# Alpha-helical regions of the protein molecule as organic nanotubes

- Anatol D Suprun
^{1}and - Liudmyla V Shmeleva
^{1}Email author

**9**:200

https://doi.org/10.1186/1556-276X-9-200

© Suprun and Shmeleva; licensee Springer. 2014

**Received: **4 December 2013

**Accepted: **8 April 2014

**Published: **1 May 2014

## Abstract

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.

### PACS

92C05

### MCS

36.20.Ey

## Keywords

## Background

### Hydrolysis of ATP and amide I excitation

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.

*in vivo*. The degree of helicity in different proteins varies from 12% to 96%. As can be seen from Figure 1, the alpha-helical fragment of protein molecules is structurally a nanotube. The same is true for its physical properties. Therefore, to such regions of protein molecules in their excited states, it is natural to apply methods that are specific for nanotubes.

As a result of hydrolysis of ATP molecule, energy is realized in the range 0.2 to 0.4 eV^{a}. 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.

## Methods

### Amide I excitation in the simplest model of alpha-helical region of protein

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.

*n*,

*m*, etc. The number of such cells is three times less than the number of peptide groups, i.e.,

*N*

_{0}/3. Peptide groups within a single cell will be enumerated by indices

*α*,

*β*, etc. that may take values 0, 1, 2. The general functional for the alpha-helix in this model has the form [7]

*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 ${\mathit{M}}_{\mathit{n\alpha},\mathit{m\beta}}{\mathit{A}}_{\mathit{\beta}\phantom{\rule{0.1em}{0ex}}\mathit{m}}^{*}{\mathit{A}}_{\mathit{\alpha n}}$ 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
} ≡ |**R**_{n + 1,α} − **R**_{n,α}|, *ρ*_{
nα
} ≡ |**R**_{n,α + 1} − **R**_{n,α}|.

*ρ*

_{ nα }=

*ρ*

_{0}is always supposed fulfilled. Factor

*R*

_{ nα }is the only value that takes into account the response of the alpha-helix on excitation. Thus, we will denote its equilibrium value as

*R*

_{0}. Values

*ρ*

_{0}and

*R*

_{0}are shown in Figure 2. Taking into account the normalization condition

Here, *w*_{⊥} ≡ *w*(*ρ*_{0}), *D*_{⊥} ≡ *D*(*ρ*_{0}), *M*_{⊥} = *M*(*ρ*_{0}), and *M*_{||} = *M*(*R*_{0}). 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.

*w*(

*R*

_{0}) ≡

*w*

_{||},

*D*(

*R*

_{0}) ≡

*D*

_{||}, $\frac{{\left|{\mathit{D}}^{/}\left({\mathit{R}}_{0}\right)\right|}^{2}}{\left|{\mathit{w}}^{//}\left({\mathit{R}}_{0}\right)\right|}\equiv \mathit{G}$, and introduce convenient re-designation:

*M*

_{||}= −|

*M*

_{||}| ≡ −2

*Λ*,

*M*

_{⊥}= |

*M*

_{⊥}| ≡ 2

*Π*, which take into account the true signs. Then for the functional (2), finally, the following formula will be obtained:

*E*

_{осн}= (

*w*

_{⊥}+

*w*

_{||})

*N*

_{0}+

*D*

_{⊥}+

*D*

_{||}, and the following is taken into account:

*N*_{0} is the number of amino acid residues in the alpha-helical region of the protein molecule, which is under consideration.

*A*

_{ αn }, it is necessary to create a conditional functional: ${\mathit{E}}^{\mathit{\u0443\u043c}}\left(\left\{\mathit{A}\right\}\right)=\mathit{E}\left(\left\{\mathit{A}\right\}\right)+\mathit{\u03f5}\left(1-{\displaystyle \sum _{\mathit{n\alpha}}{\left|{\mathit{A}}_{\mathit{\alpha n}}\right|}^{2}}\right)$. From a mathematical point of view, parameter

*ϵ*is an indefinite Lagrange multiplier, and physically, it is the eigenvalue of the considered system. The minimization procedure $\frac{\partial {\mathit{E}}^{\mathit{\u0443\u043c}}\left(\left\{\mathit{A}\right\}\right)}{\partial {\mathit{A}}_{\phantom{\rule{0.1em}{0ex}}\mathit{\alpha}\phantom{\rule{0.1em}{0ex}}\mathit{n}}^{*}}=0$ produces the equation

