Irreversible
Theoretical
adsorption of macromolecules
at a liquid-solid
interface:
studies of the effects of conformational
change
Paul R. Van Tassel
Centre Europeen de Calcul Atomique et Moleculaire, Ecole Nor-male Superieure de Lyon, Aile LR5, 46,
Allee d’ltalie, 69364, Lyon Cedex 07, France and Laboratoire de Physique Theorique des Liquides,
Universid Pierre et Marie Curie, 4, Place Jussieu, 75252, Paris Cedex 05, France
Pascal Viot and Gilles Tarjus
Laboratoire de Physique Theorique des Liquides, Universite’ Pierre et Marie Curie, 4, Place Jussieu, 7.5252,
Paris Cedex 05, France
Julian Talbot
School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907
(Received 29 April 1994; accepted 28 June 1994)
The effects of particle conformational changes on the kinetics and saturation coverage of
irreversible macromolecular adsorption at liquid-solid interfaces are investigated by computer
simulation of a modified random sequential adsorption model. In this model, macromolecules
(modeled as disks of diameter a,) adsorb onto a surface at a rate k, . Once adsorbed, the particles
spread symmetrically and discretely to a larger diameter cfi at a rate k, . Adsorption or spreading
events which result in the overlap of particles on the surface are not allowed. We investigate the
effects of changes in spreading magnitude 2 (= ~$a,> and relative spreading rate K, ( = k&k,). We
observe that the saturation coverage of spread particles decreases while that of unspread particles
increases with spreading magnitude. This dependence is most pronounced for small spreading: the
derivative of the surface coverage of both spread and unspread particles with respect to I: diverges
logarithmically when c + 1. An increase in the rate of spreading increases the saturation coverage
of spread particles while decreasing that of unspread particles. The dependence of the coverage on
spreading rate is weaker than its dependence on spreading magnitude: a four order of magnitude
change in K, results in a factor of 2 change in the partial coverages. The coverage of unspread
particles may become nonmonotonic in time for certain values of x and K, . The total density of
particles on the surface decreases and the average particle size increases with K,, in accordance
with recent protein adsorption experiments.
I. INTRODUCTION
fects into a mesoscopic view of the adsorption process. A
comprehensive review of the RSA model and its applications
Recently, interest has grown in the adsorption/deposition
has recently appeared:t
of macromolecules (large particles with diameters between
Feder and Giaever’ were the first to suggest using RSA
100 and 1000 A) such as proteins’-‘7 and colloids’8-23 at
to model macromolecular adsorption. They noted that the
liquid-solid interfaces. A variety of applications exist,3*4.637 experimental saturation coverage of the protein ferritin on a
including the separation and purification of proteins.24-27 To
solid surface was consistent with the RSA jamming limit
fully understand these processes, and to facilitate their use in
coverage of disks (19=0.547). Similar observations were later
technological applications, accurate models are needed to
made for the adsorption of colloidal particles (Onoda and
predict the kinetics of adsorption, the saturation coverage,
Linige?
and Adamczyk et &.22V23). More recently,
and the adsorbed particle arrangement as a function of parRamsden15*16has studied the adsorption kinetics of the proticle concentration, temperature, etc. Under many experitein transferrin by measuring changes in the effective refracmental conditions, the rate of desorption and surface diffutive index of guided light modes. These experiments verified
sion of macromolecules is much slower than their rate of
two major features predicted by the RSA model: (i) a short
adsorption. In these cases, equilibrium isotherm models
and intermediate time kinetics which differs from that of the
which assume reversible adsorption are inappropriate.
Langmuir model, and (ii) a slow asymptotic approach to
Alternatively, random sequential adsorption (RSA)
saturation governed by a power law behavior.
models28-41 more accurately reflect the physics of macromoDespite its success, the assumptions required by the RSA
lecular adsorption. In RSA, particles are represented as geomodel may limit its applicability to a number of systems.
metric objects and deposit on a surface irreversibly, one after
While the neglect of surface diffusion and desorption in the
another, at random positions. Once adsorbed, a particle reRSA model are justified for many systems, the rigid particle
mains fixed. Overlap with other particles is not allowed; a
assumption may be invalid for the adsorption of proteins.
particle placed in a position of overlap is immediately reIndeed, a number of experiments indicate that proteins unmoved. This process continues until new particles are unable
dergo an irreversible conformational change following adto fit on the surface. While neglecting atomic level detail,
sorption onto a surface.7-9Y11*39-41
Lundstrom,
Walton,43
RSA incorporates irreversibility and many-body blocking efand JennisenM have conjectured that a protein will initially
7064
J. Chem. Phys. 101 (B), 15 October 1994
0021-9606/94/101(8)/7064/1
O/$6.00
CQ1994 American Institute of Physics
Van Tassel et
a/.: Irreversible adsorption of macromolecules
adsorb in its globular, solution conformation and will later
undergo a change in tertiary structure due to energetic interactions with the surface. A number of instances of surface
induced conformational change in blood proteins have been
observed and are the topic of a recent review by Andrade and
Hlady.7 Schaaf and Dejardin,8*9 using scanning angle reflectometry, have observed a conformational change for the protein fibrinogen via a tilting process: adsorption to the surface
is end-on, and after a period of time the protein shifts to a
side-on orientation. Ramsden15 recently has reported a dependence of the surface protein concentration on the adsorption rate, providing evidence that a conformational change
following adsorption can increase the surface coverage of
individual adsorbed particles.
