MESOSCOPIC IRREVERSIBLE THERMODYNAMICS OF ELASTIC

RESPONSE AND AGING KINETICS OF ALPHA POLYPEPTIDES [DNA]

UNDER VARIOUS CONSTRANTS: SPECIAL REFERENCE TO THE SIMPLE

SPRING MECHANICS

Tarik Omer Ogurtani1 and Ersin Emre Ören2

1 Department of Metallurgical and Materials Engineering, Middle East Technical University, Ankara, 06531Turkey

2 Bionanodesign Laboratory, Department of Biomedical Engineering, TOBB University of Economic and Technology, 06560 Ankara, Turkey

Abstract

The mesoscopic irreversible thermodynamic treatment of α-polypeptides furnishes

two sets of analytical expressions, which allow us not only to analyze the reversible

force-extension experiments performed by atomic force microscopy (AFM) but also

to predict the irreversible ‘’aging’’ kinetics of the single- stranded [ssDNA] and

double stranded [dsDNA] helical conformation exposed to aqueous solutions and

applied constant stresses under the various constraints. The present physicochemical

cage model emphasizes the facts that the global Helmholtz free energy of the helical

conformation acts not only under the stored ‘’intrinsic’’ (unusual) torsional and

bending elastic energies inherited by the amino-acid backbone unfolded helical

structure but also reveals the importance of the interfacial Helmholtz free energy

density associated with the interaction of the side-wall branches with the surrounding

aqueous solutions. The analytical expression obtained for the unfolding force versus

extension (FE) shows strong nonlinear elasticity behavior under the twist angle

constraint when an interfacial Helmholtz energy term is incorporating into the

scenario. This behavior is in excellent quantitative agreement with the AFM test

1

results obtained by Idiris et al. [1] on the Poly-L-glutamic acid exposed to aqueous

solutions having various levels of acidity [pH3-8]. However, in the absence of the

unusual torque term [2], the total pitch height of helical alpha-polypeptide having the

usual torque and bending terms, spontaneously increases exponential fashion without

showing any aging behavior under the total twist angle constraint, regardless of its

initial states, and blows up at the infinity, unless one has a negative Poisson’s ratio.

Keywords: spring mechanics; α-polypeptide; torsional rigidity; DNA aging;

unusual torsion.

Correspondent Author:ogurtani@stanfordalumn.org; ogurtani@metu.edu.tr

URL: http//www.csl.mete.metu.edu.tr

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I. Introduction

The mechanical behavior of polypeptides or the single stranded ssDNA

having alpha-helical conformations relies on the modified1 Gaussian [3,4,5] statistical

mechanics treatments of the linear chains, which have been extensively reviewed by

Dill and Bromberg [6], Phillips et al. [7] and the most recently by Yamakawa and

Yoshizaki [8]. The theoretical interest on this topic has been restarted immediately

after the Second World War, through the successful applications of the fluctuation

theory by Flory [9,10] or the more sophisticated statistical bending theory by Landau

and Lifshitz [11] for the flexible long polymer chains or entangled long

macromolecules.

The foremost of the late continuous models, which is originated from the

discrete freely-joined chain (FJC) proposed first by Kratky and Porod (KP) [12], is the

wormlike chain (WLC) model. Initially this model is suffered from disagreements

between the theory and experiments by allowing infinite extensions at high applied

shear forces. This awkward situation is partially improved by Harris and Hearst [13],

and Hearst et al. [14] by introducing a constraint on the contour length by

incorporation a quasi-elastic bending term such as into the

Lagrangian operator with a Lagrangian multiplier . This approach then gives

required stiffness to the continuous stiff chain model of polymers, and hinders the

overstretching beyond the full contour length L. Where denotes the unit

1 Without any modifications in the Gaussian statistics, its results expansions on the macromolecule and its segments towards the infinite length, if it is subjected to larger and larger shear stresses.

3

tangent vector, and expresses the extension along the z-axis, while the

whole chain represented by the ensemble of tangent vectors . 2 This model

similar to the one derived by Freed [15], and becomes invalid for high stiffness

domain near the rod limit, it cannot give correct higher wormlike moments expressed

as . In particular, the contour length L increases indefinitely if an external force

is applied and increased. The complete picture is also lacking, which is now

demanded by ssDNA stretching experiments [1].

One of the major breakthroughs in KP model is made by Fixman [16,17] to

account for the polymer chain stiffness with the use of the modified Gaussian

probability function for the intrachain distances as such that they depend on the

internal forces associated with the intrinsic viscosity, as a function of the ‘’excluded

volume’’ and the shear rate. The numerical results deduced by Fixman [16] are

tabulated for the intrinsic viscosity and the equilibrium expansion of the mean-square

end-to-end distance . These early versions of the Gaussian modifications is

further improved by Fixmann and Kovac [18] for the freely rotating chain by

introducing an external potential acting on the end-to-end vector, so that the

Lagrangian function becomes a quadratic form having an 2 In this model tangent vectors and their line derivative with respect to contour length constitute a

continuous set to describe dynamics of the long polymer chain in 3D space.

£ = , looks like a potential energy term, and the kinetics

energy in this model is replaced by the analog bending energy term , and delta is the variation operator imposed on the extremum problem. α is bending stiffness, β becomes force constant in harmonic term.

4

additional linear constraint. Fixmann and Kovac [18] using a discrete chain model,

where identical backbone atoms are connected by bond vector set , and the

backbone potential is a function of . They have managed to furnish

tabulated values of the mean end-to-end vector versus external force for

the freely- joined chain in terms of the exact and modified Gaussian solutions, by

assuming that there is no constraint on the bond angle that means the bending

energy term may be taken to be zero. However, they are able to recover this

deficiency for the wormlike model in the limit and still keeping total

contour length invariant, and the bond angle , which

results finite persistence length and Kuhn statistical segment length by certain

limiting procedure.3,4

Saito, Takahashi and Yunoki [19], and later Freed [20] realized that there is a

close analogy between the wormlike chain and a quantal trajectory of a particle,

where the chain contour length regarded as ‘’time’’. In this formalism, the Fourier

transform of the Green’s function of Fokker-Planck equation may be represented by

the Feynman path integrals [21]. Another serious attempt for the statistical mechanics

of the continuous wormlike chain as a differentiable space curve of fixed length has

been introduced by Yamakawa [22], who first realized that this Markovian process [23]

3 , where is persistence length and the Kuhn segment length.

4 There is a repeating sign error in Eq. (51) and Eq.(52) in Ref.[17]. The correct formula should be:

.See: CRC Standard Mathematical Table. 21st ed: p. 342 (1973). Table of Integrals, Series, and Products by IS. Gradshteyn and LM. Ryzhik, Academic Press, p.36 (1980). Eq. 1.421.4

5

may be formulated in tangent vector space , which mimics the velocity space for

the Brownian motion of a free particle that may be described by a Fokker-Planck

type diffusion equation [24] as first realized by Daniels [25] and subsequently derived

by Herman and Ullman [26]. The Fourier transform of the Green’s function deduced

from the Fourier transformed Fokker-Planck equation with the constraint is

the conditional distribution function designated as , which doesn’t have closed

analytical solutions. Yamakawa [22], by using the potential energy

as proposed first by Fixmann and Kovac [18] in his Lagrangian

function defined by the expression in the Fourier

transformed of Fokker-Planck equation, Yamakawa has managed to determine the

unconditional characteristic function for the finite force . Where the

elements in the arguments correspond, respectively, to the temporal and

initial velocity sets having no constraint on their sizes, and t is the time, which should

be regarded as the contour length in the unstretched state.

By Fourier inversion, Yamakawa [22] obtained for the normalized trivariate

distribution function in a closed analytical format. is the temporal

end-to-end vector. This solution involves only two parameters . Where the

bending force term described by is given by in term of Kuhn

segment length5 , and the stretching constant is determined as a function of

5 For worm-like chain, Kuhn length is equal two times the persistence length. The persistence

length can be expressed using the bending stiffness and the Young’s modulus by 6

and . Unfortunately, still a manageable direct connection between the stretching

force and the extension is missing in their work to give us an opportunity to test their

theory with the experimental results. [7] Hoshikawa et al. [27 ] and Saito and Namiki

[28] have introduce a coupling term between the bending and

stretching in the absence of the force term into the Lagrangian function, where

and . This model gives us the correct rod limits of the

moments, but not correct first-order corrections to the limits. Similarly, the physical

meaning of this coupling is obscure for real chain. [7]

In 1986, Helfrich [29] proposed a theory to explain the helical structure of the

strips wound around as cylinders mimic the solid bilayer of chiral molecules, where

he employed a spontaneous torsion of the bilayer edges and the bending stiffness of

the bilayer. The parametric representation of one of the helical edges has been used

to calculate its curvature and torsion

, where is the radius of helix, and is

pitch height. The tilt angle of the helix is found 45o, in good

agreement with experiment.6 The later Helfrich and Prost [30] generalized the bending

free energy for anisotropic bilayers to account for the lack of mirror reflection

symmetry of bilayers of chiral molecules by adding a linear term

to the elastic free energy. Where is a right-hand set of unit vectors, where

where. .6 45o corresponds to a maximum value in the torsion as can be seen from above given expression.

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is the unit layer normal and the curvature tensor, denotes the tilt direction, and

coincides with the axis of the ferroelectric polarization to be expected for tilted

layers if the molecules are chiral.7 Its molecular origin in chiral layers is easy to

understand by looking at rows of molecules. However, this last term vanishes with

non-chiral layers because for them and – are physically equivalent. In the

presence of curvature, there is twist along these rows which is physically

inequivalent along and .

Chung et al., [31] by looking at the explicit form of the linear term between

parenthesis , which is introduced previously by Helfrich and

Prost [29], realized that this term is nothing but the intrinsic torsion of the helix,

and the corresponding quadratic energy term is associated with the torsional elastic

energy.8 Where chirality constant and it has a positive value for right-handed

structures. are, respectively, the radius and the tilt angle of the helix.

Chung et al. [31] have reported experimentally observed helical ribbon structures in

cholesterol-supersaturated bile as metastable intermediates with either of two distinct

pitch angles about : 54o or 11o. Their analytic theory also furnished the fact that for

7 Actually, this set coincides with the intrinsic tripod vector set , which is associated, respectively, with principal normal, tangent and binormal unit vectors of the edges of

the helical ribbon by definitions. Where [50] that means

. This term is misinterpreted by Helfrich& Prost by saying that it is closely connected with the tilted chiral molecules. According to Dahl and Lagherwall’s [39] formulation this term appears in two different occasions. Namely in the deformation and chiral

parts of the free energy. Similarly, one has for the ribbon; and .8 There is a typographic error in Chung et al.’s formula Eq.(5) that should be corrected as

.However, this doesn’t affect their calculated tilt angle, β.8

isotropic case the tilt angle is 45o, identical to Helfrich and Post’s results [30].

Komura and Zhong-can [ 32] have observed two types of helices with high and low-

pitch angles in the pathway for cholesterol crystallization (ChLC) correspond to

parallel and antiparallel packing of molecules at the edges of the ribbon. Their claims

rely on the complete free energy density per unit area of ChLC as proposed by

Gennes and Prost [33], which involves the splay, twist and bending elastic energies.

Komura and Zhong-can [32] obtained a simple formula for high pitch angle

, where is the angle between ribbon surface normal,

and the unit longitudinal axis vector, of the molecules. They assumed that

takes the threshold value of below which the transition from high-

pitch helices to the low pitch helices starts. This figure in above relationship gives

the optimal high-pitch angle is about 52.1o, which in excellent agreement with

the observed value by Chung et al.[31]. However, Komura and Zhong-can [32]

produced the same result by employing completely different relationship than above

mention formula, which is not derived in their paper, but rather referred to in their

forthcoming paper.

Tu and Ou-Yang [34] recently have presented governing equations on lipid

vecides, lipid membranes with free edges, and chiral lipid membranes with analytical

solutions. All these equations rely on the work of Helfrich and Prost’s [30] curvature

energy as a Lagrangian function for the extremum problem

incorporated with the Lagrange multipliers , which constrain the area and

volume , where is the Helfrich curvature function.

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Marko and Sigma [35, 36], and later Bouchiat et al., [37] have recognized the

fact that the rigorous solution of the partition function for the WLC

for any arbitrary contour length would be in formidable task, and rather tried to

handle the chain problem for extremely short length; . That reduces

the probability distribution function to a linear

Schrödinger-like equation designated by , where is the

angle between z- axis ‘along which external force is applied’ and tangent vector,

and is the angular momentum operator. The minimum eigenvalue9 of this

problem designated by turns out be the main interest to calculate the end-to-

end extension versus force (FE) connection by the following relationship;

. Marko and Siggia [35] tried to solve the numerical problem

by using spherical harmonics, but later realized that because of the axial symmetry,

no magnetic quantum number is involved. However, their work produced very useful

interpolation formula for WLC model,10 which is asymptotically exact in the large

and small force limits. Bouchiat et al. [37] is used successfully completely different

9 , this expression is given by Marko & Siggia, Eq.14 in Ref.[26], which can be easily handled, and it gives

, where minimizes the

desired eigenvalue g from which the following may be deduced exactly: . This parametric exact solution developed here is very convenient for plotting in MatCad environment.