*Λ*(

*A*

_{α,n + 1}+

*A*

_{α,n − 1}) +

*G*|

*A*

_{ αn }|

^{2}

*A*

_{ αn }−

*Π*(

*A*

_{α + 1,n}+

*A*

_{α − 1,n}) +

*ϵA*

_{ αn }= 0. After dividing this equation by

*Λ*and introducing the notations,

*A*

_{ αn }is complex. Therefore, the common solution of the system (6) has the form

*A*

_{ αn }=

*a*

_{ αn }· exp(

*iγ*

_{ αn }). Amplitude

*a*

_{ αn }and phase

*γ*

_{ αn }are real functions of the variables

*α*and

*n*. We confine ourselves to the Hamiltonian-Lagrangian approximation in phase [8]. Due to the stationarity of the solved problem, this approximation has the simplest form:

*γ*

_{ αn }≡

*kn*. If the alpha-helical part of the molecule is long enough,

^{b}a Born-Karman condition gives $\mathit{k}=\frac{2\mathit{\pi}}{{\mathit{N}}_{\mathit{c}}}\mathit{j}$. Here, ${\mathit{N}}_{\mathit{c}}\equiv \frac{{\mathit{N}}_{0}}{3}$ is the number of turns in the considered alpha-helical region of the protein molecule. It plays the role of the dimensionless length of the helical region of the protein in units of an alpha-helix step. Parameter

*j*has the values $\mathit{j}\approx 0,\phantom{\rule{0.5em}{0ex}}\pm 1,\phantom{\rule{0.5em}{0ex}}\pm 2,\dots ,\pm \frac{{\mathit{N}}_{\mathit{c}}}{2}$. Then

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).

*a*

_{α,n + 1}−

*a*

_{α,n − 1}= 0 (if not to restrict solutions by using the condition

*k*= 0), which does not depend on any symmetry of the alpha-helix: whether it is the symmetry of the model or the symmetry of the real molecule. Viewing of other conditions can appear useful on account of the real structure of the alpha-helical region. In the simplest case, it may be reduced to the equation

*a*

_{ αn }=

*P*

_{ α }. The system (8) now degenerates in the system of three nonlinear equations:

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.

*P*

_{0},

*P*

_{1}, and

*P*

_{2}and the eigenvalue

*x*. By adding and subtracting the first two equations and some transformation of the third equation, the system (10) can be reduced to the form

This transformation does not affect the solutions of the system.

*P*

_{0}+

*P*

_{1}= 0 should be used. This condition together with the condition

*P*

_{2}= 0 turns into an identity the second and third equations. After some simple transformations, we obtain the

*antisymmetric*excitations:

Next, we use the condition *P*_{0} − *P*_{1} = 0, which turns into an identity the first equation in (12). After some analysis, we can find two types of excitation:

*Symmetrical*

*Asymmetrical*

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 *a*_{α,n + 1} − *a*_{α,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.

## Results and discussion

### The analysis of the energetics of the protein excitation

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, ${\mathit{E}}_{\mathit{c}}\left(\mathit{k}\right)-{\mathit{E}}_{\mathit{a}}\left(\mathit{k}\right)=3\mathit{\Pi}+\frac{\mathit{G}}{6{\mathit{N}}_{\mathit{c}}}$ and ${\mathit{E}}_{\mathit{a}}\left(\mathit{k}\right)-{\mathit{E}}_{\u043d}\left(\mathit{k}\right)=\frac{\mathit{G}}{6{\mathit{N}}_{\mathit{c}}}$. 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 3*Π* = 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.

### Conformational response to the excitation of the alpha-helical region of the protein molecule

*R*

_{ nα }=

*R*

_{0}· (1 −

*β*|

*A*

_{ αn }|

^{2}), where designation is entered: $\mathit{\beta}\equiv \frac{\left|{\mathit{D}}^{/}\left({\mathit{R}}_{0}\right)\right|}{{\mathit{R}}_{0}\xb7\left|{\mathit{W}}^{//}\left({\mathit{R}}_{0}\right)\right|}$. If we consistently apply the model of dipole interaction between the peptide groups, then $\mathit{\beta}~\frac{\mathit{\Delta d}}{\mathit{d}}$, where, as mentioned above, Δ

*d*~ 0.29

*D*and

*d*~ 3.7

*D*. Therefore, in this dipole model [14],

*β*~ 10

^{−1}. Taking into account the definitions of coefficients

*A*

_{ αn }, given in (7), it is possible to get following:

