Section on Physical Biochemistry, Laboratory of Biochemistry and
Genetics, National Institute of Diabetes and Digestive and Kidney
Diseases, National Institutes of Health, Bethesda, Maryland 20892-0830 USA
Equilibrium and kinetic models for nonspecific adsorption
of proteins to planar surfaces are presented. These models allow for
the possibility of multiple interconvertible surface conformations of
adsorbed protein. Steric repulsion resulting in area exclusion by
adsorbed molecules is taken into account by treating the adsorbate as a
thermodynamically nonideal two-dimensional fluid. In the equilibrium
model, the possibility of attractive interactions between adsorbed
molecules is taken into account in a limited fashion by permitting one
of the adsorbed species to self-associate. Calculated equilibrium
adsorption isotherms exhibit apparent high-affinity and low-affinity
binding regions, corresponding respectively to adsorption of ligand at
low fractional area occupancy in an energetically favorable side-on
conformation and conversion at higher fractional area occupancy of the
side-on conformation to an entropically favored end-on conformation.
Adsorbate self-association may lead to considerable steepening of the
adsorption isotherm, compensating to a variable extent for the
broadening effect of steric repulsion. Kinetic calculations suggest
that in the absence of attractive interactions between adsorbate
molecules, the process of adsorption may be highly "stretched"
along the time axis, rendering the attainment of adsorption equilibrium
in the context of conventional experiments problematic.
 |
INTRODUCTION |
When the potential of interaction between a
soluble macromolecule ("ligand") and a surface depends strongly
upon position within the plane of the surface, it is conventional to
analyze interactions between ligand and the surface in the context of models for binding of ligand to discrete sites (see for example, Boeynaems and Dumont, 1980
). In the present work, as in the preceding work in this series (Chatelier and Minton, 1996), referred to herein as
CM, we shall be concerned with interactions between surface and ligand
that are to a good approximation independent of position in the plane
of the surface, i.e., depend primarily upon the distance between the
surface and the ligand molecule, and, in the present work, on the
orientation of the ligand molecule relative to the surface (Roth and
Lenhoff, 1993; Roush et al., 1994). Interactions of this type are
commonly assumed to underlie phenomena such as the nonspecific
adsorption of macromolecules to synthetic or naturally occurring
phospholipid membranes (Heimburg and Marsh, 1995), the surfaces of
particles of synthetic polymers (Al-Malah et al., 1995), or a
"molecularly flat" (see Note 1) surface of any large array of
molecular species whose cross-sectional area in the plane of the
surface is significantly smaller than that of an adsorbed ligand
molecule. Prior theoretical analysis and modeling of such systems have
indicated that steric repulsions between adsorbed ligands lead to
marked broadening of the equilibrium adsorption isotherm relative to
that characteristic of homogeneous independent site binding (Heimburg
and Marsh, 1995; Chatelier and Minton, 1996), as well as greatly
increasing the length of time required to attain steady state at high
fractional surface coverage (Jin et al., 1994; Kurrat et al., 1997).
Experimental studies of protein adsorption have suggested the
possibility that proteins may adsorb in more than a single
conformation, and that the probability of adsorbing in a given
conformation may vary with the surface density of adsorbed protein
(Brynda et al., 1986; Wahlgren et al., 1995). Talbot has recently
presented a hard particle equilibrium model for such a phenomenon
(Talbot, 1997). In the present paper we present a model that is similar in concept to that of Talbot, but which has been extended to treat the
kinetic as well as equilibrium aspects of multiple adsorbate conformations, and also to explore the consequences of self-association of one of the adsorbate conformations.
| |
THEORY AND RESULTS: EQUILIBRIUM |
The chemical potential of a single ligand species behaving ideally
in solution is given by
|
(1)
|
where µsoln,o denotes the standard state chemical
potential of ligand, c its concentration in solution,
R the molar gas constant, and T the absolute
temperature. Let us postulate that adsorbed ligand ("adsorbate")
may be present in any of i possible conformations. The
chemical potential of the ith adsorbate conformation is
given by
|
(2)
|
where µisurf,o* denotes the standard state of
the ith conformation of adsorbate, 
i the
fraction of surface area occupied by adsorbate in conformation
i, and 
i the activity coefficient of the
ith conformation, here indicated as a function of the
fractional area occupancies of all adsorbate conformations. We shall
refer to the cross section of the ith adsorbate conformation
in the plane of the surface as its "footprint," and denote the area
and circumference of the footprint by ai and
si, respectively. Then the number density of
species i will be given by
|
(3a)
|
and Eq. 2 may be rewritten as
|
(3b)
|
where µisurf,o 
µisurf,o* + RT ln ai. The condition of adsorption
equilibrium is given by
|
(4)
|
for all i. By combining Eqs. 1, 3, and 4 we obtain a
set of equations
|
(5)
|
where
|
(6)
|
is an "intrinsic" coefficient for the partitioning of ligand
between solution and surface species i in the limit of low
surface density of adsorbed ligand (i = 1 for all
i).
