COMPUTER SIMULATION OF POLYAMIDOAMINE

COMPUTER SIMULATION OF POLYAMIDOAMINE
DENDRIMERS AND THEIR COMPLEXES WITH
CISPLATIN MOLECULES IN WATER ENVIRONMENT
N.K.Balabaev
1
, V.V.Bessonov
1
, I.M.Neelov
2,3
, M.A.Mazo
4
1
Institute of Mathematical Problems of Biology RAS, Pushchino,
142290 Russia
2
School of Physics and Astronomy, University of Leeds, Leeds,
LS2 9JT United Kingdom
3
Laboratory of Polymer Chemistry, University of Helsinki,
P.O.Box 55, Helsinki, Finland
4
Institute of Chemical Physics RAS, Moscow, 119991 Russia
ABSTRACT. Molecular dynamics simulations with explicit water were
carried out for guest-host systems on the base of PAMAM-4.5
dendrimers and cisplatin PtCl
2
(NH
3
)
2
molecules. Single dendrimer

molecule and cisplatin molecules chemically attached to dendrimer
terminal groups or adsorbed on the macromolecular surface were
considered. AMBER force field, TIP3P water molecules and periodical
boundary conditions were used for calculations. It is no protonated
amines of PAMAM that correspond to pH ≥ 10. Computer experiments
were conducted at temperatures 293, 310, and 350 K and pressure 1 bar.
The structure and dynamics of guest-host systems was analysing. In all
considered cases the dendrimers form a compact globule, which shape is
far from spherical. Moreover the dendrimer cores dispose on the
molecule surface in all considered cases. The chemically attached
cisplatin penetrate into dendrimer deeper then non-attached one and
decrease a large-scale intramolecular mobility.
1. Introduction
The compound cis-PtCl
2
(NH
3
)
2
(cisplatin), has become one of the
most widely used drugs for the treatment of cancer [1,2]. Its mechanism
of interaction with cells has been studied on a molecular level and it is
well established, that death of the cell is induced by complexes of the
platinum compound to two adjacent guanine bases [2]. But its remarkable
anticancer properties can be accompanied by marked toxic effects as well
as the development of resistance to the drug. Recently a polyamidoamine
(PAMAM) dendrimer generation 4.5 was conjugated to cisplatin giving a
dendrimer-platinate (dendrimer-Pt) which was highly water soluble and
released platinum slowly in vitro [3,4]. In these works was shown that
the dendrimer-Pt improved cisplatin efficiency and was less toxic (3- to

15-fold).
4 - 24
PAMAM dendrimers are the polymers with unidispersed and well-
defined molecular structures. These molecules can be synthesized in
large quantities and have a large number of potential biomedical
applications [5,6]. Highly branched, functionalized polymers have
potential to act as a gene delivery and as efficient drug carrier systems
[5,7-9]. There are a lot of publications devoted to the study of structure
and mobility both single dendrimers, and dendrimer-guest systems.
However till now our knowledge of molecular structure of this systems
are rather fragmentary.
The essential contribution to our understanding of the dendrimer
molecules structure was achieved by molecular dynamic (MD)
simulation [10-13]. Recently, more exhaustive atomistic MD simulations
of dendrimers were carried out [14-25] including simulation PAMAM
molecules [17,19,21,24] and research a solution behavior of dendrimers
in explicit solvent [20,22,24]. Unlike simple coarse-grain dendrimer
models the simulation of detailed molecular systems is rather
complicated, as requires a substantiation for the large number of used
parameters, very expensive, the received results, as a rule, and do not
suppose wide generalization. However now it is the only way to receive
the detailed information on spatial structure and mobility of separate
molecules in solvent.
In this study, we used atomistic MD simulations with explicit water
to study the guest-host systems on the base of PAMAM-4.5 and cisplatin
molecules chemically attached to dendrimer terminal groups or adsorbed
on the macromolecular surface were considered.
2. The model and simulation details
Calculations were performed for three systems: a) water solutions of
PAMAM-4.5, b) water solutions of PAMAM-4.5 with cisplatine, and c)