10 , which differs from the exact results in extension

7% for . A is the persistence length. 10

numerical method to compute the normalized force versus scaled

extension with respect to the unstretched contour length.

All of the studies cited up to now, including Yamakawa and Fuji [38], and

Kovac and Crabb [39] have been accounted only for the bending and stretching types

of deformations but not the twisting or rotation, with the exceptions of those authors

[28, 29,30, 31,32] dealing with the helical ribbons and/or the tilted nematic or

smectic liquid crystals. However, all their efforts have been concentrated on how to

calculate the expectation values of the end-to-end distance or its projection

over the applied force unit vector , and to evaluate their dependence

on the magnitude of the applied force denoted by using the Schrödinger-like

equation obtained from the Fourier transform of the Fokker-Planck equation [7].

As one expects, all of these formulations don’t contain any unusual torque

term, which is few orders of magnitude larger than the combination of the usual

torsion and bending terms as elaborated by Ogurtani [40,41], since no one has

considered the skew symmetric part of the elastic deformation tensor. This unusual

torsional rigidity embedded in the skew symmetric part of the deformation tensor, is

associated with the nonvanishing circumferences force field along the wall of the

hypothetical cylindrical shaped backbone structure is quadratically depending on the

contour length, but not like the usual torsional and bending rigidities, which are

constant and uniform. In addition to the bulk elastic energy considerations, the

stability requires not only the local but the global Gibbs or Helmholtz free energy

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formulation as an extremal problem that depends on the whether one deals with the

isothermal isobaric or the isochoric systems, respectively.

Cluzel et al.[42] and Smith et al.[43] have reported almost at the same time their

results on the single molecule double-stranded dsDNA stretching experiments,

performed with the force-sensors, which are either monomode optical fibers or with

the optical laser tweezers, respectively. The stretching of dsDNA was modeled by

Cluzel et al.[42] with use of the JUMNA molecular mechanics program developed

for studying nucleic acid conformations. [44, 45,46] Where, dsDNA polymer studied

with the use of helical symmetry constraints and a repeat of 10 nucleotide pairs.

Stretching involved minimizing the energy per turn of the polymer as a function of

the length of one of its strands (imposed with a quadratic distance constraint between

C5’ and C3’ atoms separated by 10 nucleotides). The left DNA before stretching and

right DNA after stretching by a factor of 1.7. The both groups observed, respectively,

that the force saturates at a plateau around, under a longitudinal stress of 70 and 65

piconewtons (pN), which ends when the dsDNA has been stretched about 1.7 times

of B-form contour length or 70% longer than B-form dsDNA. This is interpreted as

the cooperative state transition of the B-DNA to the S-DNA. However, addition of an

intercalator suppresses this plateau completely, and the rise of the force with

extension is smoother before and after the plateau than the without intercalator.

Cluzel et al.[42] suggested properly that the stretching similar to the

molecular modeling experiments that used only one C-end terminal of the duplex

strands, which causes a reduction in helical radius and strong base pair inclination

that maintains both base stacking and pairing until a relative length of 2.0 Lo. This

finding correlates with early spectroscopic studies by Fraser and Fraser [47] on 12

stretched DNA fiber. During the stretching-relaxation cycle dsDNA undergoes an

over stretching at some critical tension, and two possible forms appear. The bases

might unwind and unstack into a parallels ladder, or the bases could retain partial

stacking by slanting as in the skewed latter. If the overstretched dsDNA remains in

low salt buffer for more than a few seconds, the nicked strand melts and frays back

from both sides of the nick. When tension is released, the regions which have

remained base-paired rapidly contract to B-DNA but frayed regions take several

minutes to reanneal, perhaps due to secondary structure in the frayed strands.

Marko [48] has made a final attempt to extend the worm-like chain model11 with

fixed contour length to include the twist deformation and its coupling with the

stretching in order to treat the overstretching, undertwisting of the DNA under high

tension. A close examination of Marko’s [48] paper shows that actually twist

deformation is only included symbolically in his elastic energy Eq. (1).

Later it is argued that DNA under tension often has no constraint on its twist, and the

molecule may freely swivel, and the fluctuation may be summed over to yield

the effective energy, where the average twist becomes: , and

then may be incorporated into the stretching energy , then the stretched

modulus is replaced by . That means the twist affects only in the

softening the stretching elasticity. Marko [48] has also realized further that the

experimentally observed overstretching [43,48], which manifest itself with a plateau

11 Marko’s elastic energy of DNA:

13

on the FE plot at a force of 70 pN, where changes from about 1.1 to 1.6, over

only few pN, can’t accounted by the simple relationship such as

obtained from above arguments since for large

forces , the mean extension is no longer small. Marko [48] then has

decided that one should consider adding a nonlinear elasticity potential such as

to the effective elastic energy function, assuming that this provides a

barrier over which must pass to reach the overstretched state with freely

fluctuating twist. He proceeded in using a Schrödinger–like equation, converted from

the statistical path integral { }

for the evolution of the distribution function over the short distance along

the chain, to deduce an eigenvalue equation , where . The

largest eigenvalue gives the partition function;

. Marko [48] after chosen required

parameters, which show agreement with the entropic elasticity data [49] as well as B-

DNA to S-DNA transformation data [42], obtained a force extension plot showing

the desired ‘’plateau’’ as observed by Cluzel et al. [42] for the overstretching

transition, which corresponds to the coexisting regions of normal and overstretched

DNA.

According to Gore et al., [50] simple physical intuition predicts DNA should

unwind under tension, as it is pulled towards a denaturated structure. They used rotor

bead tracking directly to measure twist-stretching coupling in single DNA molecules.

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They show that for small distortions, contrary to intuition, DNA over- winds under

tension, reaching a maximum at a tension of 30 pN. As tension is increase above this

critical value, DNA begins to unwind. The observed twist-stretched coupling predicts

that DNA should lengthen when overwound under constant tension, an effect that

quantitative confirm by Gore et al.,[50]. In Section 2c of this paper, all these basic

effects, namely; the winding and unwinding of the alpha-polypeptide (ssDNA) under

the stretching testing can be simulated if the steady pulling is performed on the twist

free C-end terminal under the constant volume constraint. However, if the same test

is performed on the dsDNA, which holds its cage diameter invariant during first

stage of the unfolding because of the high rigidity of the interlinks between the base-

pairs of strands, only the monotonous unwinding can be observed up to the unzipping

stage of dsDNA.(See: Figure. 20)

Noda and Hearst [51] applied different modification on the Kratky-Porod [11]

Gaussian model known as ‘’wormlike chain, WLC’’ in the current literature.[17,

1,52,53,54]. Early models [34,35,36] are based on the ad hoc phenomenological theory

proposed by Bell, [55] that relies on the well-known transition state or the activation

complex theories [56,57] of chemical kinetics modified by the applied stress systems.

Similarly, the role of the stored elastic torsional energy during the formation of the

helical conformation is miscalculated or completely over looked because of the lack

of knowledge in this highly controversial field of elasticity [58]. Ackbarow et al [57]

derived their rate equation for the bond breaking using some ad hoc mechanical

arguments relying on the activated-complex theory ‘’transition state theory’’[61]

modified originally by Bell [60] to take care of the effect of the applied stress on the

Gibbs free energy barrier. In later stage, they switched to so call the hierarchical Bell

15

model, which deals with statistics of the chemical bonds based on the robustness

parameter postulated by Kitano[59], whether they are breaking sequentially or

collectively.

In the majority of the molecular dynamic studies [60,61,62], the elastic network

model (ENM) is constructed by connecting nearby Cα atoms lying within a cutoff

distance Rc using harmonic springs. The more sophisticated model is proposed by

Flechsig et al.,[63] by employing semiflexible modeling for the single strand of DNA,

where each bead represented a nucleotide and is connected to its neighboring bead by

flexible links. The total elastic energy of the chain is involving two terms. The first

term, a harmonic bonding potential, takes into account the elastic deformation energy

of the links in the chain. The second term is the elastic bending energy of the chain,

which involves the angle between two adjacent links in the chain. In the case of

duplex dsDNA strands, they proposed Morse potentials for the interaction between

the strands for the additional links that bridge opposite beads.

For a long time, no one has bothered to introduce a torque term neither in the

molecular dynamics simulation studies[64,65,66] nor in the classical worm-like chain

(WLC) models [17,38], even though they are some authors [67,68] like Marko[47]

trying to improve them by adding bending, twisting and stretching elements in their

models. An extensive tensile testing, and modeling has been performed by Gross et

al., [69] by introducing two important features of the duplex stranded dsDNA into the

scenario, the helical structure and base-pairs sequence. The structural aspect of their

twistable worm like-chain (tWLC) model follows very closely Marko’s footsteps

[47] and Gore et al.,[54] by including the twist and the twist-stretching coupling

terms into the scenario, which allow them to tract down the experimental force-16

extension regime up to about 65 pN. Above which another striking feature of dsDNA

elasticity reveals itself with the overstretching transition or melting: ‘unpeeling’ of

one strand from the other, and a transition from B-DNA to an elongated double-

stranded ‘S-DNA’ form.[70]. The length stretches up to 1.7 times its B-form length

and DNA conformation changes to the S-form.[71] The elastic stored energy of DNA

molecule12 regardless whether it is single or multiple stranded contains same terms. In

Gross et al., [68] theory, extremum values of the ‘’enthalpic’’ end-to-end extension

and the torsion are obtained from the stored elastic energy minimization with respect

to torsion and extension. These finding are incorporated into the classical worm-like

chain model [72] to get the required relationship between the end-to-end distance and

the force acting on the tWLC model of DNA. This formulation allows them to

determine the absolute value of the tension-dependent twist-stretch coupling, g(F)

from magnetic tweezer studies [73], which is negative at low forces and changes sign

at about 35 pN. However, none of these models concerns with the elastostatic

properties of the helical conformation rigorously, which inherently involves large

amount of stored elastic torsional deformation energy, which is the main source for

the instability unless embedded by the severe acidic aqueous solutions [1] in the

range of pH3-4.

12 This ad hoc expression is presented as the elastic storted energy by Gross et al., and is defined as the end-to-end distance of the molecule. Then one can show that there are typological errors in their formulas entitled as (S2), (S3) and (S4) in the supplements, where F should be replaced by (F+S). Similarly, in Eq.(1) in their main text, and (S4) in the supplement are erronously called as extension. They combined their findings with WLC and write:

17

All these show clearly that some type mechanical constrain necessary on the

total twist angle or on the radius of the helix during the testing. The most practical

constrain in the case of polypeptide is to clamp the N-end terminal to the substrate

firmly, and the other C-end terminal to the grip of the testing machine with or

without the twist constraint. In the case of Young modulus measurement by

Terahertz Spectroscopy [77] of a crystalline state of DNA Poly-L-Proline helices,

because of the inherent confinement associated with the solid state rigidity, the radius

of helix may be taken as invariant and the total twist angle as the auto- adjustable

internal parameter dictated by the instantaneous value of the extension, which is

driven by the external agent. This case is thoroughly formulated previously by

Ogurtani [40], and also it is simulated in the present work. It has been observed in the

present study that with the total polar angle constraint or ‘’the number of turns is

fixed’’, the helical radius varies while total twist angle stays invariant, and the

polypeptide shows reversible simple spring response, aging and unfolding kinetics

even in the presence of the acidic environment and applied stresses.

In Section II, the fundamental result of the linear finite elasticity problem

associated with the torsion and bending in the ‘’multi-connected’’ helical simple

spring, which has been rigorously formulated by the author using the elegant dyadic

calculus, [74] is combined with the global Gibbs free concept [46,75] in connection

with the rate of internal entropy production hypothesis as advocated by Prigogine.[76,

78]. Here, we have introduced not only the unusual torsional elastic stored energy

as main driving force for the spontaneous morphological evolution but also the

interfacial Gibbs free energy density function to take care of the multi-polar

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electrostatic interactions with the surroundings aqua solutions under the isothermal

isobaric conditions.