- 1.It is possible to obtain the following formula for symmetric excitations: ${\mathit{R}}_{\mathit{n\alpha}}^{\left(\mathit{c}\right)}={\mathit{R}}_{0}\xb7\left(1-\frac{\mathit{\beta}}{3{\mathit{N}}_{\mathit{c}}}\right)$. That is, all three chains are reduced equally and evenly in the space. Then the length of every peptide chain can be appraised, so${\mathit{L}}_{\mathit{\alpha}}^{\left(\mathit{c}\right)}={\displaystyle \sum _{\mathit{n}=1}^{{\mathit{N}}_{\mathit{c}}}{\mathit{R}}_{\mathit{n\alpha}}^{\left(\mathit{c}\right)}}\equiv {\mathit{N}}_{\mathit{c}}{\mathit{R}}_{0}-\frac{1}{3}\mathit{\beta}{\mathit{R}}_{0}\equiv {\mathit{L}}_{0}-\frac{\mathit{\beta}{\mathit{R}}_{0}}{3}.$

- 2.For antisymmetric excitations, it is possible to obtain ${\mathit{R}}_{\mathit{n}0}^{\left(\u0430\right)}={\mathit{R}}_{\mathit{n}1}^{\left(\u0430\right)}={\mathit{R}}_{0}\xb7\left(1-\frac{\mathit{\beta}}{2{\mathit{N}}_{\mathit{c}}}\right)$, ${\mathit{R}}_{\mathit{n}2}^{\left(\u0430\right)}={\mathit{R}}_{0}$. Respective lengths are as follows:${\mathit{L}}_{0}^{\left(\mathit{a}\right)}={\mathit{L}}_{\phantom{\rule{0.5em}{0ex}}1}^{\left(\mathit{a}\right)}={\mathit{L}}_{0}-\frac{\mathit{\beta}{\mathit{R}}_{0}}{2};\phantom{\rule{5em}{0ex}}{\mathit{L}}_{2}^{\left(\mathit{a}\right)}={\mathit{L}}_{0}\equiv {\mathit{R}}_{0}{\mathit{N}}_{\mathit{c}}.$

*R*

_{ k }and angle

*φ*:

*β*~ 10

^{−1},

*R*

_{0}= 5.4 Å, and

*d*

_{ α }= 4.56 Å in (16) gives $\mathrm{\Delta}~\frac{{\mathit{N}}_{\mathit{c}}}{5}\phantom{\rule{0.12em}{0ex}}\left(\mathrm{\AA}\right)$. For the typical number of turns in many enzymes and membrane squirrel (

*N*

_{ c }> 10), displacement will have an order Δ > 2 Å. This is consistent with the observed values [11].

- 3.For asymmetrical excitation, the following values are implemented: ${\mathit{R}}_{\mathit{n}0}^{\left(\u043d\right)}={\mathit{R}}_{\mathit{n}1}^{\left(\u043d\right)}={\mathit{R}}_{0}\xb7\left(1-\frac{\mathit{\beta}}{6{\mathit{N}}_{\mathit{c}}}\right)$, ${\mathit{R}}_{\mathit{n}2}^{\left(\u043d\right)}={\mathit{R}}_{0}\xb7\left(1-\frac{2\mathit{\beta}}{3{\mathit{N}}_{\mathit{c}}}\right)$. The corresponding lengths of peptide chains equal${\mathit{L}}_{0}^{\left(\u043d\right)}={\mathit{L}}_{1}^{\left(\u043d\right)}={\mathit{L}}_{0}-\frac{\mathit{\beta}{\mathit{R}}_{0}}{6},\phantom{\rule{0.72em}{0ex}}{\mathit{L}}_{2}^{\left(\u043d\right)}={\mathit{L}}_{0}-\frac{2\mathit{\beta}{\mathit{R}}_{0}}{3}.$

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.

*N*

_{ c }, it falls down yet more. Second, a conformational response for this type of excitation is the biggest for

*N*

_{ c }≤ 14. This is typical for enzymatic proteins only.

## Conclusions

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 [11].

## Endnotes

^{a}Off-system unit of energy: 1 eV = 1.602 × 10^{−19} J.

^{b}For example, in the myosin protein, the helical region has about 200 turns or up to 700 amino acids.

## Declarations

## Authors’ Affiliations

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