Vol.2, No.1, 1-8 (2012) Open Journal of Immunology
http://dx.doi.org/10.4236/oji.2012.21001
Do membrane proteins cluster without binding
between molecules?
Xin Wang, Toshihiko Fukamachi, Hiromi Saito, Hiroshi Kobayashi*
Department of Biochemistry, Graduate School of Pharmaceutical Sciences, Chiba University, Chiba, Japan;
*Corresponding Author:
Received 19 December 2011; revised 20 January 2012; accepted 15 February 2012
ABSTRACT
Clustering is a basic event for the initiation of
immune cell responses, and simulation analy ses
of clustering of membrane proteins have been
performed. It was claimed that a cluster is
formed by the self-assembly induced by protein
dimerization with a high binding speed (Woolf
and Linderman, Biophys. Chem. 104, 217-227,
2003). We examined the cluster formation with
Monte Carlo simulation using two algorithms.
The first was that simulation processes were
divided into two substeps. All proteins were
subjected to movement in the first substep, fol-
lowed by reaction in the second substep. The
second algorithm was that proteins were first
selected to react and proteins which did not re-
act were subjected to movement. The self-as-
sembly induced by dimerization was simulated
only with the second algorithm. In this algorithm,
monomers dissociated from dimers do not move
because these monomers are not selected for
movement, and a la rge proportion of s uch mono-
mers are selected to form dimers in the next
step. The self-assembly was again simulated
with the first algorithm cont aining the conditions
that monomers dissociated from dimers did not
move in the next movement substep. This algo-
rithm seems to be far remov ed from natural con-
ditions. Thus, it is inferre d that the self-assembly
induced by dimerization is unlikely in situ, and
that some interaction between proteins is re-
quired for cluster formation. In contrast to algo-
rithms in previous simulations, our results sug-
gest that it is more appropriate that proteins
move to the same direction for a while and re-
flect when the collision occurs.
Keywords: Cluster Formation; Monte Carlo
Simulation; Self-Assembly; Immune Cells;
Membrane Proteins
1. INTRODUCTION
Cellular signal transduction is initiated by the binding
of a ligand to its receptor. The receptor generally func-
tions in complex forms including homo- and hetero-
multimers before and after the ligand binding [1-5].
Clustering of transmembrane proteins on the cell surface
was proposed in the lipid raft model of the plasma mem-
brane [6]. Cholesterol, unsaturated sphingolipids and
lipid modified proteins etc. do not distribute uniformly in
the plasma membrane [7]. It is suggested that proteins
may exist in “protein islands” connected to the cy-
toskeleton molecules (protein island model) [8]. Foreign
antigens are recognized by T cell antigen receptors (TCR)
on the cell surface, and the T cells become activated to
initiate an immune response [9]. The membrane organi-
zation of TCR on the T cell surface has been investigated
[10-12]. Similarly, a linker of activated T cells (LAT)
was also proposed to exist in “protein islands” on the
surface of mast cells and T cells [13]. Microscopic tech-
niques have shown separate clusters of TCR and LAT in
pre-activated T cells, and these clusters transiently con-
catenate into microclusters upon antigen recognition [14].
The co-stimulation of TCR with CD28 was reported to
require co-localization of TCR and CD28 at the plasma
membrane [15].
It has remained unclear why such complex formation
is required for signal initiation and how the complex is
formed. It is hard to answer these questions experimen-
tally because we have still few useful methods to manipu-
late the complex formation without affecting the function
of the proteins themselves. One method to facilitate such
examination would be kinetic analysis with the aid of a
computer.
Two types of computer simulation techniques are now
available: numerical integration of differential equations
and Monte Carlo simulation. The former method can
address average behavior involving a large number of
molecules and stochastic variation. In contrast, the latter
can simulate both population behavior and single mole-
cule dynamics. Monte Carlo simulation can also evaluate
time-dependent fluctuations involving noise as well as

