Research Article
Open Access
The Production of Bio-energy and its Properties
of Transport in the Living systems
Pang Xiao- feng*
Institutes of Physical electron and Life Science and Technology, University of Electronic Science and Technology of
Chengdu 610054,China.
*Corresponding authors address: Pang Xiao-feng, Institutes of Physical electron and Life Science and Technology, University of Electronic Science and Technology of Chengdu 610054,China; E-mail:
pangxf2006@aliyun.com
Received: May 5, 2017; Accepted: May13, 2017; Published: July 03, 2017
Citation: Pang Xiao- feng (2017) The Production of Bio-energy and its Properties of Transport in the Living systems.Int Struct Comput Biol 1(1):1-21.
We here introduced the form of bio-energy in living system
and eluciduted again the new theory of bio-energy transport along
protein molecules in living systems based on the changes of structure
and conformation of molecules arising from the energy, which is
released by hydrolysis of adenosine triphosphate (ATP). In this theory,
the Davydov’s Hamiltonian and wave function of the systems are
simultaneously improved and extended. A new interaction have been
added into the original Hamiltonian. The original wave function of the
excitation state of single particles have been replaced by a new wave
function of two-quanta quasicoherent state. In such a case, bio-energy
is carried and transported by the new soliton along protein molecular
chains. The soliton is formed through self- trapping of two excitons
interacting amino acid residues. The exciton is generated by vibrations
of amide-I (C=O stretching) arising from the energy of hydrolysis of
ATP. The properties of the soliton are extensively studied by analytical
method and its lifetime for a wide ranges of parameter values relevant
to protein molecules is calculated using the nonlinear quantum
perturbation theory. The lifetime of the new soliton at the biological
temperature 300 K is enough large and belongs to the order of 10-
10 second orτ/τ0≥700. The different properties of the new soliton
are further studied. The results show that the new soliton in the new
model is a better carrier of bio-energy transport and it can play an
important role in biological processes. This model is a candidate of the
bio-energy transport mechanism in protein molecules.
Keywords: Form; Living system; Bio-energy; protein; Biological
energy; Soliton; ATP hydrolysis; Amide; Exciton; Life time; Amino acid;
Quasi-coherent state.
I. The Phosphorylation and de-Phosphorylation reactions in
the Cell and the features of energy released in hydrolysis of
ATP molecules
As it is known, Kal’kar first proposed the idea of aerobic
phosphorylation, which is carried out by the phosphorylation
coupled to the respiration. Belitser studied in detail the
stoichometric ratios between the conjugated bound phosphate
and the absorption of oxygen and gave further the ratio of the
number of ionorganic phosphate molecules to the number of
oxygen atoms absorbed during the respiration, which is not
less than two. He thought also that the transfer of electrons
from the substrate to the oxygen is a possible source of energy
for the formation of two or more ATP molecules per atom of
absorbed oxygen. Therefore Belitser and Kal’kar’s research
results are foundations establishing modern theory of oxidative
phosphorylation of ATP molecules in the cell [1-3].
In such a case we must know clearly the mechanism and
properties of the oxidation process, which involves the transfer of
hydrogen atoms from the oxidized molecule to another molecule,
in while there are always protons present in water and in the
aqueous medium of the cell, thus we may only consider the transfer
of electrons in this process. The necessary number of protons to
form hydrogen atoms is taken from the aqueous medium. The
oxidation reaction is usually preceded inside the cell under the
action of special enzymes, in which two electrons are transferred
from the food substance to some kind of initial acceptor, another
enzymes transfer them further along the electron transfer chain
to the second acceptor etc. Thus a water molecule is formed in
which each oxygen atom requires two electrons and two protons.
The main initial acceptors of electrons in cells are
the oxidized forms NAD+ and NADP+ of NAD (nicotine amide
adenine dinucleotide or pyridine nucleotide with two phosphate
groups) molecules and NADP(nicotine amide adenine nucleotide
phosphate or pyridine nucleotide with three phosphate groups)
as well as FAD (flavin adenine dinucleotide or flavoquinone) and
FMN (flavin mononucleotide).The above oxidized forms of these
molecules serve for primary acceptors of electrons and hydrogen
atoms through attaching two hydrogen atoms [3], which is
expressed by
Where NADP molecule becomes the reduced molecule
NADPH. The NAD molecule has also the same active center as
the NADP molecule; it can be converted to the reduced molecule
NAD. H under combining with two atoms of hydrogen according
to the reaction [3]:
The NAD
+ and NADP
+ are the enzymes, which can
perform the reaction of dehydrogenation on compounds
containing the group of atoms through removing two hydrogen
atoms.
In the presence of enzymes, such as pyridine-dependent
hydrogenases, and with the participation NAD
+ and NADP
+
molecules two hydrogen atoms, including two protons and two
electrons, are removed from this group of atoms. One proton
and two electrons combine with the NAD
+ and NADP
+ molecule
converting them to the reduced forms NADP . H or NAD . H and
the second proton is released. This mechanism can be also used
to oxidize lactic acid (lactate) with the formation of pyruvic acid
(pyruvate) and NAD H, in which the reduced molecules NADP . H
and NAD . H serve as electron donors (reducing agents) in other
reactions. They are involved in a large number of biosynthetic
processes, such as in the synthesis of fatty acids and cholesterol.
Therefore, the molecule NAD • H can serve as an
electron donor in the process of oxidative phosphorylation, then
the phosphorylation reaction is of [3]
Where ADP is called the adenosine diphosphate. The abbreviated form of this reaction can be written as
Thus three ATP molecules are formed in the reaction, in which the
synthesis of ATP molecule are carried out through the transfer
of two electrons from the NAD • H molecule along the electron
transport chain to the oxygen molecule in the mitochondria. In
this way the energy of each electron is reduced by 1.14 eV. The
reaction is called the phosphorylation of ADP molecules.
However, an ATP molecule can reacts with water in an
aqueous medium, which results in the energy release of about
0.43eV under normal physiological conditions by virtue of some
special enzymes. The reaction can be represented by
Its abbreviated form is of
In this process ATP molecules are transformed as ADP molecules
and the bio-energy of about 0.43eV is also released. Then it is
referred to as de-phosphorylation reaction of ATP molecules.
We know from the above representations that an
increase in free energy
in reaction and its decrease in reaction
depend on their temperatures, concentrations of the ions Mg
2+
and Ca
2+ and on the pH value of the medium. Under the standard
conditions
=0.32 eV ( ~7.3 kcal/mole). If the appropriate
corrections are made taking into consideration the physiological
pH values and the concentration of Mg
2+ and Ca
2* inside the cell
as well as the normal values for the concentrations of ATP and
ADP molecules and inorganic phosphate in the cytoplasm we can
obtain a value of ~0.54eV ( ~12.5 kcal/mole) for the free energy
in the hydrolysis of ATP molecule. Hence the free energy for the
hydrolysis of ATP molecules is not constant. But it is impossibly
the same at different sites of the same cell if these sites have
different concentrations of ATP, ADP, Pi, Mg
2+, Ca
2+.
On the other hand, cells contain a number of
phosphorylated compounds the hydrolysis of which in the
cytoplasm is associated with the release of free energy. Then the
values for the standard free energy of hydrolysis for some of these
compounds are also different.
The enzymes carrying out the above synthesis of ATP
molecules from ADP molecules and inorganic phosphate in the
coupling membranes of mitochondria are the same as in the
cytoplasmic membranes of bacteria, which are mainly composed
of F and F, which are joined to each other by the small proteins F5
and F6 . These proteins form the F - F complex or the enzyme ATPase,
in which F is composed of five protein subunits and has the
shape of a sphere with a diameter of about 9nm which projects
above the surface of the membrane in the form of a protuberance.
In the coupling membrane of mitochondria and the cytoplasmic
membrane of bacteria the complex F - F is positioned so that the
enzyme F is on the inside of the membrane [1-3].
The enzyme F can extend from one side of the membrane
to the other and has a channel which lets protons through. When
two protons pass through the complex F - F in the coupling
mitochondrial membrane one ATP molecule is synthesized inside
the matrix from an ADP molecule and inorganic phosphate.
This reaction is reversible. Under certain condition the enzyme
transports protons from the matrix to the outside using the
energy of dissociation of ATP molecules, which may be observed
in a solution containing isolated molecules of enzyme F and ATP.
The largest two proteins in F, which is composed of five protein
molecules, take part in the synthesis and dissociation of ATP
molecules, the other three are apparently inhibitors controlling
these reactions.
After removing enzyme F molecules from mitochondria
the remaining F enzymes increase greatly the permeability of
protons in the coupling membranes, which confirms that the
enzyme F has really a channel for the passage of protons which is
constructed by the enzyme F. However, the complete mechanism
for the synthesis of ATP molecules by the enzyme ATP-ase is still
not clearly known up to now.