water solutions of dendrimer-Pt molecule. PAMAM have tetrafunctional
core –NCH
2
CH
2
N– with three radiating branches of –
CH
2
CH
2
CONHCH
2
CH
2
N– and 64 terminal groups
–CH
2
CH
2
COOH (Fig.1a). The dendrimer–Pt made up by joining 7
cisplatines
–PtCl(NH
3
)
2
to random choosen termal groups (Fig.1b). The molecular
weights of cisplatine, PAMAM-4.5 and dendrimer-Pt are
correspondingly 299 a.u., 11380 a.u. and 13164 a.u The calculation cell
contained one dendrimer molecule, explicit water molecules, and 8
cisplatin molecules in system b (Tabl.1).

4 - 25
Pt
Cl
NH
3
O
O
O
O
O
O
O
O
O
O
O
O
O
O O
O
O
O
O
N
N
N
N
O
N

N
O
N
N
N
N
O
O
NH
3
NH
3
O
Pt
Cl
O
Pt
Cl
NH
3
NH
3
NH
3
O
O
O
O
O
O

O
O
O
O
O
O
O O
O
O
O
O
N
N
N
N
O
N
N
O
N
N
N
N
O
O
a b
Fig. 1. Schematic drawing of PAMAM-4.5 (a) and dendrimer-Pt
molecule (b).
Tabl. 1. Some details of considered systems.
System

Number
of water
molecules
Density
at 293
K,
g/cm
3
Dendrimer
weight
concentrati
on
Cisplatin
weight
concentrati
on
a) PAMAM-4.5 3230 1.05 16.0% -
b) PAMAM-4.5 +
cisplatin
3274 1.07 18.6% 3.9%
c) Dendrimer-Pt 3230 1.07 18.5% 2.6%
a)
a)
for –PtCl(NH
3
)
2
.
The AMBER force field [26] was used for calculation. The potential
energy comprising potential terms of bond U

b
, angle U
a
, torsion U
t
, van
der Waals U
vdw
and electrostatic U
e
interactions was used:
∑
−=
,)(
2
0
llKU
lb
,)(
2
0
∑
Θ−Θ=
Θ
KU
a
∑
+=
)],cos(1[
0

ϕδ
ϕ
nKU
t
∑∑
=
)()(
ijijLJvdw
rWrUU
,
where
])/()/[(4
612
ijijijijijLJ
rrU
σσε
−=
and W(r
ij
) is the switching function in interval 0.9 ≤ r
ij
≤ 1.05 nm,
∑∑
=
),()]/([
ijeijjie
rWrqqU
ε
4 - 26
where W

e
is the screening function (R
e
= 1.05 nm),










>
≤−
=
eij
eijeij
ije
Rr
RrRr
rW
,0
,)/1(
)(
2
In this equations the following notations are used: l is the bond
length, Θ is the bond angle, ϕ is the torsion angle, l
0

, Θ
0
are equilibrium
values for the bond lengths and angles; K
l
, K
Θ
, K
ϕ
are force constants for
the bonds, angles, dihedrals angles, respectively; n
0
is the dihedral
multiplicity; r
ij
is the distance between nonbonded atoms i and j; ε
ij
, σ
ij
are Lennard-Jones parameters for the atom pairs, q
i
, q
j
are the partial
charges on atoms i, j, ε is the dielectric constant, R
e
is the screening
radius. It is no protonated amines of PAMAM that correspond to pH ≥
10.
The H

