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Biophys J, June 1999, p. 3031-3043, Vol. 76, No. 6
*Theoretical Biology and Biophysics, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, and #Department of Chemistry, Northern Arizona University, Flagstaff, Arizona 86011 USA
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ABSTRACT |
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 TOP
 ABSTRACT
 INTRODUCTION
THEORY
METHODS
RESULTS
DISCUSSION
REFERENCES
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Steric effects can influence the binding of a cell surface receptor to a multivalent ligand. To account for steric effects arising from the size of a receptor and from the spacing of binding sites on a ligand, we extend a standard mathematical model for ligand-receptor interactions by introducing a steric hindrance factor. This factor gives the fraction of unbound ligand sites that are accessible to receptors, and thus available for binding, as a function of ligand site occupancy. We derive expressions for the steric hindrance factor for various cases in which the receptor covers a compact region on the ligand surface and the ligand expresses sites that are distributed regularly or randomly in one or two dimensions. These expressions are relevant for ligands such as linear polymers, proteins, and viruses. We also present numerical algorithms that can be used to calculate steric hindrance factors for other cases. These theoretical results allow us to quantify the effects of steric hindrance on ligand-receptor kinetics and equilibria.
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INTRODUCTION |
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TOP
ABSTRACT
INTRODUCTION
THEORY
METHODS
RESULTS
DISCUSSION
REFERENCES
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For many cell surface receptors, interaction with
a multivalent ligand is essential for signal transduction (Metzger,
1992
). Multivalent ligand-receptor binding leads to aggregation of
receptors, phosphorylation of intracellular receptor domains, and
activation of cytoplasmic regulatory molecules. These receptors include
multichain immune recognition receptors (Keegan and Paul, 1992), such
as the B cell receptor, and receptor tyrosine kinases (Pazin and Williams, 1992; Fry et al., 1993), such as the epidermal growth factor
receptor. Multivalent ligand-receptor binding is important not only in
signal transduction, but also in a variety of other phenomena, such as
antibody-mediated activation of the complement cascade (Burton and
Woof, 1992) and viral attachment and entry into cells (Haywood, 1994).
Because of the importance of multivalent ligand-receptor binding,
significant effort, involving both experimental and theoretical work,
has been directed at understanding the interactions of multivalent ligands with cell surface receptors. Theoretical work on bivalent ligands (Dembo and Goldstein, 1978; Perelson and DeLisi, 1980) has been
particularly influential. The theory for these ligands has been refined
over a number of years (Posner et al., 1995b) and applied to a number
of problems, particularly the analysis of Fc
RI aggregation on the
surface of rat basophilic leukemia cells (Goldstein, 1988; Goldstein
and Wofsy, 1994). Theoretical work on multivalent ligands (Gandolfi et
al., 1978; DeLisi, 1980; Perelson, 1981) has also been applied to a
number of problems in immunology and virology (Hlavacek et al., 1999;
Sulzer and Perelson, 1997; Dee and Shuler, 1997; Goldstein and Wofsy,
1996; Wickham et al., 1990, 1995; Segal et al., 1983; Vogelstein et al., 1982; Dower and Segal, 1981; Dower et al., 1981). However, models
for multivalent ligands have yet to be fully developed.
When the valence of a ligand is greater than two, models for
ligand-receptor binding are complicated by a number of factors (Perelson, 1984; Macken and Perelson, 1985; Lauffenburger and Linderman, 1993). One complication arises when a bound receptor, because of its physical size, excludes receptor binding at neighboring ligand sites. Steric exclusion of ligand sites by bound receptors is
illustrated in Fig. 1. Shown
schematically is the surface of a multivalent ligand that is bound to
two receptors. Each receptor binds a single site but physically covers
an area that encompasses more than one site on the ligand. Sites on the
ligand are bound or unbound, and unbound sites are covered, excluded,
or available. A covered site is unavailable for receptor
binding, because it is covered by a bound receptor. Likewise, an
excluded site, although it is not covered, is unavailable
for receptor binding because it lies near a bound receptor, and binding
at this site would require an overlap of receptors. A covered or
excluded site is distinguished from an available site, at
which receptor binding is possible. Steric exclusion of ligand sites by
bound receptors reduces the average reactivity of unbound ligand sites:
available sites have the potential to bind receptors, but covered and
excluded sites do not.
