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PROTEINS: Structure, Function, and Genetics 39:76 – 81 (2000)
Monte Carlo Simulation of Diffusion of Adsorbed Proteins
V.P. Zhdanov1,2* and B. Kasemo1
Department of Applied Physics, Chalmers University of Technology, Göteborg, Sweden
Boreskov Institute of Catalysis, Russian Academy of Sciences, Russia
We present the results of threedimensional lattice Monte Carlo simulations of protein diffusion on the liquid-solid interface in a wide
temperature range including the most interesting
temperatures (from slightly below Tf and up to Tc,
where Tf and Tc are the folding and collapse temperatures). For the model under consideration (27 monomers of two types), the temperature dependence of
the diffusion coefficient is found to obey the Arrhenius law with the normal value (⬇10ⴚ2 ⴚ 10ⴚ3 cm2/s)
of the preexponential factor. Proteins 2000;39:76 – 81.
2000 Wiley-Liss, Inc.
Key words: lattice model; Monte Carlo simulations;
adsorption; diffusion; conformational
changes; Arrhenius parameters
Diffusion of proteins adsorbed at solid-liquid interfaces
is of considerable intrinsic interest and also important for
understanding the kinetics of protein adsorption. The
latter process occurs via protein diffusion (in the bulk
liquid) towards the interface, actual adsorption, and subsequent conformational changes resulting often in denaturation of the native protein structure.1 The details of
simulations of these steps depend on the ratio of the rates
of adsorption and diffusion of adsorbed proteins (for a
review, see Zhdanov and Kasemo2). The conventionally
used mean-field kinetic equations describing protein adsorption are implicitly based on the assumption that
diffusion of proteins in the adsorbed overlayer is rapid
compared to adsorption. The opposite adsorption regime
when surface diffusion of proteins is slow is usually
treated in the framework of the random sequential adsorption model.
Experimental data on protein surface diffusion at the
solid-liquid interface are limited. The possibility of this
process was first addressed in the beginning of the eighties.3,4 Subsequent investigations (see Tilton et al.,5 Rabe
and Tilton,6 Thompson et al.,7 Ramsden et al.,8 and a
recent review by Tilton9) confirmed the occurrence of
protein surface diffusion for several adsorbate/substrate
systems. The results reported have been obtained in a
narrow range of temperatures near room temperature. For
example, the coefficients of bovine serum albumin diffusion on different surfaces were found5,6 to be in the range
from 10⫺9 to 10⫺8 cm2/s. For proteins adsorbed on a lipid
bilayer, the diffusion coefficient was measured7 to be of the
order of 10⫺9 cm2/s. Diffusion coefficients obtained for
other systems9 where proteins were observed to be mobile
are of the same order of magnitude, ⬃10⫺10–10⫺7 cm2/s.
Attempts to get the Arrhenius parameters for diffusion
of adsorbed proteins are in fact absent, because in a
narrow range of temperatures the accurate measurements
of these parameters are hardly possible (this obstacle does
not however seem to be principal). Nevertheless, some of
the available experimental data can be used to estimate
the activation energy for diffusion and an order of magnitude of the preexponential factor. For example, the activation energy for bovine serum albumin diffusion on polyhexylmethacrylate was found6 to be about 30 kBT (⯝18 kcal/
mol). This estimate was based on the data obtained at
temperatures between 10 and 30°C (with increasing temperature, the diffusion coefficient was reported to increase
from 0.2 to 1.4 cm2/s). Looking through those data, one
may conclude that the accuracy of the estimate is about 6
kBT. For the preexponential factor for diffusion, one can
get Do ⯝ 105⫾3 cm2/s. This value is much larger than the
“normal” value, Do ⫽ 10 ⫺ 2 ⫺ 10 ⫺ 3 cm2/s, corresponding
to surface diffusion of atoms and simple molecules.12
Physically, protein surface diffusion is expected5,6,9 to
occur via momentary disruptions of a fraction of weak
bonds between amino-acids on the protein surface, and the
substrate. This conceptual scheme of protein diffusion is
qualitatively reasonable, but it does not specify the diffusion mechanism, because it is compatible at least with
three modes of diffusion. If the structure of an adsorbed
protein is stable (e.g., close to the native one), diffusion
may occur via (i) “skating” or (ii) “rolling” of the whole
molecule. (iii) If protein adsorption is accompanied by
denaturation, diffusion is expected to occur via rearrangements of the whole protein structure due to local elementary moves of amino-acids. Detailed simulations addressing these modes of diffusion of adsorbed proteins are
lacking. In particular, it is not clear, for example, how
large the preexponential factor for diffusion might be for
the modes described. Some related simulations10,11 have
been focused on surface diffusion of short-chain alkanes,
but the results obtained in the latter studies are however
not directly applicable to proteins because the models
employed were in fact two-dimensional. To fill this gap, we
Grant sponsor: TFR; Grant number: 281-95-782; Grant sponsor:
NUTEK Biomaterials Consortium; Grant number: Contract No. 842496-09362.