In Subsection-2a, a detailed numerical analysis of the data collected by AFM

observations by Idiris et al.,[1], has showed us that there is a strong linear connection

between the interfacial Helmholtz free energy density and the pH level of the

aqueous solutions used in their studies in the range of [pH 3-8]. Similarly, it has also

been discovered that at the end of complete unfolding, the contribution coming from

the interfacial free energy term to the global Helmholtz unfolding free energy so-

called as ‘’unfolding work’’ is nil compared to the total elastic deformation energy

contributions (torsion and bending) especially one associated the unusual torsion

term, which is practically few orders of magnitudes greater than the rest. This is

solely due to the fact that AFM force-extension test measurements performed under

the total polar angle constraint by keeping the both ends of the polypeptide macro-

molecule fully clamped against possible twisting or rotational motion at the terminal

ends but only allowing them longitudinal displacements.

In Subsection-2b, the aging kinetics of helical polypeptide is studies under the

constant applied uniaxial tension assuming that one has isothermal isobaric

irreversible processes for various constrains such as ‘’the total polar angle, invariant

helical cage radius, and finally the constant cage volume.’’ Non-linear integral

equations are obtained, which relate the normalized elapse time to the scaled

temporal total pitch height as a function of applied force, and internal free energy

density. It has been shown that the difference between the total polar angle and cage

radius constraints is negligible compared to the total volume constrain, which results

much longer surviving times for the polypeptides (ssDNA and dsDNA) under the 19

acidic-aqueous environments (3-4pH), and atmospheric pressures up to 100 Atm

(1000 m deep-waters).

II. MESOSCOPIC THERMODYNAMICS OF HELICAL

CONFORMATION

In the formulation of the global Helmholtz free energy, one has to consider

not only the ensemble of the amino-acid or nucleic-acid strands as a bulk phase but

also the existence of the interfacial or the surface Helmholtz free energy associated

with the electrochemical interactions taking place between the side-wall branching

molecular species and their surrounding aqueous solutions [77]. In our model, which

is primarily intended for the single stranded ssDNA (i.e., unfolding stage of dsDNA

may be categorized as radius constraint) and alpha-polypeptides, these side-wall

branching species are embedded by a hypothetical cage, which acts like a surface

layer of the amino-acid skeleton having finite and invariant thickness, and enclosing

the whole helical conformation. During the unfolding, on the contrary to the double

stranded dsDNA, alpha-polypeptides and single stranded ssDNA, the diameter of the

cage automatically adjust itself,13 respectively, to keep either the total polar angle

or the cage volume V invariant depending on the mechanical testing setup, whether

to use the tip of AFM as a stretching grip with the twist constraint [1] or the magnetic

and optical tweezer as a force- extension tester pulling the twist free C-end terminal

of poly-peptide [42,78].

13 Because of the rigidity of interlinks between the duplex strands, In the case of dsDNA one may assume that the radius of cylindrical cage stays invariant during the unfolding or the spontaneous aging process.[41]

20

2a. Isothermal reversible work based on the constant polar angle

constraint: Spring Mechanics of Single Stranded ssDNA.

For the theoretical analysis of the experimental force extension results, we have

proposed the following analytic procedure, which relies on the Max Planck (1887)

general criteria [79] for the isothermal infinitesimal processes in closed systems in

order to reveal basic ingredients of the simple spring model. This criterion is further

supported by the successful treatment of the more complicated dynamic evolution

problems by Ogurtani and his coworkers [75] as the typical nonequilibrium isochoric

[80] and isobaric isothermal processes [81]. This criterion tells us that the infinitesimal

work done on a closed system is equal or greater than the infinitesimal change in the

global Helmholtz free energy of the system under isothermal conditions, namely:

. Here, the equality applies to reversible processes, which is the case for

the present mechanical testing, the greater implies the irreversible or ‘’natural’’

changes, which may be used for the spontaneous morphological evolution processes

taking place during the aging, and the spontaneous unwinding occurring at the last

stage of unfolding or at the onset of the unzipping (denaturation) of dsDNA [82].

The force extension (F-E) measurement performed by AFM [1] involves by

pulling the test piece with a well prescribed constant speed while simultaneously

recording the response force for a given ‘strain rate’’. One has to repeat the test by

changing the pulling speed to be sure that the F-E curves is no longer depending on

the strain rate. Therefore, this is a ‘’quasi-static measurement’’ method, and the

results obtained definitely depend on constrains and/or the ‘’boundary conditions’’

21

acting on the edges of the test piece through the grips of the test machine.14 If the

both edges of the polypeptide specimen are firmly hold by the grips as such that no

torsional deformations or twisting are allowed to be exist there, then one can be sure

that the total polar angle designated by on the test piece or the total ‘’number of

turns’ associated with the helical conformation of alpha polypeptide stays invariant

during the elongation measurements. That means the rate change in the twist angle

along the arc length of the helical conformation denoted as λ, and it is also so-called

‘’torsion’’, changes only with the pitch height according to the fundamental

geometric relationship given by for a simple helical conformation [39].

Similarly, the radius R of the helix varies spontaneously during the extension

measurements according to the relationship: , under the assumed

constraint on the total polar angle, . Where is the total contour length or arc

length of the helix, which is constant during the unfolding, unless the extension test

is continuous in the elastic stretching stage of the backbone structure in

the absence of the helical form [7]. That means no complete or genuine unfolding

can take place during testing since the total polar stays invariant due to the

prescribed constrains. The energy injected during the pulling will be released during

refolding without having any net- dissipation of energy as observed by Idiris et al. [1]

even after 200 cycles. However, the unfolding force extensions F-E curves reported

by these authors don’t show any linear behavior in the absence of the interfacial free

energy contribution as should be the case for a simple spring model derived in this

14 According to the reported experimental setup by Idiris et al., [1] these constrains are achieved, respectively, by keeping one edge of the peptide test piece connected to the silicon substrate firmly through the PEG attachment, and the other edge to the modified tip of atomic force microscopy (AFM).

22

synopsis relying on the zero torsion boundary conditions. Authors realized these

difficulties due to inconsistencies in their measurements by saying that the extension

dependent results should be taken rather cautiously by assuming to have 15% error

tolerance.[1]

The infinitesimal change in the global Helmholtz free energy for the

isothermal reversible process in closed system may be presented by the following

expression [83,79] according to Planck-1887 equality .

(Reversible) (1)

appeared on the right side is the infinitesimal reversible work15 done on the

system under the isothermal conditions, which is not an exact differential operator. In

above relationship, the first and second terms are, respectively, associated with the

unusual and usual torsional stored elastic energies, and the third term is nothing but

the bending elastic energy [58]. In addition, there is a fourth term in Eq. (1), which is

the interfacial Helmholtz free energy, which is designated by the ‘’specific surface

energy density ’’ arising from the interaction between the sidewall branching

molecules of the helical conformation and the embedding aqua-solution having well

defined pH levels. While writing above expression, we have employed the fact that

15 ; is the force acting during the extension denoted as . In the case of hydrostatic and nonhydrostatic work, one has to replace by, respectively,

, is the hydrostatic force action on the sub-

phase denoted as alpha of the composite system. are non-symmetric stress and deformation tensors generated by the applied surface traction, torque and body forces to the amino-acid skeleton.

23

the bulk chemical part of the Helmholtz free energy for a closed system in the

absence of chemical reactions stays invariant , and only the residual

deformation or the stored elastic energy part of [Appendix A Ref:84]

appears as an additive term. The entropy term in the Helmholtz free energy is

assumed unaffected by the helical elastic deformations, which is plausible for solids

according to Callen [85] but may not be valid for biopolymers [50].

As it will be shown numerically as well as graphically that the second and the

third terms are few orders of magnitude smaller than the first term, which has been

called as unusual torque term by Ogurtani. [40] This term is discovered by Ogurtani

[86] while furnishing a rigorous mathematical treatment on the torsional deformation

of an elastic bar having circular cross-section by relying on the torsional finite

displacement field introduced by Saint-Vernant [87] in 1855. This solution satisfies

the equation of equilibrium in the absence of the body forces as well as the equation

of compatibility. In the literature not only the asymmetric part of the deformation

tensor is overlooked completely but also the non-vanishing sidewall tangential force

field density has been neglected by Timoshenko and Goodier [58] during the

approximate solution of the associated boundary-value problems for non-circular

cross sections. Our analysis produces in addition to the unusual torque term, an

exactly same usual torque term given in the literature for circular cross sections [58].

Above expression may be further organized by expressing the rate of torsion,16 in

terms of the total pitch height, and the total polar angle subjected by the

16 The unusual torque term may not even vanish if both ends of the test piece are not firmly clamped, which is the case for the tweezer experiments, where the bead edge free to twist. Then the unusual torque term takes the mid-point of the back-bone structure as an initial zero twist state, and proceeds in both directions.

24

polypeptide. Then one writes from the definition of torsion of a simple helical curve

in 3D space: where or

is the inclination or ‘’azimuthal’’ angle of the tangent vectors

with respect to the generators of the cylindrical surface.17 Similarly, the curvature

[40] may be expressed by . is the

Young’s modulus of elasticity, and it is related to the shear modulus G and Poisson’s

ratio by . Then, one writes the following expression for the

total polar angle constraint system.

(2)

And the generalized response force may take as the negative gradient of the global

Helmholtz free energy from Eq.(2), which is given by:

(3)

The applied force for a given intermediate18 “extension ”,

which is defined with respect to the initial total pitch height of helix by

17 Some authors use the complementary angle , which is the orientation of tangent vector defines with respect to the base plane of helix.

18 The most authors dealing with the theoretical aspect of the force-extension connection relying on the statistical formulations, and they are using the normalized end-to-end extension

, where z is the molecular end-to-end distance. [76] 25

, may be represented by the following summation and/or integration

expressions symbolically;

, (4)

Where, the unfolding global Helmholtz free energy function denoted as

for an extension designated by x is given by the following expression in Eq. 5:

(5)

Where, is reversible work done on the system

during the extension (stretching) or the unfolding of the helical conformation from a

given initial state designated by po.19

(6)

19 One may easily anticipate from Eq.(6) that for the isotropic elastic solid the usual torsion and

bending terms may be combined into a single term as such: , where

. This negative expression indicates that the extension at the absence of the unusual torque and interfacial free energy terms, becomes a spontaneous event, and the polypeptide unfolds completely without the help of the external driving force. This is also a typical for an initially squashed spring behavior that means one needs to apply compression to stabilize the stationary state.

26

Above expressions designated by Eq.(3) and Eq. (6) clearly show that there

are strong couplings in each term between the extension and the polar angle

with the acceptation of the interfacial free energy terms. Similarly, Eq. (3) and Eq.

(6) indicate that the generalized force and the unfolding work are linear and quadratic

functions of the end-to-end extensions, respectively, in the absence of

the interfacial free energy contributions. According to Idris et al. [1], their

experimental data are recorded in terms of the extension “ ”, which is defined by

, where p and po are, respectively, the initial and intermediate total pitch

heights of the helical conformation assuming that one has hundred per cent helicity.

Otherwise some legitimate corrections should be made on the elasticity properties

such as on the shear modulus G employed during fitting of the theoretical curve to

the experimental points obtained by AFM stretching testing.20 Relying on the

reported numbers by Idiris et al., [1] one has: for the single

(Glu)n –Cyr helical chain, which has about nine full rings, that results an

alpha helical radius of Ro = 5.93 Ao. This radius is very close to the radius of alpha

peptide 3.613 studied by Pauling et al. [88], and reported as about 6 Ao. The formula

given by Eq. (5) is employed to analyze the unfolding energy data reported by Idiris

et al. [1] [ibid; Fig. 6)], which are obtained by integrating the force extension (F-X)

curves in their Fig.4B with respect to the extension length. These curves deduced

20 One can easily anticipate the following mathematical scaling connection just looking at Eq. (1).

Namely: , where one has:

. The Helicity scaling doesn’t affect the rate of torsion denoted as λ

because of its geometric definition for helical conformation. For the present case :

.27

after subtracting PEG extension from the total extension lengths reported in their Fig.

4A, which are directly recorded by AFM stretching testing setup under different pH

conditions. The reported unfolding work of stretching a single helical (Glu)n –Cyr

chain (n=80, helicity is 80%) at pH 3 versus the limited extension lengths is chosen

for the data analysis, which are presented in graphical format in Fig. 1a-1b. In order

to justify the importance of the unusual torque and surface free energy terms in the

spring mechanics, results of two sets of optimized numerical simulations are

reported, respectively, in Fig. 1a and 1b.

Fig. 1. Unfolding work of stretching of an alpha-polypeptide ''(Glu)n –Cys'' (n=80,

helicity is 80%) is marked with black solid diamond dots with respect to the limited

extension in the range of 12.5nm-25 nm, using reported values by Idiris et. al21. [1] at

pH 3.0 (See: ibid; Fig.5-6). The blue solid lines represent change in the global

Helmholtz free energy with extension in the total pitch height of the helical

conformation, using Eq.(5). Fig.1a and Fig.1b employ unusual and usual torque terms,

respectively. Solid green line in Fig.1b uses bending only. In Fig. 1a, it also

considers in addition the unusual torque. The solid red line represents unfolding work,

having no contribution from the bending, but involves, respectively, unusual and usual

torque terms at the absence of interfacial contributions.

