Lysine Dendrimers and Their Complexes with Therapeutic and Amyloid Peptides: Computer Simulation Lysine Dendrimers and Their Complexes with Therapeutic and Amyloid Peptides: Computer Simulation

Lysine dendrimers consist of natural lysine amino acid residues. Due to this reason, they are usually not as toxic as other dendrimers. Lysine dendrimers are often used in drug and gene delivery. These dendrimers penetrate blood-brain barrier and thus could be used for the delivery of drugs and other substances, for example, bioactive peptides to brain or elimination of disease-related peptides out of the brain. To do it, dendrimers should form complex with these peptides. In the present chapter, we describe computer simulation of the interaction of lysine dendrimer of the second generation with three different peptides and check does it form complexes with them. Two of these peptides (Semax and Epithalon) are nootropic peptides and third is the fragment of amyloid peptide, which forms amyloid fibrils and plaques in Alzheimer’s disease. Our simula tion demonstrates that the lysine dendrimers form complexes with these therapeutics peptides. Thus, we demonstrated that lysine dendrimer is a good candidate for the delivery of therapeutic peptides. We also have shown that lysine dendrimer destroys existing stacks of amyloid peptides and forms a stable complex with them. Thus, it looks that it could be used in future for the treatment of Alzheimer’s diseases.


Introduction
Only a few of the peptide drugs reach their targets. Most of them accumulate in nontargeted organs and produce different side-effects. Drug delivery systems help to overcome these problems and increase safety and efficacy of therapeutic peptides [1]. Traditional drug delivery systems include tablets, injections, suspensions, ointments, creams, liquids, and aerosols. Polymers are also used for this goal and protect drugs from biological degradation during delivery [2].
Dendrimers were synthesized in seventieth to eightieth of the last century [3]. They have regular branched (star-like) structure. Dendrimers have well-defined size and constant spherical shape. Many dendrimers have a large number of charged terminal groups available for functionalization [4,5]. Dendrimers could be used for drug and gene delivery, as a branched core for multiple antigen peptides (MAPs) as well as antibacterial and antiviral agents. Also, they were proposed as potent anti-amyloid agents for the treatment of different neurodegenerative diseases (Alzheimer's, Parkinson's, etc.).
In the present chapter, we describe our results of simulation of the interaction of the second generation lysine dendrimer (Figure 1) with three types of peptides: Semax, Epithalon, and the fragment of amyloid peptide.

Molecular dynamics simulation
Molecular dynamics simulation (MDS) is widely used for modeling of different polymer and biopolymer systems. In this approach, all atoms or groups of atoms in molecules in a system are represented by spherical beads. Chemical bonds between atoms are represented by springs or rigid bonds. The dynamics of such mechanical model of real molecular systems is described by classical Newton equations of motion and the system of these equations for all beads in the system is solved numerically.
where r i (t) = (x i (t), y i (t), z i (t))-the position vector of the ith particle (or its coordinates x i (t), y i (t), z i (t)), F i -the force acting on the ith particle at time t, and m i -mass of the particle.
This method was applied for the first time for the study of simple 2D model of hard disks, which represents two-dimensional analog of a monoatomic gases [6]. Later, it was extended for simulation of liquids [7,8] and coarse-grained model of linear polymer chains [9] and n-alkanes [10]. During the last 30 years, MDS was applied for the study of many molecular systems of different structure and composition. The potential energy in MDS usually includes the energy of valence bonds, the energy of valence angles, and dihedral angles as well as van der Waals energy and electrostatic energies. In the present study, the variant of molecular dynamics simulation approach realized in the framework of GROMACS 4.5.6 software package [11] and one of the most modern force fields (AMBER_99SB-ildn) was used for the representation of potential energy [12] of our system. The force field has the following form: where l i -valence bond lengths, θ i -valence angles values, l i0 and θ i0 -equilibrium values of them, and a i and b i -force constants, correspondingly, ε ij and σ ij -values of Van der Waals parameters of Lenard-Jones 6-12 potential, q i -partial charges, c i, γ i , and n-numerical coefficients in dihedral potential while summation is done through all i-beads or pairs of ith and jth beads in the system consisting of N beads.
The more detailed description of MDS method used in the present work including simulation of linear and branched polyelectrolytes as well as dendrimers has been described in . All simulations were performed at normal conditions (temperature 300 K and pressure 1 atm).
The size of dendrimer and complexes at time t was evaluated by the mean square radius of gyration R g (t) which is defined from: where R-the center of mass of subsystem, r i and m i -coordinates and masses of i-atom correspondingly, N-the total number of atoms in subsystem, and M-the total mass of dendrimer. This function was calculated using g_gyrate function of GROMACS software.
Radial distribution of density p(r) of atoms in dendrimer and complexes as well as distribution of ion pairs were calculated using g_rdf function of the GROMACS package.
To calculate the coefficient of translational mobility of dendrimer and complexes, the time dependence of the mean square displacements of the centers of inertia (MSD) of corresponding subsystem was calculated. MSD was calculated using g_msd function of GROMACS.