*Correspondence to: V.P. Zhdanov, Boreskov Institute of Catalysis,
Russian Academy of Sciences, Novosibirsk 630090, Russia. E-mail:
Received 20 April 1999; Accepted 30 Septemer 1999
present Monte Carlo (MC) simulations of diffusion of
adsorbed proteins in the case when adsorption is accompanied by denaturation.
Our work is based on the three-dimensional (3D) lattice
model, which earlier has been widely used to study the
kinetics of protein folding in solution (see a recent review
by Shakhnovich13) and also denaturation of adsorbed
proteins (see our review2). In the lattice approximation, a
protein is schematically viewed as a linear sequence of N
amino-acids. The structure of amino-acids is not analyzed
explicitly. Instead, they are replaced by monomers, which
are constrained (along the chain) to be nearest-neighbor on
a 3D simple cubic lattice (each lattice site can be occupied
at most once). The conformation of the chain is described
by the monomer coordinates ri (i ⫽ 1, 2, . . . , N). The
energy of a given conformation for an adsorbed protein is
assumed to be the sum of the energies ⑀ij associated with
topological contacts [a topological contact is formed whenever two nonbonded monomers i and j (兩i ⫺ j兩 ⱖ 3) are
nearest neighbors] and the adsorption energies (⑀ia ⬎ 0),
兩i ⫺ j兩 ⱖ 3
⑀ij ␦共rij ⫺ a兲 ⫺
⑀ia ␦共zi兲,
Fig. 1. Model protein-like molecule in the folded state. Open and filled
circles show A and B monomers, respectively.
where zi ⱖ 0 is the monomer coordinate perpendicular to
the surface, a the lattice spacing, and ␦(0) ⫽ 1 and 0
otherwise. The monomers located in the first layer (with
zi ⫽ 0) are assumed to interact with the surface. The
interaction of the surface with monomers located in the
layers with zi/a ⱖ 1 is neglected. All the interactions in
Eq. (1) are effective because the protein-solvent interaction
is not explicitly taken into account.
Expression (1) is formally applicable both for biological
and nonbiological polymers. In particular, it has been
widely employed to study the statistics of adsorption of
neutral nonbiological polymers14 (for adsorption of charged
chains, see Kong and Muthukumar15). To apply expression
(1) to proteins, one needs to introduce representative
interaction energies ⑀ij resulting in a well-defined native
state in the bulk of the solution. The results presented
below have been obtained for a model protein molecule
(Fig. 1) designed by Shakhnovich and Gutin.16 It contains
27 monomers of two types (A and B) with energies ⑀AA ⫽
⑀BB ⫽ ⫺ 3 and ⑀AB ⫽ ⫺ 1 (for energy and temperature
we use dimensionless units with kB ⫽ 1). In the bulk, this
molecule is known to have a unique folded state with E ⫽
⫺ 84. The probability of finding a protein in this state is
considerable at temperatures below the folding temperature, T ⬍ Tf ⫽ 1.3.17,18 The folded state can be reached
provided that the temperature is above the glass transition temperature, T ⬎ Tg ⫽ 1.1 (at T ⬍ Tg, a protein is
typically trapped into one of the metastable states for a
period which is much longer than any relevant time scale).
With increasing temperature above Tf, the molecule is in
the collapsed globular state at Tf ⬍ T ⬍ Tc ⫽ 2.0 (Tc is
the collapse temperature) and then (at T ⬎ Tc) in the
random-coil state.
Adsorption and denaturation of this molecule have been
explored in our earlier work19 in a wide range of temperatures from T ⫽ 1.2 ⬍ Tf up to T ⫽ 2.4 ⬎ Tc (during
adsorption, the molecule structure is more or less compact
even if T is slightly above Tc). The present simulations of
surface diffusion of this molecule have been executed in
the same range of temperatures for two sets of the
monomer-substrate interactions. The first one, ⑀A
⫽ 3 and
⑀B ⫽ 1, corresponds to the case when the interaction of A
monomers with the surface is much stronger than that of B
monomers. The second one, ⑀A
⫽ 3 and ⑀B
⫽ 2, represents
the case when the A and B interactions with the surface
are nearly equal. Physically, these two sets of interactions
may qualitatively describe the situations when the protein
structure in the adsorbed state depends primarily on
hydrophilic and hydrophobic interactions. (i) If for example the surface is hydrophobic, the monomer-substrate
interaction is not strongly affected by adsorbed water
molecules (during protein adsorption these molecules can
be easily pushed out of the first layer) and accordingly the
interaction of the surface with hydrophobic monomers will
be much stronger than that with hydrophilic monomers.