It has been demonstrated that under experimental conditions such that
intermolecular interactions other than steric exclusion are damped out
(moderate ionic strength, pH ~ pI), the concentration dependence
of colligative properties of protein solutions may be well accounted
for by simple models in which protein molecules are represented by
convex hard particles (Zimmerman and Minton, 1993). Hard particles have
also been used to model the adsorption of macromolecules (Jin et al.,
1994; Heimburg and Marsh, 1995; Chatelier and Minton, 1996; Sild et
al., 1996). In the present work we formulate a simple hard particle
model for multiple adsorbate conformations, illustrated schematically
in Fig. 1, in which the adsorbing
macromolecule is represented as a hard rectangular parallelopiped (HRP)
of dimensions r × r × Lr (axial ratio
L). This particle may adsorb in the "side-on"
conformation (species 1), with a rectangular footprint of dimensions
r × Lr, or in the "end-on" conformation (species
2), with a square footprint of dimensions r × r. It
may be readily shown that all dimensions may be scaled to r,
with no change to the above equations other than in the numerical value of µisurf,o (and hence Ki).
For convenience in calculation we shall henceforth set r = 1 with no loss in generality. Thus a1 = L,
s1 = 2(1 + L), a2 = 1, and
s2 = 4.
View larger version (7K):
[in this window]
[in a new window]
|
FIGURE 1
Model for adsorption in multiple surface conformations.
Ligand is depicted as a hard rectangular parallelopiped of dimensions
1 × 1 × L that may adsorb onto a planar surface
in a side-on (1) or end-on (2) conformation.
|
|
In this simple model, the "intrinsic" free energy of adsorption of
each conformation is assumed to be proportional to the footprint area
and to vary with the degeneracy d of the conformation (see
Note 2).
|
(7)
|
where J is the adsorption potential, in units of RT per
unit footprint area, d1 = 4, and
d2 = 2. It follows from Eqs. 4 or 5 and 7 that
one may define a constant of equilibrium between adsorbate
conformations 1 and 2:
|
(8)
|
The activity coefficient of a particular adsorbate conformation is
calculated using a relation due to Talbot (Talbot et al., 1994) derived
from scaled particle theory for a mixture of hard convex particles in
two dimensions:
|
(9)
|
where 
j, a jaj, and s
jsj.
This equilibrium model may be extended to allow for the
self-association of end-on adsorbate molecules, illustrated
schematically in Fig. 2. Let adsorbate
species 3 be the n-mer of the end-on species 2, with a
square footprint (see Note 3). According to this model,
a3 = n, s3 = 4
, and
d3 = 2. We define the equilibrium constant for
the formation of adsorbate species 3:
|
(10)
|
Given values of J, L, K23, and the
ai, si, and
di, one may calculate the equilibrium values of
i and i (and hence
tot = i and tot = i) as functions of K2c via
iterative numerical solution of Eqs. 1, 4, 5, and 8-10 (see Note 4).
View larger version (10K):
[in this window]
[in a new window]
|
FIGURE 2
Model for an adsorption in multiple surface
conformations, allowing for the possibility of self-association of
adsorbed ligand. Adsorbate species (1) and (2) are as defined in the
caption of Fig. 1. Adsorbate species (3) is an n-mer of
species (2) that is assumed to have a square footprint with an area
corresponding to n times that of the monomer.
|
|
Some effects of multiple adsorption conformations on the adsorption
equilibrium of a single solution species are depicted in Figs.
3-9.
It is instructive to plot both the relative amount and the fractional
surface coverage of each adsorbed species as functions of the
normalized concentration of free ligand. For reference, in Fig. 3 we
plot the adsorption isotherms of HRPs that are allowed to bind only in
a single conformation, either side-on (1) or end-on (2); no
equilibration between conformations is permitted. These isotherms are
identical, in the case of conformation 2, or similar, in the case of
conformation 1, to isotherms presented in Fig. 1 of CM. The side-on
conformation has an intrinsically higher affinity because it has both a
larger contact area (assumed proportional to binding potential) as well
as a greater degeneracy than the end-on conformation.