Semax peptide
Therapeutic Semax peptide with primary sequence Met-Glu-His-Phe-Pro-Gly-Pro [34] was selected in present chapter as the first peptide for study. Semax is one of the few synthetic regulatory peptides that, after all the fundamental research, have found its application in therapy as a nootropic and neuroprotective agent [35]. However, its penetration into the brain and mechanisms of action remain insufficiently studied. In addition, the peptide is rapidly destroyed by blood enzymes [36].
To check if lysine dendrimer forms a complex with Semax peptides, we prepared system containing lysine dendrimer of the second generation, 16 Semax peptides, and counterions in water and studied the time evolution of this system. Figure 2 at different time moments during MD simulation (water molecules are not shown for clarity). It is easy to see that at the beginning of the time trajectory (Figure 2a), all peptide molecules are located far from the dendrimer. After 20 ns (Figure 2b), some peptide molecules already sit on the surface of the dendrimer, and at the end of the simulation (Figure 2c), most of the peptide molecules are adsorbed on its surface.

Snapshots of this system are shown in
The time dependence of gyration radius R g (t) (Figure 3a) shows that at the beginning of simulation, the size of subsystem consisting of lysine dendrimer and Semax peptides is big because peptides are located far from dendrimer. During first 30-40 ns, the value of R g (t) decreases because peptides begin to adsorb on dendrimer surface. Stable dendrimer-peptides complex forms within 30-40 ns and after this time the size of complex fluctuates but its average value does not change with time.
Another quantity that can characterize complex formation is the hydrogen bond number (N) between dendrimer and peptides (see Figure 3b). In the beginning of simulation, there are no contacts between dendrimer and peptides and hydrogen bond number between them equals zero. The increase of this function with time during first 30-40 ns demonstrates that the number of contacts between dendrimer and peptides increases due to complex formation and after this time it fluctuates but its average value almost does not change with time. Thus, from Figure 3b, we can conclude that the system reaches equilibrium (plateau) state after 30-40 ns. This result correlates well with results for the gyration radius in Figure 3a.
We also calculated different equilibrium characteristics (see Table 1 and Figure 4) of complex averaged through equilibrium part of our simulation (t > 40 ns). In particular, we calculated the mean square radius of gyration R g of our dendrimer and complex (consisting of G2 and 16 Semax peptides) and obtained that the value of R g of the complex is nearly twice larger than the size of dendrimer itself (see Table 1). It is quite natural because the molecular weight of the complexes also increases nearly twice in comparison with the molecular weight of the dendrimer. The shape of dendrimer and complex can be characterized by main components of the tensor of inertia R g 11 , R g