II. The physical and biological foundations of construction
of new theory
As it is known, many biological processes, such as muscle
contraction, DNA reduplication, neuroelectric pulse transfer on
the neurolemma and work of calcium pump and sodium pump,
and so on, are associated with bioenergy transport through
protein molecules, where the energy is released by the hydrolysis
of adenosine triphosphate (ATP) in the living systems. Thus there
here are always biological processes of energy transport from
production place to absorption place in the living systems. In
general, the bioenergy transport is carried out by virtue of protein
molecules. Therefore, the study of the bioenergy transport along
protein molecules is a very interesting subject in biology and has
important significance in life science. However, understanding
the mechanism of bioenergy transport in biomacromolecular
systems has been a long-standing problem that remains of great
interest today. As an alternative to electronic mechanisms [1],
one can assume that the energy is stored as vibrational energy in
the C=0 stretching mode (amide-I) of a protein molecular chain of
polypeptide. Following Davydov’s idea [2], ones take into account
the coupling between the amide-I vibrarional quantum (exciton
) and the acoustic phonon (molecular displacements) in the
amino acid residues; Through the coupling, nonlinear interaction
appears in the motion of the vibrartional quanta, which could
lead to a self-trapped state of the vibrational quantum. The latter
plus the deformational amino acid lattice together can travel
over macroscopic distances along the molecular chains, retaining
the wave shape, energy, momentum and other properties of the
quasiparticle. In this way, the bioenergy can be transported as
a localized “wave packet” or soliton. This is just the Davydov’s
model of bioenergy transport in proteins, which was proposed in
the 1970s [2,3].
Davydov model of bioenergy transport work at α −
helical proteins as shown in Figure.1.

Figure 1: Structure of α − helical protein
Following Davydov idea [3], the Hamiltonian describing
such system has in the form of
where
is is the amide-I quantum energy,
-J is the dipole-dipole interaction energy between neighboring
sites,
is the creation (annihilation) operator for an amide-I
quantum excitation (exciton) in the site n, u
n is the displacement
operator of amino acid residues at site n, P
n is its conjugate
momentum operator, M is the mass of an amino acid molecule, w
is the elasticity constant of the protein molecular chains, and
is an nonlinear coupling parameter and represents the coupling
size of the exciton-phonon interaction. The wave function of the
systems proposed by Davydov is in the form of
where I0 > =I0 >
ex Io
ph I0 >
ex and I0 >
ph are the ground
states of the exciton and phonon, respectively,
is annihilation (creation) operator of the phonon with wave
vector q,
and
are some undetermined functions of
time. Obviously,
in Eq.(2) is an eigenstate
of the number operator
corresponding to a single
excitation, i.e.,
.
The Davydov soliton obtained from Eqs. (1)-(2) in the semi
classical limit and using the continuum approximation has the
from
Corresponding to an excitation localized over a scale r
/ D
μ , where
is the sound speed in the
protein molecular chains, v is the velocity of the soliton, r0 is the
lattice constant. Evidently, the soliton contains only one exciton, i.e.,
This shows that the Davydov soliton is
formed through self-trapping of one exciton with binding energy E
BD,
Davydov’s idea yields a compelling picture for the mechanism of
bioenergy transport in protein molecules and consequently has
been the subject of a large number of works [3-28]. A lot of issues
related to the Davydov model, including the foundation and
accuracy of the theory, the quantum and classical properties and
the thermal stability and lifetimes of the Davydov soliton have
extensively been studied by many scientists [7-26]. However,
considerable controversy has arisen concerning whether the
Davydov soliton is sufficiently stable in the region of biological
temperature to provide a viable explanation for bio-energy
transport. It is out of question that the quantum fluctuations and
thermal perturbations are expected to cause the Davydov soliton
to decay into a delocalized state. Some numerical simulations
indicated that the Davydov soliton is not stable at the biological
temperature 300K [7-11, 24-26]. Other simulations showed
that the Davydov soliton is stable at 300 K [10-24], but they
were based on classical equations of motion which are likely to
yield unreliable estimates for the stability of the soliton [3]. The
simulations based on the ID
2 state in Eq.(2) generally show that
the stability of the soliton decreases with increasing temperatures
and that the soliton is not sufficiently stable in the region of
biological temperature. Since the dynamical equations used in the
simulations are not equivalent to the equation, the stability of the
soliton obtained by these numerical simulations is unavailable or
unreliable. The simulation[9] based on the ID1>state in Eq. (3)
with the thermal treatment of Davydov[3], where the equations
of motion are derived from a thermally averaged Hamiltonian,
yields the confusing result that the stability of the soliton is
enhanced with increasing temperature, predicting that ID1>- type
soliton is stable in the region of biological temperature. Evidently,
the conclusion is doubtful because the Davydov procedure, in
which an equation of motion for an average dynamical state
from an average Hamiltonian, corresponding to the Hamiltonian
averaged over a thermal distribution of phonons, is inconsistent
with standard concepts of quantum-statistical mechanics in
which a density matrix must be used to describe the system.
Therefore, any exact fully quantum- mechanical treatment for
the numerical simulation of the Davydov soliton does not exist.
However, for the thermal equilibrium properties of the Davydov
soliton, there is a quantum Monte Carlo simulation [13]. In the
simulation, correlation characteristic of solitonlike quasiparticles
occur only at low temperatures, about T< 10k, for widely accepted
parameter values. This is consistent at a qualitative level with the
result of Cottingham et al. [15,21]. The latter is a straightforward
quantum-mechanical perturbation calculation. The lifetime of the
Davydov soliton obtained by using this method is too small (about
10
-12 - 10
-13Sec) to be useful in biological processes. This indicates
clearly that the Davydov solution is not a true wave function of the
systems. A through study in terms of parameter values, different
types of disorder, different thermalization schemes, different
wave functions, and different associated dynamics leads to a
very complicated picture for the Davydov model [10-12]. These
results do not completely rule out the Davydov theory, however
they do not eliminate the possibility of another wave function and
a more sophisticated Hamiltonian of the system having a soliton
with longer lifetimes and good thermal stability.
Indeed, the question of the lifetime of the soliton in
protein molecules is twofold. In Langevin dynamics, the problem
consists of uncontrolled effects arising from the semiclassical
approximation. In quantum treatments, the problem has been
the lack of an exact wave function for the soliton. The exact wave
function of the fully quantum Davydov model has not been known
up to now. Different wave functions have been used to describe the
states of the fully quantum-mechanical systems [4,5]. Although
some of these wave functions lead to exact quantum states
and exact quantum dynamics in the J=0 state, they also share a
problem with the original Davydov wave function, namely that
the degree of approximation included when
is not known.
Therefore, it is necessary to reform Davydov’s wave function.
Scientists had though that the soliton with a multiquantum
, for example, the coherent state of Brown et al. [4], the
multiquantum state of Kerr et al. [12] and Schweitzer et.al.
[15,21], the two-quantum state of Cruzeiro-Hansson [18] and
Forner [22], and so on, would be thermally stable in the region of
biological temperature and could provide a realistic mechanism
for bioenergy transport in protein molecules. However, the
assumption of the standard coherent state is unsuitable or
impossible for biological protein molecules because there are
innumerable particles in this state and one could not retain
conservation of the number of particles of the system. The
assumption of a multiquantum state (n>2) along with a coherent
state is also inconsistent with the fact that the bioenergy released
in ATP hydrolysis can excite only two quanta of amide-I vibration.
On the other hand, the numerical result shows that the soliton of
two-quantum state is more stable than that with a one-quantum
state.
Cruzeiro-Hansson [18] had thought that Forner’s twoquantum
state in the semiclassical case was not exact. Therefore,
he constructed again a so-called exactly two-quantum state for
the semiclassical Davydov system as follows [18]:
where
is the annihilation (creation) operator for an
amide-I vibration quantum (exciton),
is the displacement of
the lattice molecules,
is its conjugate momentum, and
is the
ground state of the exciton. He calculate the average probability
distribution of the exciton per site, and average displacement
difference per site, and the thermodynamics average of the
variable,
as a measure of localization of the exciton,
versus quantity
and
in the so-called twoquantum
state. Eq.(5), where
is a nonlinear coupling parameter
related to the interaction of the exciton-phonon in the Davydov
model. Their energies and stability are compared with those
of the one-quantum state. From the results of above thermal
averages, he drew the conclusion that the wave function with
a two-quantum state can lead to more stable soliton solutions
than that with a one-quantum state, and that the usual Langevin
dynamics ,whereby the thermal lifetime of the Davydov soliton
is estimated, must be viewed as underestimating the soliton
lifetime.
However, by checking carefully Eq.(5), we can find
that the Cruzeiro-Hansson wave function[18,24-26] does not
represent exactly the two-quantum state. To find out how many
quanta the state Eq.(1), indeed contains, the expectation value of
the exciton number operator has to be computed.
, in this state Eq.(5), and sum over the sites, i.e., the exciton numbers N are
where we use the relations
Therefore, the state Eq.(5), as it is put forward in Ref. [10],
deals with four excitons (quanta), instead of two excitons,
in contradiction to the author’s statements. Obviously, it is
impossible to create the four excitons by the energy released in the
ATP hydrolysis (about 0.43 eV). Thus the author’s wave function
is still not relevant to protein molecules, and his discussion and
conclusion are all unreliable and implausible in that paper [10].
It is believed that the physical significance of the wave
function, Eq. (5), is also unclear, or at least is very difficult to
understand. As far as the physical meaning of Eq.(5) is concerned,
it represents only a combinational state of single-particle
excitation with two quanta created at sites n and m;[18,26] is
the probability amplitude of particles occurring at the sites n
and m simultaneously. In general, n=≠m and in accordance with
the author’s idea. In such a case it is very difficult to imagine
the form of the soliton by the mechanism of self- trapping of the
two quanta under the action of the nonlinear exciton-phonon
interaction, especially when the difference between n and m
is very large. Hansson has also not explained the physical and
biological reasons and the meaning for the proposed trial state.
Therefore, we think that the Cruzeiro-Hansson representation
is still not an exact wave function suitable for protein molecules.
Thus, the wave function of the systems is still an open problem
today.