2
O geometry parameters, the partial charges [27], and the
force constants [28] for TIP3P water molecules are fixed. The parameters
of force field and partial charges of cisplatin were taken same, as in [29].
Periodical boundary conditions were applied and the cells size was large
enough (∼4.8
3
nm) to exclude any intaraction between dendrimers.
Molecular dynamics techniques [30] were used for the equilibration and
regular simulation. Collisional thermostat [31] and Berendsen barostat
[32] were used for temperature and pressure support. The integration
time step ∆t=0.5 fs was used and the times of regular runs were 1 ns.
Initial structure of a complex polymer system (coordinates and
velocities of all atoms) plays the key role for successful modeling of its
behavior. The preparation of representative structure of the system is
usually complex and expensive procedure. Some technology was
elaborated to construct dendrimers under consideration. At the first stage
the special procedure was used to assemble the dendrimer structure,
which was like a dandelion flower (Fig.2). A combination of constructor
and collisional dy-
4 - 27
a
b
c
Figure 2. Simulated snapshot of initial dandelion structure at the first
stage (left) and configurations of dendrimers after 1 nsec runs (right).
(a) PAMAM-4.5, (b) PAMAM-4.5 and cisplatin, (c) dendrimer-Pt.
Water molecules are not shown to not complicate a picture.
4 - 28
namics computer programs were used to build up each next generation of

isolated dendrimer molecule during this procedure. At the second stage
collissional molecular
dynamics technique was applied to equilibrate initial configuration of the
dendrimer molecule. Than the macromolecule was immersed in water
and equilibration of the total system was accomplished. The structure
received was initial one for productive run of the system.
3. Results and discussions
One of the characteristics of the dendrimer size is the radius of
gyration R
G
. Its values averaged over the whole trajectory are given in
Table 2. From this table we notice that this value of PAMAM in water
correlate well with the evidence of another authors [14,18,20]. The
radiuses of gyration of dendrimers with adsorbed and with chemically
attached cisplatin are bigger and at the same. That curiously enough so in
the former case the cisplatin molecules were not taken into account along
calcuculations.
In all considered cases the dendrimers forms rather compact globule,
which shape is far from spherical (Fig. 2). This is apparent also from
Table 2 where the values of the main radiuses of inertia
M/JR
αα
=
(J
α
are the principal moments of gyration tensor, α=1,2,3, J
1
> J
2
> J

3
) and
relative values J
2
/J
1
and J
3
/J
1
are shown. The difference of this ratios from
1 characterise the deviation of dendrimer shape from the sphericity. In
our case we see a strong asymmetrical molecules that consistent with the
another simulation data for PAMAM [14,18,20], and the deviation from
the sphericity increase in the presence of cisplatin. The dendrimer size
and shape are independent of temperature at considered interval of
temperatures.
Tabl. 2. The radiuses of gyration R
G
(nm), the main radiuses of inertia
(nm), and the relative values J
2
/J
1
and J
3
/J
1
for PAMAM-3.5 and
dendrimer-Pt molecules.

PAMAM-4.5
in water solvent
PAMAM-4.5
in water solvent
with cisplatin
Dendrimer-Pt
in water solvent
293 K 310 K 350 K 293 K 310 K 293 K 310 K
R
G
1.33 1.34 1.34 1.45 1.44 1.47 1.48
R
1
1.22 1.22 1.23 1.36 1.36 1.40 1.40
R 1.11 1.11 1.11 1.20 1.21 1.22 1.21
R 0.89 0.93 0.93 0.94 0.92 0.95 0.96
J
2
/ J
1
0.84 0.82 0.82 0.78 0.79 0.77 0.78
J
3
/ J
1
0.54 0.55 0.57 0.47 0.46 0.46 0.46
4 - 29
The internal structure of the dendrimer and the distribution of the
solvent inside of it can be seen from radial density distribution functions
for dendrimer and solvent atoms relative to the centre of mass (CM) of