This case can qualitatively be described by assuming the A
monomers to be hydrophobic, the B monomers to be
hydrophilic, and using the first set of the monomera
substrate interactions (⑀A
⫽ 3 and ⑀B
⫽ 1). (ii) If on the
other hand the surface is hydrophilic, the monomersubstrate interaction is considerably weakened by adsorbed water molecules (these molecules can hardly be
pushed out of the hydrophilic surface) and accordingly the
difference of the interactions of different monomers with
the substrate will not be very large. This case can qualitatively be described by assuming the A monomers to be
hydrophilic, the B monomers to be hydrophobic, and using
Fig. 2. Energy (top panel) and numbers of monomers in the adsorbed
overlayer and also in the two bulk layers closest to the surface (curves 1,
2, and 3 on the bottom panel)) in the course of diffusion of a model
protein-like molecule with ⑀Aa ⫽ 3 and ⑀Ba ⫽ 1 at T ⫽ 1.2 (a) and 2.4 (b).
Time is calculated in MCS (1 MCS is defined as 27 attempts to realize a
monomer move). The kinetics have been calculated after the denaturation
procedure which contained 107 and 105 MCS at T ⫽ 1.2 and 2.4,
the second set of the monomer-substrate interactions (⑀A
⫽ 3 and ⑀B ⫽ 2).
Initially, the molecule was considered to be adsorbed in
the native folded state (Fig. 1) so that one of its sides
contacts the surface. To realize denaturation of the molecule, we executed 105–107 MCS [1 MCS (MC step) is
defined as 27 attempts to realize a monomer move]. After
denaturation, we executed 2 ⫻ 105 ⫺ 2 ⫻ 107 MCS to find
displacements of the molecule along the X and Y axes, ⌬x
⫽ ¥i ⌬xi/N and ⌬y ⫽ ¥i ⌬yi/N. Both procedures (denaturation and diffusion) were run for longer times at lower
temperatures. The diffusion coefficient was calculated as
To simulate diffusion of adsorbed proteins, we have used
the standard algorithm for end, corner and crankshaft
monomer moves: A monomer is chosen at random. If it is
an end monomer, then one of the neighboring lattice sites
is also selected at random for an end move. If it is not an
end bead, then, depending on the position of its neighbors
along the chain, it can perform either a corner move or a
crankshaft move (the direction of the latter move is
selected at random). If the move chosen would violate the
excluded volume constraint by moving the monomer to an
occupied site, the trial ends. If there are no spatial
constraints, the energies of the original and new configurations are calculated, and the move is realized with the
probability given by the Metropolis rule [W ⫽ 1 for ⌬E ⱕ
0, and W ⫽ exp(⫺⌬E/T) for ⌬E ⱖ 0, where ⌬E is the
energy difference of the final and initial states]. This
algorithm does not include collective modes of motion and
accordingly it can hardly be used to describe “skating” or
“rolling” of a protein along the surface. The role of such
modes is expected to be minor provided that adsorption is
accompanied by denaturation because in the latter case
diffusion may primarily occur via the rearrangements of
the whole protein structure due to local moves.
D ⫽ 关具共⌬x兲2典 ⫹ 具共⌬y兲2典兴/4⌬t,
where ⌬t is the diffusion time measured in MCS. To get the
mean-square displacement, we used seven MC runs.
Using local elementary moves described above, one can
observe protein diffusion only at T ⬎ Tg, because at T ⬍
Tg a protein is expected to be trapped into one of the
metastable states and accordingly there will be no displacements of the whole molecule. The glass transition temperature Tg ⫽ 1.1 introduced in the section “MODEL” corresponds to the bulk. In general, the value of this temperature
is known to depend on smoothness of the energy landscape. Rough energy landscapes resulting in high values of
Tg occur in problems in which there are many competiting
Fig. 4. Arrhenius plots for surface diffusion of the model protein
molecule with ⑀Aa ⫽ 3 and ⑀Ba ⫽ 1. The average statistical error in the
results is shown by the size of the data points.
Fig. 3. Mean-square displacement of the model protein molecule as a
function of time for ⑀Aa ⫽ 3 and ⑀Ba ⫽ 1, at T ⫽ 1.2 (a) and 2.4 (b). The
inserts exhibit typical protein structures during diffusion. The monomers
contacting the surface [20 and 17 beads in cases (a) and (b), respectively]
are located in the first layer (in the plane of the figure). The other
monomers are located in the second layer.
interactions20 (in analogy with the spin glass theory, this
competition is sometimes called “frustration”). For adsorbed proteins, due to frustration, Tg might be higher
than that for the bulk. For our model parameters, this
effect seems to be negligible, because for reasonable lengths
of MS runs (up to 2 ⫻ 107 MCS) we were able to observe
diffusion at temperatures (e.g., T ⫽ 1.2) slightly above
than that corresponding to the glass transition in the bulk.
