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A Mathematical Model for Biopanning (Affinity Selection) Using Peptide Libraries on Filamentous Phage
J. theor. Biol. (1995) 176, 523–530
A Mathematical Model for Biopanning (Affinity Selection) Using Peptide Libraries on Filamentous Phage W M,† Y.-C. J C N G Viral Discovery D90D, Abbott Laboratories, 1401 Sheridan Road, North Chicago, IL 60064-4000, U.S.A. (Received on 9 December 1994, Accepted in revised form on 7 June 1995)
A mathematical model is presented for the process of selection of peptides for binding to a target. The affinity enrichment process commonly known as biopanning relies on subjecting a library of peptides on filamentous phage to a selection for binding to the target immobilized on solid support or in solution. The model is an implementation of the mass-action law to the diverse population of macromolecular assemblies. An analytical solution is presented for the enrichment process. Most parameters in the enrichment formula can be easily determined experimentally. Two examples corresponding to biopanning with epitope libraries and antibody libraries are given. The model allows for an estimation of the contribution of equilibrium and dissociative biopanning to the overall enrichment. The model can be a tool in the evaluation of the role of different biopanning parameters. Its implementation in a spreadsheet makes it possible to perform a computer simulation of a biopanning experiment. 7 1995 Academic Press Limited
target takes place: (a) binding in solution (Parmley & Smith, 1988), and (b) binding to solid phase (Barbas & Lerner, 1991). As indicated in Table 1, type (a) procedures involve conjugation of the target protein to biotin, binding of phages to the target in solution, immobilization of phages bound to targets to solid phase via biotin-streptavidin interaction, washing the solid phase to remove the non-specific binders, elution of bound phages and steps to recover and quantitate the eluted phages. In type (b) procedures phages are bound directly to targets deposited on solid support (microtiter plates) via non-specific interactions, and subsequent steps are similar to those described for type (a) procedures. There are two mechanisms for the enrichment of a phage population by affinity to a target. First, an equilibrium is established between the target molecules and the phages having different affinities to the target. The phage subpopulation within target-phage complexes will be enriched in phages having the highest affinities to the target. This type of enrichment will be referred to in this paper as the equilibrium enrichment.
1. Introduction Biopanning (affinity selection) is a procedure for enrichment for molecules that bind to a given target, which often is a protein. In a typical experiment, a filamentous phage library is employed (Parmley & Smith, 1988). Each phage in the library displays on its coat protein, gp3 or gp8, a different peptide or protein by fusion of genetic material. The diversity of the library is frequently 107–1011 different phages. The objective of a biopanning experiment is to identify from a phage library a phage or a subset of phage that bind, most tightly to the target and, eventually, the sequences of peptides displayed on the phage. Current approaches and libraries were recently reviewed by Scott & Craig (1994). Several protocols for biopanning have been described. They fall into two basic classes according to the phase in which the binding of the phages to the † Author to whom correspondence should be addressed at: Viral Discovery D90D, Bldg. R1, Abbott Laboratories, 1401 Sheridan Road, North Chicago, IL 60064-4000, U.S.A. 0022–5193/95/200523+08 $12.00/0
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Second, when the phage immobilized to the target on solid support is washed, phages with a fast off-rate to the target are preferentially washed out, thus enriching the remaining phage population in phages having a slow off-rate. The second mechanism will be referred to as off-rate or dissociative enrichment. An example for equilibrium enrichment and a brief analysis was presented by Hawkins et al. (1992). A model for a related procedure, selection of nucleic acids for affinity to a target (Selex) has been described by Irvine et al. (1991). In this paper, a general model for biopanning is presented. The model is applicable for both solution and solid phase biopanning. An analytical expression is derived for the equilibrium panning and the dissociative panning. Several examples are described, calculations performed in a spreadsheet and the results are compared with experimental data.
is assumed that each phage affinity class is represented by a sufficiently large number of phage, to which the mass action law is applicable. This condition can be satisfied experimentally by performing biopanning with large phage populations having many identical copies of each phage. List of definitions c total concentration of targets (targets are identical), concentration of unbound targets (i.e. free cf in solution), total concentration of the phage class i, pi pfi concentration of the class i of unbound phage, [cp]i concentration of the class i of phage-target complexes. The following equations need to be satisfied in equilibrium for i=1, . . . m:
2. The Model
[cp]i=Ki pfi cf
The first step in biopanning is establishing the equilibrium between the phage population and targets. The target molecules could be either in solution or on the solid support. It is assumed in the model that the process is governed by the mass-action law applied to binding of phages to targets. For simplicity, the phage population will be divided into classes, each characterized by an association constant Ki , which is the same within the class, i=1, 2, . . . m (m is the number of classes). Ki characterizes the following reaction:
(1)
m
c= s [cp]i+cf
(2)
i=1
pfi+[cp]i=pi
(3)
These equations can be condensed into:
0
m
[cp]i=Ki (pi−[cp]i ) c− s [cp]i i=1
1
(4)
Defining: m
cb= s [cp]i
[phage class i ]
(5)
i=1
Ki
+target F [target-phage class i complex] Ki describes the apparent binding constant for the phage from the ith-class, and not for the peptide mounted on the phage. Thus, Ki depends both on the true peptide-target binding constant and on the multiplicity (’’valency’’) of the display, i.e. the number of peptides on the surface of a single phage particle. It
we get from (4): [cp]i=
Ki pi 1 +Ki c−cb
Equation (6) can be rewritten for the continuous case, in which the affinity of the phage population is
T 1 The process of biopanning Solution-phase selection
(6)
Solid-phase selection
1. Equilibrium step: equilibrate biotynylated 1. Coat solid surface with receptor target with phage library 2. Capture step: pan phage-target mixture 2. Capture step: pan phage library with with streptavidin-coated solid surface target-coated solid surface 3. Washing step: wash unbound phage from solid surface 4. Elution step: elute bound phage
J G h G j _ ~
Equilibrium Biopanning Dissociative Biopanning
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described by a density function f, which is dependent on the association constant K of phage to the target. The density function f(K) describes how many phage of a given affinity K are present (in the discontinuous case above, Ki described how many phage belonged to the i-th affinity class). As a result of equilibrium biopanning, phage bound to targets can be harvested. That population is described by a density function g(K) which [on the basis of eqn (6)] is given by the following formula: g(K)=
K f (K) 1 +K c−cb
(7)
where cb=
g
a
g(K) dK
(8)
0
Equation (7) is not the explicit solution of the problem, since the numerical value of parameter cb is not known except for special cases (see below). Nevertheless, since cb is independent of K, eqn (7) provides a general type of function that governs the equilibrium panning. Parameter cb can be interpreted from eqn (8) as the total number of phages bound specifically to the support, or equivalently, the total number of targets bound to phages. The population of phages that emerges from the equilibrium biopanning experiment is enriched in phages that bind with high affinity to the target. The enrichment function h can be represented by the ratio of two density functions, corresponding to phage distribution before (function f ) and after biopanning (function g): h(K)=
g(K) f(K)
(9)
and for equilibrium panning heq (K)=
K 1 +K c−cb
(10)
The enrichment function approaches 1 (all phages retained) for sufficiently high association constant K, and for K1/(c−cb ) it can be approximated by a linear function of K. The numerical value of the denominator, c−cb can be considered the concentration of available total target (i.e. unbound to target), ca (caEc). Figure 1 presents log–log plots of the enrichment function for different ca . Two limit cases are of interest. Often, especially in the first round of biopanning with a diverse phage population, or in many instances of solution panning,
F. 1. Enrichment function in equilibrium biopanning. Variable heq given by eqn (10) is plotted as a function of the association constant K. This dependence applies if no non-specific binding takes place. Numerical values for parameter ca=c−cb are as follows: line a, 10−5; line b, 10−6; line c, 10−7; line d, 10−8; line e, 10−9 and line f, 10−10.
