Protein energy landscapes exploration by means of the activation process model

The activation process model is used to explain energy transfer in proteins for energies less than 1 eV. The model does not require the use of quasi-particles, like phonons, excitons, or solitons. As a protein molecule absorbs electromagnetic energy, electronic transitions take place by means of activation processes associated with various conformational substates of a protein. This leads to resonances in various frequency ranges, including THz, GHz, MHz, and kHz. The resonances are explored for tubulins, several telomerases and telomeres, and ion channel proteins related to pain transmission.

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Any living cell absorbs electromagnetic waves resonantly in a fairly wide frequency range. This amazing feature was studied extensively by Hans J. H. Geesink and Dirk K. F. Meijer [1]. They proposed a hypothesis of a mathematical algorithm for electromagnetic frequencies that may create stability of biological order. Using this algorithm, they developed a bio-soliton model capable of predicting non-thermal electromagnetic radiation frequency bands that could either stabilize or destabilize life conditions [2].

Another model for the calculation of electromagnetic resonances in proteins was proposed by Irena Cosič. This is the resonant recognition model (RRM) [3]. The RRM is based on the finding that certain frequencies within the distribution of energies of delocalized electrons along a protein molecule are critical for protein biological function and/or interaction with its target. The RRM enables these frequency characteristics to be calculated. First, the protein primary structure is presented as a numerical series by assigning the electron-ion interaction pseudopotential relevant to the protein’s biological activity to each amino acid. Second, the numerical series are analyzed by digital signal analysis methods, using inverse Fourier transform and wavelet transform, in order to extract information pertinent to the biological function. Each RRM frequency characterizes one particular biological function or interaction [4]. The in-depth model description and RRM frequencies for various proteins can be found in [5].

From the physics of electron transitions standpoint, both models treat optical resonances identically. That is not the case for gigahertz, megahertz, and kilohertz resonance frequencies. These are explained in terms of phonons, excitons, and solitons. However, these quasi-particles are used differently. For example, solitons transfer energy along alpha-helices only [6], or through the entire volume of a protein [7].

Apparently, it is still not entirely clear how exactly the process of energy transfer by means of electromagnetic resonances occurs in proteins [8,9]. Assume this process results from an activation process taking place on one of the protein energy landscapes. Let us consider this problem in detail.

Protein energy landscapes

It is known that proteins possess a complex energy landscape with a large number of local minima called conformational substates (CS) that are arranged in a hierarchical fashion [10]. Daniël Thorn Leeson used flash photolysis and spectral diffusion to study the dynamics of CO molecule bond restoration in Zn-myoglobin heme in the range from 100 mK to 300K [11]. He derived the following sequence for CS tiers: CS4≤ 9.1 10-4 eV, CS3≤ 0.01 eV, CS2≤ 0.1 eV, CS1≤ 1.0 eV, and CS0>1.0 eV. To understand what kind of activation processes arise, one has to calculate the frequency of thermal radiation nmax at which the exchange of energy between radiation and a molecule is most efficient [12]. The value of nmax for T=100 mK is 5.875·109 Hz, and the one for T=300 K is 1.763·1013 Hz. The corresponding values of thermal radiation quanta are 2.432·10-5 eV and 7.299·10-2 eV.

Thus, activation processes caused by thermal equilibrium radiation can occur on CS4, CS3, and CS2 tiers. Then it would be reasonable to conclude that non-equilibrium activation processes take place on CS1 and CS0 tiers. Energy released by the hydrolysis of adenosine triphosphate (ATP) to adenosine diphosphate (ADP) and inorganic phosphate is enough for CS2 and CS1 activation processes, while light activates biochemical reactions on the CS0 tier [13].

According to Aristotle, nature does nothing uselessly. Myoglobin is sometimes called a hydrogen atom in biology [14]. Using Occam’s razor, it is tempting to generalize D.T. Leeson’s energy landscape estimates to more complex macromolecules such as tubulin protein [15,16], telomerase, telomere as an example of DNA, and TERT mRNA [17], and ion channel proteins related to pain transmission [18].

The resonant recognition model was used in [16-18] to calculate RRM frequencies for the proteins under consideration. Now we use the activation process model to interpret these frequencies.

Activation process model

The model is based on two assumptions:

1. In elementary activation interaction act, potential energy of moving atoms changes discretely or in quanta. The elementary activation act appears to be a series of quantum subsystems occurring in sequence. These subsystems can also be defined as identical quantum oscillators.
2. Energy exchange between electromagnetic radiation and interacting atoms results in discrete translational motion changes of those atoms that absorb and subsequently emit an oscillation energy quantum series [19-21].

With these assumptions in mind, the average energy of activation process is calculated as

$\stackrel{-}{{\epsilon }_m}=\frac{h\nu }{e^{mh\nu /kT}-1},\text{ (1)}$

Here, m is the number of subconformations that a protein forms when passing the activation barrier on a given tier, ν is the frequency characterizing the quantum oscillator, mhν=hνa is the activation energy for the barrier. The value of ν equals the Einstein spontaneous emission coefficient A [22], so subconformations’ lifetimes are inversely proportional to it. Consequently, one can get the unimolecular reaction rate constant as

$K=\frac{\stackrel{-}{{\epsilon }_m}}{h}=\frac{h\nu }{e^{mh\nu /kT}-1}.\text{ (2)}$

Suppose that each tier is associated with several coexisting activation processes, their activation energies being within the tier’s energy range. The main model parameters in Eqs (1-2) are nand na=mn. The number of quantum subsystems m forming a barrier is of the order of 105-106 for low-temperature equilibrium fluctuations of myoglobin [23,24] as well as for non-equilibrium L-lactate dehydrogenase catalytic reactions for reversible reduction of pyruvate to L-lactate in vivo [25]. It would be reasonable to suggest the same order of m for the proteins under investigation. In addition, the radiation of interest, by its nature, is associated with dipoles. Their dipole moments can be derived using the known formula [22]:

$\overrightarrow{D}=0.5·\sqrt{\frac{\nu ·{(137)}^3}{m·\nu }} (\widehat{A}Å).\text{ (3)}$

It was previously demonstrated that these values should fall in the range 1 to 2 A [25].

