Antigen-antibody interactions (Ka, Kd, k+1, k-1 )
Ag-Ab interactions lead to a rich phenomenology that has valuable practical applications in serology and depends both on the physical state of the antigen (soluble or insoluble, etc) and on those of the antibody (IgM or IgG ). The specificity and affinity of antibody-antigen interactions are fundamental for understanding the biological activity of these proteins and these will be summarized first. These can be described as for any other bi-molecular reaction. If A denotes the antigen and B the antibody, the concentration of complex AB is proportional to the concentrations of the reactants and to a constant property of the two reactants for given reaction conditions of temperature, ionic strength and pH:
[AB] = K [A] [B] (eq 1)
This constant (known as association or equilibrium constant) has the dimensions of the reciprocal of a concentration (M-1) and equals:
Ka = [AB] / [A] [B] (eq 2)
Antibodies with high affinity have Ka > 10^7 M^-1.
A second constant (the so called dissociation constant or Kd) describes the affinity of the reverse reaction and has the physical dimension of a concentration (M):
Kd = [A] [B] / [AB] (eq 3)
The reason why Kd is often determined in preference to Ka, is that determination of Ka requires the reaction to proceed to equilibrium whereas Kd can be derived from reactions in which half the concentration of antigen is complexed with the antibody. Thus [A] = [AB] and eq 3 simplifies to
Kd = [B] (eq 4)
Antibodies with high affinity have Kd < 10-7 M.
The above equations describe events at equilibrium but it is equally important to understand the kinetics of the reactions, ie the rates of formation of the different species in which k+1 has the dimension of M-1 s-1 and k-1 has dimension of s-1.
on rate = k+1 [A] [B] (eq 5)
off rate = k-1 [AB] (eq 6)
k+1 [A] [B] = k-1 [AB] (eq 7) and
Ka = k+1 / k-1 = [AB] / [A] [B] (eq 8)
Thus Ka and Kd are simply the ratio of the two rate constants. However, it should be apparent that different reactions can have identical Kas but different rate constants. k+1 can vary over the range from 10^5 to 10^8 M-1 s.1. k-1 ranges from 1 to > 10^3.As the off rate depends on the concentration of only one species (AB) there is a very simple relation between and the time (in secconds) taken for the complex to dissociate to 50%:
t1/2 = 0.693/ k-1 (eq 9)
where 0.693 is -ln 0.5 (ie 50% dissociation; expressed in a ln form as this linearly transforms the exponential curve of the rate).
Valency and affinity
The affinity constants described above apply to single site interactions. However, all naturally occurring antibodies are multivalent and their functional affinity is dependent not only on their intrinsic affinity for antigen but also on the number of binding sites (2 for IgD,G and E and 10 for IgM). The binding of a univalent ligand to a multivalent antibody may be expressed as:
Ka = r / (n-r) c (eq 10)
where, at equilibrium, c is the free concentration of antigen, r represents the average number of antigen molecules bound per antibody molecule and n is the maximum number of antigen that can be bound per antibody molecule (the antibody valence). A set of values of r and c can be obtained from a series of experiments in which the concentration of antibody is kept constant and from these a plot (Scatchard plot) can be constructed in which r/c is plotted against r. Linearity of this plot indicates uniformity of the binding site. Absence of linearity is commonly seen with polyclonal antibodies. At saturation (c very high) the limiting value of r is 2 for IgG and 10 for IgM.
The effect of valence on antigen binding also depends on whether the antigen itself is multivalent and, if so, on the distance between epitopes. In the case of IgD, G and E the effect of the presence of multiple epitopes for example on a surface antigen becomes significant if epitope distribution enables bridging two antigen-binding sites (monogamous binding). In the case of IgM, the decavalent nature of the molecule enables efficient bridging of epitopes present even on different particles (for example red blood cells) (bigamous binding). Thus IgM are very efficient agglutinins. The term avidity is often used to indicate the overall ability of antibodies to interact with antigen. The term has practical value but does not define precisely the contribution of affinity vs valency vs epitope density to antibody-antigen binding. It merely describes the net result of the combination of these factors to antigen binding.
Estimating antibody affinity and kinetics
The first reliable procedure for measuring the affinity of antibodies for their antigens has been equilibrium dialysis (see Fig Equilibrium dialysis method for measuring free and bound hapten). The method is particularly well suited for studying the affinity of antibodies for haptens or small antigens. A solution of antibody (at a known concentration) is placed in a dialysis bag and equilibrated with a solution of radiolabelled hapten or antigen (initially placed outside the dialysis bag). At equilibrium the concentration outside the dilaysis bag corresponds to free Antigen whereas the concentration inside the bag corresponds to bound + free hapten. From this, the concentration of bound antigen can be calculated.