the concentration of phages that binds to the target will be much less than the total concentration of the targets, i.e. cb c and c−ca1c. In that case eqns (7) and (10) provide a direct solution to the problem of equilibrium panning. For efficient equilibrium panning, the target concentration should be selected at about 1/Kmin , where Kmin is the minimum association constant for the phage desired to be obtained from the experiment. Phages with affinities higher than 1/Kmin will be effectively retained (if they are present in the phage population). The remainder phage with lower affinities will be retained proportionally to their affinity, and for those phages the enrichment function approximates the identity function h(K)1K in equilibrium panning. There might be instances when the entire phage population has a measurable affinity to the target, and the total phage concentration is much higher than the concentration of targets, and essentially all targets form complexes with the phages, e.g. second and further rounds of solid phase biopanning with a large number of phages. Then cb1c, and 1/(c−cb )1/c, the enrichment function shifts towards higher K values (Fig. 1), effectively extending the linear part of the dependence towards higher numerical values of K. This provides a mechanism for increasing the selective pressure on phages having higher affinity constant to targets than K=1/c. In a typical biopanning experiment, the phage immobilization on solid support is followed by extensive washing of the support. The purpose is to
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enrich the phage population in phages that bind tightly to the support. There are two mechanisms for the enrichment. First, the phages that are not bound tightly (but are still specifically bound) are removed preferentially over those phages that have higher affinity during the course of washing. Second, washing removes phages that are not specifically bound to the target on the solid support (but rather adhere to plastic, blocking material or get entrapped by the solid support). The dissociation of molecules specifically bound to targets on solid support has been shown in many instances (Day, 1990; O’Shannessy, 1993) to be a first order process that can be described by a simple exponential decay: s=s0 e
−kr t
(11)
where s is the density function at time t for the phage subpopulation remaining on the solid support, dependent on the association constant K and off-rate constant kr , and s0 is a constant. This formula is adopted in the model. The effectiveness of washing the support is the critical element of a biopanning experiment. Little is known about the mechanism of non-specific binding of filamentous phage to support. It will be assumed in the model that non-specific binding is independent of the specific binding, in other words, the area of the solid phase can be conceptually divided into sites for specific binding to the target, and sites for non-specific binding. The fraction of input phages that are retained on the support due to non-specific binding will be given by parameter a. The fraction, b0 (K), retained on the support due to the specific binding will depend on the affinity constant K of the input phage population to the support. Typically, both a and b are much smaller than one. Phage bound non-specifically to the support will be at least partially removed during the course of washing, which typically comprises a series of buffer additions and removals. The number of washes can vary between 1 and 40, but most often is between 5 and 20. Experimental evidence (unpublished data) supports the model according to which each buffer addition produces an effect equivalent to a dilution of the non-specifically bound phage, a significant fraction † Here is an alternative method of deriving the enrichment function. We focus on the probability of binding (Pr) for an individual phage particle. The enrichment function h [eqn (13)] is the same as: Pr(bound after washes)=Pr(specifically bound after washes=specifically bound before washes)×Pr(specifically bound before washes)+Pr(nonspecifically bound after washes=nonspecifically bound before washes)×Pr(specifically bound before washes), where the symbol Pr(=) indicates the conditional probability with the condition specified at the right-hand side of the bracket.
of which is then removed. The fraction removed is approximately the same for each wash and does not depend on the duration of the wash. No experimental support was gathered to support an alternative mechanism, i.e. first order dissociation process of non-specifically bound phage. A mechanism consistent with the dilution effect is the entrapment of phage in cavities on solid surface, such that the phage is practically in solution at all times, but the solution can not be easily displaced from vicinity of the surface. Thus the phage population remaining on the support during the course of the dissociative biopanning is: s=s0 [ad w+b0 (K)e−kr t ]
(12)
where d is the dilution factor and w is the number of washes. † A typical biopanning experiment involves both the equilibrium and dissociative phases. Therefore, considering the discussion above and eqns (10) and (12), as b0 (K)=bh(K), where b is a constant, we get the formula for the time dependence of the enrichment function h for the phage population for the single round of biopanning: h=ad w+
bK e−kr t 1 +K c−cb
(13)
If the density function for the input phage is f, then the density function g after biopanning is from eqn (9): g=h ·f
(14)
If several rounds of biopanning are performed, then the total enrichment function htot will be the product of enrichment functions at each round, that is: htot=h N ,
(15)
where N is the number of rounds. The enrichment function (13) is formulated here as a function of three variables and is dependent on six parameters. Most of the parameters, however, are easy to determine experimentally for each particular experimental conditions. The next section will provide two examples of analysis of the enrichment function for selected applications. 3. Examples 1: This example is aimed at modeling of a typical panning experiment involving a peptide library.