the macromolecule (Fig. 3a). By calculation of the density distribution,
all atoms were treated as uniform spheres with corresponding van-der-
Waals diameters. It is seen that water molecules are not incorporated into
the dendrimer. Only one water molecule dispose near the dendrimer CM
in the system with adsorbed cisplatin. The density profiles are not
essentially changed with the temperature.
It was rather unexpectedly to discover that the dendrimer core
during the run was far from CM the center of mass of the
macromolecules and for PAMAM-4.5 in water even is farther, than for
another cases (Fig. 3b). Moreover as can be seen in Fig.4 the dendrimer
cores dispose on the molecule surface. So asymmetrical structure is likely
to be characteristic for small generation of PAMAM in water at high pH,
that distinguishes its from carbosilane and polyamindoamine dendrimers
[18,24,33].
0 , 0 0 , 5 1 , 0 1 , 5 2 , 0
0 , 0
0 , 5
1 , 0
1 , 5
2 , 0
2 , 5
1
2
3

W a te r
D e n d rim e r
}
R , n m
ρ ,

g /c m
3

0 , 0 0 , 2 0 , 4 0 , 6 0 ,8 1 , 0 1 , 2 1 , 4
0 , 0 0
0 , 0 1
0 , 0 2
0 , 0 3
0 , 0 4
0 , 0 5
0 , 0 6
ρ ,
g /c m
3
R , n m
1
2
3
G 0
a b
Fig. 3. Radial density distribution relative to the centre of mass of
dendrimer at T=293 K. (a) Dendrimer and water atoms; (b) the
contribution of dendrimer core atoms. 1 – PAMAM-4.5 in water
solvent; 2 – PAMAM-4.5 in water solvent with cisplatin; 3 –
dendrimer-Pt in water solvent.
4 - 30
a b
Fig. 4. Snapshots of PAMAM with adsorbed (a) and chemically attached
(b) cisplatin at 293 K. Here the dendrimers are shown as a ball-stick
model while the core and cisplatin as a spacefilling union of spheres

model where each atom is drawn as a sphere of its Van der Waals radius.
The allocation of the adsorbed and chemically attached cisplatin in
dendrimer is considerably differing: in the latter case it penetrates deeper
in macromolecule (Fig. 5). Chemically non-connected cisplatin can
desorbs as may be seen on snapshots (see, for example, in Fig.2b) and as
evidenced a high probability to find out the cisplatin on distances more
than 2.3 nm from CM (Fig.5).
Fig. 5. Radial density distribution
of cisplatin relative to the centre of
mass of dendrimer at T=293 K. 1 –
adsorbed cisplatin; 2 – chemically
attached cisplatin.
4 - 31
0,0 0,5 1,0 1,5 2,0 2,5 3,0
0,0
0,1
0,2
0,3
0,4
ρ,
g/cm
3
R, nm
1
2
Cysplatin
Intramolecular dynamics of dendrimers was assessed as a mobility
of branch centers and hydroxyl hydrogen atoms of the end groups. For
that we calculated the time dependences of the distances L(t) between
concerned atoms and CM. As would be expected the atom mobility

depend on temperature, but even at temperature 293 K L(t) of some ends
groups vary more than 0.8 nm during 1 ns for all cases. However the
mobility of chemically connected and adsorbed cisplatin is essentially
different (Fig.6). In the first case the mean value of maximum variation
of L(t) is 0.3 nm while one for chemically non-connected cisplatin is 0.9
nm. At that a chemically connection of the heavy groups to the dendrimer
ends cause some decrease of intramolecular mobility dendrimer-Pt.
0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0
1 , 0
1 , 5
2 , 0
2 , 5
3 , 0
3 , 5
T im e , p s
R , n m

0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0
0 , 5
1 , 0
1 , 5
2 , 0

R , n m
T im e , p s
a b
Fig. 6. Time dependence of the distances L(t) between adsorbed (a) and
chemically attached (b) cisplatins at 293 K.
Acknowledgment
The work was supported by ESF program SUPERNET, NWO (project 99

005 725), and INTAS (project 00-0712).
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