X. Wang et al. / Open Journal of Immunology 2 (2012) 1-8
2
cell-to-cell population heterogeneity [16].
Since one cell contains less than 100,000 molecules of
a given membrane protein and because there are varia-
tions in biological phenomena, the latter method may be
more appropriate. Receptor-ligand formation and clus-
tering of membrane proteins have already been simulated
with Monte Carlo techniques [16-21], and their results
revealed the usefulness of this technique for clarification
of biological phenomena.
Various physiological meanings of the clustering of
membrane proteins have been proposed [22-27], but the
mechanism for this cluster formation remains unclear,
although a few mechanisms have been proposed [17,18,28].
Woolf and Linderman [17] proposed that self-assembly
is induced by protein dimerization when the binding
speed is higher than the diffusion rate of proteins. We
found in this study that different algorithms for Monte
Carlo simulation gave different results concerning the
cluster formation. The self-organization proposed by
Woolf and Linderman [17] was seen in some algorithms,
while cluster formation independent of the rate of the
dimerization was simulated in other algorithms. We dis-
cussed which algorithm was more appropriate for the
simulation of complex formation of membrane proteins,
and concluded that the self-organization is unlikely in
situ.
2. METHODS
In the present study, a simplified model in which the
cell surface is represented as a 2-dimensional plane was
assumed, and the cell surface was divided into subspaces.
A single subspace was a cubic box with a volume of 166
(5.53) nm3, as described previously [16]. One molecule
per subspace corresponded to a concentration of 10 mM.
Each calculation step was assumed to take 0.02 milli-
seconds. In all events of our Monte Carlo procedure,
real-type pseudo uniform random numbers (N) with the
range 0 ≤ N < 1 were generated, as reported previously
[29]. All proteins were initially distributed into randomly
selected subspaces with equal probability. When a se-
lected subspace was occupied, the next subspace was
selected randomly. Dimer formation was assumed as
follows: The binding of two proteins was accepted when
the two proteins occupied neighboring subspaces and N
was less than exp(–ΔE1/RT), where ΔE1, R and T are the
activation energy, the gas constant and the absolute tem-
perature, respectively. Correspondingly, dimers dissoci-
ated when N was less than exp(–ΔE2/RT).
Each protein was assumed to have a movement direc-
tion (positive or negative direction on each axis), and a
diffusion rate (υM). In this study, the movement direction
was set randomly, and υM was set to υ or υ/10, where υ
had a Maxwell-Boltzmann distribution from 0 to 999.
The probability to have υ (P(υ)) was calculated as fol-
lows.
 
1000
0
P=B,where S=B ,S





and

  
2
62 22
B=2π
b
expb 2 


when
 
mm+1
0υ0
NP and N<P, was set to m.




When b was set to be 0.005, a Maxwell-Boltzmann
distribution of υ was obtained as shown in Figure 1(a),
and P(998) and P(999) were 1 and 0, respectively. Pro-
teins moved into their neighboring subspaces according
to their movement direction when υM > τ, where τ is a
pseudo uniform random number (0 ≤ τ <1000) obtained
by multiplication of N by 1000. Its integer part was used
for the rapid simulation. When υM = 0, the proteins re-
mained in the same subspace. Proteins moved to the op-
posite side based on periodic boundary conditions when
they reached the boundaries of the simulation box. If the
opposite side was occupied, the protein was reflected in
the mirror direction. If the protein was a part of a dimer,
the protein was allowed to pivot around its partner in a
random direction. If the target subspace was occupied,
the rotation was rejected and not repeated. The move-
ment and rotation of dimers occurred at the same simula-
tion step.
The present simulation included two events; move-
ment and reaction for the formation and dissociation of
dimers. We assumed the following methods for the selec-
tion of proteins subjected to movement or reaction.
MethodA = 0: Simulation processes were divided into
two substeps, reaction and movement. All proteins were
subjected to reaction and movement in the former and
latter substeps, respectively. The reaction substep was
carried out after the movement substep.
MethodA = 1: Proteins were first selected to react in
each step. Monomers and dimers were converted to dimers
and monomers, respectively, according to the reaction
probability described above. Proteins that did not react
were subjected to movement.
The movement directions of new molecules produced
by the formation or the dissociation of dimers were de-
termined randomly, and υM of such molecules was de-
termined as follows.
MethodB = 0: υM was always set to 999.
MethodB = 1: υM that had a Maxwell-Boltzmann dis-
tribution was set as described above.
MethodB = 2: υM was always set to 0.
When proteins were not reacted, the movement direc-
tion and υM of such proteins were updated as follows.
MethodC = 0: The movement direction and υM of all
molecules were updated immediately before the move-
ment in every step, and molecules moved according to

