monovalent binding according to equation 14, it is well known (and easily seen by inserting B = A/2 into equation 14) that F,, = l/K. F,, is usually called Kd in the purely monovalent situation.
Modelling of antibody binding to a cell surface antigen
The same issue can be investigated in the mixed monovalent-divalent situation. By inserting B = A/2 = NC/2 into equation 12 it is seen that
F,,=l/K.
(13
Thus, for the binding of an Ab to a cell membrane antigen, F,, is the reciprocal of the (first) association constant, K, whether or not the antigen is freely diffusible in the plasma membrane. The second association constant, L, does not influence F,,. Another important issue is what value the second equilibrium constant, L, must have to accomplish a divalent binding by Ab that amounts at least to the monovalent binding. Consider first the situation, where the concns of Ab binding monovalently and divalently are equal. Then z = Y, or with equations
(16)
5 and 3 inserted,
L =c/x.
(17)
It can be seen from Fig. 1 that the maximal bonus effect from divalent binding is obtained at Fe l/K. In this area B approaches 0 and X approaches A = NC. Thus, the fulfillment of the condition L > l/N implies
Free Ab concn
1M 1
Fig. 1. Modelling of Ab binding to a plasma membrane antigen. The association constant for monovalent binding was assumed to be lO’mol/liter, cell concn lO*cells/liter and the single cell antigen density IO-‘* mol/cell (602,300 molecules/cell). In curve A the antigen molecules were assumed to be non-diffusible in the membrane, thus abolishing divalent binding. In curves B-D divalent binding takes place in addition to the monovalent binding. The Ab binding was calculated by using equation 13a, assuming an association constant of divalent binding, L, of 5 x IO’* in curve B, 5 x 10zoin curve C and 5 x lo’* cell liter/m01 in curve D. In (a) semi-logarithmic scales were used and in (b) double-logarithmic scales were used.
a higher divalent than monovalent
737
binding, provided
FK>>l. In this paper equation 13 was used for modelling of the Ab binding, with F as the independent variable at different values of L.The results were compared to those obtained with equation 14, implying no lateral diffusion. Alternatively, the Ab binding was modelled with N’ as the independent variable at a fixed value of F. RESULTS
In cell surfaces, where the antigen molecules are free to diffuse in the plane of the membrane, Ab molecules are bound divalently as well as monovalently. Equation 13a was used for calculation of the total bound Ab concn. The cell concn was assumed to be lo8 cells/liter (lo5 cells/ml), the single cell antigen density lo-‘* mol/cell and the first association constant, K, 10’ liter/mol. The results of such simulations are shown in Fig. 1. In curve B the single cell association constant of divalent binding, L, was assumed to be 5 x lo”, in curve C 5 x 10” and in curve D 5 x 10” liter cell/mol. For comparison, curve A shows the bound Ab concn in a situation where no divalent binding takes place. Bound Ab then was calculated by the use of equation 14. When the binding curves are drawn semi-logarithmically, they evidently are all symmetrical (Fig. la). From equation 12 it can be concluded that at F = F,, the B value is always A/2,which means that F,, = l/K = Kd even if there is lateral diffusion. In these simulations this F value was lo-’ M. At lower values of F the possibility of divalent binding caused an increase in total Ab binding compared to the purely monovalent situation. This excess binding because of divalent binding by Ab increased with the value of L and became conspicuous at Feel/K and L >>l/N. At F values > Kd the purely monovalent situation resulted in the highest Ab binding, but the relative differences in bound Ab were not as pronounced as the relative differences in the opposite direction at the very low Ab concns (Fig. lb). In Fig. 2 the single cell antigen density, N’, was instead subjected to variation between the limits lO&lO* molecules/cell using equation 13b. K was lo-’ liter/mol, L' was 8.302 x 10e4 cells/molecule (in molar units, L = 5 x 1O-‘8 liter cells/mol) and F was kept constant at lo-” M. At the lowest N’ values the bound Ab is very low and is on average less than 1 Ab molecule/cell. The increase in average bound Ab is approximately proportional to N’. Then there is an area of steeper slope. As an example, an increase of N’ from 239,OOOmolecules by a factor of ten increases the bound antibody concn by a factor of 89.5. Then again, the binding curve approaches a proportional phase. This behaviour reflects the fact that the chance of a monovalent complex to hit a free antigen molecule is small at very low single cell antigen densities. Consequently, there is essentially no diva-
.&CELARSSON
738
Single cell antigen density (molecules/cell)
Fig. 2. Modelling of Ab binding to a plasma membrane antigen using equation 13b. The monovalent association constant, K, was assumed to be lO’liter/mol, the divalent association constant, L’, 8.302 x 10m4 cells/molecule (L = 5 x 10”’ liter cells/mol) and the cell concn lO*cells/ liter. The free Ab concn was constantly lo-” A4 and the single cell antigen density, N’, varied from 100 to 10’ molecules/cell.