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Steric exclusion of ligand sites is likely to play a role in many
ligand-receptor interactions, as suggested by the interactions of
antibodies with various types of antigens. Steric effects due to ligand
site exclusion can be important in antibody binding to viruses. For
example, the protein coat of tobacco mosaic virus (TMV), a nonenveloped
virus, consists of 2130 repeating subunits, but subunit-specific,
excess monoclonal antibody binds TMV with a stoichiometry of 800:1,
which suggests that a bound antibody Fab arm covers approximately three
viral subunits (Pellequer and Van Regenmortel, 1993). Steric effects
due to ligand site exclusion also can be significant in antibody
binding to bacterial polysaccharides. Galactan from Prototheca
zopfii is composed of ~1240 galactosyl residues.
Galactan-specific antibodies, which can bind at sites along the entire
polysaccharide chain, each cover at least 10 and as many as 30 sequential galactosyl residues when bound (Glaudemans et al., 1986).
Steric effects can also be important in antibody binding to protein
antigens, because the whole surface of a protein is potentially
antigenic (Davies and Cohen, 1996) and domains important for
protein-protein binding are smaller than protein-protein interfaces
(Wells, 1996). Interactions of antibodies with densely haptenated
carrier molecules are also likely to involve steric effects (Macken and
Perelson, 1986; Hlavacek et al., 1999).
Here we develop a theoretical framework for modeling multivalent
ligand-receptor binding when steric effects due to ligand site
exclusion are important. In this framework, steric effects on
cross-linking reactions are characterized by a steric hindrance factor.
This factor indicates how the accessibility, and thus the average
reactivity, of unbound ligand sites depends on ligand site occupancy.
Steric hindrance factors are closely related to insertion probabilities
(Widom, 1963). An insertion probability is the probability of inserting
a particle onto a surface without overlap when the surface is partially
covered with other particles. By adapting the method of Andrews (1975,
1976) for calculating insertion probabilities, we are able to derive
expressions for steric hindrance factors for various types of ligands
and receptors.
In our derivation of steric hindrance factors, we focus on receptors
that cover a compact region on the ligand surface when bound, but we
consider different types of ligands with binding sites distributed
regularly or randomly in one or two dimensions. The one-dimensional
results, which complement earlier work (Macken and Perelson, 1986), are
relevant for linear polymers to which haptens have been conjugated at
random positions or a polysaccharide with regularly spaced epitopes.
The two-dimensional results are relevant for ligands such as
multisubunit proteins, haptenated proteins, surface proteins on
enveloped viruses, or whole nonenveloped viruses. Our expressions for
steric hindrance factors are approximate, except for special
one-dimensional cases. To determine the accuracy and usefulness of
these approximations, we compare approximate results with those
calculated via Monte Carlo or combinatoric methods.
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THEORY |
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TOP
ABSTRACT
INTRODUCTION
THEORY
METHODS
RESULTS
DISCUSSION
REFERENCES
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Here we develop a model for ligand-receptor binding that includes
steric effects. We focus on a multivalent ligand that interacts with a
monovalent receptor. The ligand is assumed to be symmetrical and much
smaller than a cell, and binding sites on the ligand are assumed to be
chemically identical. We allow for various arrangements of ligand
sites. The receptor is mobile and disperse on the cell surface. We
assume that receptor trafficking is inhibited such that the total
amounts of ligand and receptor are each constant (Lauffenburger and
Linderman, 1993). Steric effects arise when bound receptors hinder
access to ligand sites, as illustrated in Fig. 1. In the absence
of steric effects, the model reduces to the well-studied equivalent
site model for multivalent ligand-receptor binding (Perelson, 1984;
Macken and Perelson, 1985; Lauffenburger and Linderman, 1993).
Valence
We define the following ligand valences (Fig.