X. Wang et al. / Open Journal of Immunology 2 (2012) 1-8 3
their movement direction and υM as described above. If
the subspace was occupied, the movement was rejected
and not repeated.
MethodC = 1: The movement direction and υM of
0.1 % of molecules selected randomly were updated at
every step. If the subspace was occupied, the movement
was rejected and not repeated.
MethodC = 2: If the subspace was occupied, the mole-
cule was reflected in the mirror direction. υM was not
updated.
MethodC = 3: If the subspace was occupied, the mole-
cule was reflected in the mirror direction, and υM was
updated.
MethodC = 4: This method included the conditions of
both MethodC = 1 and MethodC = 2.
MethodC = 5: This method included the conditions of
both MethodC = 1 and MethodC = 3.
The trajectories of the membrane proteins are shown
in Figures 1(b) and (c).
Cluster size was defined as follows: All proteins pre-
sented in neighboring subspaces were defined as be-
longing to the same cluster, and the cluster size was
measured by counting all kinds of proteins in the cluster.
The source code of the computing program was im-
plemented using the C-language with Visual Studio
Figure 1. Distribution of diffusion rates and trajectories of mem-
brane protein movement. (a) Distribution of diffusion rates. For
details, see text. ((b) and (c)) The positions of a given protein were
plotted for 1 × 105 steps (2 sec) at intervals of 10 steps (0.2 msec).
MethodC = 0 (b) and MethodC = 5 (c) were used. The numbers of
subspaces and proteins set in this simulation were 80 × 80 and 960,
respectively. For details, see text.
C++.net (Microsoft Co.), and the program was run on a
personal computer under Windows XP or 2000 (Micro-
soft Co.). The source code is available from the corre-
sponding author upon request.
3. RESULTS AND DISCUSSION
In the present simulations, the binding probabilities
(exp(–E1/RT)) were set as shown in Tab le 1. To simu-
late the binding rate constant (k), the simulation surface
was assumed to contain 80 × 80 subspaces and the num-
ber of proteins was set to 960. The binding rate constant
was calculated from 100 simulated values with 10 dif-
ferent E1 values. The constants were obtained with two
other simulation surfaces consisting of 50 × 50 and 100 ×
100 subspaces containing 750 and 500 proteins, respec-
tively. The average values are shown in Table 1. The
dissociation probabilities (exp(–E2/RT)) were set to
one-tenth of the binding probability in all simulations.
The average diffusion coefficients calculated from the
moving distances of 1000 proteins as described previ-
ously [16] are shown in Table 2. In this calculation, pro-
teins are allowed to move even if the target subspace is
occupied.
Table 1. Binding rate constants.
E1/RT*

binding
probability
binding rate
constant (k)**log(k)
0.01 0.99
(1.04 ± 0.32) × 107 7.02
0.11 9.0 × 10–1 (9.75 ± 2.90) × 106 6.99
1.20 3.0 × 10–1 (4.51 ± 0.68) × 106 6.65
2.41 9.0 × 10–2 (1.61 ± 0.07) × 106 6.21
3.50 3.0 × 10–2 (5.45 ± 0.42) × 105 5.74
4.71 9.0 × 10–3 (1.58 ± 0.06) × 105 5.20
5.81 3.0 × 10–3 (5.38 ± 0.51) × 104 4.73
7.01 9.0 × 10–4 (1.60 ± 0.07) × 104 4.20
8.11 3.0 × 10–4 (5.66 ± 0.31) × 103 3.75
9.32 9.0 × 10–5 (1.64 ± 0.05) × 103 3.21
*See Methods, **M–1·sec–1.
Table 2. Diffusion coefficients.
MethodC 
M Diffusion coefficient (m2/s)
0  0.166 ± 0.005
0   0.0169 ± 0.0004
1  12.8 ± 0.5
1   0.145 ± 0.003

X. Wang et al. / Open Journal of Immunology 2 (2012) 1-8
4
In the first simulation (Figure 2), a simulation surface
consisting of 80 × 80 subspaces and 960 monomers were
set. Fifteen percent of subspaces were occupied by pro-
teins under these conditions. The average cluster size
was the same at all binding rate constants in Method
0-1-0 (This means MethodA = 0, MethodB = 1, MethodC
= 0) as shown in Figure 2(a). This cluster may be
formed without interaction between molecules at a high
protein density due to proteins not being distributed uni-
Figure 2. The average cluster size and the number of dimers
when the molecular density was 4950 proteins per μm2 and υM
was υ/10. The cell surface consisted of 80 × 80 subspaces, and
the number of proteins was initially set to 960. Fifteen percent
of subspaces were initially occupied with proteins. The diffu-
sion rate of proteins (υM) was set to υ/10. Methods used are
indicated in the figures. After the reaction reached equilibrium
stage, the total number of monomers and dimers was calculated
in each cluster at each step, and average values were obtained
(closed circles). The number of dimers at each step was calcu-
lated and the average percentage of proteins that formed dimers
was obtained (open circles). Each point represents the average
values obtained from 100 measurements, and standard devia-
tions were less than 5% in all measurements. The horizontal
line represents the binding rate constant (k).
formly at a given moment. The size of such clusters in-
creases as the protein density increases as shown in Fig-
ures 2(a), 3(a) and 4(a).
Woolf and Linderman [17] proposed that the cluster-
ing increased when the binding rate constant was high. In
their simulation, molecules were first subjected to reac-
tion and molecules that were not reacted were subjected
to movement. The cluster size increased as the binding
rate constant increased under their conditions (Method
1-1-0, Figure 2(b)). The same results were obtained in
Method 1-0-1, Method 1-1-1, and Method 1-1-5 (data
not shown). In this method (MethodA = 1), monomers
dissociated from dimers do not move because these
Figure 3. The average cluster size and the
number of dimers when the molecular density
was 9900 proteins per μm2 and υM was υ/10.
The simulation conditions were the same as in
Figure 2 except that the cell surface consisted
of 50 × 50 subspaces and the number of pro-
teins was initially set to 750. Thirty percent of
subspaces were initially occupied by proteins.