View larger version (14K):
[in this window]
[in a new window]
|
FIGURE 3
Equilibrium adsorption isotherms for ligand that can
adsorb in either side-on or end-on conformation exclusively
(solid and dashed curves, respectively),
calculated as described in the text with J = 2, L = 3, and K23 = 0. Isotherms in this figure
differ from those plotted in subsequent figures in that adsorbate is
present in only a single conformation that is not allowed to
equilibrate with any other conformation. Results are presented as
relative amount adsorbed (top panel) and fraction of surface
area occupied (bottom panel). Normalized free ligand
concentrations Kc in this figure as well as Figs. 4-9 are
set equal to K2c.
|
|
View larger version (14K):
[in this window]
[in a new window]
|
FIGURE 4
Equilibrium adsorption isotherm of a ligand adsorbing
in two interconverting conformations. Model parameters as in Fig. 3.
Plotted curves indicate the relative amount of adsorbate (top
panel) and fraction of surface area occupied (bottom
panel) for total adsorbate (solid curve) and individual
adsorbate species (dashed curves).
|
|
View larger version (19K):
[in this window]
[in a new window]
|
FIGURE 5
Effect of variation in intrinsic binding potential per
unit footprint area on total adsorption. Isotherms are calculated for
L = 3, K23 = 0, and J = 0, 1, 2, 3, 4, and 5 (rightmost to leftmost
curves). Panels as in preceding figures.
|
|
View larger version (17K):
[in this window]
[in a new window]
|
FIGURE 6
Effect of variation in shape of ligand shape on total
adsorption. Isotherms are calculated for J = 2, K23 = 0, and L = 1, 2, 3, 4, and 5 (solid to dotted curves, respectively).
|
|
View larger version (15K):
[in this window]
[in a new window]
|
FIGURE 7
Equilibrium adsorption of a ligand with three
interconverting adsorbate conformations, calculated for J = 2, L = 3, K23 = 1, and n = 4.
Plotted curves indicate the relative amount of adsorbate (top
panel) and fraction of surface area occupied (bottom
panel) for total adsorbate (solid curve) and individual
adsorbate species (dashed curves).
|
|
View larger version (17K):
[in this window]
[in a new window]
|
FIGURE 8
Effect of variation in adsorbate self-association
equilibrium constant upon total equilibrium adsorption, calculated for
J = 2, L = 3, n = 4, and
K23 = 10 16, 104, 1, 104, 108 and 1012 (curves
shift progressively leftward with increasing
K23).
|
|
View larger version (16K):
[in this window]
[in a new window]
|
FIGURE 9
Effect of variation in degree of self-association upon
total equilibrium adsorption, calculated for J = 2, L = 3, K23 = 1, and n = 1, 2, 4, 8, 16,
and 32 (curves shift progressively leftward and become steeper with
increasing n).
|
|
In Fig. 4 the side-on and end-on conformations are allowed to
equilibrate, and the activity coefficients of each conformation depends
upon the number densities of both conformations, as specified by Eq. 9.
The adsorption isotherm exhibits apparent high-affinity and
low-affinity adsorption regions. The high-affinity region corresponds
to the adsorption of ligand in the side-on conformation that is
energetically favored at low fractional surface occupancy. As
fractional surface coverage increases with increasing ligand concentration, the activity coefficient of the side-on conformation (larger footprint) increases so much more rapidly than that of the
end-on conformation (smaller footprint) that the equilibrium between
side-on and end-on (Eq. 8) shifts in favor of the now entropically
favored end-on mode. The conversion of side-on to end-on and subsequent
addition of ligand in the end-on conformation is manifested as a
lower-affinity adsorption process. In Figs. 5 and 6 the effects of
increasing the adsorption potential and increasing the axial ratio of
ligand are plotted. These two variations have qualitatively similar
effects, because increasing either parameter results in an enhancement
of the intrinsic affinity of the side-on conformation relative to that
of the end-on conformation. We note that when the energetic difference
between side-on and end-on configurations becomes sufficiently great, a
region appears in the adsorption isotherm where the fractional surface
coverage is predicted to decrease, rather than increase, with
increasing free ligand concentration. This counterintuitive result,
previously noted by Talbot (1997), occurs whenever an increase in free
ligand concentration results in more conversion of larger-footprint
conformation 1 to smaller-footprint conformation 2 than adsorption of
additional ligand from solution.
In Fig. 7 the effect of self-association of end-on conformation 2 on
the adsorption isotherm is illustrated. Since the n-meric conformation 3 excludes even less area per molecule of adsorbate than
monomeric conformation 2, area exclusion tends to promote the formation
of 3, and conformation 2 does not accumulate to a major extent at
equilibrium. In Fig. 8 the effect of altering K23 for a constant degree of oligomerization
(n = 4) is illustrated, and in Fig. 9 the effect of
altering the degree of oligomerization for a constant value of
K23 is illustrated. As the value of n increases, the maximal equilibrium abundance of end-on monomer becomes
vanishingly small, and the adsorption isotherm approaches the behavior
characteristic of a first-order phase transition; all ligand adsorbed
in excess of the "solubility limit" (in this example,
sol ~ 0.2) is incorporated into a two-dimensional
quasi-crystalline array (i.e., conformation 3 in the limit of large
n). Similar behavior was predicted earlier by CM using a
model in which hard circular adsorbate particles could equilibrate with
an oligomeric adsorbate modeled by a larger hard circle.
| |
THEORY AND RESULTS: KINETICS |
The following is a simple model for the time dependence of
adsorption in the absence of adsorbate oligomerization (see Note 5). We
individually consider the following elementary processes.