The purpose of this work is to investigate the effects of
changes in particle conformation on adsorption kinetics,
saturation coverages, and particle distributions. A modified
RSA model is used which considers the geometrical aspects
associated with conformational changes. As in standard
RSA, macromolecules will be modeled as disks which adsorb sequentially and irreversibly onto a surface. Following
adsorption, however, a particle will be allowed to grow symmetrically to a larger size if space allows. The former will be
referred to as an alpha particle (with diameter a,> and the
latter as a beta particle (with diameter O-&. They differ only
in size. The time evolution of this system of adsorbing particles obeys the following kinetic equations:
7065
b)
0@
@
0@
FIG. 1. A schematic illustrating the available surface function @A for (a)
incoming beta particles and (b) incoming alpha particles. The small and
large striped disks representalpha and beta particles,respectively.The white
outer disks representadditional surface area excluded to the centers of incoming particles. The diameter of an “exclusion disk” for a A particle
blocking a p particle is (q+a&/2.
where ph is the number density of A particles (where A= cxor
/?), p=p,+pP
is the total number density, k,c is the rate of
adsorption per unit area (c represents the particle concentration in solution), and k, is the rate of spreading. <p, is the
available surface function 29V33*38
for alpha particles which
represents the fractional surface available for adsorption.
More specifically, it is the probability that a point on the
surface chosen at random will be at least a distance a, from
the center of all alpha particles and at least a distance
(a,+ oP)/2 from the center of all beta particles. This function
will in general be a function of the total density p, the ratio
of particle sizes C( =r~dcrJ, and the ratio of the rate of
spreading to the rate of adsorption:
KS=
ks
k,c( m&4)
*
(3)
q\I is the probability for a given alpha particle on the surface
to have space available to spread, i.e., to be at least a distance
(o=+o&/2 from all other alpha particles and at least a distance o;o from all beta particles.
A special case of this model occurs when the particle
spreading is instantaneous. This model has recently been
solved analytically in one dimension,45 so a qualitative comparison can be made between theory and simulation. The
kinetic equations become
% =kac[@,(p,C)- cpp(p,c)-j,
where @ ‘pis the available surface function for the adsorption
of beta particles. The functions Qa and a, are illustrated in
Fig. 1. We will refer to the instantaneous case as model I and
the general case as model II. If the functions a’, , a,,, and
W, were known for all densities, the complete time evolution
of the system could be obtained by integrating the above
equations. As these functions can only be approximated, we
instead resort here to numerical simulation. An analytical
approximation of the kinetics for these two models, valid at
low to moderate densities, may be achieved by expressing
the functions a,, , Cp,, and W, in powers of the density. This
will appear elsewhere.46
To our knowledge, an RSA model incorporating conformational change has been applied only to particles which
spread without regard to steric interactions (i.e., the particles
may overlap while spreading), and only the saturation coverage has been investigated.47 In our study, surface geometric
exclusion effects are accounted for in two ways: adsorbed
particles serve to block both the adsorption of new particles
and the spreadingof neighboringparticles.Furthermore,the
J. Chem. Phys., Vol. 101, No. 8, 15 October 1994
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Van Tassel et al.: irreversible adsorption of macromolecules
kinetic approach towards saturation and the adsorbed particle
radial distribution functions are presented along with saturation coverage data.
II. SIMULATION
In the first of two models we shall employ (model I), the
proceeds as follows:
simulation
alpha particles
(diameter=a,) are deposited sequentially, at random positions, and with rate k, per unit area, onto a square surface of
area A = L2 (where LZ=lOOaJv?) with periodic boundary
conditions. If, during a trial insertion, the new particle overlaps another, previously adsorbed particle, the former is removed and another trial is attempted. If no overlap occurs,
the particle is permanently placed in this trial position. Following adsorption, if sufficient space is available such that
the new particle may spread, with its center position fixed, in
a symmetric manner to a larger diameter op, this is done so
immediately. If sufficient space is not available, the particle
maintains its original diameter a,. The simulation time is
incremented by an amount At= (k,cA)-‘.
Model II differs in that the rate of attempted spreading of
alpha particles is governed by the following equation:
r as=kspn.