2): 
, f, n, and 
(i)
for i = 1, ... , f. The valence is the
number of sites on a ligand in solution at which a receptor can bind,
and the effective valence f is the maximum number
of sites on a ligand that can be bound simultaneously by receptors
(Perelson, 1981, 1984). The exposure valence n is
a new concept: we define it as the number of sites on a ligand that are
exposed to receptors when the ligand is anchored to the cell surface.
As illustrated in Fig. 2, a bound ligand may expose only a fraction of
its sites to receptors. We also define for the first time the valence
of the ith bound state (i) for i = 1, ... , f. Each (i) is the number of
ways in which a ligand bound at i sites can be converted to a ligand bound at i + 1 sites, i.e.,
(i) is the number of sites that are available
for receptor binding on a ligand that is bound at i sites
averaged over all possible microscopic states of the ligand. The
quantity (i) will also be called the number of
available sites. Note that (f) = 0, because no further binding can occur once f sites
are bound, and that (i) 
n 
i. If each
exposed site on a ligand is always available for receptor binding, then f = n and (i) = n i.
However, if ligand sites can be covered or excluded by bound receptors,
as illustrated in Figs. 1 and 2, then f < n and
(i) < n i.
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Reaction scheme
Our model for ligand-receptor binding is based on the reaction scheme shown in Fig. 3. In this scheme, ligand-receptor binding proceeds through a series of reversible reactions. The initial reaction involves the binding of a solution-phase ligand to a receptor, and each subsequent reaction involves the addition of a receptor to a ligand-receptor complex on the cell surface. As indicated in Fig. 3, the model is developed in terms of ligand states. Variables in the model include the concentration of ligand in solution, which is denoted as L0; the surface density of ligand that is bound at i sites, which is denoted as Li; and the surface density of free receptors, which is denoted as R.
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Equilibria
For the initial reaction in Fig. 3,
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(1) |
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(2) |
Kinetics
The kinetics of ligand-receptor binding are difficult to model
when steric effects influence binding, because to determine exact time
courses of binding, we must write a kinetic balance equation for each
microscopic ligand state (Epstein, 1979a,b; Munro et al., 1998).
Nevertheless, exact tractable models can be developed for limiting
cases (Epstein, 1979a,b; Schaaf and Talbot, 1989; Evans, 1993). Below,
we present a model that is exact for the limiting case in which
microscopic equilibrium is established instantaneously and approximate
otherwise. Instantaneous establishment of microscopic equilibrium
(IEME) occurs when ligands bound at i sites effectively
redistribute themselves immediately among all possible microscopic
states whenever a ligand enters or leaves the ith bound
state. Microscopic equilibrium, which is a necessary condition for
binding equilibrium, allows us to characterize the number of available
sites on a ligand in a particular bound state at a particular time by
the expected number of available sites at equilibrium. The IEME
approximation improves as the rate of receptor dissociation increases,
and this approximation is least accurate when receptor binding is
irreversible. The usefulness of the IEME approximation can be
determined by comparing approximate and exact results, which can be
determined with numerical methods (Epstein, 1979b; Reiter and Epstein,
1990; Sild et al., 1996).
Given IEME, a kinetic description of ligand-receptor binding is
provided by
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(3) |
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x is the corresponding reverse rate constant. The constant C is a factor that converts surface densities
to concentrations, e.g., C is the cell concentration if
surface densities are expressed on a per cell basis. Equation 3 is
appropriate for reaction-limited binding. If diffusion of ligand to the
cell surface were limiting, then the rate constants in Eq. 3 would have
to be modified (Berg and Purcell, 1977; DeLisi and Wiegel, 1981; Shoup
and Szabo, 1982; Goldstein, 1989; Goldstein et al., 1989; Lauffenburger
and Linderman, 1993).