lent binding. As the antigen density increases, so does the chance of a hit between a monovalent complex and a free antigen molecule, which makes the formation of divalent complexes essential. At very high antigen densities the divalent binding by Ab becomes predominant and the binding curve again returns to an approximately proportional rate. DISCUSSION It cannot be expected that every monoclonal Ab directed against non-repeated epitopes of a plasma membrane antigen also accomplishes a noticeable divalent binding. The association constants (Kand L) are the ratios between the respective association rate and dissociation rate constants. The association rate constant is highly dependent on the diffusion. Consequently, a high lateral diffusion of the protein antigen would favour a high value of L and thus the divalent binding. Generally, proteins with no or a small internal part diffuse faster than those with a large internal part. A particular group of proteins are those that strongly interact with the cytoskeleton and consequently have little lateral diffusion (Darnell et al., 1986). Such antigens are poor candidates for divalent binding. Recently, it has also been found that certain cell surface proteins are anchored to the plasma membrane by glycosyl-phosphatidylinositol (reviewed by Low and Saltiel, 1988). The Thy-l antigen, which was shown to participate in divalent cell surface binding by Ab (Mason and Williams, 1980) is attached in this way (Tse et al., 1985). For proteins that are inserted into the plasma membrane in a way that makes them keep at least a predominant orientation, the direction of their epitopes may be critical. Certainly, IgG Abs have a segmental flexibility between the two Fab parts,
facilitating divalent binding at different epitope angles. However, at least the human IgG subclasses seem to have a predominant angle between their Fab parts (Gregory et al., 1987). The inter-Fab angle in the xy plane (perpendicular to the plane of symmetry of the Fc part) was estimated to vary between the subclasses from 120” (IgGl) to 240- (IgG2). In addition, all subclasses seem to have a predominant inter-Fab angle out of this plane amounting to 90”. Of the mouse IgG subclasses IgGl has the lowest segmental flexibility (Oi et al., 1984). The combination of a rigid protein molecule, inserted by a transmembrane amino acid sequence, giving it a certain orientation, and a predominant FabFab angle would accomplish a high divalent binding only at certain epitope angles (Fig. 3). The best fit for the divalent binding is obviously obtained when the angle between the epitope and the membrane surface, together with the predominant FabFab angle, amounts to 180-. The utility of a theoretical model depends on its power to suggest experimental approaches and predict experimental outcomes. In the field of tumor immunolocalization and immunotherapy the present model implies the following lines of action. One major obstacle in immunotargeting in vitro is that tumor associated antigens often exist in the circulation in addition to the surface of the tumor cells. It has been suggested previously (Price et al., 1987) that, in such situations, Abs directed against single epitopes are preferable, a view that is sup-
B
w
w
Fig. 3. Schematic drawing illustrating the importance of proper direction of the epitopes of a plasma membrane antigen. When the angle between the epitope and the membrane surface, together with the predominant FabFab angle of the Ab molecule, amounts to 180” the binding is most efficient. (A) A combination of angles favoring a high divalent binding; (B) a combination of angles not favoring divalent binding.