On the basis of the work of Cruzeio-Hansson
[12,13,18,22], Schweitzer [21] and Pang [24-26] proposed a new
model of the bioenergy transport in the protein molecules, in
which both the Hamiltonian and the wave function of the Dovydov
model [24] have been improved. A new coupling interaction
between the acoustic and amide-I vibrational modes was added
to the original Davydov’s Hamiltonian which takes into account
relative displacement of the neighbouring amino acids resulting
from dipole-dipole interaction of the neighbouring amide-1
vibrational quanta. Davydov’s wave function has been also
replaced with a quasi-coherent two-quanta state to exhibit the
coherent behaviors of collective excitations of the excitons and
phonons [25-26] which are a feature of the energy released in
ATP hydrolysis in the systems. The equation of motion and the
properties of the new soliton in the new model are different from
those in the Davydov model and as a result the soliton lifetime and
stability are greatly enhanced. It is suggested that this model can
resolve the controversy on the thermal stability and lifetime of the
soliton excited in the protein molecules. The quantum properties
of the new soliton will be studied here, but here attention is paid
also to the problem of its lifetime and thermal stability at biological
temperature 300 K and the lifetime of the new soliton at 300K is
calculated in detail by using the generally accepted values of the
parameters appropriate to -helical protein molecules in terms
of the quantum perturbation theory developed by Cottingham et
al. [15], which can take simultaneously into account the quantum
and thermal effects. It can be seen that the lifetime of the new
soliton at 300 K is long enough to provide a viable explanation of
the bio-energy transport in the proteins. The plan of this paper
is as follows. In Section 2, the new model, including the extended
Hamiltonian and the wave function, is presented. The equations
of motion and the new soliton solution in this model are given in
Section 3. In Section 4, the properties and thermal stability of the
new soliton are discussed, and the possibility of the soliton being
a suitable candidate for the mechanism of bioenergy transport
in protein molecules is predicted on the basis of results obtained
in this paper. In Section 5, the properties of the new soliton
are described and its lifetime is calculated by using quantummechanical
perturbution method. The detailed discussion of the
properties and changes of the lifetimes of the soliton for a large
range of parameter values is presented. The conclusions of this
investigation are also given in this section.
III. Eatablishment of new theory of Bio-energy transport in
the Protein molecules
Results obtained by many scientists over the years
indicate that the Davydov model, whether it is the wave function
or the Hamiltonian, is indeed too simple, i.e.., it does not denote
the elementary properties of the collective excitations occurring
in protein molecules, and many improvements of it have been
unsuccessful, as mentioned above. What is the source of this
problem? It is well known that the Davydov theory on bioenergy
transport was introduced into protein molecules from an excitonsoliton
model in generally one-dimensional molecular chains
[24]. Although the molecular structure of the alpha-helix protein
is analogous to some molecular crystals, for example acetanilide
(ACN) (in fact, both are polypeptides; the alpha-helix protein
molecule is the structure of three peptide channels, ACN is the
structure of two peptide channels. If comparing the structure of
alpha helix protein with ACN, we find that the hydrogen-boned
peptide channels with the atomic structure along the longitudinal
direction are the same except for the side group), a lot of properties
and functions of the protein molecules are completely different
from that of the latter. The protein molecules are both a kinds
of soft condensed matter and bio-self-organization with action
functions, for instance, self-assembling and self-renovating. The
physical concepts of coherence, order, collective effects, and
mutual correlation are very important in bio-self- organization,
including the protein molecules, when compared with generally
molecular systems [25.26]. Therefore, it is worth studying how
we can physically describe these properties. It is noted that
Davydov operation is not strictly correct. Therefore, it is believed
that a basic reason for the failure of the Davydov model is just
that it ignores completely the above important properties of the
protein molecules.
Let us consider the Davydov model with the present
viewpoint. First, as far as the Davydov wave functions, both
are concerned [3], they are not true solutions of the
protein molecules. On the one hand, there is obviously asymmetry
in the Davydov wave function since the phononic parts is a
coherent state, while the excitonic part is only an excitation state
of a single particle. It is not reasonable that the same nonlinear
interaction generated by the coupling between the excitons and
phonons produces different states for the phonon and exciton.
Thus, Davydov’s wave function should be modified [24-26], i.e.,
the excitonic part in it should also be coherent or quasicoherent
to represent the coherent feature of collective excitation in
protein molecules. However, the standard coherent [4] and
large-n excitation states [12,22] are not appropriate for the
protein molecules due to the reasons mentioned above. Similarly,
Forner’s and Cruzeiro-Hansson’s two-quantum states do not
fulfill the above request. In view of the above discussion, we
proposed the following wave function of the protein molecular
systems:
where
and
are boson creation and annihilation operators
for the exciton,
are the ground states of the
exciton and phonon, respectively
and
are the displacement
and momentum operators of the amino acid residue at site n
respectively. The
are there sets of unknown functions,
is a normalization
constant. It is assumed hereafter that
for convenience of
calculation, except when explicitly mentioned.
A second problem arises for the Davydov Hamiltonian
[24-26,28]. The Davydov Hamiltonian takes into account the
resonant or dipole-dipole interaction of the neighboring amide-I
vibrational quanta in neighboring amino acid residues with an
electrical moment of about 3.5D, but why do we not consider the
changes of relative displacement of the neighboring amino acid
residues arising from this interaction ? It is reasonable to add the
new interaction term
into the Davydov’s
Hamiltonian to represent correlations of the collective excitations
and collective motions in the protein molecules, as mentioned
above [24-26]. Although the dipole- dipole interaction is small as
compared with the energy of the amide-I vibrational quantum,
the change of relative displacement of neighboring peptide
groups resulting from this interaction cannot be ignored due
to the sensitive dependence of dipole-dipole interaction on the
distance between amino acids in the protein molecules, which is
a kind of soft condensed matter and bio-self-organization. Thus,
the Davydov Hamiltonian is replaced by
Where
is the energy of the exciton (the C=0
strechiong mode). The present nonlinear coupling constants
are
They represent the modulations of the on-site energy
and resonant (or dipole-dipole) interaction energy of excitons
caused by the molecules displacements, respectively .M is the
mass of a amino acid molcule and w is the elasticity constant of
the protein molecular chains. J is the dipole-dipole interaction
energy between neighboring sites. The physical meaning of
the other quantities in Eq.(6)are the same as those in the above
explanations.
The Hamiltonian and wave function shown in Eqs. (9)-
(10) are different from Davydov’s. We add a new interaction
term
into the original
Davydov Hamiltonian. Thus the Hamiltonian now has better
correspondence between the interactions and can also represent
the features of mutual correlations of the collective excitations
and of collective motions in the protein molecules. We should
point out here that the different coupling between the relevant
modes was also considered by Pang [24-26] and others [27-28] in
the Hamiltonian of the vibron-soliton model for one-dimensional
oscillator-lattice and protein systems, respectively, but the wave
functions of the systems they used are different from Eqs. (9)-
(10).
Evidently , the present wave function of the exciton in
Eq.(9) is not an excitation state of a single particle, but rather a
coherent state, more precisely, a quasicoherent state, because
it retain only for three terms of the expansion of a standard
coherent state, which can be viewed as an effective truncation of a
standard coherent state. When
is small, i.e.,
Pang
represented the wave function of the excitons,
, in Eq.(9) as
The last representation in Eq.(11) is a standard coherent state.
Therefore, the state of exciton denoted by the new wave function
has a coherent feature, thus the wave function inEq.11) is
normalized at
Since
required in thecalculation,
then this condition of
naturally satisfied for the
roteins consisting of several hundreds of amino acids. Just so, the
wave function denoted in Eq.(9) represents exactly the coherent
features of collective excitations of both the excitons and phonons
caused by the nonlinear exciton - phonon interaction, this
indicates that the wave function in Eq.(9) is justified for the
proteins. However, it is not an eigenstate of the number operator,
because of
Therefore, the
represents a coherent superposition of the
excitonic state with two quanta and the ground state of the
exciton, but it has a determinate numbers of quanta. From the
expectation value of number operator
we find that this state
contains the number of exciton is
namely, it contains only two quanta. Where we utilize Eq.(8) and
the following relation [24] is:
Therefore, the new wave function is completely different from
Davydov’s. The latter is an excitation state of a single particle with
one quantum and an eigenstate of the number operator, but the
former is not. The new state is a quasicoherent state. It contains
only two excitons, which come from the second and third terms
in Eq.(9), in which each term contributes only an exciton, but
it is not an excitation state of two single parties. Hence, as far
as the form of new wave function in Eq.(9) is concerned, it is
either two-quanta states proposed by Forner [21] and Cruzeiro-
Hansson [10,18] or a standard coherent state proposed by
Brown et al. [4,2] and Kerr et al’s [13] and Schweitzer et al’s
15,[21] multiquanta states. Therefore, the wave function, Eq.
(9), is new for the protein molecular systems. It not only exhibits
the coherent feature of the collective excitation of excitons and
phonons caused by the nonlinear interaction generated by the
exciton-phonon interaction, which , thus, also makes the wave
function of the states of the system symmetrical, but it also agrees
with the fact that the energy released in the ATP hydrolysis
(about 0.43 eV) may only create two amide-I vibrational quanta
which, thus, can also make the numbers of excitons maintain
conservation in the Hamiltonian, Eq.(10). Meanwhile, the new
wave function has another advantage, i.e., the equation of motion
of the soliton can also be obtained from the Heisenberg equations
of the creation and annihilation operators in quantum mechanics
by using Eqs. (9) and (10), but the wave function of the states of
the system in other models could not, including the one-quanta
state [3] and the two-quanta state [12,22]. Therefore, the above
Hamitonian and wave function, Eqs. (9) and (10), are reasonable
and appropriate to the protein molecules.