(6)
Note that this equation differs from Eq. (2) by a factor equal
to the fraction of alpha particles on the surface which have
space to spread. In practice, an exponentially distributed
time, t, , is assigned to each newly adsorbed particle:
b)
t
h
FIG. 2. (a) A schematicof a small target of availablesurfacefor adsorption
of an alpha particle (larger shaded region) and a beta particle (smaller
shadedregion) in the asymptotic regime when the spreadingmagnitude is
small, i.e., x= 1 fe where E< 1. (b) The enlargedtarget region showing the
dependenceon size of the parametere
overlap with a new particle and only those within ap of each
other are checked for overlap with a spreading particle.
where k; ’ is the characteristic time of spreading and R is a
uniformly distributed random number between 0 and 1. It is
easy to show that the times rs generated by Eq. (7) are exponentially distributed with mean k; t . When the simulation
time following the adsorption of this particle reaches a value
of t,, the particle spreads discretely, symmetrically, with
fixed center, to a larger diameter ap if space is available. If
space is unavailable, the particle remains forever in its original state with diameter a,. We expect that the density of beta
particles will be lower in model II, since during the time
delay, new particles may adsorb which could block the
spreading of alpha particles which, immediately following
adsorption, had space to spread.
To enhance computational efficiency, the surface is assumed to be a square grid whose elements are just small
enough to house only a single particle (each element is a
square of length a&?!). Attempted placement of new particles occurs only in unoccupied grid spaces. To compensate
for the biased insertions, the simulation time is incremented
by an amount:
At= - 1 . ~Ns, total
k,cA
Ns, vamnt’
(8)
where N,, total is the total number of square grid elements
and N,, vacantis the total number of these elements which are
vacant. To further enhance efficiency, only particles in grid
squares within (a,+@&/2 of each other are checked for
Ill. ASYMPTOTIC BEHAVIOR
A. Saturation
coverage
Before giving the results of our simulations, we present
here an analysis of the adsorption of spreading particles in
the limit of small spreading magnitude. Consider a system of
alpha particles which adsorb as described above and spread,
if possible, to a size crfi= a,( 1 + E), with e+ 1. Up until some
critical time, lc, essentially all of the adsorbing particles
have space in which to spread the small amount oae. At
some time I,, the surface will be quite crowded and the
following statements will hold true: the only positions on the
surface available for additional adsorption will be small “islands” or “targets” created by gaps among the adsorbed particles, as suggested by Pomeau3’and Swendsem3’ these targets will possess a constant, characteristic length
h<h,,+a,
(it is assumed that the surrounding particles
will have had time enough to spread). Since spreading requires that a particle have an additional distance a,e available in all directions, only alpha particles may occupy targets
with h<crae. For a target with h>a,e, an alpha particle may,
depending on its location in the target, spread to become a
beta particle. Figure 2 illustrates such a target. The probability that a target of length h >a,~ is eventually filled by a beta
particle is
J. Chem. Phys., Vol. 101, No. 8, 15 October 1994
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Van Tassel et al.: Irreversible adsorption of macromolecules
(h-q&
qe=
*
h2
At time t,, it is indistinguishable whether we are conducting (i) a standard RSA simulation, without spreading, of
particles of diameter cPa, or (ii) a modified RSA simulation
of alpha particles which adsorb and later spread to become
beta particles. At infinite time, the difference between the
saturation coverage of (i) and (ii) is just the number of particles which land in a target of size h>~~e, yet do not have
room to spread, i.e., they land in the lightly shaded outer
region of the target illustrated in Fig. 2(b):
pRS*(t=q-pg(E,t=q
h
-I
maxn(h,tJ[l
Uaf
-Pp(h)]dh+O(e).
(10)
In Eq. (lo), pRSA(t=m) and p&e,t=m)
are the saturation
number densities of beta particles for the cases (i) and (ii)
described above and n(h,t) dh is the density of targets of
size h at time c as introduced by Swendsen.31 Assuming that
n(h,t,) has a nonzero limit when h -0 and inserting Eq. (9)
into Eq. (IO), one obtains
In E,
pRSA(f=a)-pS(e,t=m)----
e-0.
(11)
The right-hand side of Eq. (11) goes to zero, as it should,
as e+O. The derivative with respect to 6,
dp,G,t=w)
de
-In
E,
n(h,t)--n(t,)e-Rah2(‘-cc).
(9)
E+O,
02)
diverges as In E for small E. A similar argument shows that
the alpha particle density diverges in the same manner. We
expect, then, that the density of both spread and unspread
particles to be very sensitive to changes in the spreading
magnitude when the magnitude is small.
These same arguments hold in any dimension. In a recent work,45 we have shown analytically that similar behavior exists in the one-dimensional version of this problem.
(15)
After integration, one obtains the following kinetic behavior
when k,( w,)2r>>l:
06)
P&JkPaw-~-1’2
and
(17)
Asymptotic kinetic equations may also be derived for
model II. When KS%-t?, the kinetics of adsorption behave in
a manner similar to the case of instantaneous spreading and
so are described by Eqs. (16) and (17). When K,G%, the
asymptotic kinetics of beta particle adsorption are controlled
by the spreading rate and are described by
pa(q-ppp(t)-e-kJr,
k,t+l.