The data obtained for the first set of the monomera
substrate interactions (⑀A
⫽ 3 and ⑀B
⫽ 1) are displayed
in Figures 2– 4. In this case, the monomers are located
(Fig. 2) primarily in the first and second layers both at low
and high temperatures (T ⫽ 1.2 and 2.4, respectively).
During diffusion, the fluctuations in the number of monomers in these layers are seen to be appreciable and
accordingly the changes of the protein structure are considerable. The typical mean-square displacements (Fig. 3) are
proportional to time as it should be for diffusion. The
dependence of the diffusion coefficient on temperature is
found to be linear in the Arrhenius coordinates (Fig. 4),
i.e., it can be represented in the standard form, D ⫽ Do
exp(⫺Ea/T), where Do ⫽ 0.18 ⫻ 10 ⫾ 0.2 a2/MCS and
Ea ⫽ 12.6 ⫾ 0.6 are the preexponential factor and
activation energy, respectively. Near Tg, one could expect
deviations from the Arrhenius behavior because the dynamics of relaxation of the protein structure in this case is
non-exponential19 (see also relevant simulations21,22 of
the relaxation of adsorbed nonbiological polymers), but in
our case the Arrhenius representation seems to be applicable down to Tg.
To convert the preexponential factor calculated in MC
units into real units, we can compare the value obtained
with that corresponding to an ideal random walk of a
single monomer. In the latter case, the prexponential
factor calculated in MC and real units is respectively given
by 0.25 a2/MCS and Do ⬇ 10 ⫺ 2 ⫺ 10 ⫺ 3 cm2/s (the latter
“normal” value directly follows from the transition state
theory12). The fact that for our model the preexponential
factor calculated in the MC units is close to 0.25 indicates
that in real units the preexponential factor should be close
to the normal value, i.e., Do ⬇ 10 ⫺ 2 ⫺ 10 ⫺ 3 cm2/s.
For the second set of the monomer-substrate interaca
tions (⑀A
⫽ 3 and ⑀B
⫽ 2), the dependence of the protein
structure on temperature is stronger. At temperature
below Tf, almost all the monomers are located in the first
layer (Fig. 5a). The occupation of the second layer is
considerable at higher temperatures (Fig. 5b). The former
finding is quite different compared to that obtained for the
first set of the monomer-substrate interactions (cf. Figs. 2a
and 5a). Despite this difference, the Arrhenius parameters
found for the second set of the interactions, Do ⫽ 0.15 ⫻
10⫾0.2 a2/MCS and Ea ⫽ 12.6 ⫾ 0.6, are very close to
those obtained for the first set.
Fig. 5.
As Figure 2 for ⑀Ba ⫽ 2.
Our MC simulations indicate that for the model under
consideration the preexponential factor for diffusion of
adsorbed proteins is close to the normal value (about
0.25a2/MCS in MC units or about 10⫺2 ⫺ 10⫺3 cm2/s in
real units) typical of monomer diffusion. The rationalization of the results obtained is not straightforward. Taking
into account that the preexponential factor found is close
to 0.25, one might suggest that diffusion is limited by the
movement of certain monomers, e.g., by corner moves. The
maximum change in energy involved in such moves is 12 (4
bonds of strength 3 are broken). The latter value is close to
the apparent activation energy (12.6) obtained in simulations. We are not however quite sure that this reasoning is
correct because in our simulations the monomers are
typically located in two layers (Fig. 3). For such configurations, the maximum change in energy during the corner
moves is 9 (3 bonds are broken). A more plausible explanation seems to be that diffusion is controlled by a combination of monomer moves complicated by spatial constraints.
The preexponential factor obtained in our simulations is
higher than that ( ⯝ 0.25/N) predicted for diffusion in a
random-coil state but much lower compared to that estimated for bovine serum albumin diffusion on polyhexylmethacrylate (see the Introduction). The former is not
surprising because in our simulations the typical structures of adsorbed protein molecules are rather compact
even at high temperatures (see e.g., Fig. 3b for T ⫽ 2.4).
The latter might indicate that conformational changes in
adsorbed bovine serum albumin are qualitatively different
compared to those predicted by our model (the structure of
adsorbed bovine serum albumin is much more stable than
that in our simulations).
Finally, we may conclude that diffusion of adsorbed
proteins merits additional theoretical studies. In particular, it is of interest to execute simulations for the same
model but with other dynamics of elementary moves or
explore other modes of diffusion including collective moves.
V.P. Zh. is grateful for the Waernska Guest Professorship at Göteborg University.
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