1. Adsorption of conformational species i from the
supernatant solution. The overall rate is the product of the
intrinsic encounter rate, kaoc,
the degeneracy of species i, di, and the
probability that a randomly selected element of surface area with the
dimensions of the footprint of i is vacant,
Pi. For a hard particle model, Pi = 1/i({}) (Lebowitz et
al., 1965) (see Note 6). Hence
|
(11)
|
2. Desorption of conformational species i. The
overall rate is the product of an intrinsic desorption rate constant
times a Boltzmann factor, Bi = exp(
Jai), reflecting the additional energy
required for conformation i to escape the adsorption
potential well.
|
(12)
|
3. Adsorbate "flipping," i.e., conversion of
conformation i to conformation j (1 
2 or 2 1) without total desorption. We estimate this rate using simple
transition state rate theory (Hill, 1960), according to which
|
(13)
|
where T is the steady-state concentration of
transition state T, kdeco is the intrinsic
decay rate of T, and fj is the fraction of T that decays to adsorbate conformation j rather than back to
i. Let the adsorption energy of the transition state T,
which is smaller in magnitude than that of either conformation 1 or 2, be denoted by x. Then T = iexp( (Jai x)) =
iexp(Jai)exp(x). fj is the product of two independent probabilities,
PT
j(1) = dj/(di + dj), representing the probability of T decaying
to j in the absence of surface area exclusion (low occupancy
limit), and PTj(2) 
Pj/(Pi + Pj), representing the effect of excluded area on
the relative likelihood of successful surface placement of an
additional molecule in each of the conformational states. Substitution
of these terms into Eq. 13 yields the approximate
expression
|
(14)
|
where k*flip kdeco exp(x).
The overall time dependence of binding is given by
|
(15)
|
where
|
(16)
|
and
|
(17)
|
To simplify calculation, let us define the scaled time t*
kdot. Then it follows from Eqs. 11,
12, 14, and 17 that
|
(18)
|
where K = kao/kdo and
kflip k*flip/kdo
(see Note 7). 1(t*), 2(t*),
and tot(t*) are obtained as functions of
J, L, Kc, and kflip by numerical
solution of Eqs. 15, 16, and 18 using the commercially available
modeling program MLAB (Civilized Software, Bethesda, MD) (see Note 8).
The time course of 1, 2, and
tot calculated as described above will be compared with
the conventional description of reaction kinetics in terms of
sums of decaying exponentials:
![&rgr;<SUB><UP>tot</UP></SUB>(t*)=<LIM><OP>∑</OP><LL><UP>j</UP></LL></LIM> &agr;<SUB><UP>j</UP></SUB>[1−<UP>exp</UP>(<UP>−</UP>t*/t<SUB><UP>j</UP></SUB>)]](/content/vol76/issue1/fulltext/176/img024.gif)
|
(19)
|
where 
j and tj denote,
respectively, the amplitude and characteristic decay time of the
contribution of the jth exponential term to total adsorption
(see Note 9).
We first consider the special reference case in which ligand is
permitted to adsorb in only one of the two conformations permitted by
the general model, and all adsorbate remains in the single selected
conformation (cf. Fig. 3). For this case, Eq. 15 reduces to
|
(20)
|
where i is either 1 or 2, j(t) = 0 for j 
i, and Eq. 18 reduces to
|
(21)
|
In the top panel of Fig. 10,
adsorption progress curves are plotted for ligand adsorbing in the
end-on conformation exclusively (top panel) and in the
side-on conformation exclusively (bottom panel) over a wide
range of free ligand concentration. In both cases, the adsorption
progress curve may be described by Eq. 19 with a single term only in
the limit of low free ligand concentration, i.e., low fractional
surface area coverage at equilibrium. With increasing ligand
concentration the progress curves become progressively broader, and at
high ligand concentration the total adsorption isotherm may not be
described even approximately by one or two exponential processes. We
refer to adsorption progress curves such as these, in which the
duration of the approach to equilibrium is substantially prolonged
relative to that characteristic of an exponential process, as
"stretched" progress curves. Ramifications of stretched adsorption
kinetics will be discussed below.