Modelling
of antibody
binding
ported by the modellings of this paper. The binding to an antigen free in solution should be approximately what is predicted by equation 14 (DeLisi and Metzger, 1976). When the circumstances are favorable, the binding in solution approximately predicted by equation 14 is far exceeded by the divalent binding at the plasma membrane (Fig. 1). That occurs when LN >>1 and FK <<1. The fulfillment of the first conclition is very much a matter of using Abs directed against the right epitopes of the right antigen. For fulfilling the second condition, Ab of low affinity in the first equilibrium should be used, and the Ab should be infused slowly to avoid high concns of free Ab. It should be kept in mind that the cell concn assumed in the present modellings (lO’/ml) is one suitable to in I;itro experiments. In tissues the cell concn is approximately lo9 per ml. Such a cell concn would optimally increase the bound Ab concns by four magnitudes as compared to those shown in Fig. 1. In order to overcome the problem of circulating antigen, an Ab should have a low antigen binding in the absence of the cell surface bonus effect, i.e. have a low K value. In order to obtain monoclonal Abs with these properties the selection procedure is essential. Thus, if an ELISA or RIA procedure, employing the antigen coated to a plastic solid phase, is used, the desired monoclonal Abs of low K values are easily lost. Accordingly the monoclonal Abs, giving an efficient divalent cell surface binding in the studies of Roe et al. (1985) and Mason and Williams (1980), were selected using living cell assays (Embleton et al., 1981; McMaster and Williams, 1979). Another obstacle in immunotargeting is the presence of tumor associated antigens also on normai cells, although at lower densities than on the tumor cells. As is demonstrated in Fig. 2, theoretically it should be possible to find conditions that also consider this complication. By selecting proper values of F, K and L in relation to the relevant range of N, a much higher difference in Ab binding is obtained than that which is proportional to the antigen amount of the cells. However, whether this is a feasible approach in practice also remains to be elucidated. Acknowledgements--I and Staffan Pauhe manuscript.
wish to thank Drs Goran Olofsson for their critical reading of the
to a cell surface
antigen
739
REFERENCES
Darnell J., Lodish H. and Baltimore D. (1986) In Molecular Cell Biology, p. 589. Scientific American Books, New York. DeLisi C. and Metzger H. (1976) Some physical chemical aspects of receptor-ligand interactions. Immun. Commun. 5, 417436. DeLisi C. and Wiegel F. W. (1981) Effect of nonspecific forces and finite receptor number on rate constants of ligand-cell bound-receptor interactions. Proc. natn. Acad. Sci. U.S.A. 78, 5569-5572. Dower S. K., Ozato K. and Segal D. M. (1984) The interaction of monoclonal antibodies with MHC class I antigens on mouse spleen cells. I. Analysis of the mechanism of binding. J. Immun. 132, 751-758. Embleton M. J., Gunn B., Byers V. S. and Baldwin R. W. (1981) Antitumour reactions of monoclonal antibody against a human osteogenic-sarcoma cell line. Br. J. Cancer 43, 582-587. Gregory L., Davis K. G., Sheth B., Boyd J., Jefferis R., Nave C. and Burton D. R. (1987) The solution conformations of the subclasses of human IgG deduced from sedimentation and small angle X-ray scattering studies. Molec. Immun. 24, 821X329._ Low M. G. and Saltiel A. R. (1988) Structural and functional roles of glycosyl-phosphatidylinositol in membranes. Science 239, 268-275. Mason D. W. and Williams A. F. (1980) The kinetics of antibody binding to membrane antigens in solution and at the cell surface. Biochem. J. 187,-l-20. McMaster W. R. and Williams A. F. (1979) . , Identification of Ia glycoproteins in rat thymus and purification from rat spleen. Eur. J. Immun. 9, 426433. Oi V. T., Vuong T. M., Hardy R., Reidler J., Dangl J., Herzenberg L. A. and Stryer L. (1984) Correlation between segmental flexibility and effector function of antibodies. Nafure 307. 136140. Parham P., Androlewic; M. J., Brodsky F. M., Holmes N. J. and Ways J. P. (1982) Monoclonal antibodies: purification, fragmentation and application to structural and functional studies of Class I MHC antigens. J. Immun. Meth. 53, 133-173. Price M. R., Edwards S., Jacobs E., Pawluczyk I. 2. A., Byers V. S. and Baldwin R. W. (1987) Mapping of monoclonal antibody-defined epitopes associated with carcinoembryonic antigen, CEA. Cancer lmmun. Immunother. 25, l&15. Reynolds J. A. (1979) Interaction of divalent antibody with cell surface antigens, Biochemistry 18, 264269. Roe R., Robins R. A., Laxton R. R. and Baldwin R. W. (1985) Kinetics of divalent monoclonal antibody binding to tumour cell surface antigens using flow cytometry: standardization and mathematical analysis. Molec. Immun. 22, 11-21. Tse A. G. D., Barclay A. N., Watts A. and Williams A. F. (1985) A glycophospholipid tail at the carboxyl terminus of the Thy-l glycoprotein of neurons and thymocytes. Science 230, 1003~1008.