The dynamic equation of Bio-energy transport
We now derive the equations of motion from Pang’s
model. First of all, we give the interpretation of
and
in Eq.(9).We know that the phonon part of the new wave function
in Eq.(9) depending on the displacement and momentum
operators is a coherent state of the normal model of creation and
annihilation operators. A coherent state for the mode with wave
vector
is [3,12,24-26]
Utilizing the standard transformations
we can get [12,23]
,where
is in Eq.(9), and
is the distance between neighboring
amino acid molecules, and
is the annihilation (creation)
operator of the phonon with wave vector
,where
Utilizing again the above results and the formulas of the
expectation values of the Heisenberg equations of operators
and
, in the state
.
We can obtain the equation of motion for the
as
From Eq.(19) we see that the presence of two quanta for the
oscillators increases the driving force on the phonon field by that
factor, when compared with the Davydov theory.
We now derive the equation of motion for the
. A basic assumption in the derivation is that
in Eq. (9) is a
solution of the time-dependent equation [24-26]:
The left-hand side of Eq. (16) has [12,23]
Now left-multiplying the both sides of Eq.(21) by
,the lefthand
side of Eq.(21) can be
Similarly, for the right-hand side of Eq. (20) we can have [12,23]
where
and utilizing Eqs.(8) and (12)-(14) and the relationships can be
obtained:
From Eqs.(20)-(23) we can obtain
In the continuum approximation we get from Eqs.(19) and (26)
and
here
and
. The soliton solution of Eq.(27) is thus
with
Although forms of the above equations of motion and
the corresponding solution, Eqs.(27)- (30), are quite similar to
that of the Davydov soliton, the properties of new soliton have
very large differences from the latter because the parameter
values in the equation of motion and the solution Eqs.(27) and
(29), including
and
have obvious distinctions
from that in the Davydov model. A straightforward result of
Pang’s model is to increase the nonlinear interaction energy
and the amplitude of the new
soliton and decrease its width due to an increase of [24-26] when
compared with Davydov soliton [3], where
, and
are the corresponding values in the
Davydov mode [3-8]. Thus the localized feature of the new
soliton is enhanced. Therefore its stability against the quantum
fluctuation and thermal perturbations increased considerably as
compared with the Davydov soliton.
V. The properties of carrier of Bio-energy transport
The energy of soliton in Pang’s model becomes [24-26]
The rest energy of the new soliton is
where
is the energy of deformation of the
lattice. The effective mass of the new soliton is
We utilize Eqs.(8) and (12)-(14) in the above calculations.
In such a case, the binding energy of the new soliton is
is larger than that of the Davydov soliton. The latter is
.They have the following relation:
We can estimate that the binding energy of the new
soliton is about several decades larger than that of the Davydov
soliton .This is a very interesting result. It is helpful to enhance
thermal stability of the new soliton. Obviously, the increase of
the binding energy of the new soliton comes from its two-quanta
nature and the added interaction
in the Hamiltonian of the systems, Eq.(10). However, we see from
Eq.(35) that the former plays the main role in the increase of the
binding energy and the enhancement of thermal stability for the
new soliton relative to the latter due to
. The increase of
the binding energy results in significant changes of properties of
the new soliton, which are discussed as follows.
In comparing various correlations to this model, it is
helpful to consider them as a function of a composite coupling
parameter like that of Pouthier and Spatchek et al. [32-35] and
Scott [6] again, it is convenient to define another composite
parameter [3,24-26] that can be written as
where
is the band edge for acoustic phonons
(Debye frequency). If,
it is said to be weak. Using widely
accepted values for the physical parameters for the alpha helix
protein molecule [2-23],
We can estimate that the coupled constant lies in the
region of
, which is not a weakly coupled theory,
the coupling strength is enhanced as compared with the Davydov
model, the latter is
Using the notation of
Bullough et al. [29,30], Teki et al. [31,32], and Pouthier et al. [33-
35]
In terms of the two composite parameters,
and
,the
soliton binding energy for Pang’s model can be written by
From the above parameter values, we find
Utilizing this value, the
versus
relations in Eq.(39) are
plotted in Fig.1. However,
for the Davydov
model (here
then the
versus
relation is also plotted in figure 2 from
this figure we see that the difference of soliton binding energies
between two models becomes larger with increasing
[24-26].

Figure 2: Binding energy
of the solitons in our model and the
Davydov model in units of dipole-dipole interaction energy (
J ) vs The
coupled constant,
, relationship
Also, we see clearly from Eqs. (28)-(32) and (35) that the
localized feature of the new soliton is enhanced due to increases
of the nonlinear interaction and of the binding energy of the new
soliton resulting from the increases of exciton-phonon interaction
in Pang’s model. Thus, the stability of the soliton against quantum
and thermal fluctuations is also enhanced considerately [24-26].
As a matter of fact, the nonlinear interaction
energy forming the new soliton in Pang’s model is
,and it is larger than
the linear dispersion energy
,i.e., the nonlinear interaction in Pang’s model is so large that it can actually cancel
or suppress the linear dispersion effect in the equation of motion
,thus the new soliton is stable in such a case according the soliton
theory [2,33-35]. On the other hand, the nonlinear interaction
energy in the Davydov model is only
and it is about three to four times smaller than
.Therefore,
the stability of the Davydov soliton is weaker as compared
with the new soliton. Moreover, the binding energy of the new
soliton in Pang’s model is
in Eq.(31),
which is somewhat larger than the thermal perturbation energy,
at
and about four times larger
than the Debye energy
(there
is the Debye frequency). This shows that transition of the new
soliton to a delocalized state can be suppressed by the large
energy difference between the initial (solitonic) state and final
(delocalized) state, which is very difficult to compensate with the
energy of the absorbed phonon. Thus ,the new soliton is robust
against quantum fluctuations and thermal perturbation, therefore
it has a large lifetime and good thermal stability in the region of
biological temperature .In practice, according to Schweitzer et
al. s studies (i.e the lifetime of the soliton increases as
and
increase at a given temperature)[15] and the
above obtained results, an inference could roughly be drawn
that the lifetime of the new soliton will increase considerably as
compared with that of the Davydov soliton due to the increase of
and
because the latter are about three times larger than
that of the Davydov model. On the other hand, the binding energy
of the Davydov soliton
and it
is about 23 times smaller than that of the new soliton, about 22
times smaller than
, and about 6 times smaller than
, respectively. Therefore, the Davydov soliton is easily destructed
by the thermal perturbation energy and quantum transition
effects. Thus it indicates that the Davdov soliton has a very
small lifetime, and it is unstable at the biological temperature
.This conclusion
is consistent at a qualitative level with the result s of Wang et al
[13,14] and Cottingham et al.[15,21].
One can sum up the differences between Pang’s model
and Davydov’s model, Eqs.(1)-(4), as follows. First, the parameter
μp is increased (μp =2μD
Secondly the nonlinear
coupling energy becomes Gp =
(G
p=2G
D
, where
is the nonlinear interaction in the Davydov model, resulting from
the two-quanta nature and the enhancement of the coupling
the coefficient
. For
helical protein molecules,
and using the parameter values listed in Eq. (37) the values of
the main parameters in Pang’s model can be calculated. These
values and the corresponding values in the Davydov model are
simultaneously listed in table 1.
Table 1: Comparison of parameters used in the Davydov model and
our new model
Parameters
Models |
μ |
G
(×10-21J) |
Amplitude of soliton
A’ |
Width of soliton Δ X(×10-10m) |
Binding energy of soliton EB
(×10-21J) |
Our
Model |
5.94 |
3.8 |
1.72 |
4.95 |
-7.8 |
Davydov
model |
1.90 |
1.18 |
0.974 |
14.88 |
-0.188 |
From table 1 we can see clearly that the new model
produces considerable changes in the properties of the new
soliton, such as large increase of the nonlinear interaction, binding
energy and amplitude of the soliton, and decrease of its width as
compared to that of the Davydov soliton. This indicates that the
soliton in Pang’s model is more localized and more robust against
quantum and the thermal stability has been enhanced [2,27,28]
which implies an increase in lifetime for the new soliton. From Eq.
(19) it can also be found that the effect of the two-quanta nature
is larger than that of the added interaction. We can therefore refer
to the new soliton as quasi-coherent.
In the above studies, the influences of quantum and
thermal effects on soliton state, which are expected to cause
the soliton to decay into delocalized states, we postulate that
the model Hamiltonian and the wave function in Pang’s model
together give a complete and realistic picture of the interaction
properties and allowed states of the protein molecules. The
additional interaction term in the Hamiltonian gives better
symmetry of interactions. The new wave function is a reasonable
choice for the protein molecules because it not only can exhibit
the coherent features of collective excitations arising from the
nonlinear interaction between the excitons and phonons, but
also retain the conservation of number of particles and fulfill
the fact that the energy released by the ATP hydrolysis can only
excite two quanta. In such a case, using a standard calculating
method [2,26]and widely accepted parameters we can calculate
the region encompassed of the excitation or the linear extent
of the new soliton,
to be greater than the lattice
constant r0, i.e.,
as shown in table 1. Conversely, we can
explicitly calculate the amplitude squared of the new soliton
using Eq. (29) in its rest frame as
Thus the
probability to find the new soliton outside a range of width r0 is
about 0.10. This number can be compatible with the continuous
approximation since the quasi-coherent soliton can spread over
more than one lattice spacing in the system in such a case. This
proves that assuming the continuous approximation used in the
calculation is valid. Therefore we should believe that the above
calculated results obtained from Pang’s model is correct.