X. Wang et al. / Open Journal of Immunology 2 (2012) 1-8 5
monomers are not selected for movement. In the next
step, a large proportion of such monomers are selected
again to form dimers when the binding rate constant is
high. This means that a large proportion of dissociated
monomers form dimers again without moving at the next
step. Therefore, the cluster size is larger at a high binding
rate constant. In contrast, when the reaction and move-
ment events are repeated at every step (MethodA = 0),
dissociated monomers have the same potential to associ-
ate as monomers formed at previous steps. The same
results were obtained with different protein densities
except that the cluster size and number of dimers in-
creased as the density increased ((a) and (b) in Figures
2-4). To confirm this explanation, the moving energy of
dissociated monomers was set to zero in MethodA = 0,
i.e., such monomers do not move in the next step. As
shown in Figure 2(c), the cluster size increased as the
binding rate constant increased.
The increase in the cluster size accompanies a decrease in
entropy. In the simple model used in this simulation, no
additional energy was supplied for the decrease in en-
tropy when the binding rate constant increased. Therefore,
it is reasonable to assume that cluster size is constant at
any binding rate constant, suggesting that MethodA = 1 is
inadequate. MethodB = 0 and MethodB = 2 seem to be
far removed from natural conditions. Therefore, MethodA
= 0 and MethodB = 1 seem to be adequate.
The next point is which method is more appropriate in
MethodC. In the method described above, the diffusion
rates and movement directions of all molecules were
updated immediately before the movement in every step.
However, it is more reasonable to assume that each
molecule has a different molecular activity, namely a
different diffusion rate, and keeps the same energy for a
while. Therefore, MethodC = 0 is less likely.
The question is thus when does the molecular activity
change? We first assumed that 0.1 % of molecules se-
lected randomly were updated in every step (MethodC =
1). The average cluster size decreased as the binding rate
constant increased in these conditions, while the decrease
in the number of dimers was small (Figure 2(d)). This
decrease was similar when the density of proteins in-
creased 2-fold (Figure 3(c)) and small at a low density of
proteins (Figure 4(c)). In this simulation, when a protein
ran against another protein, its movement was cancelled
and its movement direction was not updated. When the
binding rate constant was high, each such protein formed
a dimer with its neighboring protein immediately, and the
movement direction of the dimer was newly assigned.
Consequently, the dimer moved away, resulting in a de-
crease in the cluster size. In contrast, when the binding
rate constant was low, proteins that ran against another
protein in cluster stayed in the same subspaces for a long
time until the proteins were subjected to reaction. This
Figure 4. The average cluster size and
the number of dimers when the molecu-
lar density was 1650 proteins per μm2
and υM was υ/10. The simulation condi-
tions were the same as in Figure 2 ex-
cept that the cell surface consisted of
100 × 100 subspaces and the number of
proteins was initially set to 500. Five
percent of subspaces were initially oc-
cupied by proteins.
may be the reason for the increase in cluster size at a low
binding rate constant.
It is likely that the molecular energy is changed when
a collision between molecules occurs in the natural case.
In the next simulation, the diffusion direction was up-
dated only when a protein ran against another protein
(MethodC = 2). The average cluster size was the same
for all binding rate constants (Figure 2(e)). The same
results were obtained when both diffusion rate and direc-
tion were updated only when a molecule ran into another
molecule (MethodC = 3, Figure 2(f)). It is likely that
energy is released in a open space even if there is no col-