View larger version (22K):
[in this window]
[in a new window]
|
FIGURE 10
Kinetic progress curves for ligand adsorbing
exclusively in the side-on conformation (open symbols) or in
the end-on conformation (filled symbols), calculated for
J = 1, L = 3, and K2c = 0.1, 1, 10, 103, and 105 (right
to left). The progress curves for K2c = 0.1 are plotted together with the best fit of a single-term
exponential expression of the form of Eq. 19. The progress curves for
K2c = 105 are plotted together
with the best fit of a four-term exponential expression (dotted
curve) and a five-term exponential expression (solid
curve). Also plotted for comparison (dot-dashed curve)
is a single exponential having the total amplitude of the progress
curve for K2c = 105 and a decay
time equal to that of the initial exponential term in the
multiexponential best fit.
|
|
The most important qualitative effect of area exclusion on the kinetics
of multiple-mode adsorption is illustrated in Fig. 11, where the kinetics of adsorption at
low fractional saturation and at high fractional saturation are
contrasted. At low ligand concentration (top panel), the
reaction starts slowly, and side-on conformation 1, with a higher
intrinsic affinity than end-on conformation 2, accounts for the bulk of
adsorbed ligand throughout. The total amount of adsorbed ligand may be
well-described as a function of time by a single decaying exponential
function, that is, Eq. 19 with a single term. At high ligand
concentration, and in the absence of adsorbate flipping (bottom
panel), the reaction starts rapidly, and side-on conformation 1 binds preferentially, as would be expected. However, as the fractional
surface coverage increases, ratesoln1 begins to decrease
relative to ratesoln2, due to the enhanced increase of
1 relative to 2 with increasing surface
coverage noted above, and becomes infinitesimal after about a
hundredfold increase in elapsed time. Although at this point 1 is
unstable thermodynamically relative to 2, it can have a very long
lifetime if rate1soln and rate12flip
are small. As long as a significant fraction of the surface is occupied by 1, ratesoln2, and hence the total rate
of adsorption, will be greatly reduced relative to the rate expected in
the absence of area exclusion. Under such conditions, even though
conformation 1 represents a negligible fraction of total adsorbate at
equilibrium, it can constitute a substantial kinetic barrier to the
attainment of equilibrium. If the lifetime of 1 is sufficiently long,
the adsorption progress curve may exhibit a pre-equilibrium plateau (see below).
View larger version (17K):
[in this window]
[in a new window]
|
FIGURE 11
Kinetic progress curves for ligand adsorbing in
side-on and end-on conformations from solution at low concentration
(K2c = 0.001, top panel) and
moderately high concentration (K2c = 10, bottom panel), calculated for J = 1, L = 3, and kflip = 0. The relative amounts of
each adsorbate conformation are plotted (dashed lines)
together with total adsorbate (solid lines).
|
|
The effect of variation in flip rate on the kinetics of adsorption is
illustrated in Fig. 12. As the flip
rate increases, the lifetime of the transient side-on conformation is
shortened and equilibrium is attained more rapidly. In the example
shown, the bottom panel represents an asymptotic limit in which
adsorbate conformations 1 and 2 equilibrate essentially instantaneously throughout the process of adsorption. Although adsorption under these
conditions might be phenomenologically described as a combination of
"fast" and "slow" phases, the two apparently kinetically
distinguishable phases are in fact inextricably linked, and, as shown
in the figure, neither can be well-modeled individually as an
exponential function.
View larger version (16K):
[in this window]
[in a new window]
|
FIGURE 12
Effect of intrinsic conformational flipping rate on
kinetic progress curves, calculated for J = 1, L = 3, K2c = 30, and Kflip = 0 (top panel), 102 (middle panel), and
104 (bottom panel). The bottom panel corresponds
to the asymptotic limit in which side-on and end-on adsorbates are in
rapid equilibrium with each other throughout the adsorption process.
Also plotted in this panel (circular symbols) is the
calculated best-fit to the total progress curve of a two-term
exponential expression of the form of Eq. 19.
|
|
The effect of variation in free ligand concentration is illustrated in
Fig. 13 in the limits of no flipping
(top panel) and rapid flipping (bottom panel). In
both limits, an increase in free ligand concentration enhances the
separation of the "fast" and "slow" phases on the time scale,
but in the fast flipping limit, the slower of the two phases is less
retarded relative to the faster phase, and its apparent amplitude
increased.