The life time of the carrier of Bio-energy transport
at biological temperature
Partially Diagonalized form of the Model Hamiltonian
The lifetime of the soliton in the protein molecules is
an centre problem in the process of bioenergy transport because
the soliton possess certain biological meanings and can play an
important role in the biological process, only if it has enough long
lifetimes. Therefore, to calculate the lifetime of the new soliton in
Pang’s model has important significance.
For convenlence of calculation, we here represent the
wave function of the system in Eq. (9) by [24-26]
where
where we assume
,where n is an integer, denotes the
number of particle. The wave function, Eq.(40), does not only
exhibit coherent properties, but also agrees with the fact that the
energy released in the ATP hydrolysis (about 0.43eV) excites only
two amide-I vibrational quanta, instead of multiquanta (n>2)[24-
26]. Therefore, the Hamitonian and wave function of the systems,
Eqs. (9)-(10), or (40) are reasonable and appropriate to the
protein molecules. Using the standard transformation in Eq.(16),
where
,Eq.(10) becomes
where
In a semi classical and continuum approximations, from Eq.(41)
we can obtain the envelope soliton solution Eq.(29) in Pang’s
model, we now represent Eq.(29) by the following form[24-26]
where
The energy of the new soliton is
This treatment yields a localized coherent structure
of the excitons with size of order 2 r0/μp that propagates with
velocity v and can transfer energy ES01< . Unlike bare excitons
that are scattered by the interactions with the phonons, this soliton
state describes a quasi-particle consisting of the two excitons plus
a lattice deformation and hence a priori includes interaction with
the acoustic phonons. So the soliton is not scattered and spread
by this interaction of the vibration of amino acids (lattices), and
can maintain its form, energy, momentum and other quasiparticle
properties moving over a macroscopic distance. The bell-shaped
form of the soliton Eq. (43) does not depend on the excitation
method. It is self-consistent. Since the soliton always move with
velocity less than that of longitudinal sound in the chain they do
not emit phonons, i.e., their kinetic energy is not transformed into
thermal energy. This is one important reason for the high stability
of the new soliton. In addition the energy of the soliton state is
below the bottom of the bare exciton bands, the energy gap
being for small velocity of propagation. Hence there is an energy
penalty associated with the destruction with transformation
from the soliton state to a bare exciton state, i.e, the destruction
of the soliton state requires simultaneous removal of the lattice
distortion. We know in general that the transition probability to
a lattice state without distortion is very small, in general, being
negligible for a long chain. Considering this it is reasonable to
assume that such a soliton is stable enough to propagate through
the length of a typical protein structure. However, the thermal
stability of the soliton state must be calculated quantitatively.
The following calculation addresses this point explicitly [24-
26].We now diagonalize partially the model Hamiltonian in
order to calculate the lifetime of the soliton, Eq. (43), using the
quantum perturbation method [14] Since one is interested in
investigating the case where there is initially a soliton moving
with a velocity v on the chains, it is convenlent to do the analysis
in a frame of reference where the soliton is at rest. We should
then consider the Hamiltonian in this rest frame of the soliton,
where P is the total momentum, and
,where
. Also, in order to have simple analytical
expressions we make the usual continuum approximation. This
gives
where
represents now the field operator corresponding
to Bn in the continuum limit (whereas before it only indicated
a numerical value). Here L=Nr0, -π< kr0< π, and
≈(w/M)
1/2 r0•|q| ,x=nr
0. Since the soliton excitation is connected with the
deformation of intermolecular spacing, it is necessary to pass in
Eq.(47) to new phonons taking this deformation into account.
Such a transformation can be realized by means of the following
transformation of phonon operators [29]
which describe phonons relative to a chain with a particular
deformation, where b
q (b
q+) is the annihilation (creation) operator
of new phonon. The vacuum state for the new phonons is
which is a coherent phonon state[30], i.e.,
. The Hamiltonian
can now be rewritten[ 24-26] as
where
To describe the deformation corresponding to a soliton in the
subspace where there is
from Eq(45) in such a case. From the above formulae we can
obtain
In order to partially diagonalize the Hamiltonian Eq.(50) we
introduce the following canonical transformation[14,23]
where
The operators A
s+ and A
k+ the creation operators for the bound
states C
s(x) and delocalized state C
k(x), respectively. The detailed
calculation of the partial diagonalization and of corresponding
C
s(x) and C
k(x) are described in Appenix A. The partially
diagonalized Hamiltonian obtained is as follows
and
with
where
where
is determined by V(x) and the
condition
which is required to get the factor,
in the
in Eq.(55). Thus we find
and
For this
the
in Eq.(49) is just the coherent phonon
state introduced by Davydov. However, the bound state C
s(x) in
Eq.(56a), unlike the unbounded state C
k(x) in Eq.(56b), is selfconsistent
with the deformation. Such a self-consistent state of
the intramolecular excitation and deformation forms a soliton
which in the intrinsic reference frame is stationary. For the new
soliton described by the state vector
he average energy of
in Eq.(55) is
Evidently, the average energy of
in the soliton state
Eq.(60), is just equal to the above soliton energy E
sol, or the sum of
the energy of the bound state in Eq.(56a), Es, and the deformation
energy of the lattice,W’, i.e.,
This is
an interesting result, which shows clearly that the quasi-coherent
soliton formed by this mechanism is just a self-trapping state
of the two excitons plus the corresponding deformation of the
amino acid lattice. However, it should be noted that
is not
an exact eigenstate of
owing to the presence of the terms in
with
and
Transition probability and decay rate of the new
soliton
We now calculate the transition probability and decay
rate of the quasi-coherent soliton arising from the perturbed
potential by using the first-order quantum perturbation theory
developed by Cottingham, et al. [14], in which the influences of
the thermal and quantum effects on the properties of the soliton
can be taken into account simultaneously.
For the discussion of the decay rate and lifetime of the
new soliton state it is very convenlent to divide
in Eq.(55) into H
0+V
1+V
2 , where
where H
0 describes the relevant quasi-particle
excitations in the protein. This is a soliton together with phonons
relative to the distorted amino acid lattice. The resulting
delocalized excitations belongs to an exciton-like band with
phonons relative to a uniform lattice. The bottom of the band of
the latter is at the energy
relative to the soliton, in which
the topological stability associated with removing the lattice
distortion is included.
We now calculate the decay rate of the new soliton
along the following lines by using Eq.(61) and V2 in Eq.(63) and
quantum perturbation theory. Firstly, we compute a more general
formula for the decay rate of the soliton containing n quanta in
the system in which the three terms contained in Eq.(40a) is
replaced by (n+1) terms of the expression of a coherent state
Finally we find out the decay rate
of the new soliton with two-quanta. In such a case H0 is chosen
such the ground state, |n> has energy
in the subspace of excitation number equal to n, i.e.,
In this subspace the eigenstates have the simple form |n-m,k1k2…
km, {nq}>
where
with
The corresponding energy of the systems is
is the energy of a bound state with one exciton,
is the
energy of the unbound(delocalized) state with one exciton. When
m=0 the excitation state is a n-type soliton plus phonons relative
to the chain with the deformation corresponding to the n-type
soliton. For m=n the excited states are delocalized and the phonons
are relative to a chain without any deformation. Furthermore
except for small k, the delocalized states approximate ordinary
excitons. Thus the decay of the soliton is just a transition from the
initial state with the n-type soliton plus the new phonons:
With corresponding energy Es{nq} =w+n
n
q to the
final state with delocalized excitons and the original phonons:
with corresponding energy E
k{n
q}=n
n
q caused by the part, V
2, in the perturbation interaction V. In this case,
the initial phonon distribution will be taken to be at thermal
equilibrium. The probability of the above transitions in lowest
order perturbation theory is given by
We should calculate the transition probability of
the soliton resulting from the perturbed potential, (V
1+V
2), at
first-order in perturbation theory. Following Cottingham and
Schweitzer [14], we estimate only the transition from the soliton
state to delocalized exciton states caused by the potential V2,
which can satisfactorily be treated by means of perturbation
theory since the coefficient
(k,q) defined by Eq.(58) is
proportional to an integral over the product of the localized state
and a delocalized state, and therefore is of order 1/
. The V
1 term in the Hamiltonian is an interaction between the delocalized
excitons and the phonons. The main effect of V
1 is to modify the
spectrum of the delocalized excitatons in the weak coupling limit
(J
μp/ K
BT
0 << 1 , the definition of T
0 is given below). As a result the
delocalized excitons and phonons will have their energies shifted
and also have finite lifetimes. These effects are ignored in our
calculation since they are only of second order in V
1.
The sum over l in Eq.(69) indicates a sum over an initial
set of occupation numbers for phonons relative to the distorted
amino acid lattice with probability distribution , which is taken to
be the thermal equilibrium distribution for a given temperature
T. Since
,which is taken to be the thermal equilibrium
distribution for a given temperature T. Since
and
using the explicit form for V
2 and the fact that the sum
over states|
contains a complete set of phonons for
each values of k′, one can rewrite
as
where
here A is a new parameter introduced to describe the rate
between the new nonlinear interaction term and the one in the
Davydov’s model.