Although parameters discussed below are only approximations of experimental values, the overall behavior of the model system is intended to parallel that of the experimental one. It is assumed that the epitope library on filamentous phage has a diversity of 109 . In the biopanning experiment, all 109 phage are applied to a single microtiter plate well (diameter 5 mm). About 1012 target protein molecules were deposited in that well and bound to solid support. Considering that the volume applied to the well is 100 ml, target concentration used in the calculations will be estimated at c=10−8 M. This is a very crude assumption, since from the perspective of the mass action law, targets bound to solid phase and targets in solution are not equivalent. Considering that the total number of phage that will bind to the target on solid support is small compared with the target concentration, especially in the first/initial round of panning, c1 will be negligibly small (c1=0). Thus, in this example, the equilibrium part of the enrichment function is close to linear up to affinities of 108 M−1 . Other parameters have been estimated on the basis of experimental data (data not shown here). The fraction of all phage that bind to support is b=0.2, and that bind non-specifically is a=0.05. The duration of all washes is assumed to be t=1000 sec, and the dilution factor is d=0.4. The numerical value of the association rate constant was assumed to be independent of the phage subpopulation, and equal to 105 M−1 s−1 (a typical association rate constant for macromolecules). Thus a direct dependence exists in the model for the association constant K and the off-rate kr , i.e. kr=105/K. A numerical model was written in a spreadsheet (Excel 4.0) to simulate the process of washing and eluting phage for several panning rounds. The input for the model is also the initial distribution of the initial phage population. For the purpose of the spreadsheet calculations it is represented as up to 15 discrete groups, each characterized by a different association constant. Three dependencies were selected as indicative of the behavior of the system: the enrichment as a function of affinity, the number of phage in the washes and eluate, and the changes of distributions after rounds of panning. The enrichment function (Fig. 2A) is independent of the initial distribution. An attempt was made to distinguish contributions from the equilibrium and dissociative phases of biopanning. It is evident from the plot (overlap of lines 2 and 3) that under conditions of this example the overall enrichment is almost exclusively due to the dissociative biopanning. The reason for this was that the total wash time was fairly long. The enrichment function is close
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to the all-or-none type, and it drops by a factor of 104 when K changes only ten-fold. Overall enrichment factor (ratio of the highest to lowest value of the enrichment function) is about 3×104 . A simple affinity distribution for the phage population is presented in Fig. 2B, line 0. In this distribution, the phage subpopulation characterized by affinity of K=10i (i=0, 1, 2, . . ., 9) has approximately 109−i members (although the precise numbers are somewhat different due to normalization). In other words, the membership of the affinity class decreases by a factor of 10 as the affinity of the class increases by one order of magnitude. This distribution would be consistent with a requirement that the highest affinity of a clone from the phage library increases proportionally to the total number of different clones (i.e. diversity) of the library. The distribution rapidly changes after subsequent rounds of biopanning (Fig. 2B, lines 1, 2 and 3). Tight binders constitute about 50% of all clones after round 2, and 99.99% after round 3. The changes in the affinity distribution of the phage population are evident in the pattern of recovery of phage in washes and eluate (Fig. 2C). When essentially all phage bind to target, i.e. after round 3 (see line 3), there is a pronounced plateau, and the number of phage in the eluate is two orders of magnitude higher than the number of phage in the last wash. Biopanning can bring desired results even faster in special cases of the initial affinity distribution. The distribution in Fig. 2D, line 0 might be characteristic of cases where a short consensus sequence is necessary and sufficient for binding to the target, for instance CXSC (single amino acid code, X is any residue). The number of peptides bearing that consensus sequence in the library will be high, hence the biphasic character of this distribution. After just one round of panning (line 1) the phage population contains 60% of tight binding clones, the number approaches 100% after rounds 2 and 3. This type of initial affinity distribution manifests itself in wash curves (Fig. 2E) as a noticeable spike of phage numbers in the eluate from round 1, and a plateau and a two-log spike after rounds 2 and 3 (compare Figs 2C and E). 2: A different type of a library is described in this example. While simple epitope libraries display peptides on the surface of filamentous phage, more complex types of molecules can be also displayed on the phage. Libraries of single-chain antibodies or Fab fragments of antibodies are frequently used to select proteins that bind to a target. Libraries of larger
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proteins can be significantly different from peptide libraries. The model can be used to evaluate effects of one of the parameters, namely loading of a protein per phage. While in peptide libraries the loading is on the order of one peptide per phage (which is still less than the maximum loading possible, i.e. three to five peptides per phage or one peptide per each of the gene-3 proteins), in single-chain antibody libraries the loading is orders of magnitude less, 10−1–10−3 per phage. This can be simulated in the model by lowering the numerical value of the parameter b (fraction of phage retained on the solid support due to the specific binding) from b=0.2 to b=0.01 (parameter a is changed from 0.05 to 0.24 to keep a+b constant, i.e. total fraction of phage bound). The results of calculations are presented in Fig. 3. Overall enrichment is much less (about 300) than in Example 1 because of the high impact of non-specific binding. What changes also is the relevance of the equilibrium panning, which now
exerts the dominant effect on the enrichment function, although the dissociative panning is also significant (lines 1, 2 and 3 in Fig. 3A). The effect is due to a much shorter total wash time, compared with Example 1. The initial affinity distribution of the phage population (Fig. 3B, line 0) is assumed to be the same as in Fig. 2B, line 0. After three rounds of panning, tight-binding phage constitute only close to 10% of all phage in the pool. In most experimental designs, a fourth round of panning would be performed. A slower progress of affinity enrichment manifests itself also in the wash/elution pattern (Fig. 3C). Lines 1, 2 and 3 corresponding to results of rounds 1, 2 and 3, are almost indistinguishable (although the eluate after round 3 contains 10% of tight-binding phage). This result supports the experimental practice of assaying clones after 2, 3 or 4 rounds despite the seemingly negative pattern in Fig. 3C.
F. 2. Example 1 for biopanning with a peptide library. (A) Enrichment as a function of the association constant K. All calculations were done in Microsoft Excel 4.0. Line 1: the equilibrium enrichment only [t=0 assumed in eqn (13)]. Line 2: the dissociative panning only. Function plotted was defined as follows: h=ad w+b exp(−kr t). Line 3: Enrichment function for both equilibrium and dissociative biopanning, as in eqn (13). Parameter values were as follows: a=0.05, b=0.2, ca=c−cb=10−8 M−1, d=0.4, t=1000 s and w=10. It was assumed that kr=105/K. (B) Distribution of the phage population between affinity classes. Nine discrete classes (K=1 to 109 ) were assumed. The distribution is normalized to 1, the sum of occupancies (or frequencies) for all classes has to be 1. Line 0: initial distribution (before panning). Lines 1, 2 and 3: distributions after rounds 1, 2, and 3 of biopanning, respectively. (C) Numbers of phage in washes and in eluate. Wash 0 corresponds to the number of input phage, normalized to 1. The eluate is indicated as wash number 11. Lines 1, 2 and 3: washes after panning rounds 1, 2 and 3. (D) Distribution of the phage population between affinity classes. Similar to (B) except that a different initial affinity distribution of phage was assumed (compare line 0 of plot B with line 0 of plot D). (E) Numbers of phage in washes and in eluate corresponding to (D). Line description as in (C).