0161-5890/89 $3.00 + 0.00 Maxwell Pergamon Macmillan plc
Printed in Great Britain.
DIVALENT BINDING OF MONOCLONAL ANTIBODY A CELL SURFACE ANTIGEN. MODELLING OF EQUILIBRIUM DATA
Department
of Immunology,
University
of Stockholm,
S-106 91 Stockholm,
TO
Sweden
(First received 9 December 1988; accepted in revised form28 February 1989) Abstract--The
principles of the binding of a monoclonal Ab to a single epitope of a cell surface antigen were deduced. The equilibrium between the Ab and the antigen was regarded as a two-step reaction, involving free Ab and antigen as well as Ab, monovalently or divalently complexed to its antigen. By a few simple equations the concn of bound Ab was calculated from the values of free Ab concn, cell concn, single cell antigen density and the two equilibrium constants. This cell surface binding was compared to the binding of an ordinary one-step equilibrium reaction. It was shown that a bonus effect due to cell surface divalent binding is obtained when the product of the first association constant and the free Ab concn is less than one and, simultaneously, the product of the second equilibrium constant and the single cell antigen density is greater than one. In addition, when these conditions are fulfilled, an increase in the single cell antigen density may cause an increase in Ab binding that is much more than proportional to the change in antigen density. The implications of these results to tumor immunolocalization and immunotherapy are discussed.
MATHEMATICAL
INTRODUCTION Consider
It
has been found repeatedly that an antibody (Ab) attaching divalently to a cell surface antigen binds more avidly than the same Ab, or the corresponding Fab fragment, attaching monovalently (Mason and Williams, 1980; Parham et al., 1982; Roe et al., 1985). A few models have been published that quantitatively deals with Ab binding to cell surfaces (DeLisi and Metzger, 1976; DeLisi and Wiegel, 1981; Dower et al., 1984; Reynolds, 1979; Roe et al., 1985). The work of Roe and coworkers (1985) is highly interesting. In their model the increased Ab binding to a cell surface antigen is attributed to the fact that the equilibrium reaction occurs in two steps. First, an Ab molecule in solution binds monovalently to a free antigen molecule on the cell surface. Then, in a second equilibrium reaction the monovalent Abantigen complex binds to another free antigen molecule by diffusion in the plasma membrane. In this model, the diffusibility of the antigen molecule, not a high local concn, is the essential point. The model gave an excellent fit to experimental data obtained by using a human osteogenic sarcoma cell line and a monoclonal Ab binding to it. However, the implicit equation describing the state of equilibrium given by Roe is quite tedious, which may explain why this model has not yet obtained the attention it deserves. In the present work an explicit equation was deduced and used for modelling of equilibrium data on Ab binding to a cell surface antigen. The possibility of exploiting the bonus effect of cell surface divalent Ab binding in immunolocalization and immunotherapy of tumours is discussed. MlMM
26,?-D
uniform
a suspension surface
THEORY
of cells
of uniform
size and
composition.
To the suspension is added a monoclonal divalent antibody (IgG) with ability to bind to a single epitope of a surface antigen. The cell surface antigen molecules are assumed to diffuse freely and randomly in the plane of the plasma membrane but are unable to diffuse out of this plane. It is further assumed that the free Ab molecules in the solution in a first step bind single antigen molecules (monovalent binding) and the association (affinity) constant of this reaction is K (first equilibrium). Divalent binding by Ab only results from diffusion of the monovalent Ab-antigen complexes and free antigen molecules on the plasma membrane surface. The equilibrium constant of this second reaction is L. The following abbreviations are used: n = Avogadro’s number = 6.023 x 102’; C = cell concn (cells/liter); B = average concn of bound Ab (M); b = bound Ab (molecules/cell); F = free Ab concn (M); N = single cell antigen density (mol/cell); N’ = single cell antigen density (molecules/cell)
=
Nn; A = NC = average
(M); X = average
antigen concn in the suspension
free antigen
concn
in the suspension
(M); Y = average concn in the suspension Ab complexes (M); Z = average concn in the suspension complexes; 135
of monovalent of divalent
Ab
136
Am
LAI~S~C~N
K = association constant of the first equilibrium reaction, in which a free Ab binds to a free antigen molecule (liter/mol); L = association constant of the second equilibrium reaction, in which a monovalent complex binds another free antigen molecule (liter cells/mol); L’ = L/n (liter cells/molecule); B, = the half-saturation concn of bound Ab, amounting to A 12; F,, = the free Ab concn that accomplishes a B value of l?,= A/2. The law of mass librium gives:
action
applied
to the first equi-
Subtraction
of equation
3 from equation
7 gives:
NC-X-2Z-FXK=Q. Substitution of equation quadratic equation, (2FKLjC)X’
5 into
(8) 8 results
in the
+ (1 + FK)X - NC = 0,
with the solution, X=[-I-FK + J( 1 + FK)* + 8FKLN]/(4FKL/C). Bound
Ab concn
-
FxX
= K.
