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MCPep is based on the Monte Carlo simulation model presented in references (1, 2) and here:
Each residue n
residues has N-1 virtual bonds. Virtual bonds are highly stiff, and their lengths are
taken here as fixed at their equilibrium values of 3.81±0.03Å. The
peptide backbone conformation is defined by the 2N-5 dimensional vector
[θ_{2}, θ_{3}, ..., θ_{n}_{-1},
φ_{3}, φ_{4}, ..., φ_{N}_{-1}]
including n-2 virtual bond angles (θ) and _{i}n-3
dihedral angles (φ). The distance between _{i}S
and C_{i}_{i}^{α}, as well as the side chain virtual bond vector pointing from C_{i}^{α}
to S_{i} named θ,
are fixed at their equilibrium values. Thus, the conformation of side chain _{i}^{S}i
is expressed by the torsion angle (φ). _{i}^{S}
φ is
the rotational angle of the _{i}i^{th} virtual bond, connecting C_{i}^{α}
and C_{i-1}^{α}. θ is the
bond angle between virtual bonds _{i}i and i+1. θ
is the side chain virtual bond vector pointing from C_{i}^{S}_{i}^{α}
to S. _{i}φ is defined by C_{i}^{S}_{i-2}^{α},
C_{i-1}^{α}, C_{i}^{α}
and S._{i}
The membrane is described
using two parameters: hydrophobicity and surface charge. The hydrophobicity of
the membrane ( (1) z=15Å.
The surface charge is located (_{m}z +5)Å from the
membrane midplane. (back to top)_{m}
η=1Å^{-1} (solid line). The location of the
surface charge is designated by the dashed lines; the hydrophobic region of the
membrane is defined by the dotted lines.
The total free energy difference between a peptide in the aqueous phase and in the membrane () can be divided into several terms as follows (5, 6): (2)
(3) where is the internal energy difference between the water- and membrane-bound states of the peptide. The internal energy is derived from a statistical potential based on available 3-dimensional (3D) protein structures (3, 7). The energy function assigns a score (energy) to each peptide conformation according to the conformation's abundance in the Protein Data Bank (PDB). Common conformations are assigned high scores (low energy), while rare conformations are assigned lower scores (higher energy). ΔS refers to
the entropy difference between the water and membrane-bound states, while the
entropy ( (4)
j to be in the interval i. As
the virtual bond rotations for the first and last two amino acids are not
defined, these are omitted from the entropy calculation. The value of p
is estimated as _{i,j}(5)
N
in which a virtual bond j is in the interval i. (back to top)
ΔG (6)
z are the actual and native
widths of a monolayer. _{0}ω is a harmonic force constant related to
the membrane elasticity and is equal to ω= 0.22 kT/Å^{2}
(9). (back to top)
ΔG (7)
(8)
ε is the permittivity in vacuum, and _{0}ε
is the dielectric constant in water (taken as 80). _{r}Φ is the
potential on the plane of smeared charges; it depends on the membrane charge density σ
and on the molarity of the solution [K^{+}]:(9)
(10)
z),
similar to the function for the membrane polarity profile p(z) (Eq. 1):(11)
i and the charged membrane surface weighted by χ(_{i}z)
can be calculated according to the following:(12)
(13)
ΔG The sum of ΔG (14) The prefactors in the two terms ( and
)
represent the hydrophobicity of the environments at the respective interaction
sites, i.e., the α-carbon site and the side chain interaction center, Δ ^{s}
are calculated using the Kessel and Ben-Tal hydrophobicity scale (Table 1) (5). The scale
accounts for the free energy of transfer of the amino acids, located in the
center of a polyalanine α-helix, from the aqueous phase into the membrane
midplane. In order to avoid the excessive penalty associated with the transfer
of charged residues into the bilayer, in the model the titratable residues are
neutralized gradually upon insertion into the membrane, so that a nearly
neutral form is desolvated into the hydrophobic core. However, as described
above, a gradual transition between the charged and neutral forms of titratable
residues based on χ(z) (Eq.11) is introduced into the
model. Therefore, for the neutral state of a titratable residue Δ_{i}g_{i}^{s}
is derived from the hydrophobicity scale (Table 1); for the charged state of a
titratable residue Δg_{i}^{s} is taken as 64 kT (16):(15) is the polarity profile of the charged side chains solvation, a sigmoidal function similar to the hydrophobicity profile (Eq. 1): (16)
^{b})
is calculated using the following set of equations:
(17) where f for residues deviating significantly from the ideal
α-helical conformation, e.g., residues that are in extended conformations.
For these residues the free-energy penalty due to the transfer of both the C=O
and N-H backbone groups are taken into account:(18)
It is noteworthy that the stretches of three residues at the N- and C-termini are treated differently than the peptide core (Eq. 17). The free energy penalty associated with the transfer of the uncompensated hydrogen bonds of the N-H groups of the three residues at the N-terminus is taken into account regardless of the peptide conformation. Likewise, the free energy penalty due to the transfer of the uncompensated hydrogen bonds of the C=O groups of the three residues at the C-terminus is also taken into account, regardless of the peptide conformation.
New conformations are
generated by simultaneous random perturbations of θ,
_{i}φ_{i}^{s}:(19)
θ
and _{i}φ_{i}^{s}. The peptide configuration is changed by external motions as described below. However, it is noteworthy that a set of randomly chosen conformational changes could also lead to slight changes in the peptide's orientation in the membrane. (back to top)
External rigid body rotational and translational motions arre carried out to change the peptide's configuration, i.e. its location in, and orientation with respect to, the membrane. These motions are represented respectively by: (20)
(21)
α,
δ_{max}β, δ_{max}γ and δ_{max}d_{max} (=5°,
5°, 5° and 0.02Å, respectively) were chosen to be maximum variations of
the random perturbations of α, β, γ and d.
These parameters were determined by trial and error, such that the system would
have enough time for internal relaxation but would not be trapped too often in
local energy minima.Additionally, the width of the membrane's hydrophobic region is perturbed: (22)
z by up to 20% (17). (back to top)_{0}
To calculate the free energy difference between
a peptide in the aqueous phase and the peptide in the membrane
(Eq. 2), peptide simulations in water and in membrane environments are
performed. A standard MC protocol is employed, and acceptance of each move
is based on the Metropolis criterion and the free energy difference between
the new and old states (18). In water, peptides are subjected solely to internal
conformational modifications. In one MC cycle the number of internal
modifications performed is equal to the number of residues in the peptide.
Therefore, the acceptance criterion is based on ΔΔ (23)
4. Flory, P. J. 1969. Statistical Mechanics of Chain Molecules. Wiley-Interscience, New-York. |