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F. 3. Example 2 for biopanning with a peptide library. Line description is as in plots (A), (B) and (C) of Fig. 1. Plots were constructed as described in the legend to Fig. 1. Parameter values were representative of higher non-specific binding and short wash times, namely: a=0.24, b=0.01, ca=c−cb=10−8 M−1, d=0.4, t=30 s (3 s each wash) and w=10. It was assumed that kr=105/K.
4. Discussion - From the mathematical perspective, the solid-phase and solution biopanning are similar, and are both described by the same form of the enrichment function, eqn (13). There are a few practical differences. First, the target concentration c is normally precisely known in solution biopanning and can be adjusted as needed, while in solid-phase biopanning only very crude and unreliable estimates are possible. Moreover, as the concentration of targets is much larger than the concentration of phage, parameter cb in eqn (13) can be neglected, improving the precision of the quantitative analysis of the solution panning. Second, kinetic constants for macromolecular binding are true in solution biopanning, while they can be affected by a number of factors in solid-phase panning (such as solid-phase material itself, or multi-valent binding). On the other hand, solution biopanning presents a few experimental challenges. Since targets are typically in huge excess over phage, the efficient harvesting of phage-target complexes requires a large streptavidincovered surface area to collect both targets and phage-target complexes. If that is not provided, solution panning becomes inefficient. This would correspond in the model to low numerical values of parameter b and low overall enrichment. Also, in true solution panning the process of collecting the phage-target complexes needs to be fast—if it is slow and a secondary equilibrium establishes itself (i.e. phage to targets-on-solid-phase equilibrium), then the intended solution panning becomes an inefficient solid-phase panning. The potential valency for display is up to five ligands in the gene-3 systems. It turns out, however, that under
most experimental conditions a very small fraction of gene-3 proteins carry peptides. The numbers vary depending on the particular system. In peptide libraries of 30 amino acids in length, on M13 phage (no phagemid employed), only 5–10% of phages carry a peptide. This translates to the average of one peptide per 50–100 gene-3 proteins (Grihalde et al., 1995). In single-chain antibody libraries or enzyme (alkaline phosphatase) display systems, that number drops to 0.01–0.1% of phages carrying the protein. The inefficiency of the display system is probably a result of proteolysis. It follows that the phage libraries used in experiments consist mostly of ‘‘naked’’ phage (no peptide displayed), a small percentage of phage carrying a single ligand, and possibly a minuscule amount of phage carrying two or more ligands. From that perspective, multiple interactions due to the multiple ligand display seem to be rather rare events in the test tube (although not insignificant). . The presented model is a tool for the analysis of biopanning. It allows one to follow the affinity distribution of phages during the course of biopanning and to investigate contributions of different parameters on the outcome of the experiment. The main tenet of the model is the classification of phage in the library into the apparent affinity classes. Thus, phage bearing one ligand might be in a different affinity class than a two-ligand phage, and phage bearing no ligands (because of, for example, proteolysis) will be in yet another affinity class. It needs to be stressed that the apparent affinity relates here to the affinity of the macromolecular assembly of the ligands (peptides) and the phage particle, and not of the single ligand. The model does not provide any relationship between the true affinity of a single ligand to the target and the apparent affinity of the macromolecular assembly. Although conceivable, such a relationship would need
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to be complex and involve the observed valency of the phage (which can vary depending on the ligand because of proteolysis), the displayed peptide length, information about the way the peptide is anchored and contribution of the phage coat to the interaction. The apparent binding constant and apparent dissociation/association rate constants are measurable parameters. They directly govern the outcome of biopanning, Biacore experiments, and several types of ELISA assays involving phage particles. The model is a tool that allows to take advantage of knowing the apparent binding constants or, alternatively, to get insights about what the constants are from experimental results. The true binding constants/rates need to be obtained from analyses involving peptides or proteins obtained through the chemical synthesis or gene expression. 5. Conclusions The general model for biopanning formulated here has included both the equilibrium and dissociative phases of biopanning. Two examples have been analyzed in detail. They correspond loosely to a simple peptide library and a single-chain antibody library. Several experimental observations could be simulated in the model, including wash/elution profiles, their dependencies on the properties of the target, as well as effects of inefficient equilibrium panning. The analysis provides insights into how the affinity distribution of phage changes during the course of panning. The ability to describe quantitatively affinity enrichment
should facilitate more efficient biopanning protocols and should allow us to gain insights into properties of phage populations in libraries. We would like to thank Steve Kakavas for sharing his unpublished experimental data with us, and Cele AbadZapatero, Chris Dealwis and Mark Hayden for reading the manuscript and constructive comments. We thank Michael Klass for support. REFERENCES B, C. F. III & L, R. A. (1991). Combinatorial immunoglobulin libraries on the surface of phage (phabs): rapid selection of antigen-specific Fabs. Methods: Comp. Meth. Enzymol. 2, 119–124. D, E. D. (1990). Advanced Immunochemistry, 2nd edn, pp. 281–389. New York: Wiley-Liss. G, N. D., C, Y.-C., G, A., G, E. & M, W. (1995) Epitope mapping of anti-HIV monoclonal antibodies and characterization of mimotopes using a filamentous phage peptide library. Gene, accepted for publication. H, R. E., R, S. J. & W, G. (1992). Selection of phage antibodies by binding affinity: mimicking affinity maturation. J. molec. Biol. 226, 889–896. I, D., T, C. & G, L. (1991). Systematic evolution of ligands by exponential enrichment with integrated optimization by non-linear analysis. J. molec. Biol. 222, 739–761. O’S, D. J., B-B, M., S, K. K. & H, P. (1993). Determination of rate and equilibrium binding constants for macromolecular interactions using surface plasmon resonance: use of nonlinear least squares analysis methods. Anal. Biochem. 212, 457–468. P, S. F. & S, G. P. (1988) Antibody-selectable filamentous fd phage vectors: affinity purification of target genes. Gene 73, 305. S, J. K. & C, L. (1994). Random peptide libraries. Curr. Opin. Biotech. 5, 40–48. S, J. K. & S, G. P. (1990). Searching for peptide ligands with an epitope library. Science 249, 386–390.
