3.1. The Global Characteristics
The size and shape of the dendrimer are important for the practical applications of dendrimer in drug and gene delivery. There are various ways to estimate the size of a dendrimer. The calculation of the instant radius of gyration
is one of them:
where
is the molecular masses of the dendrimer as a whole and of its
i-th atom correspondingly and
is the distance from the
i-th atom to the center of mass of the dendrimer. The time evolution of
for the dendrimer with neutral and protonated histidine amino acid residues are shown in
Figure 2. It is easy to see that the instant size and fluctuations of the size are significantly higher for the protonated Lys-2Hisp dendrimer than for the neutral Lys-2His dendrimer.
Characteristics of the size of the Lys-2His and Lys-2Hisp dendrimers are collected in
Table 2. As can be seen, for the Lys-2Hisp dendrimer the value of each characteristic is greater than for the Lys-2His dendrimer. We will refer to particular characteristics from this table in the rest of the paper.
In
Figure 3a, we plot the average value of
for each temperature. The average
is significantly larger for the Lys-2Hisp dendrimer with protonated histidines, and it is close to
or dendrimers with similarly charged 2Lys or 2Arg spacers in dendrimers with repeating units Lys-2Lys and Lys-2Arg correspondingly. The size of Lys-2His with neutral histidine spacers is close to
for a dendrimer with neutral double glycine spacers (with Lys-2Gly repeating units).
Dendrimer shape is another important characteristic for applications. We used the asphericity parameter
for the evaluation of the dendrimer shape [
69,
70,
71]:
where
,
and
are the principal (characteristic) axes of the inertia tensor. The two limits
and
correspond to an infinitely elongated rod and an ideal sphere, respectively. Dendrimers of small generations usually have ellipsoidal shape, however, an increase of the generation number leads to the more and more spherical shapes of dendrimers. The asphericity parameter
at the temperature of 310 K is presented in
Table 2). For both dendrimers,
is very small, therefore their shape is close to spherical. At the same time, Lys-2His has a more spherical shape than Lys-2Hisp.
As the asphericity of our dendrimers at both pHs is small the diffusion of the Lys-2His and Lys2Hisp dendrimers in the solution can be described as the diffusion of a spherical particle with the Stokes radius. The coefficient of translational diffusion of the center of mass of dendrimer can be used for the calculation of the Stokes radius (hydrodynamic radius
), which is another characteristic of the dendrimer size. Another method for calculating this characteristic is to use the Kirkwood approximation [
72,
73]:
where
is a distance between two heavy atoms
i and
j. The additional characteristic is the characteristic ratio of
and
shown in
Figure 4a. It is easy to see that for both Lys-2His and Lys-2Hisp dendrimers the characteristic ratio
does not depend on temperature. Two theoretical limits [
74] of ratio
are a Gaussian coil (lower limit) and an impenetrable rigid sphere (upper limit). They are depicted in
Figure 4a. The ratio
is smaller for Lys-2Hisp and in this case it is closer to the Gaussian coil.
There are other characteristics of size as well. For example, sometimes it is important to know the position of the dendrimer–water interface. This interface could be evaluated from the outer boundary position of the dendrimer, i.e., as the radial distance from the center of the dendrimer to its spherical surface in solvent. In many simulations of dendrimers, it was shown that the behavior of dendrimers is similar to that of a dense sphere. For this model the outer boundary can be estimated as
(see
Table 2). Furthermore, the outer boundary of the dendrimer can be calculated from the position of the slip plane (the effective radius
), i.e., the boundary between the charged dendrimer surface and the diffusion layer of the salt solution. Later, we will more accurately determine the effective radius
from the electrostatic properties of the dendrimer.
Table 2 demonstrates that
and
are larger for Lys-2Hisp, than for Lys-2His.