Another approach to the estimate of antibody affinity is fluorescence quenching. Proteins contain three amino acids that cause ultraviolet fluorescence: phenylalanine, tyrosine and tryptophan. For practical purposes, however, the fluorescence emitted by phenylalanine is not readily exploited and, therefore, estimates of protein fluorescence reflect the fluorescence of tryptophan and/or tyrosine depending on the wavelenght for excitation and emission. There are two major forms of fluorescence quanching: static and dynamic (or collisional). In the case of static quenching a complex is formed between the fluorophore (for example an antibody) and the quencher (the antigen) which is non fluorescent. In the case of dynamic quenching the quencher must encounter and interact with the fluorophore during the lifetime of the excited state. In the case of static quenching the following relation applies:
Ka = [F-Q] / [F] [Q] (eq 11)
where [F-Q] is the concentration of the complex, [F] is the concentration of the uncomplexed fluorophore and Ka is the association constant of the complex. If the complexed species is non fluorescent, the fraction of the fluorescence that remains (F/F0) is given by the fraction of fluorophore that is not complexed (f). Thus f = F/F0. Since the total conccentration of fluorophore is given by:
[F]0 = [F] + [F-Q] (eq 12)
By substituting eq 12 into eq 11,
Ka = [F]0 - [F] / [F] [Q] = [F]0 / [F] [Q] - 1 / [Q] (eq 13)
and further substituting the fluorophore concentrations with fluorescence intensitites the following equation is obtained:
F0 / F = 1 + Ka [Q] (eq 14)
A third and powerful approach to the determination of the thermodynamic and kinetic constants of antibody-antigen interactions is surface plasmon resonance (see Fig Surface plasmon resonance (SPR) analysis of antigen-antibody interactions). In this technique one species (either antigen or antibody) is immobilised onto a layer of modified dextran and the second species is allowed to flow on that surface. A sensitive detector allows estimation of the amount of the immobilised species at the start f the experiment and of the amount of mobile species that becomes bound to the former. As the changes in the mass of chemicals present on the sensor chip are recorded in real time, a series of curves can be generated (sensorgrams) from which both association and dissociation kinetics can be derived. Surface plasmon resonance is a very sensitive technique generally applicable to the analysis of molecular interactions and will have a profound impact on the study of antibody-interactions as it does not suffer from the restrictions of equilibrium dialysis (which is only applicable to small and diffusible antigens) and fluorescence quenching (which is limited to antigens with appropriate absorption and fluorescent properties).
The interactions of antibodies with soluble antigens
The complexes formed between a monoclonal antibody and its antigen or the complexes formed between a polyclonal antibody and a hapten are soluble. However the complexes between polyclonal antibodies or mixtures of monoclonal antibodies and large antigens often lead to the formation of an insoluble precipitate (the so-called precipitin reaction). Such reactions can occur both in liquid media and in semisolid (gel) media and both media are widely exploited for applications in laboratory research and practice. The amount of precipitate that is recovered when an increasing amount of Ag is added to a constant amount of Ab in solution follows a typical pattern. The amount of Ab precipitated increases with the amount of Ag (zone 1) up to maximum (zone 2), after which further increase in the amount of Ag leads to a decrease in the amount of Ab precipitated (zone 3). Zone 1 is the so-called Ab excess zone, zone 2 is the equivalence zone and zone 3 is the Ag excess zone (see Fig Precipitation reactions. Principles, immunodiffusion and immunoelectrophoresis).
Why does precipitation of Ab-Ag complexes occur ? To account for precipitation and for varrying ratios of Ab/Ag in precipitates, Marrack, Heidelberger and Kendall proposed Ab-Ag complexes in solution can form aggregates of various size and that when the volume of such aggregates exceeded a critical value, precipitation occurred [the sedimentation coefficient (S) of a particle is directly proportional to its volume (V), the difference between its density (r) and that of the solvent (r0) and the gravitational field (g): S = V (r-r0) g]. Such lattice theory explains well the different Ab/Ag ratios and the fact that monoclonal antibodies cannot precipitate Ag. After a precipitate has formed, its complexes can dissociate and re-equilibrate with new Ag. If the latter is present in sufficient excess, soluble complexes will prevail and the initial precipitate will gradually dissolve. Thus, in principle, precipitation reactions are reversible. In practice, in the majority of the cases they are are not and a striking illustration of this point is the s-called Danysz phenomenon: when an equivalent amount of diphteria toxin is added to an anti-toxin serum, the mixture is non-toxic; however, if the same amount of toxins is added to the same amount of antiserum in aliquots and at intervals (of 30 mins, for example) the mixture is toxic. This is readily explained by the fact that the first addition of the toxin leads to the formation of complexes with a high Ab/Ag ratio and very slow dissociation rates (days or months). As a result, insufficient Ab is left to react with new toxin added in relatively short periods of time. There are anomalous precipitation reactions (flocculation) in which precipitation is observed in a narrow range of Ab/Ag ratios. Such reactions probably require the presence of high-affinity, non precipitating antibodies (monogamous binders) which must be saturated with Ag before remaining, 'conventional' Abs can bind and precipitate the additional Ag.
