Alpha-helical regions of the protein molecule as organic nanotubes
© Suprun and Shmeleva; licensee Springer. 2014
Received: 4 December 2013
Accepted: 8 April 2014
Published: 1 May 2014
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© Suprun and Shmeleva; licensee Springer. 2014
Received: 4 December 2013
Accepted: 8 April 2014
Published: 1 May 2014
An α-helical region of protein molecule was considered in a model of nanotube. The molecule is in conditions of quantum excitations. Such model corresponds to a one-dimensional molecular nanocrystal with three molecules in an elementary cell at the presence of excitation. For the analysis of different types of conformational response of the α-helical area of the protein molecule on excitation, the nonlinear response of this area to the intramolecular quantum excitation caused by hydrolysis of adenosine triphosphate (ATP) is taken into account. It has been established that in the simplest case, three types of excitation are realized. As estimates show, each of them ‘serves’ different kinds of protein. The symmetrical type of excitation, most likely, is realized in the reduction of traversal-striped skeletal muscles. It has the highest excitation energy. This well protects from casual actions. Antisymmetric excitations have intermediate energy (between symmetrical and asymmetrical). They, most likely, are realized in membranous and nucleic proteins. It is shown that the conformational response of the α-helical region of the protein is (in angstroms) a quantity of order N c /5, where N c is the number of spiral turns. For the number of turns typical in this case: N c ~ 10, displacement compounds are a quantity of order 2 Å. It qualitatively corresponds to observable values. Asymmetrical excitations have the lowest energy. Therefore, most likely, they are realized in enzymatic proteins. It was shown that at this type of excitation, the bending of the α-helix is formally directed to the opposite side with respect to the antisymmetric excitations. Also, it has a greater value than the antisymmetric case for N c ≤ 14 and smaller for N c > 14.
A protein molecule has a rather unique structure not only in the chemical-biological point of view but also as an interesting physical and mathematical object. If we consider it as a physical object, then such object may be referred to as a nanostructure without any doubt. Thus, the alpha-helical region of a protein molecule simultaneously may be considered both as a nanotube and as a nanowire: this depends on the considered level of structure.
Here, the alpha-helix is considered at the level of secondary structure where it is a nanotube. It is in the conditions of quantum excitation which is stimulated by reaction of hydrolysis of adenosine triphosphate (ATP). As a result of this reaction, energy in the form of quanta of infrared range is released. It is considered that they are absorbed by a group of energy states known in an alpha-helix as amide I, etc. It is considered also that these absorbing states have an internally molecular oscillating nature. The results obtained here allow giving a definite answer to this question, because in the infrared range, absorption can also have the nature of electronic transitions between states with the main quantum number equal to 2.
The alpha-helix is interesting as a mathematical object too. Due to the high sensitivity of its ‘crystalline lattice’ in relation to excitation, we are coming to a necessity to solve a nonlinear system of the so-called eigen type, i.e., actually, we are coming to a necessity to search for the eigenvalues and eigenvectors of a nonlinear system of algebraic equations. Such a problem, as it is known to us, is a scantily explored mathematical problem.
As a result of hydrolysis of ATP molecule, energy is realized in the range 0.2 to 0.4 eVa. It depends on the charge state of the ATP molecule, in which the composition of the environment influences mainly (pH, etc.). The energy of hydrolysis is absorbed by an alpha-helical region of the protein molecule. It takes place due to internal vibrational excitations of the peptide groups (HCNO) in the state amide I. Its energy is also varied within the limits of 0.2 to 0.4 eV. These excitations induce a significant increase of dipole moments of the peptide groups, which is equal to 3.7 D, on 0.29 D[4, 5].
There exists another point of view. Excitation of amide I may have an electronic nature. It may correspond to transitions between energy bands with principal quantum numbers that are equal to 2. The physical nature of excitation is inessential for further calculations, but further it will be shown that their nature may be determined experimentally.
Foremost, we need to determine the model of description of the spatial structure of the alpha-helix. Since it is considered as a molecular crystal, the nearest neighbor approximation is used, which is typical for such crystals. However, as seen from Figure 1b, the nearest neighbors for some peptide group with number n are not only group n ± 1 but also group n ± 3.
w(R nα − R mβ ) in this functional is the basic energy of interaction between peptide groups nα and mβ. It is independent on the presence of excitation and exists always. D(R nα − R mβ )|A αn |2 is an additional energy to the w(R nα − R mβ ) energy of interaction related only to excitation but considerably smaller. Factor A αn is the wave function that describes the excited state of the examined alpha-helical region of the protein molecule. It determines the spatial-temporal distribution of excitation in this region. The energy D(R nα − R mβ )|A αn |2 leads to the breaking of the equilibrium of the alpha-helix and stimulates its conformational response to excitement. Energy is also an additional energy of interaction. However, it is much less than D(R nα − R mβ )|A αn |2 but important because it provides the propagation and transfer of excitation along the alpha-helix.