08)
Equation (16) tells us that the approach to saturation for
alpha particles will occur algebraically (as in the standard
RSA of disks). The approach to saturation for beta particles,
on the other hand, will be essentially exponential [Eqs. (17)
and (18)]. This behavior mimics that of RSA of a binary
mixture of large and small particles.35V40The implication is
that the beta particles will in general reach a saturation density much more rapidly than the alpha particles. The magnitude of spreading, though, will dictate the time required to
observe this algebraic approach. If K,>2,
only for
t> llk,(e~,)~ will the kinetic approach toward saturation occur in the manner predicted by Eqs. (16) and (17). When
K,<z, asymptotic behavior will occur when t> Ilk, and be
independent of the spreading magnitude. Thus, for small values of e, a greater amount of time will be required to observe
true, asymptotic kinetics. We expect, then, difficulty in observing the asymptotic behavior of alpha particles which
spread only a small amount.
C. Saturation
coverage
(14)
When both the spreading rate and the spreading magnitude are large, i.e., model I with large Z, we can construct an
approximate kinetic model for determining the saturation
coverage using reasoning similar to that used for the RSA of
a binary mixture.35 Consider the requirements for a successful adsorption event. The incoming alpha particle must certainly avoid all previously placed beta particles. Since only a
very small amount of the available surface is blocked collectively by two or more beta particles (because of their large;
size relative to the alpha particle), we may approximate the
probability of avoiding the beta particles as one minus the
sum of the fractional areas excluded by each adsorbed beta
particle. Now, given that an incoming alpha particle avoids
the beta particles, the probability that it avoids overlap with
one or more alpha particles is just the available surface function Q, for the RSA of disks applied only to the area left open
by the beta particles. The kinetic equation for the time evolution of the total density is then
where again, n(h,t) dh is the density of surface targets of
linear size h at time t:31
(19)
B. Approach
to saturation
It is well known that in standard RSA (the limit of E=O
in model I), the approach towards equilibrium
is
algebraic.30V31For nonzero E, we may obtain the asymptotic
approach to saturation by considering the kinetic equations,
valid when t >t, , that describe the filling of remaining small
targets by both alpha and beta particles. In model I, these
may be written as
h*n(h,t)dh
hma[h2-(hqJ%W>dk
+kaI ena
(13)
J. Chem. Phys., Vol. 101, No. 8, 15 October 1994
Van Tassel et a/.: Irreversible adsorption of macromolecules
7068
TABLE I. The predicted saturation coveragesof alpha and beta particles
when I% 1 basedon the numerical solutions to Eqs. (23) and (24). Note that
for P=m, alpha particles are points and thus cover no area.
z
L9,(r=m)
e&t=+
4.0
6.0
8.0
0.2150
0.2435
0.2559
0.328 1
0.3290
0.33 17
0.3338
0.3390
0.3430
0.3456
10.0
0.2629
20.0
50.0
m
0.2762
0.2836
...
0.4
0.2
0.1
_____--__0.0
0.0
1.0
2.0
t*
3.0
4.0
where
e;=8, 1-pg
i
7r(u,+up)2
-l
4 i
(20)
with 0, being the fractional surface coverage of alpha particles and the other symbols have their previously described
meanings.
Now, consider events leading to beta particle adsorption.
The incoming particle must of course land at least a distance
aP from all previously placed beta particles. The probability
of this occurring is just @(ep>. Given that the incoming particle avoids all beta particles, the conditional probability of
avoiding all alpha particles is just the probability that all
previously placed alpha particles are located outside of a
circle of diameter oh+ up. We assume the alpha particles are
roughly uncorrelated, so this probability is just
P,M~af@~41
Na
Fa= i l-N,{1-Pg[~(Oa+(rp)2/4]}
i
.
(21)
Allowing N, to become macroscopically large, we can write
Eq. (21) as
p,[4ua+
q312/41
Fa=exp i - 1 -pB[?T(a,+a,)2/4]
1.
(22)
The kinetic equation for the adsorption of beta particles becomes
The kinetic equation for the adsorption of alpha particles is
obtained by subtracting Eq. (23) from Eq. (19):
f$=k,
i
1-pp
du,+
4
uTpJ2
~(~h)-km~,)L
1
(24)
Equations (22) and (23) are easily evaluated numerically
using the previously determined analytical solution to Cp up
to third order in density.33 Table I shows saturation coverages
derived from this simple model.
It is easy to show that the saturation coverage of beta
particles goes to zero in model II in the limit of large spreading. This can be seen as follows: For any finite k, , there will
be a short time (e l/k,) in which a small density of particles
adsorb but no spreading has yet occurred. At this time, there
will be a characteristic separation between the particles of
FIG. 3. The surface coverage Bi= ( 7ro?/4)pi of alpha and beta particles as
a function of time for different values of Z=gda’aOin model I. (5,= 1.0
correspondsto standardRSA.)
approximately lldp. We simply choose up~lll~p, rendering
spreading events impossible. This argument is valid for any
finite rate of spreading.