The average position of the terminal groups
of the dendrimer is another way to evaluate this boundary.
can be calculated as the mean square radial distance from the center of the dendrimer to the nitrogen atoms in the terminal NH
groups:
Here, the summation is performed over all nitrogen atoms in the terminal NH
groups. The values of
are provided in
Table 2. The average position of the terminal groups
is again larger for Lys-2Hisp, than for Lys-2His. This is due to the different spacer length in both dendrimers, because the contour path from the dendrimer center to the ends consists of the spacers. In order to illustrate the effect, we calculated and plotted in
Figure 4b the distributions of the spacer length. It is easy to see that the distribution for the Lys-2Hisp dendrimer is shifted towards longer length of the spacer and is narrower due to electrostatic repulsion of protonated histidines in its spacers.
3.3. The Imidazole and Guanidine Pairing
An unusual slowdown of the orientational mobility in 2Arg arginine spacers of the peptide dendrimer with Lys-2Arg repeating units has been found recently in the NMR experiment [
36]. It was suggested that this is due to the arginine–arginine paring effect [
51,
52,
84] previously studied for dimers and short arginine peptides. However, we have shown in our last paper that the pairing effect in the Lys-2Arg dendrimer is too small and another explanation of the slowdown was found [
41]. In this paper, we focus on the possibility of pairing between imidazole groups [
84] in the Lys-2His and Lys-2Hisp dendrimers and on comparing this with the guanidine pairing in the Lys-2Arg dendrimer obtained in our recent work. All possible pairs of plane groups in each system can be divided into pairs: (i) between neighboring His (from the same 2His spacer) and (ii) between non-neighboring His (belonging to different spacers). The matrix element
is the number of imidazole–imidazole (or guanidine–guanidine) pairs at a distance
r between their centers, and with the angle
between the planes of these plane groups.
We studied pairing using the simple geometric model shown in
Figure 9. Each plane group (imidazole or guanidine) lies completely in one plane. Therefore, the angle between these groups is the angle between the normals
and
to the planes of corresponding groups (see
Figure 9). The coordinates
of the normal vector are the coefficients of the equation
for the plane and, as is known, can be found by solving the equation
where
are the coordinates for three atoms forming the plane.
Let
be the coordinates of the normal vector
to the plane of the first group (imidazole or guanidine). The coefficients
are obtained by solving the Equation (
10) for the first group. Using the same scheme, we obtained the coordinates
of the normal vector
to the plane of the second group (imidazole or guanidine). Then, the angle
between the two planes can be found as the angle between the vectors of the normal to these planes
The distance r between two groups can be found as the distance between the centers of these groups. Thus, for each plane group, the distribution matrix can be calculated.
The matrices
for neighboring and non-neighboring imidazole–imidazole pairs in Lys-2His and Lys-2Hisp are shown in the
Figure 10a–d. The angle between the planes can range from 0 to 90 degrees. In both cases, a cut-off at a distance of 1.8 nm was chosen. Pairs located at larger distance are of no interest for analysis. The neighboring imidazole groups (from the same spacer) cannot go beyond this limit due to topological constraints. For comparison, we have added similar maps (see
Figure 10e,f) for Lis-2Arg.
Comparison of two dimensional maps in
Figure 10a,c,e for the neighboring His–His (a), Hisp–Hisp (c) or Arg–Arg (e) belonging to the same spacer (2His, 2Hisp or 2Arg, respectively) side groups shows that
(a) neighboring uncharged imidazole groups of histidines (His) form stable His–His pairs at three possible distances r between the centers of the imidazole groups: the first narrow region is located at distances r of about 0.4 nm (at angles between their planes from 15 to 80 degrees), the second is the wide range of distances r from 0.7 to 0.9 nm (with possible angles from 0 to 90 degrees) and the third narrow region is at a distance nm (with angles from 5 to 90 degrees) with a small number of pairs;
(c) neighboring protonated imidazole groups of histidines (Hisp) do not form pairs at distances (r about 0.4 nm); the region of Hisp–Hisp pairs from 0.7 to 0.9 nm become smaller than that for His–His pairs and the pairs in this region are more structured (at the angles are close to 20 degrees, and at nm the angles are close to 80 degrees) and the third narrow region with a small number of pairs at nm is similar to that for uncharged His;
(e) for neighboring charged guanidine groups of arginine pairs (Arg-Arg), there is only the region of existence of Arg–Arg pairs at large distances r from 1.1 to 1.5 nm with a wide range of possible angles from 30 to 90 degrees with the most probable distance r near 1.3 and angles from 70 to 90 degrees.