21 file:///G:/WALTER/Protein%20Eng.-2000-Idiris-763-70.pdf28

Where various alternative combinations of the shear modulus and the interfacial free

22 energy densities are tried using Eq.(6) to get the best matching to the data points

presented by Idiris et al.[1] by employing the unusual and usual torque terms,

respectively, in Fig. 1a and Fig.1b. In each figure, we have three alternatives

combinations: i) the blue solid line, which includes the bending as well as the

surface free energy contributions for best fit. ii) the red solid lines involve no

bending and surface free energy terms, but use, respective, full torque terms in

Fig.1a, but only usual torque term in Fig. 1b. iii) the green lines doesn’t have any

contribution from the interfacial free energy in both cases, but involves the bending

and torque terms having a shear modulus of G=13.6 MPa in Fig. 1a, and only a

bending term having an unusually high shear modulus of G=3950 GPa in Fig.1b;

assuming that one has isotropic elasticity with Poisson’s ratio of , and the

surface free energy density of gs =-24 erg/cm2 .

These two plots show very clearly that the contributions of the bending and

the usual torque terms to the unfolding scenario are almost nil compared to the

unusual torque and interfacial free energy terms, if one uses reasonable values for the

elastic shear G and Young E modulus in the bending and the usual torsion

rigidities , respectively, for a rod of circular cross section [40,58]. Where, the

unusual torsional rigidity varies along the backbone structure of the polypeptide

starting from the firmly held N-end terminal where it is zero, and changing

22 For the best curve fitting, the following parameters are used for good fitting: G (1a)=13.6MPa and G(1b) =3950 GPa, and gs =-24 erg/cm2. Data: =18 (9 Rings); a=0.75 Ao; po = 10 nm; Initial total pitch height of (Glu)n-

Cys is reported as 10 nm, and the total stretchable length is =34 nm after subtracting PEG extension. The helicity 80% correction factor is about 2, which results GCorr. =27 MPa.

29

quadratically with the arc distance of the cross sections; [40,41]. In Fig.

1b all those curves given in Fig. 1a are replotted by replacing unusual torque by the

usual torque term observing the similarity transformation between the usual and

unusual torques as appeared explicitly in Eq.(1), namely: .

Where the shear modulus is employed for the usual torque term in Fig. 1b, and

is the shear modulus used for the corresponding unusual torque in Fig 1a. These

transformation results exact replications of two curves associated with the usual and

unusual torque terms, including the interfacial free energy contributions at the

absence of the bending terms. Above mentioned mapping results an enormously high

shear modulus of elasticity such as 3750 GPa compared to 13.6 MPa obtained in

Fig.1a to get the best fitting by employing exactly the same interfacial specific

Helmholtz free energy density of -24 erg/cm2.

In fact, if one compares the initial stored elastic energies associated with the

bending and usual torque terms with the unusual torque term, one would realize that

they are, respectively, five and four orders of magnitude smaller than the stored

elastic energy due to unusual torque. In the numerical calculations, if one employs Ga

=13.6 MPa in connection with the initial length of po=10 nm, the elastic stored

energies associated with the unusual, and usual torque terms in addition to the

bending term appearing in Eq. (6) can be computed by taking extension is equal to

zero, respectively. They are found to be, respectively, {4.575 eV, 1.575×10-5 eV and

2.659×10-4 eV} for the polypeptide designated in Figs. 1a-1b. These figures show

clearly that the unusual torsional stored elastic energy is few orders larger than the

30

usual and bending energies. In fact, the bending stored energy dominates the usual

torque term initially.

Fig. 2. The changes in the cage radius and the associated variations in the interfacial

Helmholtz free energy are calculated at pH-3 and G=13.6 MPa from the master equation

Eq. 5 using the best value of gs=-24 erg/cm2 estimated in Fig. 1a. Double red and blue

dots indicate the boundary of the regions at which the backbone structure of the helical

conformation reaches the straight circular bar shape, having a cage radius of 1.5 Ao, and

still involves the initial torsional deformation intact but nil bending.

In Fig. 2, the changes in the interfacial free energy (x) as well as the cage

radius variation of the helical conformation up to x=23.5 Ao are plotted as a

function of the extension using the following relationships deduced from Eq. (6) and

the geometric connection between the total polar angle and the total pitch

height as presented previously:

, (7)

And

31

(8)

The cage radius shows monotonic decrease, and reaches to zero if we wouldn’t

interrupt it at x=23.5 Ao, where . This fictitious last number

corresponds to a state at which the full unfolding of the helical conformation takes

place, and the polypeptide turns into a straight cylindrical wire in shape while

keeping its original twist deformation invariant . The cage in this

configuration has a hypothetical radius, which is composed of the radius of the amino

acid skeleton (denoted as ), and plus the effective thickness (

) of the cylindrical shell that enclosing the whole sidewall branching molecules at

x=15 nm. . The interfacial free energy has a negative in sign and it shows well

defined minima

Fig. 3. Effects of the interfacial free energy density variations on the unfolding work are

presented for the positive and the negative specific volumetric free energy densities,

which show that the positive density enhances the unfolding work, and the negative one

decreases. Data points are extracted from Figs.5 & 6 in Reference [1]

32

Fig.3 shows that at the start there is an exceptionally large deviation in the

observed unfolding free energy (red solid line) in the acidic and basic environments

from the simple spring behavior dictated by the torsional and bending elastic energies

(blue solid line), which is finally recovered completely in the vicinity of the full

extension of 25 nm. There is a systematic negative contribution coming solely from

the interfacial free energy as seen clearly in Fig.2, and that amounts to about [-10 to -

5] eV for those extensions vary in the range of 12.5 nm to 24 nm. However, the

difference arising from the negative interfacial free energy reduces drastically,

according to the cofactor in the interfacial free energy Eq. (6)

, above the limiting extension length of 23.5 nm.

Fig. 4. The effect of the pH level on the unfolding work exhibits itself by the

consecutive variations in the interfacial free energy densities as well as in the shear

33

modulus of the back-bone structure of alpha-polypeptide [1]. Data points are extracted

from Figs.5 & 6 in Reference [1]

In Fig.4, effects of the aqueous solution having various pH levels of acidities

on the unfolding work of an alpha-polypeptide are presented in a systematic fashion

by identifying a proper set of interfacial free energy densities and shear modulus

for each pH level, which are producing the best fit to the experimental data

obtained by Idiris et al. [1]. The similar experimental procedures have been

performed by Idiris et al [1] at higher pH levels show systematic decrease in the

unfolding energies for the given limited extensions (SEE: ibid. Fig.6).

However, the final unfolding energy is independent from the interfacial energy

term as can be seen above given Fig. 3 and Fig. 4 because of the above-mentioned

analytical form Eq.(7), the interfacial free energy contribution becomes zero due to

vanishing of the cage radius, Eq. (8). One should be reminded that at the end of

quasi- unfolding state alpha-polypeptide contains the whole torsional deformation

elastic energy represented by =18, which now spread all along the contour length

of 35 nm, and having only the torsional elastic energy of 56 eV. That means the full

extension of the helical conformation is a conservative process, if one releases the

edge of the test piece slowly but still keeping it firmly by the grip of AFM test

machine, polypeptide returns its original length of 10 nm, without losing anything

from its initial stored elastic energy of 4.575 eV. That means the reversible unfolding

work done on the system by the test machine is about 51 eV, which is exactly equal

to the unfolding work reported in the literature [1].

34

Fig. 5. Theoretical force-extensions plots of Alpha-Polypeptide (Glu)n-Cyr (n=80),

which is exposed to aqueous solutions having various pH-levels are tested by AFM

under the constant total polar angle constraint. Where the shear modulus varies with

pH according to the list given in Fig. 4 by taking Go=13.6MPa, The contour length 35

nm, = 9 turns.

In Fig.5, the pulling force applied to the polypeptide for a given

instantenous extension is calculated from Eq. (3) for different pH-leves using the

proper combinations of the interfacial free enery densities and the shear modulus

presented in Fig.4. As it may be seen from the last term of the cited equation, the

force blows-up at termination of the unfolding process because of the intefacial

free energy term. This is a general behaviour observed in the literature [1,78,89]

for single stranded ssDNA as well as the double stranded dsDNA molecules just

at the onset of the ‘’untearing of the interlinks’’ between the basepairs.[42]

35

Fig. 6. Effects of the pH level on the shear modulus and the interfacial Helmholtz free energy density are presented for (Glu)n –Cys. (n=80,helicity 80%). Initial total pitch height is 10nm, the total arc length is 35 nm. The shear modulus and İnterfacial free energy density are, respectively, Go =13.6 MPa and gs =-24 erg/cm2 at pH3. DATA: fs(x) ={3.8x -36.5} erg/cm2, and G(x)/Go={-0.225x+1.647}, where x=pH value.

In Fig. 6, the shear modulus and the interfacial free energy densities, which

are found to produce the best fitting curves to experimental points in Fig.3, are

plotted with respect to the pH levels of the aqueous solutions used for the single

(Glu)n –Cys chain experiments by Idiris et al. [Ref 1: Fig.6]. The interfacial free

energy density shows straight-line behavior, which may be represented by the

expression . The shear modulus also shows

straight line behavior with a knee at pH6, which almost corresponds to the

transition threshold of the helical conformation to the random-coil structure

according to the experiment done on the pH dependence of the degree of helicity

of (Glu)n –Cys, based on the results of CD spectroscopy.[1]

36

2b. Evolution kinetics under the constant total polar angle constraint:

isothermal isobaric irreversible process

In the case of an irreversible process under the applied constant loading, one

can recast the global Gibbs free energy in the following form [40]. Here, the fourth

term between braces is related to the interfacial free energy associates with the

embedding cage of the side-wall molecular branches, and the fifth term is due to the

applied constant pressure on the cage volume , while keeping the total

polar angle invariant by clamping the C-end terminal constraint against twisting.

(9)

Where N-terminal is assumed to be attached firmly to the substrate base as such that

no twist can place there. At this point one can use the connection between the rates of

internal entropy production and the global Gibbs free energy advocated by Ogurtani

[40], which is valid under the isothermal and isobaric closed systems. Namely:

>0

(10)

By incorporating the generalized driving force expression given by Eq.(9) into Eq.

(10), then one may write the connection between the conjugate flux and the

generalized force without omitting the usual torque and bending terms, which are

actually few orders of magnitude smaller than the unusual torque term by a factor of

37

[a /l o ]2≪1. Here, the conjugate flux corresponds to the displacement velocity of

helix, which may be given by the following linear expression between the conjugate

flux and the generalized force after some legal decomposition of the entropy

production, [90] and rearrangements of terms, namely;

(11)

The second term represents the combination of the usual torsion and the bending

contributions. Where, for the isotropic solid one may replace cofactor by

, where because of the well connection E=2(1+)G

in the theory of elasticity of solids [50].23 Consequently in the absence of the unusual

torque term, the bending incorporated into the generalized force on the right hand

side of Eq. (11) dominates the usual torsion term, regardless the magnitudes of the

usual torsion and the curvature along the length of the helical conformation. Then, it

may encourage increasing the total pitch height spontaneously , and

terminates the unfolding process even at the absence of the external positive

interventions such as the applied uniaxial tension (thermal expansion) or the positive

interfacial free energy arising from the strong basic aqueous solution surroundings.

All these arguments show us that how important is the existence of the unusual

torsion contribution in the energetics and kinetics of the helical polypeptides (i.e.,

23 That is a negative quantity because of the fact that Poisson’s ratio even it is defined in general as

, is a positive quantity for the most stable and isotropic linear elastic materials with the exception of auxetic materials (e.g., some polymeric foams, origami folds, and certain cells) can exhibit negative Poisson’s ratio because of their internal structures.

38

ssDNA and dsDNA), Here the instantaneous value of the total pitch height is

normalized with respect to the contour length designated as . The relaxation time

of the evolution process may be given by the expression; .

Then, the morphological evolution behavior of the helical conformation may be

deduced by the integration procedure as presented below in Eq. (12) by combining

unusual torsion term with the combination of usual torsion and bending under the

first term appeared below, assuming that we have isotropic elasticity.