IV. RESULTS
A. Model I
We first consider the simplest case where the spreading
occurs instantaneously and the system obeys the kinetic
equations of Eqs. (4) and (5). Figure 3 shows the surface
coverage of alpha and beta particles as a function of simulation time for various values of z (=u2/ul). Since spreading
is prevented only by blockage from previously adsorbed particles, the alpha particle density increases as the square of the
time for short times. In contrast, the spreading is not prevented for the earliest adsorbing particles, so the beta particle
density initially increases linearly with time. It is clear that,
particularly for large 2, the beta particles approach saturation
much more quickly than do the alpha particles as predicted
by the above arguments. Figure 4 illustrates the t-1/2 dependence of the alpha particle approach to saturation as predicted by Eq. (16). Note that for small spreading magnitudes,
it is difficult to observe the asymptotic kinetics due to poor
statistics at long times.
Figure 5 shows the saturation surface coverage and number density vs spreading magnitude 2. A number of points
are of interest. Fist, the density of alpha particles increases
and the density of beta particles decreases when 2 is increased. This reflects the unlikeliness (for steric reasons) of
spreading a large distance. In addition, the total surface coverage increases with 2, indicating the surface is more efficiently filled when the particles differ greatly in size. Finally,
we observe that the surface coverage of beta particles approaches a constant value for large 2,. This implies that the
number density of beta particles goes to zero and the total
number density approaches the density of the alpha particles
as CM2.
Of particular interest is the steep slope of the saturation
coverage of both alpha and beta particles for 2 not much
J. Chem. Phys., Vol. 101, No. 8, 15 October 1994
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Van Tassel et al.: irreversible adsorption of macromolecules
8.0
0.3
6.0
0.2
CDs
=
d
1
4.0
0.1
4_
/J
0.0
*
1
z11.01
I “.......I
0.0
2.0
w.06
-
0.1
93
0.4
0.3
t*
0.0, I
I.5
1.5
2.0
2.5
3.5
3.0
r/o,
FIG. 4. The asymptotic kinetics of the surface coverageof alpha particles
for various spreadingmagnitudesin model I where the I-‘~ dependenceis
displayed. The arrows indicate an estimate of the earliest time in which
asymptotic kinetics is expected to be described by Eq. (16). Dotted lines
representextrapolation of coverage to infinite time. Adsorption events become rare at long times, leading to poor statistics.
FIG, 6. The a-a, cu-fl, and p-/3 radial distribution functions at saturation
in model I for Z = 1S. Notice that both g nn and ga8 diverge at contact while
gPBremains finite.
dard RSA,32 we see that the correlations between particles
decay very quickly at large distances.
greater than 1. As indicated by Eq. (12), the slope diverges as
In E as e-+0 (C-+1). Indeed, a plot of the saturation surface
coverages vs l In E is linear for small E and its slope is 0.62,
or approximately twice the known analytical value in one
dimension.45 The physical implications of this divergence
will be discussed below.
Figure 6 shows the CY--(Y,a-/3, and p-/3 radial distribution functions at saturation for Z= 1.5. It is observed that
both goa and gap diverge at contact while gp,., remains finite.
Pomeau3’ and Swendsen3’ have given arguments showing
that the divergence of g(r) in standard RSA reflects the high
probability for creation of neighboring pairs at close separation by the insertion of particles in small surface targets during the asymptotic regime. Similarly, as only alpha particles
adsorb close to saturation, we observe, as expected, the divergence of g,, and gap at contact. In addition, as in stan-
B. Model II
We now consider model II, the more general case with
finite spreading rate whose time evolution obeys Eqs. (1) and
(2). Figure 7 shows the surface coverages as a function of
time for different values of spreading magnitude C for a
moderate spreading rate (K,= 1). For sufficiently small X,
the density of alpha particles may become nonmonotonic
with time. This occurs when the rate of alpha particle spreading exceeds the rate of deposition [QP,<K,paVr, in Eq. (l)].
Nonmonotonic behavior is most favored when K, is far from
zero or infinity and c is small.
Figure 8 shows the coverage evolution for various relative spreading rates at a small spreading magnitude (c
= 1.05). As expected, for large K, , both alpha and beta particle coverages approach those of model I. For small K, , the
alpha particle coverage approaches the profile observed in
standard RSA and the beta particles coverage approaches
1I
I
%+%
0.6I/,
0.5
,
I
T
&
/
0.4
,,L------
1
[
Q 0.2
0.0 ’
1.0
/,'
____----@!t' *0.
e, (L1.6)
4,'
I
1.5
2.0
2.5
z
3.0
3.5
4.0
0.0
i 8'
-
0.0
FIG. 5. The partial and total surface coverage and the partial and total
reduced number density at saturation as a function of spreadingmagnitude
in model I. 6’,is defined as in Fig. 3 and p: = ( m$4)pi .
a, @=l.oe)
/ /‘.