22
, R g 33 , (see Table 1). It can be roughly estimated by ratio R g 33 /R g 11 of largest and smallest eigenvalues of inertia tensor of the corresponding subsystem. The ratio is equal to 1.68 for dendrimer and 1.45 for dendrimer with peptides. Thus, we obtained that complex has more spherical shape than dendrimer.
The radial density distribution functions are shown in Figure 4a. It demonstrates that atoms of dendrimer (curve 2, Figure 4a) are located mainly in the center (i.e. at small distances r from the center of mass), whereas peptides (curve 1, Figure 4) are mainly on the surface of the  complex. At the same time, some fraction of peptides could slightly penetrate into dendrimer but not to its inner part.
The average number of hydrogen bonds between peptides and dendrimers in the equilibrium state (which was determined from second part (t > 40 ns) of Figure 3b) shows how tightly peptides are connected with dendrimer. From our calculations, it follows that average hydrogen bonds number in equilibrium state for the complex is close to 19. It means that each peptide is connected in average by one hydrogen bond with dendrimer. We could also check how the number of hydrogen bonds in the equilibrium state fluctuates. It is easy to see from Another equilibrium characteristic of interaction between dendrimer and peptides in the complex is the distribution of ion pair numbers between their oppositely charged groups. Figure 4b shows the dependence of ion pair number on the distance between dendrimer charges and peptides charges (curve 1) in the complex as well as between dendrimer charges and counterions (curve 2). It is seen, that there is a sharp peak, corresponding to the direct contact between positively charged groups (NH 3 + ) of dendrimer and negatively charged groups (COO − ) of the glutamic acid in peptides (curve 1) and peak between dendrimer charges and counterions is less pronounced (curve 2). It means that dendrimer and peptides are strongly connected by this ion pairs with each other, whereas dendrimer charges and counterions contacts are not so strong.
To determine diffusion coefficient of the dendrimer-peptide complex, we calculated its mean square displacement (MSD) as a function of time. It was obtained, that the time dependence of MSD function is close to linear in double logarithm coordinates (not shown). It means that the translational motion of complex is the diffusion-like motion. The diffusion coefficient of the complex was determined from the slope of the time dependence of MSD and it is equal to (0.12 ± 0.03) × 10 5 sm 2 /s.
The results of simulation of the system consisting of lysine dendrimer of the second generation and 16 Semax peptides confirm that lysine dendrimer quickly forms complex with these peptides and thus lysine dendrimers could be a good candidate for using as a vehicle for delivery of Semax peptides.

Epithalon peptides
Therapeutic peptide Epithalon was selected in present chapter as the second model peptide.
Biologically active regulatory peptides are promising and effective medicines in a wide range of diseases treatment. Such peptides are effective at low concentrations, selective, multifunctional, and completely biodegradable and do not produce side-effects [37]. Epithalon is a regulatory tetrapeptide with amino acid sequence of alanine-glutamate-asparagine-glycine (AENG). It is synthesized as analogs of the epithalamine peptide preparation and isolated from the epiphysis of the animal brain [38].
One of the most important properties of this peptide is its ability to activate the telomerase enzyme in patients' body and to prolong human cells life. The most well-known pharmacological properties of Epithalon are the following: 1. Regulation of neuroendocrine system.

2.
Increased sensitivity of hypothalamus to endogenous hormonal effects.

4.
Strengthening of the immune system.

5.
Inhibition of spontaneous and induced carcinogenesis.

Improves rheological properties of blood and reduces formation of blood clots.
To check if lysine dendrimer could form the complex with Epithalon peptides, we prepared system consisting of lysine dendrimer of second generation, 16 Epithalon peptides, counterions, and water and studied time evolution of this system. Snapshots of a system consisting of dendrimers, peptides, and ions during simulation are shown in Figure 5 (water molecules are not shown for clarity). It is easy to see that at the beginning of process (Figure 5a) peptide molecules located far from dendrimer. After 20 ns (Figure 5b), some part of the peptide molecules are already sitting on the surface of the dendrimer, and in the end (Figure 5c) most of the peptide molecules adsorbed on its surface. This behavior is very similar to that for the system consisting of dendrimer and Semax peptides discussed above.
The time dependence of radius of gyration R g (t) describes the process of transition of the system to equilibrium state with the formation of a complex between dendrimer and Epithalon peptides (see Figure 6a). From Figure 6a, one can see that at the beginning of simulation the size of subsystem consisting of dendrimer and peptides is big because peptides are far from dendrimer. During first 20-30 ns, the value of R g (t) decreases because peptides begin to adsorb on dendrimer surface and after this time the average value of R g practically does not change with time. It means that stable dendrimer-peptides complex forms within 20-30 ns.
Another quantity that can characterize complex formation is the hydrogen bond number N(t) between dendrimer and peptides (see Figure 6b). In the beginning of simulation, there are no contacts between dendrimer and peptides and the number of hydrogen bonds between them equals zero. The increase of this function with time during first 20-30 ns demonstrates that the number of contacts between dendrimer and peptides increases due to complex formation. From Figure 6b, we can conclude that the complex goes into an equilibrium (plateau) state after 30-40 ns. This result correlates well with the results for the gyration radius R g (t) in Figure 6a.
After equilibration and complex formation, all characteristics of complex fluctuate but their average values practically do not change with time. We calculated different equilibrium characteristics of complex averaged through these equilibrium part of our simulation (t > 30 ns).
It was obtained that in equilibrium state the size R g of the complex (G2 and 16 Epithalon peptides) is near 2.18 times larger than the size of dendrimer itself. The shape of the complex can be characterized by main components of the tensor of inertia R g 11 , R g 22 , R g 33 (see Table 2). In the simplest case, anisotropy can be characterized by R g 33 /R g 11 . For dendrimer, this ratio is equal to 1.69 and for the complex-1.28. Thus, the anisotropy of dendrimer with adsorbed Epithalon peptides is less than anisotropy of dendrimer itself as it was shown earlier for the complex of dendrimer with Semax peptides.
The equilibrium (t > 30 ns) radial density distribution functions p(r) are shown in Figure 7a. It demonstrates that atoms of dendrimer (curve 2, Figure 7a) located in the center of the complex (i.e. at small distances r from the center of mass) and Epithalon peptides (curve 1, Figure 7a) located mainly on the surface of the complex. At the same time, some fraction of peptides could slightly penetrate into dendrimer but not to its inner part.
The average number of hydrogen bonds between dendrimer and Epithalon peptides in the equilibrium state (which was determined by averaging through t > 30 ns of Figure 6b) shows how tightly peptides are associated with dendrimer. From our calculations, it follows  that average hydrogen bond number in equilibrium state for the complex is close to 20. It means that each peptide is connected in average by one hydrogen bond with dendrimer.
We also could check how the number of hydrogen bonds in the equilibrium state fluctuates. Fluctuations in hydrogen bonds number between dendrimer and peptides in the complex (see Figure 6b) are in the range of 9-28. It means that the average number of hydrogen bonds with dendrimer per peptide fluctuate between 0 and 2.