(12)

Integration of Eq. (12) at absence of the interfacial free energy and the stress terms

can be performed analytically, which yields following very useful exponential

expression:24

(13)

The second expression given in Eq.(13) is important in the isotropic elasticity theory,

which tells us that at the absence of the unusual torque term [86 92], the total pitch

height of helical alpha-polypeptide increases spontaneously without showing any

aging behavior under the total twist angle constraint regardless of its initial state. 24 The unusual torque term is represented by figure “1” in the argument of the exponential

function in Eq.(13) and in the first term of the integrand of Eq.(12).39

That means the helical configuration is unstable even if one keeps the both end

terminals of the polypeptide are clamped against free twisting, but allowing it only

for the free longitudinal extension with and without an applied force. Therefore,

without the unusual torsion term, polypeptides can’t demonstrate a proper spring

mechanics behavior, but rather spontaneously extents to the full arc length, unless

they have negative Poisson’s ratios.

Fig. 7. Aging behavior of polypeptide Glun-Cys (n=100) is demonstrated under the

positive [pH6 ] and negative [pH≤4] interfacial Gibbs free energy densities25 without of

the applied stress.

In Fig.7 aging behavior of alpha-polypeptide (Glu)n –Cys is presented using

Eq. (12) for various interfacial free energy densities by taking Poisson’s ratio ν=1/2

and the shear modulus Go=1GPa. [1]. The high acidic environment represented by a

large negative interfacial free energy density does indeed stabilizes the polypeptide

25 Idiris et al.[1 ] Fig. 2 pH dependence on the degree of helicity of (Glu)n –Cys based on CD spectroscopy.

40

not only opposing the transition to the random-coil structure but also avoids the

complete degradation by aging, which happens in neutral and alkali environments.

Actually, a realistic stability analysis of aging behavior described by Eq. (12)

could be done by incorporating the pressure term into the scenario. In fact that shows

us that there are two distinct sectors in the total pitch height versus time diagram as

we have discussed in our previous work, which deals with aging of dsDNA under the

constant cage radius constraint.[40] In the absence of the interfacial free energy term,

the normalized generalized force given in Eq. (11) becomes a quadratic function of

the total pitch height , and it has two roots, one of which is negative, and the other

is positive in sign. If this positive root denoted by happens to be in the open

interval of 0 1, the global Gibbs free energy represented by the expression

between curly-braces in Eq. (9) has an extremum (i.e., a free energy barrier) in that

region. According to Eq.(11) this can be controlled by the normalized applied

pressure with respect to shear modulus, . That means any initial state in

front of the barrier may have gone through a complete unfolding or the

‘’cross-coiled transition’’ under the ‘’properly selected’’ normalized

positive hydrostatic pressure. If the initial state would be behind the barrier

, it is still possible to overcome that barrier by applying very high

negative hydrostatic pressures (thermal expansion stresses). The threshold value of

hydrostatic normalized pressure can be calculated by equating the initial

pitch height of the polypeptide to one of those roots of the quadratic equation

41

obtained from Eq.(11). That reads: , which reduces to the

following connection , where . These

last two expressions may be combined to get the threshold pressure:

( P/G )|Thr=(2 a2 Θ4/9 lo2 ) [ p̄o−(3 p̄o )

−1 ]−1. The trajectory drawn by this normalized

pressure in the pitch height- time diagram results a horizontal straight line as a

boundary between those two sectors. [40 ] It is very clear from the last expression

that the initial state should satisfy the following inequality, (the

twist angle β ≤ cos−1 (1 /√3 )≃5 4o) in order to be in the positive hydrostatic pressure

sector.

Fig. 8. The aging kinetics of an alpha-polypeptide having an initial state 0.835, which is

above the critical total pitch height given by in the presence of the applied positive and negative pressures without interfacial free energy term.

42

To test our theory, we have chosen ( ) as an initial

state for the (Glu)n–Cyr polypeptide in helical form having a contour length of 35nm

with the total polar angle /2=9 turns with shear modulus G=1GPa, and the

employed Eq. (12). Using above given connection, the threshold normalized pressure

obtained, and it is found to be , and for various positive

hydrostatic stresses are selected above and below of this threshold pressure level, and

the results are plotted in Fig. 8. It is clear from this study that if the total polar angle

of the helical conformation can be constraint at both edges of the polypeptide

(ssDNA), namely by fully clamping both N-end and C-end terminals as such that no

free rotation take place there, then the life time can be decreased by increasing the

applied hydrostatic pressure up to 22.79 GPa. This aging behavior is in accord with

the case of constraint helical radius problem elaborated in our previous work [40],

where any increase in positive hydrostatic pressure decreases the aging time of

dsDNA. However, above the threshold pressure the helical conformation switches to

the anomalous sector by showing accelerated decimation towards the quasi-

unfolding, while keeping its torsion intact. That clearly shows that the boundary

conditions at the edges of the helical conformation are vital importance in

determining the kinetics of aging, and evolution processes.

Fig. 9, shows another interesting case, where the initial total pitch height

is less than the critical pitch height , which results

negative hydrostatic stress solutions for the upper region of the diagram. The

calculated threshold level is found to be for the (Glu)n–Cyr

polypeptide has initial pitch height 10 nm and the contour length 35nm having total

43

polar angle /2=9 turns. That means it allows a helical polypeptide under the

negative pressures increase its pitch height if it is above this threshold level

otherwise decreases to the new stationary state levels, respectively.

Fig. 9. Aging kinetics of alpha-polypeptide having an initial state below the critical total

pitch height in the presence of the normalized applied positive and negative pressures with respect to the shear modulus G=1GPA, without interfacial free energy term.

Furthermore, the pitch heights of the upper stationary states approach to the critical

pitch height asymptotically with increase in the magnitude of the

negative hydrostatic pressure. In the present case, still the positive hydrostatic

pressure operates genuine aging sector, and any increase in its value shortens the life

time of the polypeptide, similar to the case studies in our previous work [40], where

the helical radius of alpha-peptide is invariant but the total polar angle is varying

(dsDNA).

44

In Fig.10. the aging kinetics trajectories of the alpha-polypeptide for various

normalized values of the interfacial free energy density are presented at the absence

of the applied pressure. The normalization is done with respect to the inverse-

cofactor of the interfacial free density in Eq. 12, which is given by .

Above this threshold value an anomalous behavior of unfolding kinetics takes place,

which shows after a long incubation region almost sudden decimation of the

polypeptide at the absence of the external applied stresses. This threshold is very

sensitive function of the shear modulus of the backbone structure of the polypeptide

skeleton and the amount of torsional energy stored in the system as characterized by

the total twist turns , since the other factors wouldn’t change very much in practice.

Fig. 10. Aging kinetics of alpha-polypeptide having an initial state above the critical

total pitch height for positive and negative interfacial free energy densities normalized with respect to the critical interfacial free energy

density: , where =9.687x103 erg/cm2. Data:

G=1GPa and =35nm., =9x2.

45

The initial total pitch height in Fig.10 is selected especially as ,

which is well above the critical pitch height designated as ,

at which according to Eq. 11, the interfacial free energy contribution to the

generalized force becomes identically equal to zero regardless its value.26 At

the same time according to Fig. 11, this critical total pitch height corresponds

the lower bond of the total pitch height at which not only the elastic torsional

energy but also the global Gibbs free energy shows maximum value, where the

inclination angle becomes β=450 and the torsion takes its maximum value

.

Fig. 11. Shift in the position of the Global Gibbs free energy maxima according to the variations in the interfacial free energy densities while keeping the theta torsion angle Θ constant. The maxima are designated by the generalized force versus zero-line

intersections. The lower limit is set by where gs .

26 According to the cofactor of the third term in Eq.(11), which becomes identically zero if

normalized pitch becomes .46

Inspection of Fig.11 shows that the maximum point of the global Gibbs free

energy is situated in the open interval of [ 1], at the absence of the

external pressure. Its exact position depends on the normalized interfacial energy

density having monotonic decrease with the increase in the density that approaches to

, asymptotically. The aging kinetics of the polypeptide (Glu)n –Cyr (n=80)

having an initial total pitch height in that interval as demonstrated previously exhibits

accelerated unfolding kinetics without applied force if its normalized interfacial

energy density is greater than units, gn >1, otherwise it shows usual quasi-

exponential decay aging kinetics. This behavior is very similar to the positive

hydrostatic pressure case given in Fig.8. However, the switch over value of the

interfacial free energy density (severe alkaline environments) here is extremely high

about 104 erg/cm2 for (Glu)n -Cyr having contour length of 35nm and G=13.6 MPa,

compared to the pressure threshold value, which is about 300 MPa .

In the case of negative interfacial free energy densities, the maximum peaks in the

global free energy occur in the lower open interval [0 ].

2c. Evolution behavior of helical conformation based on the constant volume

constraint: Isothermal isochoric irreversible process.

In this section the global Helmholtz free variations under the constant cage

volume constraint is developed for the helical conformation of alpha-

polypeptide to see how the energetic and kinetics behavior vary when it is exposed to

the unfolding. The extremum problem of a grand global Helmholtz free energy

47

function will be designated as, , which includes not only the surface free

energy and the stored elastic energy terms but also the fictitious mathematical term,

which is the volumetric change Z δV=Z δ (π R2 p )⇒0 moderated by the Lagrange

multiplier Z that might be identifies as the external applied pressure P acting on the

system. All these may be represented by the following expressions using the fact that

no external worked done on the system because of the volume constrain, during the

evolution process: Isochoric or workless isothermal process. [90]

{¿δΔF=δ {¿ π3

Ga2 λ2lo3+ π

4G a4 λ2 lo

¿+ π8

E a4 κ2 lo+2 πR gs p }≤ 0

¿And¿

¿V o=(π R2 p )δ V o=0}⇒ { ¿ π

3G a2 λ2 lo

3+ π4

G a4 λ2l o

¿+π8 E a4 κ2lo+2πR gs p+Z π R2 p}≤ 0 ,

(14)

The last expression in Eq.(14) may be further reduced using definitions of the rate of

torsion and bending for the helical conformations such as; , and

. Where one also has and . These

connections allow us to substitute in above equation, which may be

further simplified by eliminating the total polar angle by using the geometric

identity that is valid for the simple helix. Finally, one would have

the following relationship for the grand global Helmholtz free energy variations.

48

δΔ~FV ( p , R )=δ {¿ π3

G a2 p2 R−2 (lo2−p2 ) lo

−1+ π4

G a4 R−2 p2 (lo2−p2) l o

−3

¿+ π8

E a4 R−2 (lo2−p2)2 lo

−3+2 πR gs p+Z π R2 p }=0 (15)

For the solution of the extremum problem under the constant volume constraint, the

expression between curl braces may be differentiated with respect to the free

variables to obtain two independent equations:

∂∂ p

Δ~FV={¿ π3

G a2 R−2 (2 p lo2−4 p3 ) lo

−1+ π4

G a4 R−2 (2lo2 p−4 p3 ) lo

−3

¿−π2

E a4 R−2 p (lo2−p2 )l o

−3+2πR gs+Z π R2 }=0 (16)

And

∂∂ R

Δ~FV={¿−2 π3

G a2 R−3 p2 ( lo2−p2 ) lo

−1− π2

G a4 R−3 p2 ( lo2−p2 ) lo

−3

¿− π4

E a4 R−3 ( lo2−p2 )2lo

−3+2 π gs p+2 Z πRp }=0 (17)

By rearranging the terms by multiplying above equations, respectively, by 2 p and R,

and subtraction, the following equation may be obtained:

(18)

After rearranging and normalizing the terms with respect to the contour length, l0

the following equation may be written:

49

(19)

Here, one has an additional mathematical connection or constrain in term of initial

state variables, namely Vo = , which will be used to eliminate R from

Eq. (19) to get an explicit expression that involves the instantaneous value of the

total pitch height as the only independent variable. Where, and , respectively,

are the initial radius and total pitch height of the helix. In above normalized

expression, the unusual and usual torque terms are collected in the first term, which

involves a cofactor that shows their total contributions. Finally, the connection

between the Shear and Young modulus, in isotropic solid may be

used in the second term to obtain following expression, which involves normalized

constant volume explicitly rather than initial cage radius:

(20)

50

Fig. 12. The generalized force versus pitch height profile is presented for the constant volume constraint. The intersection with zero line that is marked by diamond dot, corresponds to the optimal configuration under the volume constraint, which yields the position of the extremal or ‘maximum’ point in the global Helmholtz free energy

function as. . Data: G=13.6 MPa, gs=10 erg/cm2, and

It is also clear that the expression on the left-hand side of Eq. (19) is nothing

but the gradient of the global Helmholtz free energy in the manifold of , which

also corresponds to the negative generalized thermodynamics force. The global

Helmholtz free energy takes the following expression for the helical conformation in

the case of constant cage volume constrain, , where the total pitch

height becomes the only independent variable. Then one reads:

51

(21)

Under the present constrain, the negative generalized force may take the following

form:

(22)

The rate of change in the total pitch height with respect to time may be derived by

using the connection between the rate of entropy production and the rate of global

Helmholtz free energy variation under the constant volume.[78]

(23)

Where, the equal sign corresponds to the isothermal reversible isochoric process.