Comparison of two dimensional maps in
Figure 10b,d,f for non-neighboring His–His (b), Hisp–Hisp (d), Arg–Arg pairs and (f) pairs belonging to different spacers (2His, 2Hisp or 2Arg, respectively) shows that
(b) non-neighboring uncharged imidazole groups of histidines (His) form stable His–His pairs at two possible distances r between their centers: the first narrow region is at distance r close to 0.4 nm (at angles between their planes from 10 to 90 degrees, mainly from 70 to 90) with a small number of pairs, the second one is a very wide unstructured region at distances from 0.6 to 1.6 nm;
(d) for non-neighboring protonated imidazole pairs (Hisp–Hisp), there is no the narrow region at r about 0.4 nm, but there is a wide region of existence of Hisp–Hisp pairs at distances r from 0.7 to 1.75 nm with a wide interval of possible angles from 25 to 90 degrees (with the most probable range of distances from 1.3 to 1.6 nm and angles from 65 to 90 degrees).
(f) for non-neighboring charged guanidine groups of arginine pairs (Arg–Arg), there are two regions: the first region is very narrow and compact (around r = 0.4 nm, with angles from 0 to 30 degrees; the second one is a wide region of existence of Arg–Arg pairs at distances r from 0.8 to 1.8 nm, and with a wide range of possible angles from 30 to 90 degrees (with the most probable distances r from 1.6 to 1.8 nm and angles from 70 to 90 degrees). It is interesting to note that this region does not exist for neighboring guanidine groups (belonging to the same spacer) in the Lys–2Arg dendrimer, and it appears in these dendrimers only for non-neighboring pairs.
Another important characteristic is the radial distribution obtained by averaging the
over all possible orientational angles between two plane groups:
The radial distributions for neighboring and non-neighboring pairs are shown in
Figure 11 at different temperatures. It is easy to see that the results practically independ of temperature, with the exception of some relatively small irregular fluctuations in
Figure 11b.
In the case of neighboring pairs of the studied plane groups, a sharp peak of exists in the region of the smallest possible distances nm between plane groups only for neutral His–His pairs. For charged Hisp–Hisp and Arg–Arg pairs, a peak at this distance also exists but its height is negligible. There is a more intense and broader second peaks for His–His and Hisp–Hisp pairs at distances r from 0.6 to 0.9 nm and a smaller third peak at distances close to nm. For Arg–Arg pairs, there is only a peak trace at and a rather wide and flat elevated area of the number of Arg–Arg pairs between and 1.5 nm.
In the case of non-neighboring pairs, a strong peak of
in the region of the minimum possible distances
nm is observed for neutral His–His pairs. For Hisp–Hisp pairs, the peak at this distance (
nm) is negligible while for Arg–Arg non-neighboring pairs it exists but it is three times smaller than for His-His pairs. Thus, the largest first peak at
nm for His–His pairs between non-neighboring groups is in Lys-2His (see
Figure 11b). This means that in this case there is the largest number of non-neighboring pairs of plane groups.
We integrated the area under the first peak in all systems to estimate the average number of these closest pairs
where the upper limits were chosen to take into account the number of pairs under peak at
nm. The choice of the integration limits corresponds to the restrictions chosen by the authors of other works [
51,
52,
84].
The dependences of the number of pairs
on temperature are shown in
Figure 12. It is easy to see that for almost all types of the studied plain groups (His–His, Hisp–Hisp) there are practically no temperature dependences, with the exception of fluctuation due to the finite time of the simulation. There is a slight systematic decrease in
for Arg–Arg pairs (see
Figure 12f). Due to this reason, we compared the values of
for different dendrimers only for T = 310 K.
