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Clark: Part2 Page 1

Antibody-Antigen Reactions Mike Clark

(adapted with permission from Geoff Hale's Lectures) Department of Pathology, University of Cambridge

Affinity of antibody antigen reactions The binding of antibody to antigen is a reversible process, involving non-covalent bonds. It obeys the law of mass action which states that the rate of a reaction is proportional to the concentrations of the reactants. For a single antibody combining site binding to a single antigenic determinant, it is a simple matter to define an equilibrium constant which describes the tightness of binding, or "affinity". For the reaction: Ab + Ag = AbAg

the rate of formation of the AbAg complex = kforward[Ab][Ag] the rate at which it breaks down = kbackward[AbAg] (Square brackets indicate concentrations. The rate constants kforward and kbackward will depend on temperature, pH and other conditions. ) At equilibrium, the rate of formation of the AbAg complex equals the rate of its breakdown and an equilibrium or affinity constant K is defined: K = kforward kbackward = [AbAg] [Ab][Ag]

Notice that the concentrations of free antibody [Ab] and free antigen [Ag] will not be the same as the starting concentrations because some has been used up in forming the complex. Affinity constants for antibodies usually lie in the range 106 to 109 M-1. You can think of the affinity constant as the reciprocal of the concentration of antibody at which the antigen is half saturated. So for a typical antibody this might occur at 0.1 to 100 mg/ml. Real antibody-antigen interactions are more complicated than this because the antibody has multiple binding sites (two for IgG) and so does the antigen (many binding sites in the case of cells or microorganisms). Once the antibody has bound by one arm, it is possible for the other to bind, which results in an overall increase in the tightness of binding. The second reaction, however, is an internal rearrangement of a single complex and so it is not decribed by the same rate equations. The extra tightness achieved by binding though multiple sites is called "avidity" or "functional affinity" and it can be considerably greater than the single-site affinity. In practice, when we measure the binding of an antibody to a cell surface antigen, it is a sort of average "functional affinity" which is measured and we do not usually pay much attention to the exact number of binding sites. This functional affinity constant has an important bearing on the appropriate concentration of antibody needed for a particular application eg diagnostic assay or therapy.

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In the case of Fab fragments and monovalent or bifunctional antibodies, it is important to remember that the affinity of binding will usually be significantly less than the functional affinity of the parental bivalent antibody, and this might be a limitation to their use for therapy. The relationship between affinity and specificity During the course of the immune response to a foreign antigen, there is selection for antibodies with increased affinity as a result of somatic mutation. It is often wrongly stated that this increase in affinity is accompanied by an increase in specificity of the antibodies. In fact, the opposite is true. Let me try to explain what I mean. By specificity, I mean the functional ability of an antibody to discriminate between the target antigen and other, chemically similar structures. In practice, we can only measure binding which is above a certain affinity (the exact threshold will depend on the assay), so we say that an antibody non-specifically binds homologous antigens when the affinity for those antigens is sufficient for the binding to be detected by our assay (eg immunofluorescence, immunohistology etc.). If an antibody is mutated to bind its target antigen with greater affinity it will, on average, bind homologous antigens more tightly. This means that binding to some homologous antigens, which was formerly too weak to be detected, will now exceed the sensitivity threshold and be counted as an extra non-specific reactivity. This concept is nicely demonstrated by the data of Roberts, Cheetham and Rees (Nature 328: 731-734, 1987), which I summarise here. A monoclonal antibody (Gloop2) was generated which binds to a peptide from hen egg-white lysozyme. By site directed mutagenesis mutants were made having altered amino acids in the complementarity-determining regions of light or heavy chains. One such mutant had significantly higher binding affinity for the free peptide. Affinity constants for binding of wild type and mutant antibodies to peptide and variant lysozymes. Gloop2 50 4.3 8.6 0.3 0.03 0.01 x 106 M-1 mutant 210 36 5 1.2 0.1 0.03 x 106 M-1

peptide hen egg lysozyme turkey egg lysozyme plover egg lysozyme quail egg lysozyme human lysozyme

The affinity for some of the various lysozymes relative to the peptide is reduced in the mutant but, in general, the absolute affinity is still higher. Is this an increase in specificity or not? In a polyclonal antiserum, there will be a mixture of different monoclonal antibodies. Many will recognize different parts of the antigen and so will have completely different patterns of cross-reactivity with unrelated (but chemically similar) molecules. Because of the relatively low concentration of each individual type of antibody, many of these cross-reactions will be diluted out and undetected in our assays. So the polyclonal antiserum usually seems to increase in both affinity and specificity during the immune response. It is important to remember that this is not the case with monoclonal antibodies. Although there has been very little success to date in improving the affinity of monoclonal antibodies by genetic engineering, there is still considerable optimism that this will be useful. However, it may well be that increases in affinity will be accompanied by unwanted extra reactivities, as cross-reactions exceed the threshold of detection.