Tubulin

Tubulin is a protein that exists in all eukaryotic cells. It is polymerized as a dimer of alpha and beta molecules. These dimmers are then polymerized into long filaments called microtubules. Their conductivity depends on whether they are filled with water or not. Microtubules are essential for cell movement, intracellular transport, cell division, and synaptic plasticity in brain neurons. However, the mechanisms by which microtubules organize intracellular dynamics are still not known.

A meticulously conducted, comprehensive experimental study of microtubules derived from pig brain uncovered several resonant frequencies in various regions of the electromagnetic spectrum [15]. In [16], the RRM method was applied to a variety of tubulins, namely rat, pig, human, mouse, and bovine ones. The RRM frequencies analysis revealed a striking commonality in the functioning of microtubules regardless of the mammal species.

Mean values of these RRM frequencies are now used to calculate all necessary parameters for the activation process model. They are presented in Tables 1-4. Column 6 shows tiers for activation barriers. Columns 7 and 8 demonstrate unimolecular reaction rates for physiological temperatures of 298 K and 314 K. They vary greatly from 10-23 Hz to 106 Hz. For example, row 1 of Table 1 presents a CS1-tier chemical reaction that is likely a tubulin polymerization [26,27]. In general, the calculated reaction rates can be divided into two ranges. Ultraslow reactions with reaction rates from 10-23 Hz to 10-14 Hz are apparently associated with cell division. Normal ones with reaction rates from 101 Hz to 106 Hz describe tubulin transformations. It is known that labile (that is, forming and then disintegrating) microtubules are found most often in the cell cytoplasm [27]. This is the reason for the diversity of chemical reactions.

Telomerases and telomeres

These proteins protect the end of the chromosome from DNA damage or from fusion with neighboring chromosomes. Several groups of them were studied in [17] by means of the RRM model:

1. A first group consisted of five TERT telomerase proteins: Q27ID4 – TERT_BOVIN, O14746 – TERT_HUMAN (1-230), O14746 – TERT_HUMAN (325-550), O70372 – TERT_MOUSE and Q673L6 – TERT_RAT.
2. A second one included six telomerase TERT mRNA coding sequences, Homo sapiens telomerase reverse transcriptase isoforms: JF896280.1 – Delta2, JF896283.1 – Delta2-8, JF896285.1 – Delta4-13, JF896281.1 – Delta3p-12, and JF896286.1 – INTR1.
3. A third one contained ten homo sapiens telomeric repeat region isolates: HQ167745.1 – AE, HQ167740.1 – DVW, HQ167744.1 – KA, HQ167746.1 – KM, HQ167741.1 – KP, HQ167742.1 – LH, HQ167748.1 – SA, HQ167747.1 – SE, HQ167743.1 – VDEC, and AF0207783.1 – chromosome 20.

The activation barrier characteristics are calculated as in the previous case. The results are presented in Tables 5-8. Row 11 of Table 5 fits the experimental data of [28]. Still, a thorough experimental data search remains to be done.

Ion channel proteins related to the pain transmission

Pain is a sensation associated with human suffering. Therefore, scientific research into its elimination is a priority.

In [18], the RRM model was used to study pain transmission, which is an electrical signal along the nerve (axon). This electrical signal is formed by complex activation (opening and closing) of specific pain-related ion channels and the redistribution of electrically charged ions at the nerve cell membrane. The activity of sodium and calcium ion channels is the most critical for pain transmission along nerves. The authors of [18] analyzed human, mouse, and rat ion channels only, as they are mostly investigated and mostly related to human pain. The following protein sequences were used: twelve sodium ion channel proteins related to pain; three pain toxin proteins that influence opening and closing of sodium ion channels; fifteen mu-conotoxin proteins that also influence opening and closing of sodium ion channels; eight calcium ion channel proteins related to pain and eight omega-conotoxin proteins that influence opening and closing of calcium ion channels. Two characteristic RRM frequencies were identified: fn1=0.1465 for sodium ion channels and fc2=0.1021 for calcium ones. When different modalities of charge transfer through the protein backbone were introduced, the resonant frequencies for sodium and calcium ion channels were estimated to be in various frequency ranges, including THz, GHz, MHz, and kHz. Then it was concluded that different modalities of charge transfer can produce different resonant frequencies, which are not necessarily related to the protein’s biological function, but could be related to ion channel resonances in general.

The activation barrier characteristics are shown in Tables 9-12. These are the number of quantum subsystems forming a barrier, dipole moments values, barrier activation energies, what tiers they belong to, and reaction rates at several temperatures. Row 1 of Table 9 fits the experimental data of [29]. More experimental confirmations are expected to be found.

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