To estimate the lifetime of the soliton we are interested
in the long-time behavior of
. By straightforward calculation,
the average transition probability or decay rate of the soliton is
given by
where the thermal average is
with
This rather unusual expression of Γn occurs because the
phonons in the final state are related to a different deformation.
However, the analytical evaluation of
is a critical step in
the calculation of the decay rate Γn. It is well known that the trace
contained in
can be approximately calculated by using
the occupation number states of single-particles and coherent
state.
However the former is both a very tedious calculation,
including the summation of infinite series, and also not rigorous
because the state of the excited quasiparticles is coherent in
Pang’s model. Here we use the coherent state to calculate the
as it is described in Appendix B. The decay rate obtained
finally is
Where
This is just a generally analytical expression for the
decay rate of the soliton containing n quanta at any temperature
within lowest order perturbation theory. Note that in the case
where a phonon with wave vector k in Eq.(75) is absorbed, the
delocalized excitation produced does not need to have wave
vector equal to k. The wave vector here is only approximately
conserved by the sech
2
term. This is, of course,
a consequence of the breaking of the translation symmetry by
the deformation. Consequently, we do not find the usual energy
conservation. The terms, R
n(t) and
, occur because the
phonons in the initial and final states are defined relative to
different deformations[24-26].
We should point out that the approximations made
in the above calculation are physically justified because the
transition and decay of the soliton is mainly determined by the
energy of the thermal phonons absorbed. Thus the phonons with
large wave vectors, which fulfill wave vector conservation, make
a major contribution to the transition matrix element, while the
contributions of the phonons with small wave vector, which do
not fulfill wave vector conservation, are very small, and can be
neglected.
From Eqs.(74) and (75) we see that the Γn and R
n(t)
and
and
mentioned above are all changed by
increasing the number of quanta, n. Therefore, the approximation
methods used to calculate Γn and related quantities (especially
the integral contained in Γn) should be different for different n.
We now calculate the explicit formula of the decay rate of the new
soliton with two-quanta (n=2) by using Eqs. (74)-(75) in Pang’s
model. In such a case we can compute explicitly the expressions
of this integral and R
2 (t) and
contained in Eqs.(74)-(75)
by means of approximation. As a matter of fact, in Eq. 75) at n=2
the functions R
2 (t) and
can be exactly evaluated in terms
of the digamma function and its derivative. In the case when the
soliton velocity approaches zero and the phonon frequency q ω
is approximated by
|q|r0, as it is shown in Appendix C. For
(because we are interested in the long-time steady
behavior) the asymptotic forms of R
2(t) and
are
where
At R
0< 1 and T
0< T and R
0T/T
0< 1 for the protein molecules,
one can evaluate the integral including in Eq.(74) by using the
approximation which is shown in Appendix C. The result is
where
The decay rate of the soliton, in such an approximation,
can be represented, from Eqs.(74) and (80), by
This is the final analytical expression for the decay rate
of the quasi-coherent solition with two-quanta. Evidently, it is
different from that in the Davydov model [15,21]. To emphasis
the difference of the decay rate between the two models we
rewrite down the corresponding quantity for the Davydov soliton
[15,21]
where
Equation (83) can also be found out from Eq.(74) at n=1
by using the Cottingham et al’s approximation.
The two formulae above, Eqs. (82) and (83), are
completely different, not only for the parameter’s values, but also
the factors contained in them. In Eq.(82) the factor,
is added, while in Eq.(83) the factor,
replaces the term
in Eq.(82) due to the two-quanta nature of the new wave function
and the additional interaction term in the new Hamiltonian. In
Eq. (82) the η, R0 and T0 are not small, unlike in the Davydov
model. Using Eq.(72) and table 1 we find out the values of
,R0 and To at T=300K in both models, which are listed in Table
2. From this table we see that the
, Ro and To for Pang’s model
are about 3 times larger than the corresponding values in the
Davydov model due to the increases of μp and of the non-linear
interaction coefficient Gp. Thus the approximations used in the
Davydov model by Cottingham, et al. [14] cannot be applied in our
calculation of lifetime of the new soliton, although we utilized the
same quantum-perturbation scheme. Hence we can audaciously
suppose that the lifetimes of the quasi-coherent soliton will
greatly change.
Table 2: Comparison of characteristic parameters in the Davydov
model and in our new model
|
Ro |
To (K) |
|
New model |
0.529 |
294 |
6.527 |
Davydov model |
0.16 |
95 |
2.096 |
Discussion for the Lifetime of the New Soliton
and Results
The above expression, Eq. (82), allows us to compute
numerically the decay rate, , and the lifetimes of the new soliton,
τ= 1/ , for values of the physical parameters appropriate to
-helical protein molecules. Using the parameter values given in
Eq.(37), tables 1 and 2 , v=0.2v0 and assuming the wave vectors
are in the Brillouin zone, the values of between 1.54×1010S-1
- 1.89×1010S-1 can be obtained. This corresponds to the soliton
lifetimes τ, of between 0.53×10-10S- 0.65×10-10S at T=300K, or τ/
τ0=510-630, where τ/τ0=r0/v0 is the time for travelling one lattice
spacing at the speed of sound, equal to (M/w)1/2=0.96×10-13S. In
this amount of time, the new soliton, travelling at two tenths of
the speed of sound in the chain, would travel several hundreds of
amino acid lattice spacings, that is several hundred times more
than the Davydov soliton for which τ/τ0< 10 at 300K [15,21] (i.e.,
the Davydov soliton traveling at a half of the sound speed can
cover less than 10 lattice spacing in its lifetime) The lifetime is
sufficiently long for the new soliton excitation to be a carrier of
bio-energy. Therefore the quasi-coherent soliton is a viable mechanism
for the bio-energy transport at biological temperature in
the above range of parameters.
Attention is being paid to the relationship between the
lifetime of the quasi-coherent soliton and temperature. Fig.3
shows the relative lifetimes τ/τ0 of the new soliton versus temperature
T for a set of widely accepted parameter values as shown
in Eq.(37). Since one assumes that v< v0, the soliton will not travel
the length of the chain unless τ/τ0 is large compared with L/r0,
where L=Nr0 is the typical length of the protein molecular chains.
Hence for L/r0≈100,τ/τ0>500 is a reasonable criterion for the
soliton to be a possible mechanism of the bio-energy transport
in protein molecules. The lifetime of the quasi-coherent soliton
shown in Figure.4 decreases rapidly as temperature increases,
but below T=310K it is still large enough to fulfill the criterion.
Table 3: Comparison of features of the solitons between our model and Davydov model
Model |
nonlinear interaction G(10-21J) |
Amplitude |
Width 10-10m |
Binding energy
(10-21J) |
Lifetime at 300K
(S) |
Critical temperature
(K) |
Number of amino acid traveled by soliton in lifetime |
Our model |
3.8 |
1.72 |
4.95 |
-7.8 |
10-9-10-10 |
320 |
Several handreds |
Davydov model |
1.18 |
0.974 |
14.88 |
-0.188 |
10-12-10-13 |
<200 |
Less than 10 |
Thus the new soliton can play an important role in biological processes.
For comparison, log versus the temperature relationships
was plotted simultaneously for the Davydov soliton and the
new soliton with a quasi-coherent two-quanta state in Figure 4.
The temperature-dependence of log ( τ/τ0) of the Davydov soliton
is obtained from Eq. (83). We find that the differences of values
of ( τ/τ0) between the two models are very large. The value of
( τ/τ0) of the Davydov soliton really is too small, and it can only
travel fewer than ten lattice spacings in half the speed of sound
in the protein chain. Hence it is true that the Davydov soliton is
ineffective for biological processes [3-23].
Figure 3: Soliton lifetime τ relatively to τ0 as a function of the temperatureT for
parameters appropriate to the α-helical molecules in Pan’s model in Eq. (9)
Figure 4: log(τ/τ0) versus the temperature. The solid line is the result of Pang’s
model, the dashed line is the result of the Davydov model.
The dependency of the soliton lifetime on the other parameters
can also be studied by using Eq. (82). Parameter values
near the above accepted values shown in Eq. (37) are chosen. In
Pang’s model we know from Eq. (82) that the lifetime of the soliton
depends mainly on the following parameters: coupling constants
(χ
1+χ
2),, M, w, J, phonon energy
, as well as well as on
the composite parameters μ(μ=μp ), R
0 and T/T
0. At a given temperature,
τ/τ
0 increases as μ and T
0 increase. The dependences of
the lifetime τ/τ
0 ,at 300K on ((χ
1+χ
2)) and μ are shown in Figs.5
and 6, respectively . Since μ is inversely proportional to the size of
the soliton, and determines the binding energy in the new model,
it is an important quantity. It is regarded as an independent variable.
In such a case the other parameters in Eq. (82) adopt the
values in Eq. (37). It is clear from Figs.5 and 6 that the lifetime of
the soliton, τ/τ0, increases rapidly with increasing μand ((χ
1+χ
2)).
Furthermore, when μ≥5.8 and ((χ
1+χ
2))≥7.5×10
-11N, which are
values appropriate to the new model, we find
τ/τ
0>500.
For comparison, the corresponding result obtained using
Eq. (83) is shown for the original Davydov model as a dashed
line in Fig.6. Here we see that the increase in lifetime of the Davydov
soliton with increasing μ is quite slow and the difference between
the two models increases rapidly with increasing μ. The
same holds for the dependency on the parameter (χ
1+χ
2), but the
result for the Davydov soliton is not drawn in Figure5. These results
show again that the quasi-coherent soliton in Pang’s model
is a likely candidate for the mechanism of bio-energy transport
in the protein molecules. In addition it shows that a basic mechanism
for increasing the lifetime of the soliton in the biomacromolecules
is to enhance the strength of the exciton-phonon interaction.