View larger version (26K):
[in this window]
[in a new window]
|
FIGURE 13
Effect of solution ligand concentration on kinetic
progress curves, calculated for J = 1, L = 3, and
the indicated values of K2c in the limits of no
conformational flipping (top panel) and fast conformational
flipping (bottom panel). Also plotted in the upper panel are
the amounts bound at equilibrium (filled circles), obtained
from equilibrium results plotted in Fig. 5, a single exponential
progress curve calculated according to the (ideal) Langmuir model for
K2c = 1, a single exponential best fit to
the progress curve at K2c = 103, and a four-term exponential best fit to the
progress curve at K2c = 102.
|
|
The effect of variation in J for constant L and
Kc is illustrated in Fig.
14. As J increases, the
"slow" phase is progressively retarded relative to the "fast"
phase and the overall approach to equilibrium becomes increasingly
stretched. The extent of stretching is enhanced in the absence of
adsorbate flipping because the lifetime of conformation 1 is increased
in accordance with Eq. 12. At sufficiently large values of J
a nonequilibrium plateau of long duration appears in the adsorption
progress curve. In the absence of kinetic data covering many orders of
magnitude in time, such a plateau could be reasonably (but mistakenly)
taken to represent an equilibrium end-point. In the fast flipping
limit, when the two adsorbate conformations are in continuous
equilibrium, enhanced stretching is the result of an increase in the
equilibrium constant K21 (Eq. 8) and a
consequent shift to higher values of the fractional area occupancy at
which conformation 2 becomes stable relative to conformation 1. For
similar reasons, an increase in L at constant J
is also expected to enhance stretching of the adsorption progress
curve.
View larger version (16K):
[in this window]
[in a new window]
|
FIGURE 14
Effect of variation in intrinsic adsorption potential
per unit surface area upon the kinetic progress curve in the limits of
no conformational flipping (top panel) and rapid
conformational flipping (bottom panel), calculated for
L = 3, K2c = 30, and J = 0, 1, 2, and 3 ("slow" phase of calculated curve shifts to right
with increasing J).
|
|
| |
DISCUSSION |
Although it is recognized that both repulsive and attractive
interactions between adsorbed macromolecules play major roles in
determining the shapes of equilibrium adsorption isotherms and kinetic
progress curves (Nygren, 1993; Heimburg and Marsh, 1995; Kurrat et
al., 1997; Wahlgren and Elofsson, 1997), relatively few attempts have
been made to model such effects quantitatively. These attempts have
fallen into two general classes, namely lattice simulations (Stenberg
and Nygren, 1991; Jin, et al., 1994, Sild, et al., 1996) and continuum
models based upon theories of two-dimensional fluids (Talbot, et al.,
1994; Heimburg and Marsh, 1995; Chatelier and Minton, 1996; Talbot,
1997). Lattice model simulations provide straightforward means for
calculating the probability of a particular configuration of the system
(and hence estimating system entropy) as well as for the introduction
of near-neighbor interactions between adsorbed particles (Stenberg and
Nygren, 1991). The major disadvantage of such simulations is that a
change in any parameter of the system requires a repetition of the
entire simulation, which may be computationally costly and
time-consuming. Moreover, lattice simulations may be subject to
artifacts arising from the quantized representation of the system being
modeled (see below). While continuum fluid models rely upon approximate
theories of the fluid state, the numerical calculations are
comparatively simple and quite rapid, and for the case of purely
repulsive interactions between adsorbate molecules, the results of
continuum model calculations of adsorption kinetics have been shown to
agree quite well with the results of lattice simulations carried out
using identical parameter values (Jin et al., 1994; Talbot et al.,
1994).
Jin et al. (1994) have calculated (t) for uniformly sized
hard disks adsorbing to a surface by using a semiempirical expression for P() proposed earlier by Schaaf and Talbot (1989).
Only very limited results are presented by these authors, and they are
plotted as fractional surface area versus time rather than log
t, hindering comparison over a broad time scale with the
results presented here. However, we observe that in both studies the
rate of slowing of the adsorption reaction with increasing surface
occupancy is gradual rather than stepwise, i.e., does not exhibit
multiphasic behavior that might be characteristic of separable
exponential processes. Thus both the present results and those of Jin
et al. (1994) stand in contrast to results presented by Sild et al.
(1996), obtained via simulation on a lattice. These authors reported
that the time-dependent adsorption of square ligands on a square
lattice could be described a sum of two exponentials representing a
relatively rapid process and a very slow process, the second of which
was suggested to correspond to a "shuffling" of adsorbed ligands on the surface to make room for additional ligand. An exponential description of our own results for high ligand concentrations according
to Eq. 19 requires four or five exponential terms rather than two (Fig.
10), and we can assign no physical significance to any individual term.
We suggest that the "fast" exponential process reported by Sild et
al. is, at least in part, an artifact arising from the unphysical
orientational registration of square particles on a square lattice,
which would tend to substantially reduce the area excluded by one
adsorbed particle to another.