X. Wang et al. / Open Journal of Immunology 2 (2012) 1-8
6
lision. The same results were obtained in MethodC = 4
and 5 which included the updating of 0.1% of molecules
at every step (MethodC = 1) in addition to the conditions
of MethodC = 2 and 3, respectively (Figures 2(g) and
(h)). The cluster sizes were the same again in MethodC =
2 to 5 at the protein densities described in Figures 3 and
4 (data obtained with MethodC = 2 to 4 are not shown).
The binding rate constants measured experimentally
were less than 1 × 107 M–1·sec–1 [30-33]. When the bind-
ing rate constant was less than 5 × 106 M
–1·sec–1, all
methods used in the present simulation gave similar re-
sults except Method 0-1-1 (Figures 2-4). However, it
may be better to use Method 0-1-5.
In MethodC = 1, 4, and 5, 0.1% of proteins selected
randomly were updated in every step. It remains unclear
whether or not this setting is the most appropriate. The
trajectories in MethodC = 5 (Figure 1(c)) were similar to
those observed experimentally [34]. Although more de-
tailed experimental data are required for more proper
setting, 0.1% is probably appropriate.
When the diffusion coefficient of molecules was in-
creased 10 fold (υM = υ), similar results were obtained
except that the cluster size decreased more dramatically
as the binding rate constant increased as compared with
the lower diffusion coefficient (Figure 5(c)). The diffu-
sion coefficient of membrane proteins observed experi-
mentally was 0.1 to 0.3 μm2·sec–1 in prokaryotes [35] and
eukaryotes [36,37], and the diffusion coefficient in the
setting of υM = υ was 12.8 μm2·sec–1 in MethodC = 1.
Therefore, this setting may be less appropriate. It should
be noted that the same diffusion coefficient was obtained
in MethodC = 1 to 5 because proteins were allowed to
move even if the target subspace was occupied when the
diffusion coefficient was calculated.
Our present simulations with appropriate algorithms
demonstrated that the cluster size was dependent on nei-
ther the diffusion coefficient nor the binding speed of
proteins at all protein densities tested. Thus, the self-as-
sembly induced by protein dimerization with a high
binding speed is unlikely in situ.
GPCRs have been shown to form not only dimers but
also oligomers [23,38-40], but structural studies of these
receptors have suggested them to have only one pro-
tein-protein binding site [41]. It may be possible for a
membrane protein complex to be formed without such
binding site. One possibility is that the hydrophilic sur-
face regions of membrane proteins might bind each other
in the membranes. Another possibility is that matrix pro-
teins in the outer or inner cell surface trap membrane
proteins in a local area to increase the protein density. It
was observed that membrane proteins undergoing Brownian
diffusion were confined within a limited area, probably
by the binding to a membrane-associated cytoskeleton
network [42]. In any case, some interaction between
Figure 5. The average cluster size and
the number of dimers when the molecu-
lar density was 4950 proteins per μm2
and υM was υ. The simulation conditions
were the same as in Figure 2 except that
the diffusion rate of proteins (υM) was
set to υ.
proteins may be required for the cluster formation of
membrane proteins on the cell surface at a low protein
density observed experimentally.
4. CONCLUSIONS
We examined the cluster formation with Monte Carlo
simulation using two algorithms. The first one was that
simulation processes were divided into two substeps. All
proteins were subjected to movement in the first substep,
and then subjected to reaction in the second substep. The
second algorithm was that proteins were first selected to
react and then proteins that did not react were subjected
to movement in each step. The self-assembly induced by
protein dimerization with a high binding speed, which
was claimed by Woolf and Linderman [17], was simu-

X. Wang et al. / Open Journal of Immunology 2 (2012) 1-8 7
lated with the second algorithm, while the cluster size
was dependent on neither the diffusion coefficient nor
the binding speed of proteins with the first algorithm. In
the second algorithm, monomers dissociated from dimers
do not move because these monomers are not selected
for movement, and a large proportion of such monomers
are selected to form dimers before their movement in the
next step. The self-organization was again simulated in
the former algorithm containing the conditions that the
monomers dissociated from dimers did not move in the
next movement substep. This algorithm seems to be far
removed from natural conditions. Thus, it is inferred that
the self-assembly induced by protein dimerization is
unlikely in situ, and that some interaction between pro-
teins is required for the cluster formation.
The second algorithm has been used in many previous
works, but the present simulation suggests that the first
one is more appropriate. We also examined which algo-
rithm was more appropriate for the molecular movement.
It has been assumed in many previous simulations that
molecules move to the neighboring subspace randomly
in each simulation step. In this study, it was shown to be
more appropriate that molecules continued to have the
same movement direction for a while and the direction
was changed at the step selected randomly. Many previ-
ous studies adopted the algorithm that the movement was
cancelled when the collision occurred. The present study
demonstrated that this algorithm was less appropriate,
and molecules should change their movement direction
in a mirror manner when the neighboring subspace was
occupied. It should be clarified in future simulations
which interaction is required for clustering of membrane
proteins observed experimentally using these appropriate
methods.
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