Conservation
Conservation of ligand can be expressed as
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(4) |
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(5) |
Steric effects
To account for steric effects, we must determine how the number of
available sites (i) varies with ligand site occupancy i. The fraction of exposed ligand sites that are available
for receptor binding is given by (i)/n, which can be
interpreted as the probability that an exposed site is available for
receptor binding. This probability, which we denote as
Pi(
), is called the insertion
probability, because if we were to attempt to add an additional
receptor to a ligand-receptor complex at a randomly chosen site on the
ligand, Pi() is the probability that this insertion attempt is successful. Several approaches are available for
calculating insertion probabilities. Here we follow an approach of
Andrews (1975, 1976) to develop expressions for
Pi() for ligands with sites distributed
randomly or ordered regularly in one or two dimensions.
Ligands with sites distributed randomly on a two-dimensional surface
Let us focus on the case illustrated in Fig. 1. A ligand is bound at i sites, and the ligand exposes n sites, which are distributed randomly over an area A, to receptors on the cell surface. A bound receptor covers a circular area a on the ligand surface. Later, we will consider cases in which a is noncircular. We assume that a
A, such that edge
effects are negligible.
As illustrated in Fig. 1, an exposed ligand site is in one of four
states: it is bound (
), covered (
), excluded (
), or available
(). The probability that a site is available can be expressed as the
following product:
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(6) |
) is the probability
that a site is not bound;
Pi(
|) is the
conditional probability that a site is not covered, given that it is
not bound; and
Pi(
|)
is the conditional probability that a site is not excluded, given that
it is neither bound nor covered. We can determine
Pi() and
Pi(|) exactly, and
we can determine
Pi(|) approximately.
The probability that a site is not bound,
Pi(), is related to the probability
that a site is bound Pi():
Pi() = 1 Pi(). Because Pi()
is equivalent to the fraction of exposed sites that are bound,
i/n,
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(7) |
|), is
related to the probability that it is covered given that it is not
bound Pi(|):
Pi(|) = 1 Pi(|). Because
Pi(|) is equivalent to the
fraction of the exposed ligand surface that is covered by bound
receptors ia/A,
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(8) |
)Pi(|) = (1 i/n)(1 ia/A) is the probability that a
site is neither bound nor covered.
If a site is neither bound nor covered, then it lies somewhere in an
area of size A ia. Sites in this area are either
excluded or available. A site is excluded if it lies within the
"exclusion area" of a bound receptor. As illustrated in Fig. 1,
the exclusion area of a receptor that covers a circular area
a is an annular region of area 3a that surrounds
a. Thus the probability that a site is excluded by a given
bound receptor is 3a/(A ia), and the probability
that the site is not excluded by this particular receptor is
1 3a/(A ia). Following Andrews (1975, 1976),
we estimate
Pi(|),
the probability that a site is not excluded by any of the i
receptors, as the ith power of the probability that a site
is not excluded by a given receptor. Thus,
|
(9) |
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(10) |
(i),
which can be substituted into the model equations (Eqs. 1-5), because
(i) = nPi(). The general method that we
have used to derive Eq. 10 is applied below to derive expressions for
several other cases.
Equation 10 can be generalized for receptors that cover any convex area
a by using the results of Boublík (1975). These
results indicate that the exclusion area of a receptor that covers an area a is (2
+ 1)a if a is convex
and if receptors bind in random orientations. The shape factor 
1 is defined as s2/(4
a), where s is
the perimeter of a. This factor has a value of ~1 for many
shapes. If a is disk shaped, then = 1, and the exclusion
area is 3a, as expected (Fig. 1). If a is
equilateral triangular, = 3
/ 
1.6. If a
is square, = 4/ 1.3. If a is hexagonal, = 2/ 1.1. In general, if a is a regular
polygon with k sides, = (k/)tan(/k). If a is rectangular with
aspect ratio 
= 2 (the length of the longer side is twice that of
the shorter side), = 9/(2) 1.4. In general, if
a is rectangular with aspect ratio 1, = ( + 1)2/(). These results indicate that insertion
probabilities are somewhat insensitive to the shape of a as
long as a is compact. The generalized form of Eq. 10 is
given in Table 1 (Case 1).