Talbot (1997) has very recently used scaled particle theory in a manner
quite similar to that employed here to estimate the effect of area
exclusion on the adsorption isotherm of a single molecular species that
may adsorb in side-on and end-on conformations. Results qualitatively
similar to those shown in the bottom panel of Fig. 4 were obtained. The
present model represents an extension of Talbot's approach to allow
for self-association (i.e., attractive as well as repulsive
interactions between adsorbate molecules) and to provide a description
of rate processes as well as equilibria.
The existence of multiple interconvertible adsorbate conformations with
different adsorption energies and surface footprints has the potential
to significantly broaden the equilibrium adsorption isotherm along the
concentration axis and the kinetic progress curve along the time axis.
The progress curves plotted in Fig. 13 exhibit regions where binding
increases linearly with the logarithm of time (to a good approximation)
over as much as three orders of magnitude in time. Such nonclassical
kinetics have been termed "fractal" (Nygren, 1993) although there
does not seem to be any fractal aspect to the underlying mechanism in
this particular instance.
Comparison of the results of model calculations, such as those
presented here, with results obtained from experimental measurement of
protein adsorption rates and equilibria is hindered by several complicating factors. An often-reported conclusion that a particular protein is absorbed irreversibly in part or in whole may be based upon
the observation that the protein fails to dissociate substantially when
exposed to protein-free solvent for a limited time period (see, for
example, Brynda et al., 1986 and Schmitt et al., 1983). This
interpretation neglects the "retention effect," i.e., the buildup
and subsequent resorption of newly desorbed ligand in the unstirred
layer immediately adjacent to the surface (Silhavy et al., 1975), which
can profoundly slow the overall process of desorption. Even if means
can be found to eliminate complications due to mass transport
limitation of adsorption and desorption rates (Schuck, 1997),
reversibility and/or attainment of equilibrium of a particular
adsorption process may be extremely difficult to establish in a system
exhibiting stretched kinetics resembling those indicated in Figs.
10-13. This is because kinetic data are ordinarily acquired and
displayed at uniform intervals of time, where the size of the interval
is dictated by the most rapidly changing (i.e., initial) part of the
process (see, for example, Wahlgren, et al., 1995 and Kurrat et al.,
1997). When represented on a constant time axis, even highly stretched
kinetics appear to be describable (to within reasonable experimental
error) over a limited range of time by an empirical expression of the
form of Eq. 19 with one or two exponential terms, and the equilibrium
adsorption estimated as the sum of the coefficients j.
As shown in Fig. 15, this common procedure can lead to significant (20-40%) underestimates of the equilibrium value of .
View larger version (12K):
[in this window]
[in a new window]
|
FIGURE 15
Replot on a linear time axis of a short-time subset of
the time points plotted in the top panel of Fig. 10
(K2c = 105). Also plotted
(solid line) is the best fit to this truncated data set of a
two-term exponential expression of the form of Eq. 19, which appears to
the eye to be a satisfactory description of the data. The sum of the
amplitudes of the two exponential terms is 0.505, whereas the total
amount adsorbed at equilibrium (see Fig. 10) is 0.67.
|
|
The results presented here also suggest the possibility that in
contrast to the kinetic behavior predicted by the Langmuir model, large
increases in free ligand concentration do not necessarily appreciably
shorten the time required to attain adsorption equilibrium, which may
in unfavorable cases be much larger than the duration of any reasonable
experiment (see Note 10).
The adoption of surface-induced (i.e., nonsolution) conformations by
adsorbed proteins is a very real possibility (Hummel and Anderson,
1965; Slack and Horbett, 1995). The present equilibrium model is
compatible with such conformational change so long as surface-induced
conformations are in reversible equilibrium with the solution
conformation. In the present instance, any or all of the adsorbate
conformations may be different from the solution conformation. No
property of the equilibrium model depends upon the solution
conformation, since the protein is assumed to behave ideally in
solution, and any energy of conformational change upon adsorption may
be subsumed into the adsorption potential. In contrast, the kinetic
model would have to be generalized to allow for a rate (or rates) of
conformational change following adsorption, and such generalization
could conceivably stretch the progress curves still further.
Several globular proteins have been reported to attain maximum surface
densities that are close to that calculated for hexagonal close packing
of quasi-spherical particles with the same molar mass and density as
the protein (Fig. 16). These
observations suggest that at least some globular proteins retain a
native or nativelike globular conformation upon adsorption, and that
these proteins form large two-dimensional quasi-crystalline arrays.