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A more detailed analysis of antibody valency and affinity (adapted from HN Eisen, Immunology, 2nd edition, pub Harper & Row)

In the simplest reaction a small univalent ligand binds reversibly to a specific site on the antibody. The antibody may have n identical sites and at saturating concentration of the ligand they will all be occupied. The number of bound ligand molecules per antibody molecule will then equal n, the antibody valence. If all the binding sites are identical and independant, the binding reaction is represented:







where S is a binding site on the antibody, and L is a ligand. At equilibrium the rates of the forward and back reaction are equal, ie kforward [S] [L] = kbackward [SL] (2)

where [SL] is the concentration of occupied antibody sites, [S] is the concentration of free sites and [L] is the concentration of unbound ligand. The ration of the rate constants gives the association constant K which represents the affinity of the antibody for the ligand. K is measured in units of litres/mol (ie M-1) K = kforward kbackward = [SL] [S][L] (3)

If the total concentration of ligand molecules [L]+[SL] and the total concentration of antibody sites [S]+[SL] are known, then K can be determined by measurement of any one term on the right side of equation (3). The most important technique historically is equilibrium dialysis, by which the free ligand concentration [L] can be measured. Note that the total concentration of antibody sites, [S] +[SL] is n times the total concentration of antibody [Ab]. Equation (3) can be rearranged in terms of measurable parameters by dividing top and bottom by the antibody concentration [Ab]. This gives: [SL] [Ab] (4) K = [S]+[SL] [SL] [L] [Ab] [Ab]



We can tidy this up by relabelling the number of ligand molecules bound per antibody molecule, [SL]/[Ab] as r and then get: K = r (n-r) [L] (5)

which is the same as: r [L] = Kn - Kr (6)

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Results are oftern plotted on a graph of r/[L] versus r, the so-called Scatchard plot. This should give a straight line with slope -K, provided the original assumption was correct that all the binding sites are identical and independent. If not, the line will be curved, and the degree of curvature can give information about the heterogeneity of binding sites in the sample of antibody molecules. When the concentration of free ligand is very high, r/[L] approaches 0 and r approaches n, ie the number of bound ligand molecules per antibody molecule equals the antibody valence.

Direct plot n = valency r 0.5n r [L]

Scatchard Plot

slope -K

intercept n 1/K [L] r

An example of measured affinity constants for a hapten and analogues The reagent fluorodinitrobenzene reacts with amino groups in proteins to give a dinitrophenylated protein. This was used to immunise an animal and the affinity of the antibodies for different ligands was measured. Ligand Affinity constant K (× 105 M-1)


NO 2 NO 2


NO 2 COOH HC-(CH 2) 3-NH NH2 NO 2 H2N NO 2 NO 2



NO 2 NO 2



NO 2


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The theoretical treatment of antibody affinity and specificity is covered in most basic textbooks of Immunology, but the quality varies. Some of the best accounts I have seen are in the following: Eisen, H.F. Immunology, published by Harper and Row The best account of thermodynamics of antibody-antigen interactions. Scatchard plots with plenty of examples. Other aspects are also well covered. Karush, F. in Comprehensive Immunology (edited by R.A. Good and S.B. Day) volume 5, chapter 3. "The affinity of antibody: Range, variability and the role of multivalence" A detailed account of avidity (functional affinity); its importance and distinction from intrinsic affinity. Richards, F.F. et al in Comprehensive Immunology (edited by R.A. Good and S.B. Day) volume 5, chapter 4 "Antibody combining regions". A good discussion of the structural work with useful rough calculations on the possible range and specificity of antibodies. Weir, D.M (editor) Handbook of experimental immunology, 4th edition. Volume 1 Immunochemistry, published by Blackwell. A very practical manual of techniques. Much of the book is concerned with antibody-antigen interactions. Each chapter has an account of the background and theory of the technique, and chapter 25 is an overview of the study of antibody affinity. Most of the information will also be found in earlier editions.

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Annex I Some miscellaneous figures of use in estimating antibody concentrations and affinities. Relative molecular mass (Mr) of an immunoglobulin domain is approximately 12,500 thus Mr of an Ig light chain (2 domains) is approximately 25,000 and Mr of an IgG heavy chain (4 domains) is approximately 50,000 (hinge and CHO extra) implies Mr of an IgG is approximately 150,000 (2H + 2L) Concentration of human IgG in plasma is approximately 10 g l-1 (10mg ml-1) So for an adult plasma volume of 5 litres there are approximately 3.33 x 10-4 moles of antibody and this means that there are approximately 2 x 1020 molecules of antibody. An antibody concentration of 1nM is 1.5 x 10-4 g l-1 which is approximately 6 x1014 molecules per litre. A cell concentration of 1 x 106 per ml means that there are 1 x 109 cells in a litre. So 1nM of antibody would be enough to provide for 6 x 105 molecules of antibody for each cell at these concentrations. For IgG 10 g l-1 is approximately 6.7 x 10-5M which is approximately 67x103 nM i.e you could have approximately 100,000 different IgGs present in human plasma each at an average concentration of about 1nM


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