(11)
(1)
The law of mass action used for the second equilibrium at the cell surface of uniform cells can be written as: Z/C (X/C) x (Y/C)
(2)
= L*
From this equation, Z is first eliminated using equation 6 and then Y using equation 3. This results in B = NC/2 f FXKJ2 - X/2.
(12)
Finally, when X is substituted using equation 10, an equation is obtained that describes B as a function of F in one step:
B=NC+[-l-FK+~(l+FK)2+8FKLN]xCx(FK-1) 2
(13a)
8FKL For calculation of the number molecules per cell the form
b=N’+[-l-FK+j(l+FK)2+8FKL’N’]x(FK-l) 2
where X/C is the single cell density of free antigen, while Y/C and Z/C are the single cell densities of Ab, binding antigen monovalently and divalently, respectively. Equation 1 can be written as Y=FXK
(3)
2 as
Y=Z. Combining gives:
these
two equations
and solving
Z = X*FKLIC. The antigen
concn
can be described
for Z
(5) as:
A=NC=X+Y-+22,
(6)
Y=NC-X-222.
(7)
or
of
bound
Ab
(13b)
8FKL’ is a proper
and equation
(10)
is B=Y+Z.
Y
(9)
one.
For calculation of a state where only monovalent cell surface binding takes place the Langmuir binding isotherm, AKF B=----l+KF’ is used. It is merely a rearrangement of the law of mass action. It corresponds to a situation where the antigen cannot diffuse in the surface and where the sites facilitating divalent binding, as well as the sites where one bound Ab molecule obstructs the binding of another, can be neglected. If instead, the antigen is dissolved in the solution the binding is also approximately described by equation 14, with a similar value of K, provided the mol. wt of the antigen is not much less than that of the Ab (DeLisi and Metzger, 1976). The maximal concn of bound Ab is A. Half of that concn is here denoted B, and the corresponding free Ab concn F,, . With only monovalent binding according to equation 14, it is well known (and easily seen by inserting B = A/2 into equation 14) that F,, = l/K. F,, is usually called Kd in the purely monovalent situation.
Modelling of antibody binding to a cell surface antigen
The same issue can be investigated in the mixed monovalent-divalent situation. By inserting B = A/2 = NC/2 into equation 12 it is seen that
F,,=l/K.
(13
Thus, for the binding of an Ab to a cell membrane antigen, F,, is the reciprocal of the (first) association constant, K, whether or not the antigen is freely diffusible in the plasma membrane. The second association constant, L, does not influence F,,. Another important issue is what value the second equilibrium constant, L, must have to accomplish a divalent binding by Ab that amounts at least to the monovalent binding. Consider first the situation, where the concns of Ab binding monovalently and divalently are equal. Then z = Y, or with equations
(16)
5 and 3 inserted,
L =c/x.