0.1
Y
2.0
--_-___________
L
4.0
t*
6.0
-I
8.0
FIG. 7. The surface coverageof alpha and beta particles as a function of
time for differentvaluesof Z in modelII for K, = 1,
J. Chem. Phys., Vol. 101, No. 8, 15 October 1994
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Van Tassel et a/.: Irreversible adsorption of macromolecules
Saturation Density
Modelll,L-1.05
0.1
v \- /---Ll;
I/ __-)c_c% K-~
0.0 0.0
2.0
4.0
t*
6.0
6.0
001
. 0.0
0.1
1.0
10.0
100.0
10.0
100.0
Y
FIG. 8. The surface coverage of alpha and beta particles as a function of
time for different values of K, in model II for P = 1.05.
ModeIll,
L-1.6
0.6
zero. Nonmonotonic behavior of the alpha particle density is
observed for all K, ; the value of the local density maximum
decreases with increasing K,. Note that at short times the
alpha particles initially fill the surface. Indeed, the short time
behavior is the opposite to that observed in model I: the
alpha and beta particle densities increase linearly and quadratically with time, respectively. Thus, for sufficiently short
but finite times, the kinetics of adsorption for model I and
model II will differ qualitatively even for large K, .
Figure 9 shows the saturation coverage and number density as a function of the spreading magnitude. We observe the
following qualitative differences from model I: (1) The beta
particle saturation surface coverage approaches zero for large
2. This is due to the reasoning introduced in Sec. III C: for
any finite K, , a sufficiently large 2 will ensure that all particles on the surface are blocked from spreading by neighboring particles. (2) A consequence of this behavior is that
the total coverage and the total number density, monotonic
functions in model I, here display a maximum and a minimum, respectively. At low 2, both of these functions behave
as in model I because the likelihood of depositing a particle
c
FIG. 9. The surface coverageand the reducednumber density at saturation
as a function of spreadingmagnitudein model II for K, = 1.
0.5
c
&
0.4
h
b
0.3
T
g
0.2
0.1
0.0
0.0
0.1
1.0
KS
FIG. 10. The surfacecoverageand the reducednumber density at saturation
as a function of spreadingrate in model II for (a) z= 1.05 and (b) 2 = 1.6.
which would block the small spreading amount is small. At
higher 2, though (and particularly when K, is small) incoming particles will block particles which in model I would
have spread, thus decreasing the area covered but increasing
the space available for additional (alpha) particles, and thus
also increasing the total number density. As in model I, the
slope of the partial densities vs 2 diverges logarithmically as
E approaches unity.
Figures 10(a) and 10(b) show the surface coverage and
number density of adsorbed particles vs K, . As expected, the
density of alpha particles decreases and that of the beta particles increases with relative spreading rate. These changes
are more strongly pronounced when 2 is large. It is interesting to note that the partial coverages are much less sensitive
to spreading rate than to spreading magnitude: e.g., a four
orders of magnitude change in K, causes only a factor of 2
change in the partial densities, in contrast to the rapid change
in density with 2 observed above. The total area density
increases somewhat and the total number density decreases
somewhat with increasing spreading rate.
We define an average particle expansion as the ratio of
the surface coverage to the number density at saturation. Figure 1l(a) shows that this average particle size exhibits a
maximum for finite K, at a Z which increases with K, . For
J. Chem. Phys., Vol. 101, No. 8, 15 October 1994
7071
Van Tassel et a/.: irreversible adsorption of macromolecules
Average Particle Expansion
Experimental Comparison
Saturation
Density
1
1.6
7
Q.
2
1 1.4
z
CD
0.0
0.1
c I (mgkm’)
Averige
Particle
Expansion
1.6
2.20
2.10
5i&
I-i”
2.00
E
0
z
1.90
g
1.60
1.70
1.60
1.50
FIG. 11. The averageparticle expansionat saturation,definedas the ratio of
the total surface coverage to the total reduced number density, in model II
(a) as a function of spreadingmagnitudefor different values of K, and (b) as
a function of spreadingrate for different values of 2.
finite K, , in the limit of large 2, the average particle expansion goes to unity as expected since tVPapproaches zero.
Figure 11(b) shows that the average particle size increases
with relative spreading rate.
It is possible to convert our simulated data to a form
comparable to that of the protein adsorption experiments of
Ramsden,” who has measured the saturation density and the
average particle size for the protein transferrin. By using the
characteristic spreading time estimated by Ramsden and the
relation of L&ique,48 the relative spreading rate may be related to the bulk protein concentration. In Fig. 12(a), we
assume the experimental data at greatest bulk protein concentration represent the saturation coverage of K,=O. Qualitative agreement is noted between these experimental data
and the simulation data for moderately spreading particles.