It can be summarized that the total number of pairs
of plane groups is the highest for non-neighboring pairs of non-charged (His) imidazole groups in Lys-2His dendrimer (see
Table 5). This is due to the fact that the His groups in this dendrimer are uncharged and there is no need to overcome electrostatic interactions for pairing. It is interesting that the next highest number is for non-neighboring Arg–Arg pairs and then for non-neighboring Hisp–Hisp pairs. For neighboring pairs
is the highest for His–His, and then for Hisp–Hisp pairs, and the smallest
is for neighboring Arg–Arg pairs.
The pairing between neighboring plane groups belonging to the same spacer could influence only the local structure and mobility in the dendrimer. Thus, the most important results obtained in this work are the existence of the largest number of non-neighboring pairs in the Lys–2His dendrimer with neutral histidines (see
Figure 10b,
Figure 11b and
Table 5), and a significant lower number of pairs in the Lys–2Hisp dendrimer with protonated histidines and in the Lys-2Arg dendrimer. Furthermore, only in the Lys–2His dendrimer pairing could affect both the local and global structure and mobility of the dendrimer.
At the same time, we found that all plane groups do not form permanent pairs that exist during the entire simulation time and pairs can be form and break up during the calculation. Therefore, the important characteristic of pairing in the dendrimers with different plane groups is also their lifetimes.
The dynamic characteristics of imidazole–imidazole (or guanidine–guanidine) pairing can be studied using the same mathematical approach as for hydrogen bonding [
81,
82,
83]. The structural relaxation
for imidazole–imidazole (or guanidine–guanidine) pairs (pair can break and then form again) defines the following time correlation function:
where
is unity when a particular imidazole–imidazole (or guanidine–guanidine) pair forms at time
t; otherwise, it is zero. The
are shown on
Figure 13. We have extracted the relaxation time from each autocorrelation function (
from
) for pairs between neighboring groups and for pairs between non-neighboring groups and presented them in
Table 5). The main result is that neutral His–His pairs have the longest lifetimes. Taking into account the fact that the number of non-neighboring pairs is the largest among all possible pairs in the dendrimers studied, we can conclude that the pairing effect in the Lys–2His dendrimers with neutral histidines could have the greatest impact on the local and global properties of this particular dendrimers among all dendrimers studied by us.
3.4. The Interaction of Dendrimers with Molecules of Therapeutic Tetrapeptide
Dendrimers are often used to deliver drugs and genetic material to targeted cells, tissues and organs [
8,
9,
10,
12]. In this part of our article, we use the MD method to test the possibility of using the studied Lys–2His dendrimer as a delivery vehicle for small bioactive peptides in general, and for the tetrapeptide AEDG (Ala–Glu–Asp–Gly) in particular. This oligopeptide was synthesized to mimic the peptide drug “epithalamin” [
85,
86]. Each tetrapeptide molecule has a charge of −2. We assume that they should form a complex with the Lys-2His dendrimer at both normal and low pH since this dendrimer has positively charged terminal lysine groups with constant charge (+16) under these conditions. At the same time, this dendrimer has internal double histidine spacers that are neutral (2His, charge equal to 0) at normal pH and protonated (2Hisp, charge equal to +2) at low pH. All protonated spacers have the total charge equal to +28. Protonation leads to a significant increase in the total charge of the dendrimer (from +16 to +44) and to a change in its size, radial density and charge profiles. This should also lead to a change in its capacity for the transfer of oppositely charged drug molecules, for example, oligopeptides.