Figure 7b
shows the dependence of ion pair numbers on the distance between dendrimer charges and Epithalon peptides charges in the complex as well as between dendrimer charges and counterions. It is seen that there is a sharp peak, corresponding to the direct contact between positively charged groups (NH 3 + ) of dendrimer and negatively charged groups (COO − ) of peptides (curve 1) and significantly less sharp peak (curve 2) between dendrimer charges and counterions. It means that dendrimer and peptides are strongly connected at these points with each other, whereas dendrimer charges and counterions contacts are not so strong.
To determine diffusion coefficient of the dendrimer-peptide complex, we calculated its mean square displacement (MSD) as a function of time. It was obtained, that the time dependence of MSD function is close to linear in double logarithm coordinates (not shown). It means that the translational motion of complex is the diffusion-like motion. The diffusion coefficient of the complex was determined from the slope of the time dependence of MSD and is equal to (0.21 ± 0.03) × 10 5 sm 2 /s.
The results of simulation of the system consisting of lysine dendrimer and Epithalon peptides confirm that the dendrimer could form complex with these peptides and thus lysine dendrimers could be a good candidate for using as a vehicle for delivery of Epithalon peptides.

Amyloid fibrils
Fragment of amyloid peptide (LVFFAE) was selected in present chapter as a third model peptide. Many peptides and proteins have the ability to self-assembly into amyloid fibrils. Polypeptides that can form amyloid fibrils include molecules associated with neurodegenerative diseases [39,40]. For example, in Alzheimer's disease, it is a β-amyloid peptide, in Parkinson's disease, it is a-synuclein protein, and in type 2 diabetes mellitus, it is an isletamyloid peptide (IAPP, or amylin). The most studied are amyloid peptides that are also called amyloid-β peptides or Aβ-peptides.
It is known that synthetic protonated dendrimers (for example, polyamidoamine (PAMAM) dendrimers) can prevent aggregation of amyloid peptides [41]. It was also shown that they can destroy already existing amyloid fibrils in solution. Recently, it was experimentally shown that lysine dendrimers can also destroy amyloid fibrils [14].
To confirm the possibility of amyloid stack destruction by lysine dendrimer, we prepared system consisting of lysine dendrimer of second generation and 16 short amyloid peptides (LVFFAE) in water with counterions and studied the time evolution of this system. Figure 8, one can see that at the beginning of the process (Figure 8a) all peptide molecules of the stack are located rather far from the dendrimer. After 20 ns (Figure 8b), a few of peptide molecules are already detached from the stack and adsorbed on the surface of dendrimer, and in the end of calculations after 160 ns (Figure 8c), most of the peptide molecules in the system adsorbed on its surface.