A Mathematical Model for Biopanning (Affinity Selection) Using Peptide Libraries on Filamentous Phage W M,† Y.-C. J C N G Viral Discovery D90D, Abbott Laboratories, 1401 Sheridan Road, North Chicago, IL 60064-4000, U.S.A. (Received on 9 December 1994, Accepted in revised form on 7 June 1995)
A mathematical model is presented for the process of selection of peptides for binding to a target. The affinity enrichment process commonly known as biopanning relies on subjecting a library of peptides on filamentous phage to a selection for binding to the target immobilized on solid support or in solution. The model is an implementation of the mass-action law to the diverse population of macromolecular assemblies. An analytical solution is presented for the enrichment process. Most parameters in the enrichment formula can be easily determined experimentally. Two examples corresponding to biopanning with epitope libraries and antibody libraries are given. The model allows for an estimation of the contribution of equilibrium and dissociative biopanning to the overall enrichment. The model can be a tool in the evaluation of the role of different biopanning parameters. Its implementation in a spreadsheet makes it possible to perform a computer simulation of a biopanning experiment. 7 1995 Academic Press Limited
target takes place: (a) binding in solution (Parmley & Smith, 1988), and (b) binding to solid phase (Barbas & Lerner, 1991). As indicated in Table 1, type (a) procedures involve conjugation of the target protein to biotin, binding of phages to the target in solution, immobilization of phages bound to targets to solid phase via biotin-streptavidin interaction, washing the solid phase to remove the non-specific binders, elution of bound phages and steps to recover and quantitate the eluted phages. In type (b) procedures phages are bound directly to targets deposited on solid support (microtiter plates) via non-specific interactions, and subsequent steps are similar to those described for type (a) procedures. There are two mechanisms for the enrichment of a phage population by affinity to a target. First, an equilibrium is established between the target molecules and the phages having different affinities to the target. The phage subpopulation within target-phage complexes will be enriched in phages having the highest affinities to the target. This type of enrichment will be referred to in this paper as the equilibrium enrichment.
1. Introduction Biopanning (affinity selection) is a procedure for enrichment for molecules that bind to a given target, which often is a protein. In a typical experiment, a filamentous phage library is employed (Parmley & Smith, 1988). Each phage in the library displays on its coat protein, gp3 or gp8, a different peptide or protein by fusion of genetic material. The diversity of the library is frequently 107–1011 different phages. The objective of a biopanning experiment is to identify from a phage library a phage or a subset of phage that bind, most tightly to the target and, eventually, the sequences of peptides displayed on the phage. Current approaches and libraries were recently reviewed by Scott & Craig (1994). Several protocols for biopanning have been described. They fall into two basic classes according to the phase in which the binding of the phages to the † Author to whom correspondence should be addressed at: Viral Discovery D90D, Bldg. R1, Abbott Laboratories, 1401 Sheridan Road, North Chicago, IL 60064-4000, U.S.A. 0022–5193/95/200523+08 $12.00/0
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Second, when the phage immobilized to the target on solid support is washed, phages with a fast off-rate to the target are preferentially washed out, thus enriching the remaining phage population in phages having a slow off-rate. The second mechanism will be referred to as off-rate or dissociative enrichment. An example for equilibrium enrichment and a brief analysis was presented by Hawkins et al. (1992). A model for a related procedure, selection of nucleic acids for affinity to a target (Selex) has been described by Irvine et al. (1991). In this paper, a general model for biopanning is presented. The model is applicable for both solution and solid phase biopanning. An analytical expression is derived for the equilibrium panning and the dissociative panning. Several examples are described, calculations performed in a spreadsheet and the results are compared with experimental data.
is assumed that each phage affinity class is represented by a sufficiently large number of phage, to which the mass action law is applicable. This condition can be satisfied experimentally by performing biopanning with large phage populations having many identical copies of each phage. List of definitions c total concentration of targets (targets are identical), concentration of unbound targets (i.e. free cf in solution), total concentration of the phage class i, pi pfi concentration of the class i of unbound phage, [cp]i concentration of the class i of phage-target complexes. The following equations need to be satisfied in equilibrium for i=1, . . . m:
2. The Model
[cp]i=Ki pfi cf
The first step in biopanning is establishing the equilibrium between the phage population and targets. The target molecules could be either in solution or on the solid support. It is assumed in the model that the process is governed by the mass-action law applied to binding of phages to targets. For simplicity, the phage population will be divided into classes, each characterized by an association constant Ki , which is the same within the class, i=1, 2, . . . m (m is the number of classes). Ki characterizes the following reaction:
(1)
m
c= s [cp]i+cf
(2)
i=1
pfi+[cp]i=pi
(3)
These equations can be condensed into:
0
m
[cp]i=Ki (pi−[cp]i ) c− s [cp]i i=1
1
(4)
Defining: m
cb= s [cp]i
[phage class i ]
(5)
i=1
Ki
+target F [target-phage class i complex] Ki describes the apparent binding constant for the phage from the ith-class, and not for the peptide mounted on the phage. Thus, Ki depends both on the true peptide-target binding constant and on the multiplicity (’’valency’’) of the display, i.e. the number of peptides on the surface of a single phage particle. It
we get from (4): [cp]i=
Ki pi 1 +Ki c−cb
Equation (6) can be rewritten for the continuous case, in which the affinity of the phage population is
T 1 The process of biopanning Solution-phase selection
(6)
Solid-phase selection
1. Equilibrium step: equilibrate biotynylated 1. Coat solid surface with receptor target with phage library 2. Capture step: pan phage-target mixture 2. Capture step: pan phage library with with streptavidin-coated solid surface target-coated solid surface 3. Washing step: wash unbound phage from solid surface 4. Elution step: elute bound phage
J G h G j _ ~
Equilibrium Biopanning Dissociative Biopanning
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described by a density function f, which is dependent on the association constant K of phage to the target. The density function f(K) describes how many phage of a given affinity K are present (in the discontinuous case above, Ki described how many phage belonged to the i-th affinity class). As a result of equilibrium biopanning, phage bound to targets can be harvested. That population is described by a density function g(K) which [on the basis of eqn (6)] is given by the following formula: g(K)=
K f (K) 1 +K c−cb
(7)
where cb=
g
a
g(K) dK
(8)
0
Equation (7) is not the explicit solution of the problem, since the numerical value of parameter cb is not known except for special cases (see below). Nevertheless, since cb is independent of K, eqn (7) provides a general type of function that governs the equilibrium panning. Parameter cb can be interpreted from eqn (8) as the total number of phages bound specifically to the support, or equivalently, the total number of targets bound to phages. The population of phages that emerges from the equilibrium biopanning experiment is enriched in phages that bind with high affinity to the target. The enrichment function h can be represented by the ratio of two density functions, corresponding to phage distribution before (function f ) and after biopanning (function g): h(K)=
g(K) f(K)
(9)
and for equilibrium panning heq (K)=
K 1 +K c−cb
(10)
The enrichment function approaches 1 (all phages retained) for sufficiently high association constant K, and for K1/(c−cb ) it can be approximated by a linear function of K. The numerical value of the denominator, c−cb can be considered the concentration of available total target (i.e. unbound to target), ca (caEc). Figure 1 presents log–log plots of the enrichment function for different ca . Two limit cases are of interest. Often, especially in the first round of biopanning with a diverse phage population, or in many instances of solution panning,
F. 1. Enrichment function in equilibrium biopanning. Variable heq given by eqn (10) is plotted as a function of the association constant K. This dependence applies if no non-specific binding takes place. Numerical values for parameter ca=c−cb are as follows: line a, 10−5; line b, 10−6; line c, 10−7; line d, 10−8; line e, 10−9 and line f, 10−10.