We simulated a system consisting of one dendrimer at two different pH (Lys–2His or Lys–2Hisp, respectively) and 16 AEDG tetrapeptide molecules by the MD method under the same conditions as the dendrimer simulation described at the beginning of this article. At the beginning of the MD simulation, the dendrimer (Lys-2His or Lys-2Hisp) is located in the center of the periodic cubic simulation cell with a size of 9 nm, and 16 tetrapeptide molecules are located at the periphery of this cell (see
Figure 14a,b). During the first half of the trajectory the peptides becomes closer and closer to each of dendrimer due to electrostatic interactions. However, at time 90–100 ns (see
Figure 14c,d), all tetrapetides become completely adsorbed only by protonated Lys-2Hisp dendrimer while Lys-2His adsorbed only 11 of 16 tetrapeptides.
To demonstrate the formation of a complex between dendrimers and tetrapeptide molecules we calculated the instant number of hydrogen bonds
between dendrimer and peptide molecules (see
Figure 15). At the beginning of simulation in both systems,
because free tetrapeptides are far from the dendrimer and, therefore, do not come into contact with it. When the peptides become closer to the dendrimer, contacts arise between them, and their number
increases with time. Thus, the slope of
during this initial time characterizes the rate of complex formation. It is clearly seen from these plots that the first system reaches dynamic equilibrium in about 60 ns. The second system reaches an equilibrium state after in 90 ns. From
Figure 15, it follows that the average number of hydrogen bonds in the equilibrium state (at
t > 60 ns) for the first complex is close to 24 and for the second complex in equilibrium (at
t > 90 ns) it is close to 58.
After our systems reach equilibrium, we can calculate the equilibrium size and other characteristics of the complex. To do this, we need to know how many tetrapeptide molecules are in the complex with the dendrimer. For this goal we used a simple condition for a local criteria:
where
is the distance between the
i-th atom of the dendrimer and the
j-th atom of the peptide,
is the distance at which the potential of non-bonded interactions reaches a minimum for the corresponding types of atoms
i and
j, and
s is a some factor (typically it is equal to 1.0, 1.5 and 2.0). If at least one atom of the peptide exists that satisfies the condition (
15) with at least one atom of the dendrimer, then we assumed that such a peptide is complexed with the dendrimer. The instant numbers of tetrapeptide molecules in complexes with the Lys-2His and Lys-2Hisp dendrimers at different points in time are shown in
Figure 16. It is easy to see that for the Lys-2His dendrimer this value fluctuates between 5 and 13, while for the Lys-2Hisp dendrimer it ranges from 13 to 16, but at most points in time, it is close to 16. The average value of the number of tetrapeptide molecules in the first complex is close to 9 and for the second complex it is close to 16 (see
Table 6). Thus, the capacity of the Lys–2His dendrimer with neutral (2His) spacer to transport these tetrapeptide molecules is almost twice less than that of the Lys–2Hisp dendrimer with protonated (2Hisp) spacers.
It is also interesting to know how the peptide molecules are distributed in complex: on the surface of each dendrimer or in its whole volume? To answer this question, we calculated the radial distribution functions of the atomic density of the dendrimer, peptides and all atoms in the complex (see
Figure 17).
The size (the radius of gyration) of the first dendrimer with neutral histidine spacers in the complex is close to 1.2 nm (see
Table 6). From
Figure 17a, it is easy to see that most of the atoms of the tetrapeptide molecules (red curve) are located at a distance of more than
nm from the center of the dendrimer. This means that tetrapeptides are manly located on the surface of the Lys–2His dendrimer and do not penetrate into it closer than
nm from the center of the dendrimer. The size of the second dendrimer with protonated histidine spacers in the complex is ~1.6 nm but contrary to the first one, the density of tetrapeptide atoms increases in the center of the Lys–2Hisp dendrimer, and most of these atoms are inside this dendrimer (i.e., at a distance
nm from the dendrimer center). This difference between the two complexes is due to the different distribution of charges in these dendrimers. In the Lys–2Hisp dendrimer with neutral 2His spacers, the positive charges are only in the terminal lysines (i.e., manly on the surface of the dendrimer). In the Lys–2Hisp dendrimer with protonated 2His spacers, the charges are distributed through the whole volume of the dendrimer.