From snapshots in
The first part (t < 20-30 ns) of the time dependence of gyration radius R g (t) describes the process of destruction of the amyloid stack by G2 dendrimer and dendrimer-peptides complex formation (Figure 9a). From Figure 9a, one can see that second generation dendrimer forms complex with 16 peptides within 20-30 ns. After that, the complex size R g (t) fluctuates slightly, but its average value R g practically does not change systematically with time. Therefore, we can assume that after 20-30 ns the system is in equilibrium state. It correlates with snapshots that are shown in Figure 8. From Figure 9b, it can be concluded that the system reaches equilibrium (plateau) state after 20-30 ns. The distance between neighboring peptides in amyloid stack (Figure 10a, curve 1) is an important characteristic of stability of the stack. The distance between dendrimer and peptides (Figure 10a, curve 2) allows estimating the rate of dendrimer-peptides complex formation after stack was destructed by dendrimer. At the beginning of simulation, the distances between the neighboring peptides of the stack are small. During first 20-30 ns, this distance is increase. It means that the destruction of amyloid stack occurs and peptides become separated from each other.
At the same time, the distance between dendrimer and peptides (Figure 10a, curve 2) in the beginning of the simulation (at t = 0) is large because peptides are located far from the  dendrimer. During first 20-30 ns, the peptides become attracted by dendrimer and distance between them and dendrimer decreases. It means that the dendrimer of the second generation could destroy the stack of amyloid peptides and adsorb the peptides.
After complex formation (t > 20-30 ns), all characteristics of complex fluctuate but their average values do not change with time. We calculated different equilibrium characteristics of complex averaged through equilibrium part of our simulation (t > 30 ns).
In equilibrium state, the mean square radius of gyration R g of the complex is 1.7 times larger than the size of the dendrimer G2 itself (see Table 3). The ratio R g 33 /R g 11 (which allows to roughly estimating anisotropy of molecule) is equal to 1.68 for dendrimer and 1.45 for the complex of dendrimer with peptides. Thus, the anisotropy of dendrimer with adsorbed amyloid peptides is again less than anisotropy of dendrimer itself as it was for previous two complexes of lysine dendrimer with model peptides Semax and Epithalon.
Radial distribution p(r) density (not shown) obtained for this complex is similar to that for complexes of the dendrimer with two peptides (Semax and Epithalon) studied above. It demonstrates that dendrimer atoms are located mainly in the center of the complex and amyloid peptides are mainly on the surface of the complex. At the same time, some fraction of amyloid peptides could slightly penetrate into the outer part of the dendrimer.
The average number of hydrogen bonds N between dendrimer and LVFFAE peptides in our complex (determined from second part (t > 30 ns) of N(t) from Figure 9b) is close to 15. It means that in average each amyloid peptide is connected by one hydrogen bond with dendrimer in the complex. Fluctuations in hydrogen bond number are in the range of 5-27, and it means that a number of hydrogen bonds per peptide can fluctuate between 0 and 2. Figure 10b shows the dependence of number of ion pairs between dendrimer and peptide charges on the distance between them in the complex. One can see that there is a sharp peak, corresponding to the direct contact between positively charged groups (NH 3 + ) of dendrimer and negatively charged groups (COO − ) in peptides (curve 1) and significantly less sharp peak between dendrimer charges and counterions (curve 2). It means that dendrimer and LVFFAE peptides are strongly connected at these points with each other, whereas dendrimer charges and counterions contacts are not so strong.
To determine diffusion coefficient of dendrimer-peptide complex, we calculated its mean square displacement (MSD) as a function of time. It was obtained that the time dependence of MSD function is close to linear in double logarithm coordinates (not shown). It means that the translational motion of complex is the diffusion-like motion. The coefficient of translational diffusion of the complex was determined from the slope of the time dependence of MSD and is equal to (0.13 ± 0.04) × 10 5 sm 2 /s.
The results of simulation of the system consisting of lysine dendrimer of second generation and fragments of amyloid peptides confirm that this lysine dendrimer could form the complex with amyloid peptides. Thus, lysine dendrimers could be a good candidate for using as anti-amyloid agent.

Conclusion
In the present chapter, we have shown that lysine dendrimer of the second generation could form stable complexes with therapeutic Semax and Epithalon peptides. We also have shown that this dendrimer could destroy the stable stacks of amyloid peptides and form stable complex with them. Thus, our simulations confirm that lysine dendrimers could be a good candidate for vehicles for the delivery of different nootropic peptides to brain and for the destruction of fibril consisting of disease-related peptides and elimination of them from the brain.