As shown in Figure 2, the nearest neighbors for some peptide group nα will only be the peptide groups m = n ± 1, β = α and m = n, β = α ± 1. Taking into account that in the considered model all energy terms depend on the distances between amino acid residues only, the following formulae in the nearest neighbor approximation may be obtained: R nα ≡ |Rn + 1,α − Rn,α|, ρ nα ≡ |Rn,α + 1 − Rn,α|.
Here, w⊥ ≡ w(ρ0), D⊥ ≡ D(ρ0), M⊥ = M(ρ0), and M|| = M(R0). Obviously, |M⊥| ≠ |M|||. If resonance interaction has no electronic nature, inequality will be realized: |M⊥| < |M|||. If excitation has an electronic nature, inequality will be reversed: |M⊥| > |M|||. This difference may be detected experimentally, and the answer of the question about the physical nature of excitation may be obtained.
N0 is the number of amino acid residues in the alpha-helical region of the protein molecule, which is under consideration.
The solution of this system is usually determined after transition to continuous approximation. But we will analyze systems (8) and (9) without using the continuous approximation, because we are interested in very short alpha-helical regions (10 to 30 turns).
The last, fourth, equation arose out from normalization condition (1). The coefficients P α (α = 0, 1, 2) determine the excitement of each peptide chain as a whole.
This transformation does not affect the solutions of the system.
Next, we use the condition P0 − P1 = 0, which turns into an identity the first equation in (12). After some analysis, we can find two types of excitation:
The energies E a (k), E c (k), and E н (k) contain parameters Λ = |M|||/2 and Π = |M⊥|/2. As it was noted between Equations 2 and 3, the relation between these parameters makes the determination of the physical nature of excitation possible: whether they are electronic or intramolecular. Because one of them (Λ) determines the width of the excited energy bands, and the other (Π) their positions, this is the basis for the experimental analysis of the nature of excitations.
There are a few possibilities else for searching for solutions of the system (12). Preliminary analysis shows that the obtained excitations are peculiar in a more or less degree for both symmetries: whether it is the symmetry of the model or the symmetry of the real molecule. The other solutions of the system (12) need to be analyzed only in the conditions of the maximum account of the real structure of an alpha-helix. But the general analysis of this system shows that the solutions of a new quality are not present: all of them belong to the asymmetrical type. However, attention should be paid to the equation aα,n + 1 − aα,n − 1 = 0, which has led to the requirement a αn = P α . This condition is strong enough and essentially limits the solution: it is a constant in variable n, i.e., does not have the spatial distribution along an alpha-helix.
From definitions (13), (14), and (15), it ensues that received excitations are located in accordance with the inequality E c (k) > E a (k) > E н (k). Thus, and . It can be seen that for the alpha-helical region of finite length, when the number of turns N c ≠ ∞, the lowest energy is the energy of asymmetric excitation E н . Also, it is visible that energy E c is always strongly separated from energies E a and E н . Even when the number of turns N c ⇒ ∞ and the energies E a and E н practically coincide, the energy E c is separated from E a and E н on a value 3Π = 3|M⊥|/2. Amide I excitations manifested experimentally are probably E c energy.
It is possible to make the supposition that each of the examined energies executes some, expressly certain, function. For example, the main function of symmetric excitations can be activation of muscle proteins. At the same time, they can activate both membrane and enzymatic proteins that are quite often actually observed in the activation of myosin [9–11].
Antisymmetric excitation energy is not enough to excite the muscle protein because it lies below the symmetric energy. Activation of membrane proteins can be their main function. At the same time, these excitations are able to activate enzymatic proteins that are also actually observed often enough during activation of membranes [11–13].
And, lastly, asymmetrical excitations have only one function - to activate exceptionally enzymatic activity in those cases, when membrane and muscular activities are not needed. That is only for intracellular processes.
Here, Δ is the displacement for antisymmetric excitations, which is determined by Equation 16. Unlike displacement Δ, displacement Δ(н) ‘directed’ to the opposite side. Executing numerical estimates, it is possible to set that Δ(н) > Δ, if the number of turns in the alpha-helix N c ≤ 14, but at N c > 14, we will have Δ(н) < Δ accordingly.
The general methods [7, 15–17] of description of the excited states of the condensed environments were applied to the alpha-helix region of a protein molecule. The alpha-helix is considered as a nanotube, and excitations of the environment are described as quasiparticles. It is shown that three different types of excitation exist, and each of them is probably used by three different types of protein. The symmetrical type of excitation is used for muscle proteins, the antisymmetric type of excitation is used for membrane proteins, and the asymmetric type of excitation is used for enzymatic proteins. It is possible that some excitations of asymmetrical type exist, which are also used by enzymes. The estimations were done for displacements of the free end of the alpha-helix. The obtained displacements are in agreement with experimental data. Therefore, the obtained results can be the basis of the interpretation of the functional properties of proteins characterizing their activity related to their conformational changes .
aOff-system unit of energy: 1 eV = 1.602 × 10−19 J.
bFor example, in the myosin protein, the helical region has about 200 turns or up to 700 amino acids.
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