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Ligands with sites ordered regularly on a two-dimensional surface
If ligand sites are ordered regularly on a lattice instead of distributed randomly, then we can adapt the results of Stankowski (1983) to calculate the insertion probability
Pi(). These results were derived originally
for adsorption reactions, also by using the approach of Andrews (1975,
1976). Earlier, we characterized the ligand-receptor interface with an
area a, but here we characterize this interface with the
contact number 
, which represents the number of adjacent sites that
are bound or covered by a bound receptor. The pattern of ligand sites
contacted by a receptor must be symmetrical; a hexagonal contact
pattern is illustrated in Fig. 4. Exact
expressions for Pi() and
Pi(|) and an
approximate expression for
Pi(|)
are given in Table 1 (Case 2). This latter expression involves an
excluded-area parameter 
, which depends on the contact pattern of
the receptor and the lattice of ligand sites. A recipe for calculating
the excluded-area parameter is given by Stankowski (1983). For a ligand
with sites arranged on a square lattice and a square contact pattern,
= 3 (4
1)/, where 
[1, 4, 9, ...]. For a ligand with sites arranged on a hexagonal lattice and a
hexagonal contact pattern (Fig. 4), = 3 
/, where [1, 7, 19, ...].
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Ligands with sites distributed randomly along a one-dimensional array
For a ligand with sites distributed randomly along a one-dimensional array, expressions for Pi(),
Pi(|), and
Pi(|) are given in Table 1 (Cases 3 and 4). The derivation of these expressions is analogous to the derivation of Eqs. 7-9. In the
expressions of Table 1, the parameter L, which is analogous
to A, represents the total length of the array of sites
exposed to receptors, and the parameter l, which is
analogous to a, represents the length of the array that is
covered by a bound receptor. The expressions for
Pi() and
Pi(|) are exact. The
expression for
Pi(|) also is exact when L/l n, but it is approximate
otherwise. The expressions given for Case 3 in Table 1 are for a ligand
with sites distributed along a ring, i.e., a closed one-dimensional array. The expressions given for Case 4 in Table 1 are for a ligand
with sites distributed along a chain, i.e., an open one-dimensional array. The expressions for the two cases are essentially the same, except that
Pi(|)
is multiplied by a factor to account for edge effects when sites are
ordered along a chain instead of a ring. This factor is 1 l/(L il), where l/(L il) is the
probability that a site is excluded because it is close to an edge. We
assume that a site is excluded because of edge effects if a receptor is
unable to bind at that site entirely within the length L, as might be the case if the receptor, in addition to binding the ligand
site, requires nonspecific interactions over a larger contact area. If
receptor binding is possible at such a site, then the expressions for
Case 4 in Table 1 can be used to determine a lower bound on the
insertion probability.
Ligands with sites ordered regularly along a one-dimensional array
For a ligand with sites ordered along a ring and a receptor that contacts sequential sites, we can calculate
Pi(),
Pi(|), and
Pi(|)
exactly. The results are summarized in Table 1 (Case 5). When
n, the expression for
Pi(|)
reduces to (1 i/(n i( 1)))
1, which corresponds to the result of McGhee
and von Hippel (1974) for an infinite lattice. If sites on a ligand are
ordered along a chain, as depicted in Fig. 2, rather than along a ring,
then the expressions for Pi(),
Pi(|), and
Pi(|)
are essentially the same, except that
Pi(|)
is multiplied by a factor to account for edge effects (Table 1, Case
6). When a receptor is unable to bind at "edge" sites (i.e., sites
at which a bound receptor would contact fewer than sites), the
correction factor is 1 ( 1)/(n i( 1)), where ( 1)/(n i( 1)) is the probability that a site is an edge site. As before, the expressions for
Pi(),
Pi(|), and
Pi(|)
are exact. If receptors are able to bind at edge sites, the expressions
for Case 6 in Table 1 can be used to obtain exact results. We simply
replace n with n + 1, i.e., we
introduce 1 virtual edge sites.
The steric hindrance factor
To connect our model equations with the equivalent site model
(Perelson, 1984; Lauffenburger and Linderman, 1993), we introduce the
following formalism. We define H(i) for i = 1, ... , f 1 as (i)/(n i), which is
the fraction of exposed unbound ligand sites that are available for
receptor binding. Thus,
|
(11) |
|
(12) |
1 as a factor that corrects
for steric hindrance. In the absence of steric effects, H(i) = 1, because (i) = n i, as discussed earlier.