We expect the converted simulation points to be somewhat
of an overestimate since even at this small spreading rate,
some of the particles will spread, thus lowering the total
number density. Even so, the experimental data appear compatible with the present model for a spreading magnitude
somewhat below 2. Figure 12(b) shows the average particle
area per unit mass vs bulk protein concentration. Again, good
agreement is observed for c= 1.6 to 2. It should be noted,
FIG. 12. (a) The experimental saturation mass density of the protein transferrin vs concentrationreportedby Ramsden(Ref. 15). along with simulated
data converted to this form. Values of K, are converted to c using the
relation of Lev&que (Ref. 48) and the relaxation rate of the protein on the
surface reported in Ref. 15. Values of p(t=m) are converted to M, by
assuming the experimental data at greatestc representsthe saturation coverageof K,=O. (b) The experimentalarea per adsorbedprotein mass (a/m)
vs bulk concentration(Ref. 15). along with simulated data convertedto this
form. The simulated average particle size is converted by assuming that
K, =0 correspondsto the smallest a/m observedexperimentally.
however, that the experimental determination of aim was
made by data analysis which assumed that the transferrin
adsorbed as in standard RSA without spreading, a model
which the present results confirm is an oversimplification.
V. DISCUSSION
In many circumstances, the adsorption of macromolecules is essentially irreversible, i.e., surface diffusion and
desorption are slow on an experimental time scale. In the
adsorption of colloids, latex spheres, and even some proteins,
the assumption of a rigid particle, as in standard RSA, is
acceptable. However, considerable experimental evidence
suggests that many proteins undergo a change of conformation following interaction with a surface, and that this can
enhance the surface area excluded by an adsorbate. Clearly,
surface structural rearrangement must be incorporated into a
physically meaningful model of protein adsorption, In this
J. Chem. Phys., Vol. 101, No. 8, 15 October 1994
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Van Tassel et al.: Irreversible adsorption of macromolecules
work, we investigate particle conformational changes within
the context of a modified RSA model, i.e., we investigate
those aspects of changes in conformation which are geometrical in origin.
The discrete, two state spreading mechanism employed
here is justified both theoretically and experimentally. This
simple model of particle conformational change represents a
change in the tertiary structure of the particle in order to
minimize its free energy under the influence of the surface. A
free energy barrier connects the states and its height dictates
the rate at which the conformational changes occur. Schaaf
et aZ.,8*9*13
Elaissari and Pefferkorn,47 and Ramsden15 have
observed proteins which adsorb in one state and, as a function of time, change to a conformation which increases the
particles’ surface interaction. Observations such as these
have led to the assumptions put into our model.
We observe that even a small spreading magnitude may
result in a significant portion of the adsorbed particles in the
unspread state at saturation. This results from (although it is
not implied by) the divergence derived in Eq. (12) and observed in Fig. 5. One not familiar with the nuances of RSA
might have naively thought that for a sufficiently small
spreading magnitude, virtually all of the adsorbing particles
would spread upon adsorption. However, it is known that
particle configurations generated by RSA have a high probability of particle-particle contacts, evidenced by a radial
distribution function which diverges at contact.30’3’The high
probability of near contacts causes the divergence observed
here, i.e., it serves to prevent even a small amount of spreading. This result has implications for the potential use of solid
supports of enzyme catalysts. We can assume that the spreading of a particle (in this case, an enzyme) is associated with
a change in its tertiary structure, and thus may affect its
function. Our result implies that even for a small amount of
spreading, a significant fraction of the surface adsorbed protein would retain its original structure, and thus also its original function.
In contrast, a change in the relative spreading rate effects
the saturation coverages less drastically than does a change
in the spreading magnitude. A four order of magnitude
change in the relative spreading rate is required to achieve a
factor of 2 change in the partial densities, and an even
smaller change (especially for a small spreading magnitude)
in the total density. This dependence of the surface density
on spreading rate may be of greater practical value than its
dependence on spreading magnitude since the relative rate of
spreading (to that of adsorption) may be controlled through
the particle concentration in the contacting solution. The
spreading magnitude, on the other hand, may be an intrinsic
function of the specific particle and difficult to control.
An interesting feature of model II is that the alpha particle density may become nonmonotonic in time. This occurs
when K,p,qa becomes greater than @ ‘nin Eq. (1). To understand this behavior, it is instructive to consider the evolution
of ap, and K,p,W, over time. At t=O, @ ‘a is unity and
K,p,ilf, is zero. Over time, @ ‘(Iwill monotonically decrease.