the concentration of phages that binds to the target will be much less than the total concentration of the targets, i.e. cb c and c−ca1c. In that case eqns (7) and (10) provide a direct solution to the problem of equilibrium panning. For efficient equilibrium panning, the target concentration should be selected at about 1/Kmin , where Kmin is the minimum association constant for the phage desired to be obtained from the experiment. Phages with affinities higher than 1/Kmin will be effectively retained (if they are present in the phage population). The remainder phage with lower affinities will be retained proportionally to their affinity, and for those phages the enrichment function approximates the identity function h(K)1K in equilibrium panning. There might be instances when the entire phage population has a measurable affinity to the target, and the total phage concentration is much higher than the concentration of targets, and essentially all targets form complexes with the phages, e.g. second and further rounds of solid phase biopanning with a large number of phages. Then cb1c, and 1/(c−cb )1/c, the enrichment function shifts towards higher K values (Fig. 1), effectively extending the linear part of the dependence towards higher numerical values of K. This provides a mechanism for increasing the selective pressure on phages having higher affinity constant to targets than K=1/c. In a typical biopanning experiment, the phage immobilization on solid support is followed by extensive washing of the support. The purpose is to
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enrich the phage population in phages that bind tightly to the support. There are two mechanisms for the enrichment. First, the phages that are not bound tightly (but are still specifically bound) are removed preferentially over those phages that have higher affinity during the course of washing. Second, washing removes phages that are not specifically bound to the target on the solid support (but rather adhere to plastic, blocking material or get entrapped by the solid support). The dissociation of molecules specifically bound to targets on solid support has been shown in many instances (Day, 1990; O’Shannessy, 1993) to be a first order process that can be described by a simple exponential decay: s=s0 e
−kr t
(11)
where s is the density function at time t for the phage subpopulation remaining on the solid support, dependent on the association constant K and off-rate constant kr , and s0 is a constant. This formula is adopted in the model. The effectiveness of washing the support is the critical element of a biopanning experiment. Little is known about the mechanism of non-specific binding of filamentous phage to support. It will be assumed in the model that non-specific binding is independent of the specific binding, in other words, the area of the solid phase can be conceptually divided into sites for specific binding to the target, and sites for non-specific binding. The fraction of input phages that are retained on the support due to non-specific binding will be given by parameter a. The fraction, b0 (K), retained on the support due to the specific binding will depend on the affinity constant K of the input phage population to the support. Typically, both a and b are much smaller than one. Phage bound non-specifically to the support will be at least partially removed during the course of washing, which typically comprises a series of buffer additions and removals. The number of washes can vary between 1 and 40, but most often is between 5 and 20. Experimental evidence (unpublished data) supports the model according to which each buffer addition produces an effect equivalent to a dilution of the non-specifically bound phage, a significant fraction † Here is an alternative method of deriving the enrichment function. We focus on the probability of binding (Pr) for an individual phage particle. The enrichment function h [eqn (13)] is the same as: Pr(bound after washes)=Pr(specifically bound after washes=specifically bound before washes)×Pr(specifically bound before washes)+Pr(nonspecifically bound after washes=nonspecifically bound before washes)×Pr(specifically bound before washes), where the symbol Pr(=) indicates the conditional probability with the condition specified at the right-hand side of the bracket.
of which is then removed. The fraction removed is approximately the same for each wash and does not depend on the duration of the wash. No experimental support was gathered to support an alternative mechanism, i.e. first order dissociation process of non-specifically bound phage. A mechanism consistent with the dilution effect is the entrapment of phage in cavities on solid surface, such that the phage is practically in solution at all times, but the solution can not be easily displaced from vicinity of the surface. Thus the phage population remaining on the support during the course of the dissociative biopanning is: s=s0 [ad w+b0 (K)e−kr t ]
(12)
where d is the dilution factor and w is the number of washes. † A typical biopanning experiment involves both the equilibrium and dissociative phases. Therefore, considering the discussion above and eqns (10) and (12), as b0 (K)=bh(K), where b is a constant, we get the formula for the time dependence of the enrichment function h for the phage population for the single round of biopanning: h=ad w+
bK e−kr t 1 +K c−cb
(13)
If the density function for the input phage is f, then the density function g after biopanning is from eqn (9): g=h ·f
(14)
If several rounds of biopanning are performed, then the total enrichment function htot will be the product of enrichment functions at each round, that is: htot=h N ,
(15)
where N is the number of rounds. The enrichment function (13) is formulated here as a function of three variables and is dependent on six parameters. Most of the parameters, however, are easy to determine experimentally for each particular experimental conditions. The next section will provide two examples of analysis of the enrichment function for selected applications. 3. Examples 1: This example is aimed at modeling of a typical panning experiment involving a peptide library.