When H(i) = 1 for all i, the model equations
(Eqs. 1-5) reduce to the equivalent site model. In the presence of
steric effects, H(i) < 1, because (i) < n i, as discussed earlier. The smaller the value of H(i),
the larger the effect of steric hindrance on ligand-receptor binding.
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METHODS |
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TOP
ABSTRACT
INTRODUCTION
THEORY
METHODS
RESULTS
DISCUSSION
REFERENCES
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Algorithms
The insertion probability Pi() is
equivalent to (i)/n, and the steric hindrance factor
H(i) is equivalent to (i)/(n i). Thus,
Pi() and H(i) can be calculated
from (i). To determine (i) directly, we
specify the geometry of a ligand-receptor interface and the spacing of
exposed ligand sites. Then we count the number of sites that are
available on average for the different microscopic bound states of the
ligand. Here, we present two algorithms for calculating insertion
probabilities that are based on this approach: a combinatoric
algorithm, which is efficient when n is small, and a Monte
Carlo algorithm, which is efficient when n is large. The
Monte Carlo algorithm is similar to that used by Siepmann et al. (1992)
to calculate insertion probabilities for fluids of hard rods and disks.
Combinatoric algorithm
A configuration of n ligand sites is generated. A recursive procedure then is used to generate every possible configuration of i bound receptors (Taylor et al., 1991). If
a configuration of receptors is acceptable, meaning that none of the
i receptors overlap, then the number of available sites is
counted. A site is available if a receptor can be placed at that site
without overlapping other receptors. We compute the number of available sites averaged over all acceptable configurations of receptors. For
ligands with randomly distributed sites, multiple configurations of
sites are generated, and the above process is repeated for each
configuration. The efficiency of the algorithm varies inversely with
n!/[(n i)!i!], which is the number of ways that
i receptors can be distributed among n ligand
sites. A similar but more efficient algorithm has been described
(Badcoe, 1992) that can be used with one-dimensional ligands that have
regularly ordered sites.
Monte Carlo algorithm
Initialization. A configuration of n ligand sites is generated. We then attempt to distribute i receptors among these sites such that none of the receptors overlap. The receptors are placed at i randomly chosen but distinct sites. Each receptor, except the first, is then checked for overlap in sequential order. If a receptor overlaps another receptor, we attempt to move it to an unbound site. If all attempts to move a receptor to an unbound site result in overlap, we randomly redistribute the i receptors among the n sites and attempt to eliminate overlap as before. If an acceptable distribution of receptors cannot be obtained after a fixed number of attempts, we abandon the configuration of sites, i.e., we assume that this configuration of n ligand sites does not permit the binding of i receptors. Because this procedure may abandon configurations of sites that do indeed permit the binding of i receptors, extreme caution must be exercised when calculating small insertion probabilities, for which the algorithm is inefficient in any case. Execution of a Monte Carlo cycle. After an initial configuration of i nonoverlapping receptors is generated, we then generate a new configuration of receptors by executing a Monte Carlo cycle. In a Monte Carlo cycle, we sequentially attempt to move each of the i receptors once from its present ligand site sj to a neighboring ligand site sk. A move is rejected if it results in an overlap of receptors. If the move results in no overlap, it is accepted with probability min(Nj/Nk, 1), where Nj is the number of sites that neighbor site sj and Nk is the number of sites that neighbor site sk. This acceptance criterion is necessary to ensure that a move from site sj to sk is as likely as a move from site sk to sj. The neighborhood of a site is defined as the collection of sites within a fixed distance of the site. This distance is chosen so that a site's expected number of neighbors is well above 1. After each Monte Carlo cycle, the number of available sites on the ligand is determined. We perform a fixed number of Monte Carlo cycles and compute the average number of available sites. For ligands with randomly distributed sites, multiple configurations of ligand sites are generated, and the above process, including the initialization procedure, is repeated for each configuration.Calculating insertion probabilities
For ligands with n 20, we use the combinatoric
algorithm, whereas for ligands with n > 20, we use the
Monte Carlo algorithm. We use periodic boundary conditions in all
calculations. For ligands with randomly distributed sites, each
reported insertion probability is the mean of 100 computational runs.