Using the Onsager’s rule [91,91] for the decomposition of the generalized forces and

conjugated fluxes, one may obtain:

(24)

52

After rearranging the terms, one may further write the connection between the

generalized force and the conjugate flux, which is nothing but the rate of unfolding

of the alpha-polypeptide.

(25)

Equally well, one may also replace constant volume Vo by , and then gets the

following expression after some arithmetic operations; where is the initial total

pitch height of the alpha peptide, and is initial helix radius:

(26)

That may be further reduces to the following normalized forms with respect to the

initial arc length, and the relaxation time designated by ,

where the bar sign over the letters indicates normalization with respect to space and

time, respectively.

(27)

Above expression can be integrated to obtain a relationship, which gives a

connection between aging time and the instantaneous value of the total pitch height

53

as showed in Fig.13 by comparing other aging processes going on under the different

constraints.

(28)

Fig. 13. Aging behavior of alpha-peptide 3.613 under the constant volume (red line), radius (blue line), and total torsion (green line), are given with respect to their own relaxation times. Here, it has been assumed that no interfacial interaction and the applied hydrostatic pressure do exist.

Where and are scaled initial pitch height and time, respectively.

The relaxation time defined previously is equally well may be given in term of

54

volume constraint format explicitly; for the morphological

evolution of the helical conformation of alpha-peptide in the units of seconds. This

expression may be compared with the relaxation time obtained for the constraint

helical radius R case that is given in [40]. In order to compare

these three different constrains on the aging kinetics, one should employ the same

relaxation time for the scaling procedure in the graphical demonstration. This can be

done by multiplying the constant volume and radius expressions designated as Eq.

(11) and Eq.(26), respective, by{τV /τ R=4 po/ lo }≅ 0.56 and

{τΘ/ τR=[2 lo2 /Θ2 Ro

2 ]}≅ 2.104 to get the proper numerical results based on the common

scaling denominator, namely .

55

Fig. 14. Aging behavior of alpha-peptide 3.613 under the constant volume (green line), radius (blue line) and total torsion (red line), are given with respect to the relaxation time of the constant radius constrain, R. It has been assumed that no interfacial interaction and the applied hydrostatic pressure do exist.

In Fig. 14, the normalized total pitch heights obtained under the various

constrains at the absence of the interfacial and applied pressure contributions are

presented using the relaxation time of the constant radius after multiplying their

normalized elapsed times by the calculated scaling factors, namely;

0.56 × t̄ V and 2.104 × t̄ Θ. These plots given in Fig. 14 show very clearly that the

constant theta and radius constraint graphs are overlapping even though they may be

represented by different mathematical expressions rigorously, such as:

and (29)

56

Above expressions are exact mathematical forms obtained after some legitimate

rearrangements, which show that the cofactor between the curly braces in the second

equation doesn’t produce any observable difference in the semi-logarithmic plot used

in Fig.14, since the real life time for the radius constrain case is , which is very

close the scaling time calculated and used in that graphical demonstration.

On the hand, the constant volume constrain case has much sluggish aging behavior

especially in the asymptotic region, which amounts to one order of magnitude

enhancement in the delay time for the complete decimation. This is due to the

negative contributions of the bending and usual torsion terms to the generalized force

and positive contribution to the elapsed time as discussed previously dealing with

Eq.(12).

The volumetric constraint relationship designated by Eq. (28) can be integrated

exactly at the absence of the bending and interfacial free energy terms. This

isothermal isochoric aging phenomenon may be represented by the following formula

in normalized and scaled space:

(29)

That gives, (30)

or equally well,

57

(31)

Fig. 15. Aging behavior of alpha-peptide 3.613 under the constant volume in red line is given with respect to the relaxation time τV. The left scale shows that the system at the relaxation time v=1, it has already spent 29.2 % of the total expectation life time compared to the genuine exponential decay kinetics, where it should be exp(-1) =38.2%.

In Fig.15. The unfolding aging trajectory of the helical conformation associated with

alpha-peptide 3.613 for the isochoric isothermal process at the absence of the interfacial

interaction term is plotted using above given explicit formula in Eq.(30).

2d. Unfolding and stretching mechanics of alpha-polypeptide under the

constant volume constraint: Isothermal isochoric reversible process

58

In this section, the equality between infinitesimal variations in the global

Helmholtz free energy and the work done on the system is used by referring the

expression given in Eq.(15) by assuming that the cage volume is invariant quantity:

(32)

The above expression is written in such a way that the total pitch height as well as

the cage radius are assumed to be independent variables, and the total polar angle

= is automatically adjust itself. One may use the constraint on the

cage volume, to eliminate the independent variable by a simple replacement

designated by , and then the following format given previously by Eq. (20)

may be obtained. Where the total pitch height appears to be the only independent

variable:

(33)

59

Similarly, the total polar angle may be easily expressed in terms of the normalized

temporal and initial values of the total pitch heights in the presence the

normalized initial radius of the polypeptide cage:27

Θ( p)=( π (lo2−p2 ) pV o

)1/2

⇒ ( (1− p̄2 ) p̄R̄o

2 p̄o)

1 /2

and R(x; po)=¿¿ (34)

Here, one has to make an important remark that the stored elastic energy

associated with the unusual torsion term in Eq. (33) can’t be omitted or neglected

completely even though during the extension and stretching experiments. That means

the C-end terminal of polypeptide [ssDNA] might be free for twisting or rotation

during the testing as it is the case for optical [43] or magnetic tweezers experiments.

[92 ] As shown previously, this unusual torsion energy is about five to six orders of

magnitude larger than the other terms for the constraint twist angle case where both

end terminals of the polypeptide molecule are hold firmly to avoid rotation as

discussed previously. In the present case, it is assumed that the free C-end terminal of

the polypeptide directly attached to the magnetized bead or it is optically trapped that

allows it to be free to twist in the case of laser tweezer experiments. [43,93] In another

words, it is pulled by the magnetic field of the tweezer probe without exercising any

constraint on the possible twisting motion of the bead [79].

By differentiating the global Helmholtz free energy with respect to the total

pitch height, one may obtain the following expression for the response force

27 Twist free C-end terminal boundary condition under the constant volume case is not proper for dsDNA if one assumes that intra strands are stiff therefore one has to keep the radius of the cage not to vary during the pulling procedure because that violates the constant volume hypothesis.

60

associated with the polypeptide having initial total pitch height of .

Here, the extension is denoted as , which is recorded under the steady state pulling

speed (i.e., constant strain rate) applied to the twist Θ free C-end terminal of a single

molecule of polypeptide (ssDNA).

(35)

Fig. 16. Polypeptide having a positive interfacial free energy density is unfolding

under the effect of the stored torsional and bending stresses by the applied constant

pulling rate. The force- extension profile is interrupted by the flat zero force section

before reaching the full arc length to skip the irreversible spontaneous unfolding stage.

This unfolding stage is followed up by the uniform stretching of the backbone skeleton

assuming that this final stage is controlled by the strong nonlinearity. The torques curve

shows the elastic energy part, which may be combined with the straight line (i.e.,

missing) associated with the interfacial free energy contribution to produce the global

61

Helmholtz free energy profile during the unfolding stage. Data:

, Unfolding: and gs=+5 erg/cm2 ;

In Fig. 16, the unfolding energy and the associated the pulling force profile are

plotted using the data furnished for the poly-L-glutamic acid-chain against

normalized extension defined as , by using Eq. (34) and Eq.(35). Where,

the combination of torsion (unusual & usual) terms in addition to the bending and the

interfacial free energy contributions are employed. As can be seen from this figure,

this peculiar system shows twist-winding while responding to the pulling action

of the test machine with a negative response force starting from the initial state

up to, and included the critical point designated by ,

where it becomes zero. The critical point corresponds to the maxima on the global

free energy profile given by red solid curve in Fig. 16. The elastic part of the global

free energy is given by the torques line in this figure, which clearly shows that

contribution of interfacial free energy is very large positive shift and positive

inclination of the global free energy. This contribution on the response force is

almost uniform during the unfolding, and it varies from 9.24pN down to 5.05pN. An

increase in the total twist angle or the ‘’winding’’ is also taking place initially

(=11.9 Rd, that amounts to about two extra helical turns) up to the inflection point

on the global free energy profile, which almost coincide with the maximum on the

response force. Then the twist angle starts to unwind down to zero, which

indicates that polypeptide is complete in the unfolding state having no torsion and

bending left. The negative responds force opposes to the stretching initially because 62

of the up-hill climbing, and then the system moves down-hill spontaneously without

showing any resistance to the pulling of the test machine, which moves with a

constant speed. That is ‘’the strain relaxation’’ sector, where initially stored elastic

energy is completely released, only the interfacial free

energy associated with the backbone skeleton remains with some increase

5.98 eV↗10.58 eV due to enlargement in the surface area of the cage volume. 28

Therefore, there is no way of tracking down this positive response force by the static

stretching testing, but rather one might detect sudden unwinding with a great loss of

power even in the absence of external driving force (see: zero blue thin solid line in

Fig.16). Although, the kinetics studies may be performed similar to the previous

section as shown in Fig. (9). Namely by omitting the unusual torsion term

represented by unity “1” in the argument of the following expression

, one remains with a term that involves only the

combination of the bending and usual torsion terms, which tells us that the total pitch

height blows up with time. Similar phenomenon but different context is also treated

by Marko [47] under the title of “overstretching with freely fluctuating twist”,

which produces a well-defined plateau in F-E plot.

28 Calculated Initial stored elastic energy is about Ee =+0.33 eV, and interfacial energy is Ef =5.66 eV. The net initial energy is ET =5.98 eV. At the end of the complete unfolding Ee =0 and Ef

=10.58 eV. The total energy lost from the grand system is about - 5.25 eV. For a reversible

isothermal process this would be equal to the work done by the test machine on the system. Actually, the unfolding energy of the amino acid bone structure is 0.33 eV and the rest are spent as an interfacial free energy during the increase in the cage surface area.

63

Fig. 17. Polypeptide having no-interfacial free energy is unfolding in the presence of the stored torsional and bending elastic energies pulled under the uniform strain rate. This stage is followed up by the uniform stretching of the backbone skeleton assuming that this final stage is controlled by the strong fourth order elastic nonlinearity. The elastic region is intentionally aggregated by taking very high value for the Young modulus of elasticity.

In Fig. 17, we kept all the internal and external parameters of (Glu)n –Cyr test

piece used in Fig.16, only avoid the existence of the interfacial free energy

contribution. That is not only affected the maximum strength of the Global free

energy profile by reducing it drastically from 12.4 eV down ¿↘2.80 eV .because of

the lag of positive contribution of the interfacial free energy but also diminished the

whole stored folding energy almost to nil due to disappearance of the torsion and

bending deformations completely. There is also a slight shift in the energy maxima

towards the smaller pitch height sector, and the peak value of the pulling force from

64pN down to 31.5 pN. This difference 32.5 eV in the pulling force has spent

previously not only for to overcome the interfacial energy contribution to the

64

potential energy barrier but also against the uniform capillary forces arising due to

enlargement of the surface area of the cage volume.

In Fig.16 and Fig. 17, the force and extension behavior of the amino acid

backbone structure, which is completely striped from the torsional and bending

deformation used in the previous unfolding testing stages, are described. Where, the

Young modulus of elasticity E=3GPa obtained by Idiris et al [1] is employed to

simulate the elastic stretching sector in order to simulate the observed plateau region

by Cluzel et al. [42] and Smith et al. [43] in the force extension plot, [See: Fig. 1 in

Ref:47]. It is clear from Fig. 17 that this nonlinearity is not enough to generate the

plateau region but the fourth degrees polynomial used here extents the flat force

region that started during the last stage of unfolding up to 1.2 of the full arc length of

polypeptide, having free fluctuating twist constraint on the C-terminal. The

following relations are employed to simulate the elastic stretching region of single

strandant polypeptides, which includes contribution from the surface free energy

term:

(36)

And

(37)

Where is Heaviside step function, and the last terms added in Eq. (36)

and Eq. (37) are due to the interfacial Helmholtz free energy and its derivative

associated with the interaction of the amino acid backbone structure with the

surrounding aqueous solution, assuming that its volume doesn’t

65

change with the stretching (ν=1/2). Marko [48] while working on the overstretching

region with the free fluctuating twist constraint, realized that the

quadratic elasticity (Hooke’s Law) applied to the backbone structure of the

helical polypeptide doesn’t produce any plateau region in the force extension curve

beyond the normal unfolding domain. Then, he decided that the essential term

needed to generate the observed plateau is to include an additional potential energy

term , which appears in the Fokker-Planck equation as an undefined

potential, assuming that in concert with constraint , provides a barrier over

which the extension denoted as u must pass to reach the overstretching state.