The rate of this decrease grows with 2 and K, : for a given
time, the available surface area for new adsorbing particles is
greatest when the spreading is small in magnitude and slow
in rate. K,p,W, is initially zero, becomes positive, reaches a
maximum, decreases, and at very long times again approaches zero. The initial time derivative of K,p,W, increases with K, ; an intersection with Qp, will occur at a
lower density when K, is large, as seen in Fig. 8. This initial
rate of increase is independent of 2, but the maximum value
attained by K,p,‘P, is expected to decrease with larger 2 (as
the surface fills, the density of particles which can spread a
large amount diminishes more quickly than those which will
spread only a small amount). For large enough 2, K,p,‘P,
never reaches a value equal to Qp,. A reasonable rule of
thumb is obtained if one assumes the particles are uncorrelated [g(R)=1 for all R]: @ ,=K,p,‘P, when the area required for spreading is equal to that required for adsorption,
i.e., when x=fl.
The results of Fig. 7 are consistent with
this simple hypothesis. If the initial alpha particle maximum
occurs at a low enough density, we should be able to estimate
its value by expanding @ ‘nand p,Q, to first order in density
and setting Eq. (1) equal to zero. This yields a maximum
density of pE”=(4+K,)-‘,
which appears to provide a good
estimate for small Z and for KS not less than unity. A more
thorough investigation of the adsorption kinetics via a density expansion will appear elsewhere.46A nonmonotonic partial density implies that at least two beta particle densities
may be achieved for a given alpha particle density, a situation which may be exploited to achieve greater control over
surface composition in protein adsorption applications.
We derive theoretically, and observe via simulation, a
kinetic approach to saturation which is exponential for beta
particles and is algebraic for alpha particles. Furthermore, the
time required to exhibit true asymptotic behavior depends on
the spreading magnitude E as l/z when K,>z. By obeying
these kinetics, the surface tends to fill quickly with beta particles; the smaller gaps between these particles then fill principally with alpha particles. An implication for protein adsorption applications is that the rate of surface filling is not
enhanced by employing a particle which spreads, since it is
the unspread particles which will dictate the long time kinetics.
Standard RSA (without particle conformational change)
has been studied extensively.z8-41 Recently, an RSA model
of particles which spread instantaneously (as in model I) has
been solved in one dimension!5 Many similarities exist between one and two dimensions: beta particles approach saturation exponentially while alpha particles approach saturation algebraically, the alpha particle density increases and
beta particle density decreases with increased spreading
magnitude, and even the slope of the partial densities vs
spreading magnitude diverges logarithmically (although it is
difficult to observe in 1D because of a small prefactor).
However, in two dimensions, the proportion of spread to
unspread particles is overall more sensitive to the spreading
magnitude. We observe here that for a large spreading magnitude, the area densities of spread and unspread particles are
nearly equal. In lD, these differ by a factor of 2. It is quite
remarkable, though, that the one-dimensional problem captures much of the physics observed in the two-dimensional
simulation.
The parameter K, in model II serves as a link between
J. Chem. Phys., Vol. 101, No. 8, 15 October 1994
Van Tassel et a/.: irreversible adsorption of macromolecules
standardRSA and model I. When
K, is 0, model II reduces
to a standard RSA simulation of particles of size a,. When
K, is infinite, model II is identical to model I. (Note that
when K, gets large, pa*, will become small but K,p,W,
will remain finite and approach QDp.)For finite, nonzero KS,
we observe behavior which does not occur in either extreme:
(i) the existence of a local maxima and minima in the total
saturation coverage and density, respectively, vs spreading
magnitude and (ii) the possibility of a nonmonotonic partial
density.
VI. CONCLUSIONS
We present results of a modified random sequential adsorption model which allows for particle conformational
change in the form of discrete, symmetrical spreading following adsorption to a surface. The effects of the relative
spreading rate and the spreading magnitude on the saturation
coverage, the kinetic approach towards saturation, and the
adsorbed particle configurations are investigated.
We find that the saturation coverage of alpha (unspread)
and beta (spread) particles depends on both the spreading
magnitude and the relative spreading rate, although more
strongly on the former. We observe a partial density of alpha
particles which is nonmonotonic in time for certain values of
spreading magnitude and spreading rate. The kinetic approach to saturation of alpha particles obeys an algebraic
power law (as in standard RSA), while the approach of beta
particles obeys an exponential law (and thus occurs more
rapidly). The total density of particles on the surface decreases and the average particle size increases with an increase in relative spreading rate. By equating this to the decreased adsorption rate obtained by lowering the bulk protein
concentration, we show our results to be in accord with protein adsorption experiments. In addition, numerous other
practical implications concerning the adsorption of proteins
may be derived from these results.
ACKNOWLEDGMENTS
The authors wish to thank the Centre Europien de Calcul Atomique et Mol&ulaire, the Centre National de la Recherche Scientifique, and the National Science Foundation
for financial support of this work. P.R.V.T. additionally
wishes to thank the Chateaubriand fellowship program for
financial support. We are particularly grateful to Professor
Giovanni Ciccotti for helpful support and discussions during
the preparation of this manuscript. The Laboratoire de Physique Thkorique des Liquides is Unit& de Recherche No. 765
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