To obtain reproducible results, we adjust algorithmic parameters so
that the standard deviation divided by the mean is less than 0.1.
Calculating equilibrium states
Calculation of equilibrium states is aided by combining Eqs. 1, 2,
4, and 5, which yields
|
(13) |
|
(14) |
1) = 1 for j = 1, ... , i,
then the product in Eq. 14 reduces to the statistical factor
n!/[(n i)!i!].
When values for the parameters (, n, C, LT,
RT, K, and Kx)
are specified and a value for the steric hindrance factor
H(i) is specified for i = 1, ... , f 1, Eq. 13 is a nonlinear equation involving a single unknown: the
fraction of free receptors R/RT. To determine
the fraction of free receptors at equilibrium, we solve this equation
by using the method of bisection (Press et al., 1992). Once
R/RT is known, other states at equilibrium can be determined by using the relations
L0/LT = n/[n + KCRT 
i=1f
(i)] and
Li/RT = (L0/LT)(KLT/n)(i),
which are derived from Eqs. 1, 2, and 4.
Calculating time courses
To calculate time courses of ligand-receptor binding, we solve an
initial value problem that involves f differential equations and two auxiliary algebraic equations. These equations are derived from
Eqs. 3-5 by using K = kf/kr and
Kx = kx/kx and by
introducing dimensionless variables: 
= kxt, r = R/RT,
l = L0/LT, and
xi = Li/RT for i = 1, ... , f. From Eq. 3, we obtain
|
(15) |
x)[(KLT)rl x1], i)H(i)(KxRT) (i + 1)xi+1
for i = 1, ... , f 1, and
uf = 0. From Eqs. 4 and 5, we
obtain
|
(16) |
|
(17) |
= 0. These initial conditions and Eqs. 15-17 define an initial value problem, which we solve numerically by using the FORTRAN
subroutine LSODE (http://www.netlib.org/odepack; Hindmarsh, 1983).
Quantifying receptor aggregation
A ligand bound at i sites is bound to i
receptors. Thus the fraction of receptors in ligand-induced aggregates
of i or more receptors is given by
|
(18) |
; Perelson, 1981), is given by (2). Receptor aggregates of size 10 or more, termed immunons, have been
suggested to be the minimum signaling unit for B cells (Dintzis et al., 1976, 1983). The fraction of receptors in immunons is given by (10).
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RESULTS |
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TOP
ABSTRACT
INTRODUCTION
THEORY
METHODS
RESULTS
DISCUSSION
REFERENCES
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We have developed a model for ligand-receptor binding (Eqs. 1-5)
in which (i) represents the expected number of ligand
sites that are available for receptor binding when a ligand is bound at
i sites. We have related (i), which is
sensitive to steric effects (Figs. 1 and 2), to a steric hindrance
factor H(i) 1 (Eq. 11). When H(i) = 1 for all
i, Eqs. 1-5 reduce to an equivalent site model (Perelson,
1984; Lauffenburger and Linderman, 1993). The steric hindrance factor
H(i) is related to the insertion probability Pi() (Eq. 12), the probability that a site on
a ligand is available for receptor binding when the ligand is bound at
i sites. By following the approach of Andrews (1975, 1976),
we have derived exact or approximate expressions for
Pi() for different types of ligands and
receptors (Table 1). Below, we examine the accuracy of these expressions. We also examine steric effects on ligand-receptor binding
at equilibrium and steric effects on time courses of ligand-receptor binding.
Accuracy of theoretical expressions
Theoretical expressions for the insertion probability
Pi() and the steric hindrance factor
H(i) are given for six cases in Table 1. Expressions for the
first four cases are approximate, whereas expressions for the last two
cases are exact. The accuracy of these expressions is illustrated in
Figs. 5 and
6, in which insertion probabilities
calculated using the expressions in Table 1 are compared with those
calculated using the Monte Carlo algorithm.