Although parameters discussed below are only approximations of experimental values, the overall behavior of the model system is intended to parallel that of the experimental one. It is assumed that the epitope library on filamentous phage has a diversity of 109 . In the biopanning experiment, all 109 phage are applied to a single microtiter plate well (diameter 5 mm). About 1012 target protein molecules were deposited in that well and bound to solid support. Considering that the volume applied to the well is 100 ml, target concentration used in the calculations will be estimated at c=10−8 M. This is a very crude assumption, since from the perspective of the mass action law, targets bound to solid phase and targets in solution are not equivalent. Considering that the total number of phage that will bind to the target on solid support is small compared with the target concentration, especially in the first/initial round of panning, c1 will be negligibly small (c1=0). Thus, in this example, the equilibrium part of the enrichment function is close to linear up to affinities of 108 M−1 . Other parameters have been estimated on the basis of experimental data (data not shown here). The fraction of all phage that bind to support is b=0.2, and that bind non-specifically is a=0.05. The duration of all washes is assumed to be t=1000 sec, and the dilution factor is d=0.4. The numerical value of the association rate constant was assumed to be independent of the phage subpopulation, and equal to 105 M−1 s−1 (a typical association rate constant for macromolecules). Thus a direct dependence exists in the model for the association constant K and the off-rate kr , i.e. kr=105/K. A numerical model was written in a spreadsheet (Excel 4.0) to simulate the process of washing and eluting phage for several panning rounds. The input for the model is also the initial distribution of the initial phage population. For the purpose of the spreadsheet calculations it is represented as up to 15 discrete groups, each characterized by a different association constant. Three dependencies were selected as indicative of the behavior of the system: the enrichment as a function of affinity, the number of phage in the washes and eluate, and the changes of distributions after rounds of panning. The enrichment function (Fig. 2A) is independent of the initial distribution. An attempt was made to distinguish contributions from the equilibrium and dissociative phases of biopanning. It is evident from the plot (overlap of lines 2 and 3) that under conditions of this example the overall enrichment is almost exclusively due to the dissociative biopanning. The reason for this was that the total wash time was fairly long. The enrichment function is close
527
to the all-or-none type, and it drops by a factor of 104 when K changes only ten-fold. Overall enrichment factor (ratio of the highest to lowest value of the enrichment function) is about 3×104 . A simple affinity distribution for the phage population is presented in Fig. 2B, line 0. In this distribution, the phage subpopulation characterized by affinity of K=10i (i=0, 1, 2, . . ., 9) has approximately 109−i members (although the precise numbers are somewhat different due to normalization). In other words, the membership of the affinity class decreases by a factor of 10 as the affinity of the class increases by one order of magnitude. This distribution would be consistent with a requirement that the highest affinity of a clone from the phage library increases proportionally to the total number of different clones (i.e. diversity) of the library. The distribution rapidly changes after subsequent rounds of biopanning (Fig. 2B, lines 1, 2 and 3). Tight binders constitute about 50% of all clones after round 2, and 99.99% after round 3. The changes in the affinity distribution of the phage population are evident in the pattern of recovery of phage in washes and eluate (Fig. 2C). When essentially all phage bind to target, i.e. after round 3 (see line 3), there is a pronounced plateau, and the number of phage in the eluate is two orders of magnitude higher than the number of phage in the last wash. Biopanning can bring desired results even faster in special cases of the initial affinity distribution. The distribution in Fig. 2D, line 0 might be characteristic of cases where a short consensus sequence is necessary and sufficient for binding to the target, for instance CXSC (single amino acid code, X is any residue). The number of peptides bearing that consensus sequence in the library will be high, hence the biphasic character of this distribution. After just one round of panning (line 1) the phage population contains 60% of tight binding clones, the number approaches 100% after rounds 2 and 3. This type of initial affinity distribution manifests itself in wash curves (Fig. 2E) as a noticeable spike of phage numbers in the eluate from round 1, and a plateau and a two-log spike after rounds 2 and 3 (compare Figs 2C and E). 2: A different type of a library is described in this example. While simple epitope libraries display peptides on the surface of filamentous phage, more complex types of molecules can be also displayed on the phage. Libraries of single-chain antibodies or Fab fragments of antibodies are frequently used to select proteins that bind to a target. Libraries of larger
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proteins can be significantly different from peptide libraries. The model can be used to evaluate effects of one of the parameters, namely loading of a protein per phage. While in peptide libraries the loading is on the order of one peptide per phage (which is still less than the maximum loading possible, i.e. three to five peptides per phage or one peptide per each of the gene-3 proteins), in single-chain antibody libraries the loading is orders of magnitude less, 10−1–10−3 per phage. This can be simulated in the model by lowering the numerical value of the parameter b (fraction of phage retained on the solid support due to the specific binding) from b=0.2 to b=0.01 (parameter a is changed from 0.05 to 0.24 to keep a+b constant, i.e. total fraction of phage bound). The results of calculations are presented in Fig. 3. Overall enrichment is much less (about 300) than in Example 1 because of the high impact of non-specific binding. What changes also is the relevance of the equilibrium panning, which now
exerts the dominant effect on the enrichment function, although the dissociative panning is also significant (lines 1, 2 and 3 in Fig. 3A). The effect is due to a much shorter total wash time, compared with Example 1. The initial affinity distribution of the phage population (Fig. 3B, line 0) is assumed to be the same as in Fig. 2B, line 0. After three rounds of panning, tight-binding phage constitute only close to 10% of all phage in the pool. In most experimental designs, a fourth round of panning would be performed. A slower progress of affinity enrichment manifests itself also in the wash/elution pattern (Fig. 3C). Lines 1, 2 and 3 corresponding to results of rounds 1, 2 and 3, are almost indistinguishable (although the eluate after round 3 contains 10% of tight-binding phage). This result supports the experimental practice of assaying clones after 2, 3 or 4 rounds despite the seemingly negative pattern in Fig. 3C.
F. 2. Example 1 for biopanning with a peptide library. (A) Enrichment as a function of the association constant K. All calculations were done in Microsoft Excel 4.0. Line 1: the equilibrium enrichment only [t=0 assumed in eqn (13)]. Line 2: the dissociative panning only. Function plotted was defined as follows: h=ad w+b exp(−kr t). Line 3: Enrichment function for both equilibrium and dissociative biopanning, as in eqn (13). Parameter values were as follows: a=0.05, b=0.2, ca=c−cb=10−8 M−1, d=0.4, t=1000 s and w=10. It was assumed that kr=105/K. (B) Distribution of the phage population between affinity classes. Nine discrete classes (K=1 to 109 ) were assumed. The distribution is normalized to 1, the sum of occupancies (or frequencies) for all classes has to be 1. Line 0: initial distribution (before panning). Lines 1, 2 and 3: distributions after rounds 1, 2, and 3 of biopanning, respectively. (C) Numbers of phage in washes and in eluate. Wash 0 corresponds to the number of input phage, normalized to 1. The eluate is indicated as wash number 11. Lines 1, 2 and 3: washes after panning rounds 1, 2 and 3. (D) Distribution of the phage population between affinity classes. Similar to (B) except that a different initial affinity distribution of phage was assumed (compare line 0 of plot B with line 0 of plot D). (E) Numbers of phage in washes and in eluate corresponding to (D). Line description as in (C).