Such behavior would be consistent with the concept of adsorption-linked
oligomerization or cluster formation (Ramsden et al., 1994; Minton,
1995). However, the process of formation of quasi-crystalline arrays
must involve significant abundances of aggregates of intermediate size,
as Langmuir-like adsorption isotherms reported in the literature (Al-Malah et al., 1995) are not consistent with those predicted by a
simple three-state (solution monomer + adsorbed monomer + adsorbed n-mer) model when n is large (cf. Fig. 3
of CM). The kinetics and equilibria of multiassociation state
adsorption models will be treated in a subsequent report.
View larger version (20K):
[in this window]
[in a new window]
|
FIGURE 16
Experimentally measured monolayer capacity of
-lactalbumin,  -lactoglobulin, and BSA adsorbed to silanized
silica surfaces (Al-Malah et al., 1995) plotted against molar mass of
protein. Also shown are curves calculated for models in which adsorbed
protein is represented by a hexagonal close-packed array of spheres or
prolate ellipsoids of rotation (axial ratio 1.25, axis of rotation
normal to the surface), and particle density taken to be 0.73 cm3/g (i.e., equal to the average partial specific volume
of protein).
|
|
| |
NOTES |
1. A surface is considered to be "flat" within the present
context if irregularities in the direction normal to the surface plane
are small relative to the smallest external dimension of the adsorbing ligand.
2. The number of equivalent ways in which a solution species can adsorb
to the surface in a given orientation.
3. The selection of a single aggregate species with a square footprint
is arbitrary and obviously oversimplified compared to any real
aggregation process beyond dimer formation, which would probably
involve multiple species of differing stoichiometry and footprints. The
purpose of this model is simply to explore qualitative distinctions
between adsorption isotherms in the presence and absence of adsorbate aggregation.
4. Copies of the MSDOS-executable program and Pascal source code are
available upon request.
5. Consideration of the time-dependent formation/dissolution of
adsorbed oligomers (e.g., species 3 in the equilibrium model) under
highly area-excluded conditions is beyond the scope of the present
work, because of the necessity of taking into account the possible
existence of numerous additional kinetic intermediates.
6. The appropriateness of using scaled particle theory (or any other
equilibrium equation of state) to calculate the instantaneous value of
Pi({}) in a dynamically evolving system
has been discussed by Talbot and co-workers (Jin et al., 1994; Talbot
et al., 1994), who calculated that the difference between available
area in irreversible and equilibrium conformations of hard particles
randomly placed on a surface is probably smaller than the inexactness
of the approximate equilibrium theories.
7. It may be readily shown that in the limit
di/dt* = 0, Eq. 18 is consistent
with the equilibrium relations (Eqs. 5 and 8).
8. A copy of the MLAB script file is available upon request.
9. In the present context the index j is phenomenological
and does not necessarily represent the contribution of an individual molecular species or state to the overall adsorption process.
10. It is likely that at least some of the hysteresis curves reported
to characterize adsorption of proteins (Jennisen, 1985; Norde and
Haynes, 1995) are the result of a simple failure to attain equilibrium
within the duration of the experiment, rather than an intrinsically
irreversible mechanism of adsorption.
The author thanks Prof. K. Yutani and the staff and students of the
Laboratory of Solution Chemistry, Institute for Protein Research, Osaka
University, for their gracious hospitality and support during the
tenure of a Visiting Professorship, June-August 1997, during which the
work reported here was initiated.
Address reprint requests to Dr. Allen P. Minton, Building 8, Room 226, National Institutes of Health, Bethesda, MD 20892-0830. Tel.:
301-496-3604; Fax: 301-402-0240; E-mail: minton{at}helix.nih.gov.
Top
Abstract
Introduction
![+k<SUB><UP>flip</UP></SUB><FENCE><FR><NU><AR><R><C>&rgr;<SUB><UP>j</UP></SUB><UP>exp</UP>(<UP>−</UP>Ja<SUB><UP>j</UP></SUB>)d<SUB><UP>i</UP></SUB>&ggr;<SUB><UP>j</UP></SUB>({&rgr;})</C></R><R><C> <UP>−</UP>&rgr;<SUB><UP>i</UP></SUB><UP>exp</UP>(<UP>−</UP>Ja<SUB><UP>i</UP></SUB>)d<SUB><UP>j</UP></SUB>&ggr;<SUB><UP>i</UP></SUB>({&rgr;})</C></R></AR></NU><DE>(d<SUB><UP>i</UP></SUB>+d<SUB><UP>j</UP></SUB>)[&ggr;<SUB><UP>i</UP></SUB>({&rgr;})+&ggr;<SUB><UP>j</UP></SUB>({&rgr;})]</DE></FR></FENCE>](/content/vol76/issue1/fulltext/176/img023.gif)
a critical review.
In
Proteins at Interfaces II. American Chemical Society, Washington, D.C.