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In Fig. 5, results are shown for ligands with regularly ordered sites.
In each panel, we consider three ligands, which interact with the same
receptor. The ligands have different numbers of sites but are otherwise
identical. The solid and broken lines in Fig. 5 A are based
on the expression for Pi() for Case 5 in Table 1, which is exact. Thus comparison of these results with the
corresponding numerical results, which are represented by points,
provides a test of our Monte Carlo algorithm for calculating insertion
probabilities. As expected, the theoretical and numerical results are
indistinguishable. The solid and broken lines in Fig. 5 B
are based on the expression for Pi() for Case
2 in Table 1, which is approximate. Despite the approximate nature of
these results, they agree closely with the corresponding numerical
results. The potential for this level of accuracy is consistent with
earlier observations (Stankowski, 1984).
In Fig. 6, results are shown for ligands with randomly distributed sites. In each panel, we consider four ligands, three with different numbers of sites and one with ordered sites, all of which can be bound simultaneously by receptors. The theoretical results, which are based on expressions in Table 1, can be compared with the corresponding numerical results. As can be seen, the theoretical expressions are capable of predicting how insertion probabilities, and therefore steric effects, vary with the number of binding sites on a ligand.
We have examined the accuracy of Eq. 10 in more detail (unpublished results). We find that accuracy decreases as either the fraction of sites bound i/n or the fraction of ligand surface covered by receptors ia/A increases. In other words, Eq. 10 is less accurate when the surface of the ligand is tightly packed with receptors, as can be expected. Thus, under conditions that favor close packing of receptors on the ligand surface, such as a receptor concentration in excess of ligand concentration or a large cross-linking constant, the usefulness of Eq. 10 should be checked. We expect that the results of this analysis are typical for the approximate expressions in Table 1, because all of these expressions were derived by the same method.
Steric effects on ligand-receptor equilibria
Equilibrium cross-linking curves are shown in Fig.
7 for cases where steric effects do and
do not influence binding. Cross-linking, as measured by (2) or
(10) (Eq. 18), is plotted as a function of ligand concentration for
two ligands. One ligand has ordered sites, which all can be bound
simultaneously, and the other ligand has randomly distributed sites,
not all of which can be bound simultaneously, because of potential for
steric exclusion of ligand sites by bound receptors. To ensure a
controlled comparison, the two ligands are otherwise identical.
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In the comparison of Fig. 7 A, we see that steric effects on
equilibrium cross-linking, as measured by (2), are minor.
Essentially the same fraction of receptors are aggregated in the
presence or absence of steric effects. This result is typical for other cases that we have examined. However, as illustrated in Fig. 7 B, steric effects can significantly influence the
distribution of receptor aggregates. Steric effects inhibit the
formation of higher-order complexes, such as immunons. As can be seen,
the peak fraction of receptors in immunons, which is given by (10), is reduced by approximately twofold because of steric effects. This
result suggests that steric effects on ligand-receptor binding can have
different consequences for cellular responses, depending on how the
cell senses receptor aggregation. One can expect signals that are
triggered by dimeric and larger aggregates to be less sensitive to
steric effects than signals that are triggered only by oligomeric aggregates.
By comparing the broken and solid lines in Fig. 7, we can see that Eq. 10, an approximate expression, is capable of accurately modeling steric effects on equilibrium cross-linking.
Steric effects on ligand-receptor kinetics
Time courses of ligand-receptor binding are shown in Fig. 8 for cases where steric effects do and do not influence binding. As in Fig. 7, we consider a ligand with ordered sites, which all can be bound simultaneously, and a ligand with randomly distributed sites, only a fraction of which can be bound simultaneously because of the potential for steric exclusion of ligand sites by bound receptors.
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Bound receptor. The first step involves the attachment of
solution-phase ligand to a cell-surface receptor. Each subsequent step
involves the addition of a receptor to a ligand-receptor complex on the
cell surface. R indicates a receptor,
L0 indicates a ligand in solution, and
L1 and L2 indicate
ligands that are bound to one and two receptors, respectively.