Figure 5: τ/τ0versus (χ1+χ2) relation in Eq. (82)
Figure 6: τ/τ0versus μ relation. The solid and dashed lines are results of Eq. (82)
and Eq.(83),respectively
Figure 7: τ/τ0 versus η relation in Eq. (82)
In Figure 7, τ/τ
0 versus
is plotted. Since
designates
the influence of the thermal phonons on the soliton, it is also an
important quantity. Thus, it is regarded here as an independent
variable. The other parameters in Eq.(82) take the values in Eq.
(37). From this figure we see that τ/τ
0 increases with increasing
. Therefore, to enhance
can also increase the value of τ/τ
0.
In order to understand the behavior of the quasi-coherent
soliton lifetime in very wide ranges, it is necessary to study
τ/τ
0 in the limit
in Eq.(75) or Eqs.(C1) and (C3) (i.e., this is in the initial case) in
which we can evaluate analytically the values of R
2(t) and ξ
2(t). In
fact, for
both R
2(t) and ξ
2(t) have power-series expansions.
To the lowest order as
, it can be found from Eq.(75)
Thus
when T/T
0>1 and π
40T/2μT0>1. The above integral is a
generalization of the usual δ - function for energy conservation in
zero-temperature perturbation theory. Thus we can obtain from
Eqs.(74) and (87) at n=2 the decay rate of the soliton as

Figure 8: τ/τ0vs T relation in Pang”s model in Eq. (88)
Figure 9: τ/τ0versus (χ1+χ2), relations in the new model in Eq. (88)
Figure 10: τ/τ0 versus μ relation in the new model in Eq. (88)
Figure 11: τ/τ0 versus T0 relation .Here the solid and dashed lines are the results in
the new model in Eq.(88) and in the Davydov model, respectively
In Figure9 and 10 we plot τ/τ0 versus (χ1+χ2) and versus
μ at T=300K, respectively. From Figure 8-10 it can be seen
that τ/τ0 increases as T decreases and as μ and (χ1+χ2) increase.
Furthermore, it is clear from this Gaussian expression in Eq.(88)
that the lifetime of the new soliton will be large if μ and (χ1+χ2)
are larger, but the Gaussian expression is very small for k and
k’ between -π/r0 and +π/r0, i.e., in the Brillouin zero. Obviously,
the temperature dependence of the lifetime of the new soliton is
mainly due to the temperature dependence of the width of the
Gaussian, which decreases with decreasing temperature. The
dashed line in Fig.10 is the result for the Davydov soliton under
the same conditions. It is clear that the lifetime of the Davydov
soliton is lower than that of the new soliton, especially at large
, although at low the difference between them is small. Taking
Figure 4 also into account we find that the lifetime of the Davydov
soliton is indeed generally low. However this is not the case for
the new soliton. In Figure 11, τ/τ0 is plotted as a function of T0
at T=300K. T0 is related to the Debye temperature of the systems,
therefore it is also an important quantity. It is regarded here as
an independent variable and evaluate other parameters as in Eq.
(37). From this figure it can seen that the lifetime of the new soliton
is large if T0 is either large or small, because the Gaussian
expression in Eq.(88) is very small for k and k’ between -π/r0 and
+π/r0. As a point of reference, note that these parameters have
the values τ/τ0≈1.03 —1.06, JT/KBT =4.10 at 300K and μ=5.81-
5.96 depending on whether the widely accepted or the“threechannel”
parameter values for the protein are assumed. From
these results, it is clear that using widely accepted and realistic
parameter values, the new model can satisfy the relation τ/
τ00≥500 at 300K and large μ and large T0. Hence the proposed
new soliton model provides a viable candidate for the biological
processes.
Conclusions
Here a new theory of bio-energy transport is proposed
to study the properties of the nonlinear excitation and motion
of the soliton along protein molecules. In this theory, Davydov’s
Hamiltonian and wave function of the systems are simultaneously
improved and extended, a new interaction is added into the
original Hamiltonian, and the original wave function of the excitation
state of single particles is replaced by a new wave function
of a two-quanta quasi-coherent state. From this model, a lot of
interesting and new results are obtained. The soliton has sufficiently
long lifetime and can pay an important role in biological
processes. Therefore, it is an exact carrier of bio-energy in living
systems. Present problem is why the quasi-coherent soliton has
such long lifetime? From Eqs. (35) and (45) and tables 1 and 2 it
can be seen that the binding energy and localization of the new
soliton increase due to the increase of the nonlinear interactions
of exciton-phonon interaction, i.e., the new wave function with
two-quanta state and the new Hamiltonian with the added interaction
produce considerable changes to the properties of the soliton.
In fact, the nonlinear interaction energy in the new model is
Gp=8(χ
1+χ
2)
2 /(1-s
2)w=3.8×10-21J, and it is larger than the linear
dispersion energy, J=1.55×10-22J, i.e., the nonlinear interaction is
so large that it can really cancel or suppress the linear dispersion
effects in the equation of motion of this model. From this point
the soliton is stable according to the conditions of formation and
stability of the soliton in the soliton theory [27,28]. By comparison,
the non-linear interaction energy in the Davydov model is
G
D=4χ
21/(1-s
2)w≈1.18×10
-21J and it is 3-4 times smaller than
Gp. Thus the stability of the Davydov soliton is weak compared
to that of the new soliton. Moreover, the binding energy of the
quaasi-coherent soliton in Pang’s model is E
BP=4 μ J/3=7.8×10
-
21J in Eq.(19), which is about 2 times larger than the thermal energy,
KBT=4.14×10
-21J, at 300K, and about 6 times larger than the
Debye energy,
=1.2×10
-21J (here ω
D is Debye frequency),
and it is approximately equal to
, i.e., it has
same order of magnitude of the energy of the amide-I vibrational
quantum,
./T4his shows that the quasi-coherent soliton is robust
against the quantum fluctuation and thermal perturbation of the
systems due to the large energy gap between the soliton state and
the delocalized state. In contrast, the binding energy of the Davydov
soliton is only E
BD=
=0.188×10
-21J , which is about 41
times smaller than that of the new soliton, about 23 times smaller
than K
BT and about six times smaller than
=1.2×10
-21J
,respectively. Therefore, it is easily destroyed by thermal and
quantum effects. Hence the Davydov soliton has very small lifetime
(about 10-12
˜ 0-13s), and it is unstable at 300 K [15-18,24-
26]. Therefore, the quasi-coherent soliton can provide a realistic
mechanism for the bio-energy transport in protein molecules.
The two-quanta nature of the quasi-coherent soliton
plays a more important role in the increase of lifetime relative
to that of the added interaction because of the following facts.
(1) The change of the nonlinear interaction energyGP=2G
D
by μ
p produced the added interaction in the new
model are ΔG =G
P (χ2≠0) (χ2=0)=1.08GD < GP(χ2=0)=2GD and
Δμ=μ
P(χ2≠0)-μ
P(χ2=0)=1.08μD< μ
P(χ2=0) =2μD,respectively,
i.e., the roles of the added interaction on Gp and μ
p are smaller
than that of the two-quanta nature. The two parameters GP and
μ
P are responsible for the lifetime of the soliton. Thus the effect
of the former on the lifetimes is smaller than the latter. (2) The
contribution of the added interaction to the binding energy of
the soliton is about
which is smaller than
that of the two-quanta nature which is E =16E
BD. Putting them
together in Eq.(35) we see that EBP≈ 41EBD. (3)From the (χ
1+χ
2)-
dependence of τ/τ
0 in Fig.5, τ/τ
0≈100 has already been found
directly at χ
2=0 which is about 20 times larger than that of the
Davydov soliton under the same conditions. This shows clearly
that the major effect in the increase of the lifetime is due to the
modified wave function. Therefore, it is very reasonable to refer
to the new soliton as the quasi-coherent soliton [30-35].
The above calculation helps to resolve the controversies
on the lifetime of the Davydov soliton, which is too small in
the region of biological temperature. However, by modifying the
wave function and the Hamiltonian of the model, a stable soliton
at biological temperatures could be produced. This result
was obtained considering a new coupled interaction between the
acoustic and amide-I vibration modes and a wave function with
quasi-coherent two-quanta state. In such a way, the quasi-coherent
soliton is a viable mechanism for the bio-energy transport in
living systems. Therefore, it can be seen that Pang’s model is completely
different from the Davydov’s model. Thus, the equation of
motion and properties of the soliton occurring in Pang’s model
are also different from that in the Davydov’s model. The distinction
of features of the solitons between the two models is shown
in Table 3[15]. From the table 3 we know that our new model
repulse and refuse the shortcomings of the Davydov model [3],
the new soliton in Pang’s model is thermal stable at biological
temperature 300K, and has so enough long lifetime, thus it can
plays important role in biological processes.
Appendix A
The partial diagonalization of the Hamiltonian implies
the diagonalization of that part of the Hamiltonian in Eq. (50)
which does not involve the creation and annihilation operators of
new phonons Eq.(48). Thus the condition imposed into the functions
Cj(x) contained in Eq.(53) to realize such a diagonalization
are equivalent, in the continuum approximation, to the following
problems of eigenfunctions Cj(x) and eigenvalues Ej determined
by
For the above expression of V(x) in Eq.(52) there is only one
bound state in Eq.(A1)
with energy
and unbounded(delocalized) states
with energy
The energy of the lowest unbounded state is greater
than that of the bounded state by the value . The functions Ck(x)
are normalized as follows:
Therefore, A
+s is an excitation which is localized at the
lattice distortion, while A
+k creates an unbounded excitation with
wave vector k.