When Ab and Ag are introduced in different reqions of an agar or agarose gel, they diffuse toward each other and form opaque bands of precipitates in certain areas located between Ab and Ag. The simplest version of this approach is single diffusion in one dimension (Oudin) in which an antigen solution is left to diffuse over an antiserum that is 'trapped' in a gel and a precipitate forms in the agar in regions corresponding to the equivalence zone. In the double diffusion in two dimensions (Ouchterlony), a large number of geometric arrangements are possible that enable analysis of Ag mixtures (reactions of identity, partial identity, non identity). Modifications of these techniques allow quantification of Ag or Ab. For example, if a uniform concentration of Ab is incorporated in the gel and Ag is applied in a well, the distance at which the precipitin ring forms, depends on the concentration of Ag in the well (Mancini). Thus, with appropriate standards, the concentration of Ag in unknown samples can be calculated from a calibration curve of [Ag] vs d (diameter) or a (area). Further modifications combine gel diffusion and precipitation with electrophoretic separation of the Ag. This can be carried out in order to analyse complex Ag mixtures (Grabar) or in order to decrease the time required in order to achieve precipitation and increase sensitivty of the Mancini-type procedure (Laurell). Finally the principle and an example of the numerous application of enzyme-linked immunosorbent assays (ELISA), a family of antigen-antibody assays which exploit antibody-enzyme conjugates and which are widely used due to their sensitivity and their better safety and cost profiles compared to radioimmunoassays (see Fig An example of an enzyme linked immunosorbent assays (ELISA).
The interactions of antibodies with insoluble antigens
Bacterial and animal cells are usually agglutinated (clumped) when mixed with specific antisera. Such agglutination reactions are very sensitive, they often exhibit inhibition at high Ab concentrations and are carried out at physiological pH and ionic strength. Agglutination plays a very important role in the diagnosis of bacterial infections and in blood typing. An example of red blood cells agglutination of a glass slide is attached, examples of red blood cells agglutination in microtitre plates are shown (see Fig Blood typing through antibody-dependent red blood cells agglutination). The latter has been conducted for many decades with natural (IgM) antibodies. IgM are strong haemoagglutinins. In contrast other antibodies not only fail to produce haemoagglutination but positively inhibits it (blocking antibodies). Anti-Rh antibodies are a classic example.
These antibodies either bind in a monovalent maner or they bind in a divalent manner to the same cell. They can be revealed by using an antiglobulin antiserum (Coombs reagent) after Robin Coombs, the scientist who discovered the antiglobulin reaction. In the direct Coombs test, washed red blood cells are incubated directly with a Coombs reagent. Agglutination will demonstrate the presence of the 'incomplete' antibodies to the red cell. This test is carried on red blood of newborns in order to establish the presence of maternal anti-Rh antbodies. In the indirect Coombs test, serum is added to red blood cells carrying the Ag under study (for xample Rh) and, after the cells have been washed, the Coombs reagent is added. Once more the presence of anti-Rh antibodies in the test serum will be revealed by agglutination. This test therefore can be used in order to monitor the presence (and the titre) of anti-Rh antibodies in a Rh- mother carrying a Rh+ fetus.
The simplicity and sensitivity of the haemoaglutination reactions can be considerably widened in so-called passive agglutination. Red blood cells readily absorb many polysaccharides and protein antigens can be coupled covalently using bifunctional reagents. Passive agglutination assays are very important because they are simple and do not require special laboratory instrumentation. They are, therefore, particularly suited for thirld-world countries. Passive agglutination does not necessarily require blood cells. Other particles, such as polysterene beads (microspheres) can substitute red blood cells and are also used in a variety of assays, for example in pregnancy tests.
Any substance that serve by antigen or hapten can be measured by radioimmunoassay (Berson and Yalow). These assays are usually based on competition for Ab between a radioactive ligand (L*) and its unlabelled counterpart (L). The concentration of L in unknown samples can be determined from a calibration curve in which known concentrations of L are used to compete for the binding of L* to the Ab (Fig 8 below). These assays too a revery sensitive and can be carried out in a variety of formats, including solid phase-formats which simplify determination of Ab-L*. Adaptation of these assays use enzyme-linked instead of radiolabeled L*. These assays too tend to employ a solid phase and can be carried out in a variety of ways (direct or sandwich) (see Fig The principle of a simple radioimmunoassay)
Finally the use of fluorescent antibody conjugates (or alternatively anzyme-antibody conjugates) enables sensitive and accurate detection of antigens on tissue sections. This has major implications for diagnosis especially after the availability of monoclonal antibodies.
An example of this is given in the Fig Tissue staining with antibodies in which different primary antibodies and different fluorochrome-tagged secondary antibodies were carefully employed in order to obtain a faithful representation of the topographical distribution ofB cell,, T cell and macrophages in spleen.
Topics for Exam
- Specificity and affinity of antibody-antigen interactions
- Interactions between antibodies and soluble antigens
- Interactions between antibodies and insoluble antigen
- Thermodynamic and kinetic constants governing Ag-Ab interactions
- The equilibrium dialysis method for measuring Ab affinity
- Surface plasmon resonance methods for measuring Ab affinity
- Immunoprecipitation reactions
- Immunodiffusion and immunoelectrophoresis
- Enzyme-linked immunosorbent assays
- Immunofluorescence techniques
- Red blood cell agglutination reactions
- Coomb's direct test
- Coomb's indirect test
- Passive agglutination reactions