Actually, this cubic term doesn’t give him desired plateau but six degrees polynomial

could.

Fig. 18. The helical conformation of polypeptide is unfolding under effects of the stored elastic bending energy Eb =9.55x10-6 eV, and the interfacial free energy Ef =1.18 eV term with the absence of the torsion. The bending energy contribution alone is drawn by the dotted blue line that has a mean amplitude of 10 -11 pN, which illustrates that its contribution is practically zero. The interfacial free energy increases up to 2.20 eV even though the radius shows decrease from 6Ao down to 3.16Ao.

66

In Fig.18, only the bending and interfacial free energy terms in Eq. (33) and

Eq. (35) are used for the unfolding extension sector. The backbone stretching sector

is exactly the same plot presented in Fig. 17. The numerical analysis using above

equations shows that at the start of the extension test, the interfacial free energy

content of the polypeptide is 1.178 eV, which is about five orders of magnitude

greater than the bending elastic energy term that found to be about 9.552 x 10 -6 eV,

assuming one has E=3MPa and gf =5 erg/cm-2. That immediately shows that any

theory, which relies on the bending term alone in order to produce reasonable

positive response force in the range of 4 pN and higher for the initial extension, it

should have enormously large Young modulus of elasticity in the range of E=60

GPa, which is 1.5x103 time larger than the Young modulus E= (3x13.6) MPa have

used for the bending term in the energetic discussions of (Glu)n -Cys polypeptide at

pH 3 in Fig.4. Therefore, it turns out again that without having unusual torque and/or

interfacial energy terms one needs enormously high elastic constants or rigidities to

explain the experimental observations obtained on the mechanical F-E test,

quantitatively. According to the demonstration in Fig. 18, the pulling force shows

monotonic decrease from 9.39 pN down to 5 pN at the end of the full unfolding. This

force almost due to the response force associated with the steady enlargement of the

interfacial area between embedding aqua solution and the cage volume, even though

its radius is decreasing to keep the cage volume constant during the unfolding.

Actually, one can’t have bending deformation in the helical shaping without

the torsion unless the wire or the single stranded polypeptide before winding around

the cylindrical surface is exposed to an inverse torsion as a straight line to

compensate the subsequent helical torsion generated naturally during the folding.

67

This is possible in principle if one considers the following elastic energy expression

[8], for the generalized bending and torsional deformations,

. Where α=Eπ a4 /4 and β=Gπ a4/2 are

respectively bending and usual torsional rigidities, and are the additional

bending and torsional deformations applied to the back-bone structure of polypeptide

before the formation of the helical conformation. Here, the pure bending and torsion

created during the formation of helix at the initial state denoted by may be

given, respectively, by following formula and

, from which total polar angle can be calculated from the

connection , assuming further that the arc length is invariant.

During the unfolding procedure, bi-product varies with the temporal value of

the total pitch height . Therefore, the kinetic behavior of the system depends

upon what parameter can be assumed to be constrained internally or externally

during the testing. It is natural to speculate that the helical radius R stays constant for

the double strandant dsDNA, because of the very high rigidities associated with the

interlinks connecting the base pairs. This procedure at the end of the unfolding

process results complete unfolding without torsion and bending. On the hand, in the

case of single strandant ssDNA, one keeps the total polar angle as a constant by

applying mechanical constrains on the N-end and C-End terminals.

68

Fig. 19. Unfolding behavior of the helical conformation of polypeptide [ssDNA], which is exposed to the compensating torsion of nine turns with a total pitch height 10nm before folding. The stored elastic bending energy Eb =1.9.55x10-5 eV, and interfacial free energy of the cage is Ef =1.165 eV, and initially it has zero net torsional energy. The peak values of the force and the global free energy are 19.3 pN and 3.23 eV, respectively. Solid torques line presents the bending plus interfacial free energy at absence of the torsion, assuming that the initial total polar angle δΘ=0 is constrained.

In Fig. 19 we have kept the total polar angle =0 constant by keeping the N-

End and C-End terminals fully clamp against twisting. In the constant total angle

case , the variable helical radius may be obtained above given connection to get

necessary expressions for the temporal values of the bending and torsion. During the

unfolding and stay constant but the bending as well

as the torsion are changing with the extension of the total pitch height

. For the present demonstration of the hypothetical case as presented in Fig. 19,

one takes the additional bending term as zero , and then selects the inverse

69

torsion properly just to compensate the torsion generated during

helical forming operation for given initial pitch height , cylindrical radius

and the total arc length of , namely;

, where one finds . However, this

compensation is strictly valid for the initial state, while extension takes place torsion

shows linear increase, , and

after the incomplete unfolding state , the intrinsic temporal torsion takes a

new enhanced value given by . The net torsion left in the straight

‘’without bending’’ polypeptide would be . This theta constraint Θ

system mimics AFM force extension test applied to the polypeptide by Idiris et al.,

[1], which resulted a finite global energy of 4.12 eV instead of 2.12 eV appeared in

the present case at the end of the quasi-unfolding due to the nonvanishing torque.

This fact can be seen form the torques line in Fig. 19, which represents the bending

and interfacial terms in the global free energy landscape.

In Fig. 20, the results of another computer simulation are presented for Glun –

Cys polypeptide assuming that it mimics the double strandant dsDNA, and keeping

its cage radius invariant in the unfolding sector. The connection between the global

Helmholtz free energy change and the infinitesimal reversible work done on the

system may be given by the following equation Eq. (37) for the constant helical

radius . The expression between curly braces represents the global

Helmholtz free energy, which is plotted in Fig. 20 together with the elastic energy

70

associated with torsion and bending with respect to the normalized total pitch height

marked with sold red and light blue lines.

Fig. 20. Unfolding behavior of the helical conformation of double strandant polypeptide [dsDNA], which is constraint to keep its helix radius constant because of its stiff Intralinks between strands. The stored initial global free and elastic energies are Eg =1.507 eV, and Ee =0.319 eV, respectively. At fully unfolding stage elastic energies reduce to zero, only the interfacial energy stays enhanced up to 4.123 eV. The peak values of the response force reach to 29.76 pN at p= 24.5 nm.

The solid torques line shows the contribution of the interfacial free energy to the f-

global Helmholtz free energy having a density of gs=5 erg/cm2. That plot almost

shows linear dependence with the extension, and contributes to the very large portion

of the global Helmholtz free energy, which changes from 1.18 eV to 4.1 eV during

unfolding.

71

R=0 (37)

The unwinding of the total polar angle is illustrated by the green line in this plot. The

maximum value of the pulling force is about 29.75 pN, which shows steady drop

with the extension, and finally becomes negative that means spontaneous or

‘’irreversible’’ relaxation occurs in that region, which may not be detected by the

quasi-static testing. In the present treatment we haven’t considered the overstretching

regime where interlinks between the strand pairs are experienced slip type tearing,

which is indicated by the constant force regime, and 1.7-2 times stretching. [42,43]

III. DISCUSSION AND CONCLUSIONS

i. In Section 2.a, we have demonstrated that under the constant total twist angle Θ ,

one obtains rigorously a reversible work equation designated by Eq.(6), which

indicates that the elastic energy part of the Global Helmholtz free energy

behaves like a simple helical spring type of non-linear elasticity.

ii. This non-linear spring elasticity theory applied to the single (Glu)n - Cys (n=80)

poly-peptide chain tested by Idiris et al.[1] through the AFM force-extension

measurement method at the intermediate pHs level of 3.0 produced a best fit

with a shear modulus G=13.6MPa, and the interfacial specific Helmholtz free

energy density of -24 erg/cm2. This shear modulus, which is corrected for the

80%/ helicity is found to be 26.6 MPa. See: Fig. 1a.

72

iii. Idiris et al.[1] have estimated the Young’s modulus of helical Poly-L-glutamic

acid (PGA) from the slope of the F-E curve at pH 3, which is about 0.03 nN/nm

in the region of 5-10 nm extension. By assuming that the length of (Glu)n -Cys is

10 nm (n=80, helicity=80%), and the radius of the α-helix is 0.2 nm., they

obtained a value of E=3GPa for the Young modulus instead of ECorr.=2.39GPa,

which yields G=0.8 GPa if one takes the Poisson’s ratio of =½. See: Fig.1a &

Fig.5.

iv. Actually, their assumption on the radius of the amino acid residue is too large. If

one uses the following equation to get the Young’s modulus:

, then one finds E=17GPa. This value for Young’s

modulus is very close to the figure (E=14.63GPa) reported very recently for

crystalline polyethylene by Ruggiero et al.[77] using Terahertz Spectroscopy.

v. Young’s modulus calculations from the initial slopes of FE curves using

Hooke’s law, with and without the interfacial free energy contributions, yield

E=2.93GPa and E=6.03GPa, respectively, compared to the reported value of

3GPa at [pH3]. The actual shear modulus of the amino-acid skeleton

incorporated into the torsional rigidity decreases [13.6MPa 1.90 MPa]

linearly with a sharp knee at pH6, while the acidity of aqueous solutions is

changing step wise from [pH3] to [pH8]. This is also accompanied by the

linear decrease in the interfacial Helmholtz free energy density without any

knee. The calculated unfolding work for the full stretching varies from

51.25eV/molecule down to 7.13 eV/molecules, which is in excellent agreement

with the reported experimental AFM results.[1]. See: Fig. 4and Fig. 6.

73

vi. In Section 2b. aging kinetics of the single Glu)n - Cys (n=80) poly-peptide under

the constant twist angle Θ is investigated assuming that one has isothermal

isobaric irreversible processes. See: Eq.(11) & Eq.(12 Depending upon whether

the initial normalized pitch height is in the range of the certain critical values

such as p̄o ≡0.285 ≤ 1/√3, and p̄o ≡0.85 ≥√2 following scenarios might take

place: a) If one applies a positive or negative pressure and surface free

energy to the cage volume having the initial scaled pitch height below the

critical value of1/√3 , then it shows, respectively, a decrease or increase in

the life time of the alpha-polypeptides [See: Fig.7]. Especially, applications of

the moderate negative surface free energies not only stabilize the total pitch

heights towards the nonequilibrium stationary states having lower pitch values

but also towards the much higher levels above initial state for the large surface

free energy densities [See: Fig.9]. b) In the case of above the upper critical

pitch height p̄o ≡0.85 ≥√2 [See: Fig.8&Fig.10], any applications of the positive

pressures and surface free energies above the threshold levels of P23 GPa

and gs =1.00 erg/cm, the system shows dynamics instability towards the

complete unfolding regime, and the below of these critical values; the

system still shows aging behavior towards the complete decimation. c) The

effect of interfacial free energy density on the aging behavior, which is closely

correlated with the acidity ( pH-level) of the aqueous environment as

demonstrated in Fig. 4&6. found to be not so drastic as compared to the

hydrostatic pressures but still acidic environment acts as a stabilizer. See: Figs:

7-10

74

vii. In Section 2c. Evolution behavior of helical conformation based on the

constant volume is studied as an isothermal isochoric irreversible process under

the two special boundary conditions such as: the constant radius δR=0

[dsDNA] and constant total twist angle Θ=0 [ssDNA] constraints, Fig.13.

It is demonstrated that there are minor differences in radius and twist angle

constraints in Fig.14, but the volume constraint alone without any further

shape restrictions such as cylindrical cage model enhances the aging time of

the polypeptide, one orders of magnitude at least, and shows strong bias

towards the elderly populations as isochoric irreversible isothermal process.

viii. In section 2d. Unfolding and overstretching of alpha-polypeptide is studied

under the constant volume. Two special boundary conditions may be imposed

on the C-end terminal assuming that N-end terminal is fully clamped: a) the

twist free C-end terminal, Fig. 16, where the total twist angle and the cage

radius vary according to Eq. (34) to keep the volume invariant δV=0, which

mimics [dsDNA]. The second case is fully clamped C-end terminal that keeps

the total twist angle invariant Θ=0 [ssDNA] buy allows the case radius

vary while keeping the volume invariant, Fig. 19. Both cases not only show

either winding or unwinding depending on the clamp conditions, respectively,

during the unfolding but also the negative response force sectors, which

shouldn’t be observed by the usual quasi-statics stretching testing because they

belong to the dissipative irreversible process at the absence of the externally

applied work ‘zero force’. Unwinding behavior of the helical conformation of

double strandant polypeptide [dsDNA] in unclamp twist free state demonstrated

75

in Fig. 20, which is constraint to keep its helix radius constant because of its

stiff Intralinks between strands.