In getting Eq. (A1) the variable x was assumed to be
continuous and the chain length to tend to infinity L=Nr0→∞.
Thus this wave vector k has a continuous value between -∞ and
∞. In subsequent calculation we mainly use a discrete description.
The continuous description is transformed into a discrete
one according to the rules
Utilizing Eqs. (50)-(51), (53) and (54), then the partially
diagonalized Hamiltonian in the new representation is just
Eq.(55).
Appendix B
We now calculate U(k, k’’,t) in Eq.(72) utilizing the coherent state
|u> defined by bq|u>=uq|u> with
Utilizing the coherent state |u>, the U(k,k″,t) in Eq.(72) can be
represented by
Where the integration measure is defined as
Since we can show that
it follows that the first matrix element in Eq.(B1) equals
The second matrix element in Eq. (B1) can be represented
as a path integral that can be evaluated exactly. Utilizing
the general relationship between the matrix element and the
path integral:
where
We can evaluate the path integral by stardand techniques. The
result for Eq. (B2) is
Substituting above the matrix elements obtained into Eq.(B1) we
get
where
The and u” integrations can easily be finished. For instance, the
contribution from the term with the
factor, which we can
denote by
since it is associated with the absorption of a
phonon, is
where
We note that the breaking of the translational symmetry
by the deformation leads to off-diagonal terms corresponding
to violation of wave vector conservation. However, we can
prove that these terms are proportional to
which can be
neglected when either |k|or |k″| is large as compared to 4μp/πr
0
as can be seen in the definition of αk in Eq.(59). Furthermore,
when -π≤kr
0≤π and μp< π
2 the off-diagonal terms are negligible
except for a small region at the center of the Brillouin zone. Since
the small wave vector terms do not significantly contribute to Γn
due to the k-dependence of (q,k), and thus the off-diagonal terms
can be neglected in
in the calculation of Γn. The energy
of the soliton state is less than that of the unlocalized exciton in
the uniform lattice. Therefore, the parts of corresponding to the
absorption of a phonon make the major contributions to the sum
in Eq.(72) at the temperature and parameter values of interest,
and their off-diagonal terms may also be neglected just as above.
Using the result of the
obtained from the above formulae
of Eq. (72) the decay rate Eq.(74) can be obtained.
Appendix C
If the soliton velocity approaches zero we can get an
analytical expression for R
2(t) and
at n=2 defined in Eq. (75)
or Eqs. (B5) and (B7) through inserting Eq. (59) into Eqs.(B5) and
(B7) and applying the relation of
where
ψ is the digamma function, ψ ′is its derivative and x′= t=K
BT
0t/ℏ
can be easily elvaluated when v≈0 and R
0< 1 at sufficiently
high temperature T>T
0(T
0= ℏ ωα/KB). In this case it is
where we use the relation .
As t →∞ (because we are interested in the long-time steady behaviour)
the leading terms in the above asymptotic formulae of
R
2(t) and ξ
2 (t) can be represented by
(where we approximated coth
) , i.e.,
Except at low temperature, the
dependent
term in the real part of R
2(t) is small with respect to
for
parameter values of interest and can be neglected. Furthermore,
since R
0< 1 ( but it is not very small, about R
0≈0.529) and T
0< T
(but it is not too small, about T
0≈294K) and R
0 T/T
0< 1 for the
protein molecules, then one can evaluate the integral in Eq.(72)
by using the following approximation and utilizing the above results
of Eqs.(C4-C6).
where
- Szent-Györgyi A. Dionysians and Apollonians. Science Retrieved. 2016;194:93(4038)609-612.doi:10.1126/science.176.4038.96.
- Bakhshi K, Otto P, Ladik J, Seel M. On the electronic structure and conduction properties of aperiodic DNA and proteins. II. Electronic structure of aperiodic DNA. Chem. Phys. 1986;108(20):687-696. doi: 10.1016/0301-010(86)85043-1
- Davydov, A.S. Solitons in Molecular Systems, 2nd ed.; Reidel Publishing Company: Dordrecht, The Netherlands. 1991;23–79.
- Christiansen PL , Scott AC. Davydov’s Soliton Revisited. Plenum Press. New York, NY, USA. 1990; 34–103.
- Brown DW, Lindenberg K, West BJ. Nonlinear density-matrix equation for the study of finitetemperature soliton dynamics. Phys Rev B. 1987;35(12);6169–6180. doi: 10.1103/PhysRevB.35.6169
- Scott AC. Dynamics of Davydov’s soliton. Phys. Rev. A.1982 ;2:578–595. doi 10.1103/Phys Rev A. 1982;26(1):578. doi.: 10.1103/PhysRevA.26.578
- Pang, X.F.; Zhang, A.Y. Investigations of mechanism and properties of biological thermal effects of the microwave. China J. Atom. Mol. Phys. 2001, 18, 278–281.
- Christiansen.PL. and Scott. AC., Self-trapping of vibrational energy, Plenum Press, New York. 1990.45-89.
- . Forner W. Quantum and disorder effects in Davydov soliton theory. Phys Rev A. 1991;44(4) :2694–2708.
- Cruzeiro L, Halding J, Christiansen P L, Skovgard O; Scott AC. Temperature effects on Davydov soliton. Phys Rev A. 1985;37:880–887. doi: 10.1103/Phys Rev A.37.880
- Cruzeiro L. The Davydov/Scott model for energy storage and transport in proteins. J Biol Phys. 2009;35(1):43–55.
- Forner W. Quantum and disorder effects in Davydov soliton theory. Phys Rev A. 1991;44(4) :2694–2708.
- Kerr WC, Lomdahl PS. Quantum-mechanical derivation of the equations of motion for Davydov solitons. Phys Rev B 1987;35(7):3629–3632.
- Wang X, Brown DW, Lindenberg K, West BJ. Alternative formulation of Davydov theory of energy transport in biomolecules systems. Phys Rev A. 1988;37(9):3557–3566.
- Cottingham JP, Schweitzer JW. Calculation of the lifetime of a Davydov soliton at finite temperature. Phys Rev Lett. 1989;62(15):1792–1795. doi: 10.1103/PhysRevLett.62.1792
- Lomdahl PS, Kerr WC. Do Davydov solitons exist at 300 K? Phys Rev Lett. 1985;55(11):1235–1239. doi: 10.1103/PhysRevLett.55.1235
- Wang X, Brown DW, Lindenberg, K. Quantum Monte Carlo simulation of Davydov model. Phys Rev Lett. 1989;62: 1796–1799.
- Cruzeio Hansson L. Mechanism of thermal destabilization of the Davydov soliton. Phys Rev A. 1992;45(6):4111–4115.
- Mechtly B, Shaw PB. Evolution of a molecular exciton on a Davydov lattice at T = 0. Phys. Rev B. 1988;38:3075–3086. doi: 10.1103/PhysRevB.38.3075
- Pouthier V. Energy relaxation of the amide-I mode in hydrogen-bonded peptide units: A route to conformational change. J Chem Phys. 2008;128(6). doi:10.1063/1.2831508.
- chweitzer JW. Lifetime of the Davydov soliton. Phys Rev A. 1992; 45(12): 8914–8922. doi:10.1103/PhysRevA.45.8914
- Forner W. Davydov soliton dynamics: Two-quantum states and diagonal disorder. J Phys Condens. Matter. 1991; 3:3235–3254.
- Zekovi S, Ivic Z. Damping and modification of the multiquanta Davydov-like solitons in molecular chains. Bioelectrochem. Bioenerg. 1999;48(2):297–300.
- Pang XF. An improvement of the Davydov theory of bio-energy etransport in the protein molecular systems. Phys Rev E. 2000;62:6989–6998.
- Pang XF. The lifetime of the soliton in the improved Davydov model at the biological temperature 300 K for protein molecules. Eur Phys J B. 2001;19(2):297–316. doi: 10.1007/s100510170339
- Pang XF. Nonlinear Quantum Mechanical Theory; Chongqing Press, Chongqing, China. 1994;356–465.
- Popp FA, Li KH and Gu Q. Recent advances in biophoton research and its application World Scientific, Singapore. 1993;156-213
- Pang XF. Soliton physics, Chinese Sichuan Science and Technology Press, Chengdu. 2000:2-180.
- Bullough P K, Caudrey P J. Soliton , Spinger; New York. 1982: 80-160.
- Guo BL, Pang XF. Solitons. Chinese Science Press, Beijing; China. 1987:4–140.
- Teki J, Ivic Z, Zekovi S, Przulj Z. Kinetic properties of multiquanta Davydov-like solitons in molecular chains. Phys Rev E. 1999;60: 821–825. doi: 10.1103/PhysRevE.60.821
- Pouthier V, Falvo C. Relaxation channels of two-vibron bound states in alpha-helix proteins. Phys Rev E. 2004;69(4): 041906. doi:10.1103/PhysRevE.69.041906
- Stiefe lJ. Einfuhrung in die Numerische Mathematik, Teubner Verlag, Stuttgart, 1965, pp123-198
- Atkinson K. An Introduction to Numerical Analysis. Second edition. New York ; Wiley. 1987:96-145.
- Spatschek KH, Mertens FG. Nonlinear coherent structures in physics and Biology, Plenum Press, New York. 1994;76-176.