(17)
It can be seen from Fig. 1 that the maximal bonus effect from divalent binding is obtained at Fe l/K. In this area B approaches 0 and X approaches A = NC. Thus, the fulfillment of the condition L > l/N implies
Free Ab concn
1M 1
Fig. 1. Modelling of Ab binding to a plasma membrane antigen. The association constant for monovalent binding was assumed to be lO’mol/liter, cell concn lO*cells/liter and the single cell antigen density IO-‘* mol/cell (602,300 molecules/cell). In curve A the antigen molecules were assumed to be non-diffusible in the membrane, thus abolishing divalent binding. In curves B-D divalent binding takes place in addition to the monovalent binding. The Ab binding was calculated by using equation 13a, assuming an association constant of divalent binding, L, of 5 x IO’* in curve B, 5 x 10zoin curve C and 5 x lo’* cell liter/m01 in curve D. In (a) semi-logarithmic scales were used and in (b) double-logarithmic scales were used.
a higher divalent than monovalent
737
binding, provided
FK>>l. In this paper equation 13 was used for modelling of the Ab binding, with F as the independent variable at different values of L.The results were compared to those obtained with equation 14, implying no lateral diffusion. Alternatively, the Ab binding was modelled with N’ as the independent variable at a fixed value of F. RESULTS
In cell surfaces, where the antigen molecules are free to diffuse in the plane of the membrane, Ab molecules are bound divalently as well as monovalently. Equation 13a was used for calculation of the total bound Ab concn. The cell concn was assumed to be lo8 cells/liter (lo5 cells/ml), the single cell antigen density lo-‘* mol/cell and the first association constant, K, 10’ liter/mol. The results of such simulations are shown in Fig. 1. In curve B the single cell association constant of divalent binding, L, was assumed to be 5 x lo”, in curve C 5 x 10” and in curve D 5 x 10” liter cell/mol. For comparison, curve A shows the bound Ab concn in a situation where no divalent binding takes place. Bound Ab then was calculated by the use of equation 14. When the binding curves are drawn semi-logarithmically, they evidently are all symmetrical (Fig. la). From equation 12 it can be concluded that at F = F,, the B value is always A/2,which means that F,, = l/K = Kd even if there is lateral diffusion. In these simulations this F value was lo-’ M. At lower values of F the possibility of divalent binding caused an increase in total Ab binding compared to the purely monovalent situation. This excess binding because of divalent binding by Ab increased with the value of L and became conspicuous at Feel/K and L >>l/N. At F values > Kd the purely monovalent situation resulted in the highest Ab binding, but the relative differences in bound Ab were not as pronounced as the relative differences in the opposite direction at the very low Ab concns (Fig. lb). In Fig. 2 the single cell antigen density, N’, was instead subjected to variation between the limits lO&lO* molecules/cell using equation 13b. K was lo-’ liter/mol, L' was 8.302 x 10e4 cells/molecule (in molar units, L = 5 x 1O-‘8 liter cells/mol) and F was kept constant at lo-” M. At the lowest N’ values the bound Ab is very low and is on average less than 1 Ab molecule/cell. The increase in average bound Ab is approximately proportional to N’. Then there is an area of steeper slope. As an example, an increase of N’ from 239,OOOmolecules by a factor of ten increases the bound antibody concn by a factor of 89.5. Then again, the binding curve approaches a proportional phase. This behaviour reflects the fact that the chance of a monovalent complex to hit a free antigen molecule is small at very low single cell antigen densities. Consequently, there is essentially no diva-
.&CELARSSON
738
Single cell antigen density (molecules/cell)
Fig. 2. Modelling of Ab binding to a plasma membrane antigen using equation 13b. The monovalent association constant, K, was assumed to be lO’liter/mol, the divalent association constant, L’, 8.302 x 10m4 cells/molecule (L = 5 x 10”’ liter cells/mol) and the cell concn lO*cells/ liter. The free Ab concn was constantly lo-” A4 and the single cell antigen density, N’, varied from 100 to 10’ molecules/cell.