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F. 3. Example 2 for biopanning with a peptide library. Line description is as in plots (A), (B) and (C) of Fig. 1. Plots were constructed as described in the legend to Fig. 1. Parameter values were representative of higher non-specific binding and short wash times, namely: a=0.24, b=0.01, ca=c−cb=10−8 M−1, d=0.4, t=30 s (3 s each wash) and w=10. It was assumed that kr=105/K.
4. Discussion - From the mathematical perspective, the solid-phase and solution biopanning are similar, and are both described by the same form of the enrichment function, eqn (13). There are a few practical differences. First, the target concentration c is normally precisely known in solution biopanning and can be adjusted as needed, while in solid-phase biopanning only very crude and unreliable estimates are possible. Moreover, as the concentration of targets is much larger than the concentration of phage, parameter cb in eqn (13) can be neglected, improving the precision of the quantitative analysis of the solution panning. Second, kinetic constants for macromolecular binding are true in solution biopanning, while they can be affected by a number of factors in solid-phase panning (such as solid-phase material itself, or multi-valent binding). On the other hand, solution biopanning presents a few experimental challenges. Since targets are typically in huge excess over phage, the efficient harvesting of phage-target complexes requires a large streptavidincovered surface area to collect both targets and phage-target complexes. If that is not provided, solution panning becomes inefficient. This would correspond in the model to low numerical values of parameter b and low overall enrichment. Also, in true solution panning the process of collecting the phage-target complexes needs to be fast—if it is slow and a secondary equilibrium establishes itself (i.e. phage to targets-on-solid-phase equilibrium), then the intended solution panning becomes an inefficient solid-phase panning. The potential valency for display is up to five ligands in the gene-3 systems. It turns out, however, that under
most experimental conditions a very small fraction of gene-3 proteins carry peptides. The numbers vary depending on the particular system. In peptide libraries of 30 amino acids in length, on M13 phage (no phagemid employed), only 5–10% of phages carry a peptide. This translates to the average of one peptide per 50–100 gene-3 proteins (Grihalde et al., 1995). In single-chain antibody libraries or enzyme (alkaline phosphatase) display systems, that number drops to 0.01–0.1% of phages carrying the protein. The inefficiency of the display system is probably a result of proteolysis. It follows that the phage libraries used in experiments consist mostly of ‘‘naked’’ phage (no peptide displayed), a small percentage of phage carrying a single ligand, and possibly a minuscule amount of phage carrying two or more ligands. From that perspective, multiple interactions due to the multiple ligand display seem to be rather rare events in the test tube (although not insignificant). . The presented model is a tool for the analysis of biopanning. It allows one to follow the affinity distribution of phages during the course of biopanning and to investigate contributions of different parameters on the outcome of the experiment. The main tenet of the model is the classification of phage in the library into the apparent affinity classes. Thus, phage bearing one ligand might be in a different affinity class than a two-ligand phage, and phage bearing no ligands (because of, for example, proteolysis) will be in yet another affinity class. It needs to be stressed that the apparent affinity relates here to the affinity of the macromolecular assembly of the ligands (peptides) and the phage particle, and not of the single ligand. The model does not provide any relationship between the true affinity of a single ligand to the target and the apparent affinity of the macromolecular assembly. Although conceivable, such a relationship would need
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to be complex and involve the observed valency of the phage (which can vary depending on the ligand because of proteolysis), the displayed peptide length, information about the way the peptide is anchored and contribution of the phage coat to the interaction. The apparent binding constant and apparent dissociation/association rate constants are measurable parameters. They directly govern the outcome of biopanning, Biacore experiments, and several types of ELISA assays involving phage particles. The model is a tool that allows to take advantage of knowing the apparent binding constants or, alternatively, to get insights about what the constants are from experimental results. The true binding constants/rates need to be obtained from analyses involving peptides or proteins obtained through the chemical synthesis or gene expression. 5. Conclusions The general model for biopanning formulated here has included both the equilibrium and dissociative phases of biopanning. Two examples have been analyzed in detail. They correspond loosely to a simple peptide library and a single-chain antibody library. Several experimental observations could be simulated in the model, including wash/elution profiles, their dependencies on the properties of the target, as well as effects of inefficient equilibrium panning. The analysis provides insights into how the affinity distribution of phage changes during the course of panning. The ability to describe quantitatively affinity enrichment
should facilitate more efficient biopanning protocols and should allow us to gain insights into properties of phage populations in libraries. We would like to thank Steve Kakavas for sharing his unpublished experimental data with us, and Cele AbadZapatero, Chris Dealwis and Mark Hayden for reading the manuscript and constructive comments. We thank Michael Klass for support. REFERENCES B, C. F. III & L, R. A. (1991). Combinatorial immunoglobulin libraries on the surface of phage (phabs): rapid selection of antigen-specific Fabs. Methods: Comp. Meth. Enzymol. 2, 119–124. D, E. D. (1990). Advanced Immunochemistry, 2nd edn, pp. 281–389. New York: Wiley-Liss. G, N. D., C, Y.-C., G, A., G, E. & M, W. (1995) Epitope mapping of anti-HIV monoclonal antibodies and characterization of mimotopes using a filamentous phage peptide library. Gene, accepted for publication. H, R. E., R, S. J. & W, G. (1992). Selection of phage antibodies by binding affinity: mimicking affinity maturation. J. molec. Biol. 226, 889–896. I, D., T, C. & G, L. (1991). Systematic evolution of ligands by exponential enrichment with integrated optimization by non-linear analysis. J. molec. Biol. 222, 739–761. O’S, D. J., B-B, M., S, K. K. & H, P. (1993). Determination of rate and equilibrium binding constants for macromolecular interactions using surface plasmon resonance: use of nonlinear least squares analysis methods. Anal. Biochem. 212, 457–468. P, S. F. & S, G. P. (1988) Antibody-selectable filamentous fd phage vectors: affinity purification of target genes. Gene 73, 305. S, J. K. & C, L. (1994). Random peptide libraries. Curr. Opin. Biotech. 5, 40–48. S, J. K. & S, G. P. (1990). Searching for peptide ligands with an epitope library. Science 249, 386–390.