IV. ACKNOWLEDGMENTS

The author wishes to thank Professor Walter F. Schmidt of Agricultural

Research Service, USA, who gave not only valuable advice but also inspired us in

showing interest on the energetic and the macro-static stabilities of the helical

conformations of peptides. Thanks, are also due Professor Emre Ersin Ören of TOBB

University of Economic and Technology for his invaluable comments in the context

of this paper as well as for his help in producing computer-graphic forms of our

MatCad drawn figures. Professor of Oncu Akyildiz of Hittite his comments on the

variational formulation of this thermokinetics problem. This work is partially

supported by METU and “Turkish Scientific and Technological Research

Council, TUBİTAK” with a Grant No: 107M011.

V. DATA AVAILABILITY STATEMENT

Data for the experimental points, which are shown by black solid marks in Fig. 1,

Fig.3, Fig. 4, and blue and red solid diamond marks in Fig.6 of our present

manuscript, are extracted directly from the graphics as appeared in Fig. 5 and Fig.

6 of a published work by A. Idiris, M. Taufiq, A. Ikai,(2000), “Spring Mechanics of

76

α–Helical Polypeptide, Protein Eng. 13(11) : (2000) pp.763-770. This work is cited

as Reference [1] in our paper. The data that support the findings of this study are

openly available in:

https://www.google.com.tr/search?q=A.+Idiris,+M.+Taufiq,+A.+Ikai,+Spring+Mechanics+of+

%CE%B1%E2%80%93Helical+Polypeptide,+Protein+Eng.+13(11)+:+(2000)+pp.763-770.

APPENDIX

Marko [47] has some conceptual difficulties29 in recognizing the differences

between the extension or ‘’elongation’’ along the longitudinal direction of an elastic

wire. The elongation is defined by

.

Here, the pure strain tensor is given by , where is self-

conjugate dyadic, and is the Idemfactor or unit dyadic, is the displacement

vector, which are defined using the rectangular Euclidian coordinate system [94]. The

strain energy density function [95], which should appear in the global Helmholtz free

energy as follows; , where is the stress dyadic in

an isotropic elastic body, and are Lame’s coefficients of elasticity. The

29 Marko replaces the arc length with an ill-defined so-called internal coordinate , and then

presents the so-called axial ‘’longitudinal’’ strain by u= -1, where . That means

,which is nothing but the extension. Unfortunately, there is also

typographic error in the bending energy term. That should be replaced by due to the definition of curvature. and the whole bending energy should be multiplied by

because in the integration operation. 77

strain energy function may be put into the following format; ,

where is a Cauchy’s infinitesimal strain tensor, and the dilatation is given by

in rectangular coordinate system.30 The pure strain tensor appeared in

Marko’s [47] and others authors [29,30,39] treatments eliminates the appearance of

the unusual torque term, while trying to connect the stretching to the twist

deformation31, because of the fact that this can be revealed only by the skew

symmetric part of the deformation tensor

denoted as [39]. Marko’s treatment [47] of the twist has nothing to do even

with the usual torsional deformation concept since it doesn’t show any dependence

on the intrinsic torsion of the helical conformation.

According Yamakawa and Yoshizaki [7], who follows closely the treatment of

Landau and Lifshitz [96] on the bending and torsion in a helical elastic wire, can be

represented as by referring the localized coordinate system

at the contour, where is the unit tangent vector, and the others

denote the principal and binormal vectors, respectively. Where,

denotes the infinitesimal angular rotations applied,

respectively, along the corresponding directions. The variations in the three pod

30 This can be put into invariant form denoted as: , where

is the second invariant in terms of the deviatoric strain defined as

, where , is called bulk modulus.

31 , where small rotation is .78

vectors along the arc length can be defined as , which may be called

as the rate of angular displacements, and denoted as , where the

first two terms describe the bending deformations along the principal directions of

the moment of inertia of the non-circular cross section, and the third term is related to

the pure torsion. The elastic energy is given by

, for per unit length of wire. Here

and are the bending and torsional rigidities for the

circular cross section having a radius of , and they are related by

for the isotropic elastic solids. This intrinsic elastic

deformation state associated with the helical conformation may be described by

.

The most authors have overlooked to the fact that in the definition of

‘’torsion’’ as the rate of twist angle by Landau and Lifshitz [96,97], and

Funk [95], there is something missing, which is its real meaning in the differential

geometry of 3D curves. Namely, the ‘’intrinsic torsion’’ [98] involves a strong

dependence on the local inclination angle of the tangent vector with respect the

direction of the cylindrical z-axis [39] of the helical conformation.32 For a simple

helical spring form made out from a hypothetical polypeptide elastic bar with a

32 Some authors dealing with membranes are defining the inclination angle of helix with respect to the basal plane, which opposes the definition in differential geometry. [50]

,

79

diameter of 2a having circular cross sections the ‘intrinsic’’ torsion is given by

[98]. Where, is the radius of the helix, which satisfies

. Here is the total polar angle measured along

the full arc length, and is the total pitch height, which is the projection of the end-

to-end arc length along the generators of the helix in the folding plane:

, which also measures the instantaneous extensions during the

stretching testing [1]. Similarly, the curvature is defined by . All

these mathematical connections show that during the extension measurements not

only the byproduct of the total polar angle of helix and its radius , but also the

torsion and curvature of helical conformation are all changing, which prove that there

are always mutual coupling among them. Depending upon the boundary conditions

imposed at the N- and C- end terminal of polypeptide (i.e., clamped or free swinging

or swiveling) and the strength of the interlinking between double strands of (ds)DNA

whether they are stiff or soft, the force extension curves may exhibit completely

different behavior.

80

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LIST OF FIGURE CAPTIONS

Fig. 1. Unfolding work of stretching of an alpha-polypeptide ''(Glu)n –Cys'' (n=80, helicity is 80%)

is marked with black solid diamond dots with respect to the limited extension in the range of

12.5nm-25 nm, using reported values by Idiris et. al. [1] at pH 3.0 (See: ibid; Fig.5-6). The blue

solid lines represent change in the global Helmholtz free energy with extension in the total pitch

height of the helical conformation, using Eq.(5). Fig.1a and Fig.1b employ unusual and usual

torque terms, respectively. Solid green line in Fig.1b uses bending only. In Fig. 1a, it also

considers in addition the unusual torque. The solid red line represents unfolding work, having no

contribution from the bending, but involves, respectively, unusual and usual torque terms at the

absence of interfacial contributions.

Fig. 2. The changes in the cage radius and the associated variations in the interfacial Helmholtz free

energy are calculated at pH-3 and G=13.6 MPa from the master equation Eq. 5 using the best value

of gs=-24 erg/cm2 estimated in Fig. 1a. Double red and blue dots indicate the boundary of the

regions at which the backbone structure of the helical conformation reaches the straight circular bar

shape, having a cage radius of 1.5 Ao, and still involves the initial torsional deformation intact but

nil bending.

Fig. 3. Effects of the interfacial free energy density variations on the unfolding work are presented

for the positive and the negative specific volumetric densities, which show that the positive density

enhances the unfolding work, and the negative one decreases.

Fig. 4. The effect of the pH level on the unfolding work exhibits itself by the consecutive variations

in the interfacial free energy density and the shear modulus of the back-bone structure

Fig. 5. Theoretical force- extensions plots of Alpha-Polypeptide (Glu)n-Cyr (n=80), which is

exposed to aqueous solutions having various pH-levels are tested by AFM under the constant total

polar angle constraint. Where the shear modulus varies with pH according to the list given in Fig.

4 by taking Go=13.6MPa, The contour length 35 nm, = 9 turns.

Fig. 6. Effects of the pH level on the shear modulus and the interfacial Helmholtz free energy

density are presented for (Glu)n –Cys. (n=80,helicity 80%). Initial total pitch height is 10nm, the

total arc length is 35 nm. The shear modulus and İnterfacial free energy density are, respectively, Go

=13.6 MPa and gs =-24 erg/cm2 at pH3. DATA: fs(x) ={3.8x -36.5} erg/cm2, and G(x)/Go={-

0.225x+1.647}, where x=pH value.

Fig. 7. Aging behavior of polypeptide Glun-Cys (n=100) is demonstrated under the positive

[pH6 ] and negative [pH≤4] interfacial Gibbs free energy densities without of the applied stress.

Fig. 8. The aging kinetics of an alpha-polypeptide having an initial state 0.835, which is above the

critical total pitch height given by in the presence of the applied positive

pressures without interfacial free energy term.

Fig. 9. Aging kinetics of alpha-polypeptide having an initial state below the critical total pitch

height in the presence of the normalized applied positive and negative

pressures with respect to the shear modulus G=1GPA, without interfacial free energy term.

Fig. 10. Aging kinetics of alpha-polypeptide having an initial state above the critical total pitch

height for positive and negative interfacial free energy densities

normalized with respect to the critical interfacial free energy density: , where

=9.687x103 erg/cm2. Data: G=1GPa and =35nm., =9x2.

Fig. 11. Shift in the position of the Global Gibbs free energy maxima according to the variations in

the interfacial free energy densities while keeping the theta angle constant. The maxima are

designated by the generalized force versus zero-line intersections. The lower limit is set by

where gs .

Fig. 12. The generalized force versus pitch height profile is presented for the constant volume

constraint. The intersection with zero line that is marked by diamond dot, corresponds to the

optimal configuration under the volume constraint, which yields the position of the extremal or

‘maximum’ point in the global Helmholtz free energy function as. . Data: G=13.6

MPa, gs=10 erg/cm2, and

Fig. 13. Aging behavior of alpha-peptide 3.613 under the constant volume (red line), radius (blue

line), and total torsion (green line), are given with respect to their own relaxation times. Here, it has

been assumed that no interfacial interaction and the applied hydrostatic pressure do exist.

Fig. 14. Aging behavior of alpha-peptide 3.613 under the constant volume (green line), radius (blue

line) and total torsion (red line), are given with respect to the relaxation time of the constant radius

constrain, R. It has been assumed that no interfacial interaction and the applied hydrostatic

pressure do exist.

Fig. 15. Aging behavior of alpha-peptide 3.613 under the constant volume in red line is given with

respect to the relaxation time τV. The left scale shows that the system at the relaxation time v=1, it

has already spent 29.2 % of the total expectation life time compared to the genuine exponential

decay kinetics, where it should be exp(-1) =38.2%.

Fig. 16. Polypeptide having a positive interfacial free energy density is unfolding under the effect

of the stored torsional and bending stresses by the applied constant pulling rate. The force-

extension profile is interrupted by the flat zero force section before reaching the full arc length to

skip the irreversible spontaneous unfolding stage. This unfolding stage is followed up by the

uniform stretching of the backbone skeleton assuming that this final stage is controlled by the

strong nonlinearity. The torques curve shows the elastic energy part, which may be combined with

the straight line (i.e., missing) associated with the interfacial free energy contribution to produce

the global Helmholtz free energy profile during the unfolding stage. Data:

,Unfolding: and gs=+5 erg/cm2 ; lo=35nm , po=10 nm , a=0.75 Ao

Fig. 17. Polypeptide having no-interfacial free energy is unfolding in the presence of the stored

torsional and bending elastic energies pulled under the uniform strain rate. This stage is followed

up by the uniform stretching of the backbone skeleton assuming that this final stage is controlled

by the strong fourth order elastic nonlinearity. The elastic region is intentionally aggregated by

taking very high value for the Young modulus of elasticity.

Fig. 18. The helical conformation of polypeptide is unfolding under effects of the stored elastic

bending energy Eb =9.55x10-6 eV, and the interfacial free energy Ef =1.18 eV term with the absence

of the torsion. The bending energy contribution alone is drawn by the dotted blue line that has a

mean amplitude of 10-11 pN, which illustrates that its contribution is practically zero. The interfacial

free energy increases up to 2.20 eV even though the radius shows decrease from 6Ao down to

3.16Ao.

Fig. 19. Unfolding behavior of the helical conformation of polypeptide [ssDNA], which is exposed

to the compensating torsion of nine turns with a total pitch height 10nm before folding. The stored

elastic bending energy Eb =1.9.55x10-5 eV, and interfacial free energy of the cage is Ef =1.165 eV,

and initially it has zero net torsional energy. The peak values of the force and the global free

energy are 19.3 pN and 3.23 eV, respectively. Solid torques line presents the bending plus

interfacial free energy at absence of the torsion, assuming that the initial total polar angle δΘ=0 is

constrained.

Fig. 20. Unfolding behavior of the helical conformation of double strandant polypeptide [dsDNA],

which is constraint to keep its helix radius constant because of its stiff Intralinks between strands.

The stored initial global free and elastic energies are Eg =1.507 eV, and Ee =0.319 eV, respectively.

At fully unfolding stage elastic energies reduce to zero, only the interfacial energy stays enhanced

up to 4.123 eV. The peak values of the response force reach to 29.76 pN at p= 24.5 nm.