lent binding. As the antigen density increases, so does the chance of a hit between a monovalent complex and a free antigen molecule, which makes the formation of divalent complexes essential. At very high antigen densities the divalent binding by Ab becomes predominant and the binding curve again returns to an approximately proportional rate. DISCUSSION It cannot be expected that every monoclonal Ab directed against non-repeated epitopes of a plasma membrane antigen also accomplishes a noticeable divalent binding. The association constants (Kand L) are the ratios between the respective association rate and dissociation rate constants. The association rate constant is highly dependent on the diffusion. Consequently, a high lateral diffusion of the protein antigen would favour a high value of L and thus the divalent binding. Generally, proteins with no or a small internal part diffuse faster than those with a large internal part. A particular group of proteins are those that strongly interact with the cytoskeleton and consequently have little lateral diffusion (Darnell et al., 1986). Such antigens are poor candidates for divalent binding. Recently, it has also been found that certain cell surface proteins are anchored to the plasma membrane by glycosyl-phosphatidylinositol (reviewed by Low and Saltiel, 1988). The Thy-l antigen, which was shown to participate in divalent cell surface binding by Ab (Mason and Williams, 1980) is attached in this way (Tse et al., 1985). For proteins that are inserted into the plasma membrane in a way that makes them keep at least a predominant orientation, the direction of their epitopes may be critical. Certainly, IgG Abs have a segmental flexibility between the two Fab parts,
facilitating divalent binding at different epitope angles. However, at least the human IgG subclasses seem to have a predominant angle between their Fab parts (Gregory et al., 1987). The inter-Fab angle in the xy plane (perpendicular to the plane of symmetry of the Fc part) was estimated to vary between the subclasses from 120” (IgGl) to 240- (IgG2). In addition, all subclasses seem to have a predominant inter-Fab angle out of this plane amounting to 90”. Of the mouse IgG subclasses IgGl has the lowest segmental flexibility (Oi et al., 1984). The combination of a rigid protein molecule, inserted by a transmembrane amino acid sequence, giving it a certain orientation, and a predominant FabFab angle would accomplish a high divalent binding only at certain epitope angles (Fig. 3). The best fit for the divalent binding is obviously obtained when the angle between the epitope and the membrane surface, together with the predominant FabFab angle, amounts to 180-. The utility of a theoretical model depends on its power to suggest experimental approaches and predict experimental outcomes. In the field of tumor immunolocalization and immunotherapy the present model implies the following lines of action. One major obstacle in immunotargeting in vitro is that tumor associated antigens often exist in the circulation in addition to the surface of the tumor cells. It has been suggested previously (Price et al., 1987) that, in such situations, Abs directed against single epitopes are preferable, a view that is sup-
B
w
w
Fig. 3. Schematic drawing illustrating the importance of proper direction of the epitopes of a plasma membrane antigen. When the angle between the epitope and the membrane surface, together with the predominant FabFab angle of the Ab molecule, amounts to 180” the binding is most efficient. (A) A combination of angles favoring a high divalent binding; (B) a combination of angles not favoring divalent binding.
Modelling
of antibody
binding
ported by the modellings of this paper. The binding to an antigen free in solution should be approximately what is predicted by equation 14 (DeLisi and Metzger, 1976). When the circumstances are favorable, the binding in solution approximately predicted by equation 14 is far exceeded by the divalent binding at the plasma membrane (Fig. 1). That occurs when LN >>1 and FK <<1. The fulfillment of the first conclition is very much a matter of using Abs directed against the right epitopes of the right antigen. For fulfilling the second condition, Ab of low affinity in the first equilibrium should be used, and the Ab should be infused slowly to avoid high concns of free Ab. It should be kept in mind that the cell concn assumed in the present modellings (lO’/ml) is one suitable to in I;itro experiments. In tissues the cell concn is approximately lo9 per ml. Such a cell concn would optimally increase the bound Ab concns by four magnitudes as compared to those shown in Fig. 1. In order to overcome the problem of circulating antigen, an Ab should have a low antigen binding in the absence of the cell surface bonus effect, i.e. have a low K value. In order to obtain monoclonal Abs with these properties the selection procedure is essential. Thus, if an ELISA or RIA procedure, employing the antigen coated to a plastic solid phase, is used, the desired monoclonal Abs of low K values are easily lost. Accordingly the monoclonal Abs, giving an efficient divalent cell surface binding in the studies of Roe et al. (1985) and Mason and Williams (1980), were selected using living cell assays (Embleton et al., 1981; McMaster and Williams, 1979). Another obstacle in immunotargeting is the presence of tumor associated antigens also on normai cells, although at lower densities than on the tumor cells. As is demonstrated in Fig. 2, theoretically it should be possible to find conditions that also consider this complication. By selecting proper values of F, K and L in relation to the relevant range of N, a much higher difference in Ab binding is obtained than that which is proportional to the antigen amount of the cells. However, whether this is a feasible approach in practice also remains to be elucidated. Acknowledgements--I and Staffan Pauhe manuscript.
wish to thank Drs Goran Olofsson for their critical reading of the
to a cell surface
antigen
739
REFERENCES
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