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Statistical mechanics of stretching of biopolymers

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J. Stat. Mech. (2011) P05019

(http://iopscience.iop.org/1742-5468/2011/05/P05019)

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J.Stat.M

ech.(2011)

P05019

ournal of Statistical Mechanics:J Theory and Experiment

Statistical mechanics of stretching ofbiopolymers

A R Singh1, D Giri1,2 and S Kumar1

1 Department of Physics, Banaras Hindu University, Varanasi 221 005, India2 Department of Applied Physics, Institute of Technology, Banaras HinduUniversity, Varanasi 221 005, IndiaE-mail: amitarj phy@rediffmail.com, dgiri.app@itbhu.ac.in andyashankit@yahoo.com (S Kumar)

Received 19 January 2011Accepted 26 April 2011Published 18 May 2011

Online at stacks.iop.org/JSTAT/2011/P05019doi:10.1088/1742-5468/2011/05/P05019

Abstract. We developed a simple model of polymers on a triangular latticeto study the force-induced transitions related to biopolymers. Using an exactenumeration technique, we calculate various thermodynamic quantities associatedwith it. We show here, by including different parameters, e.g. bending andparing interactions in the model system, that one can understand the qualitativedifferences in the force–extension curves exhibited by different biopolymers. Ourstudy also shows that the solvent plays an important role in the unfolding ofproteins.

Keywords: mechanical properties (DNA, RNA, membranes, bio-polymers)(theory), structures and conformations (theory), protein folding (theory), singlemolecule

c©2011 IOP Publishing Ltd and SISSA 1742-5468/11/P05019+17$33.00

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Contents

1. Introduction 2

2. Model and method 4

3. Pulling polysaccharides 5

4. Modeling of unfolding of homopolymer 9

5. Protein unfolding 12

6. Conclusions 15

Acknowledgments 16

References 16

1. Introduction

Polymers from living organisms are called biopolymers. Polysaccharides, proteins andnucleic acids are some examples of biopolymers, in which the repeating units (monomers)are either mono- or disaccharides, amino acids or nucleotides, respectively [1, 2].Biopolymers usually have well-defined structures and functions. This is because of themolecular interactions involved among different monomers, which make them differentthan the polymers. In recent years, with the advent of experimental techniques likeoptical tweezers, magnetic tweezers, atomic force microscopes, etc, efforts have been madeto measure the inter- and intra-molecular forces involved in the stability of biopolymericsystems [3]–[6]. These experiments have also enhanced our understanding about therole of these forces in determining the elastic, functional and structural properties ofbiomolecules [5], [7]–[9].

In such experiments, there is a large conformational change in the reaction coordinate(e.g. the end-to-end distance (x) of a biopolymer, number of native contacts (m), etc),which has been used to follow the progress of the reaction [4, 5, 10]. One of the notableaspects of the single-molecule experiments is that the end-to-end distance is directlymeasurable or controlled by the instrumentation. Therefore, x becomes a natural reactioncoordinate for describing the mechanical processes. Such processes generally have beenmodeled by a simple two-state model [11].

Different biomolecules under an applied force exhibit different kinds of transi-tions [12]–[16], e.g. the ‘folding–unfolding’ transition of proteins, the ‘stretching’ transitionof DNA, the ‘ball–string’ transition of a polymer, the ‘boat–chair’ transition in polysac-charides, etc. The force–extension curves for all these transitions differ significantly fromeach other, although the underlying physics remains the same [11]. The applied forcetilts the free energy surface along the reaction coordinate [5] by an amount linearly de-pendent on the reaction coordinate. This provides the important information about thefree energy landscape of the system in the form of the force–extension curve. Theoreticalefforts which followed these experiments have been studied in the framework of statistical

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mechanics and they were mostly based on simple models that are amenable to analyticaltreatment, such as freely jointed chain (FJC) or worm-like chain (WLC) models [8, 9, 17].

The major shortcomings of these models are that they ignore the excluded-volumeeffect in their description and have only one characteristic length scale known as thepersistence length or Kuhn length [18]. Moreover, at low temperature the excluded-volume effect becomes more prominent as the system prefers to be in the folded/compactstate. This has been seen in the case of titin, where saw-tooth-like behavior [18] in theforce–extension curve has been observed, which corresponds to the unfolding of differentmodules of domain. These curves are characterized by the same persistent length ofdifferent chain lengths. However, in the case of polysaccharides, the force–extension curvesshow a deviation from the entropic and Hookean mechanisms of the chain elasticity (FJCor WLC models). It was pointed out that the deviation is due to the conformationaltransition within each dextran monomer, where the C5–C6 bond of the sugar ring flipsinto a new conformation (chair–boat transition) and there is a 10% elongation in thelength. The force–extension curve also shows a plateau around 250 pN, which signifies aconformational change [16]. Thus, the experiment suggests that the simple WLC modelis not sufficient enough to describe the force–extension behavior as it involves only onelength scale (persistent length) in its description.

Lattice models, despite their simplicity, provide enhanced insights into the mechanisminvolved in the force-induced transitions, where a biopolymer is represented by a chainof monomers on a lattice [9], [19]–[21]. The bond lengths between monomers are keptconstant. This model can take care of the excluded-volume effect, and for the finitesize chain the model can be solved exactly. It is interesting to note that most of thetheoretical modeling of the force-induced transitions was confined either to the squarelattice (in two dimensions) or on a cubic lattice (in three dimension) [19]–[23]. Because ofthe finite coordination number of the lattice, the entropy of the chain is underestimatedcompared to that of the off-lattice model [24]. As a result, the force–extension curve forthe lattice models of a polymer differs significantly from the counter off-lattice model [25].Monte Carlo and molecular dynamics simulations have been used extensively to studysuch transitions [26]–[28]. It is to be noted here that generally the biological solvent is apoor solvent which corresponds to a low temperature. Such a system does not equilibrateeasily, and hence, in the computer simulations, reliable statistics are difficult to obtainat low temperature. Therefore, a complete phase diagram is not easy to obtain throughnumerical simulations. In this respect, the lattice models with a higher coordinationnumber may provide a better platform to study such transitions as they are close to theoff-lattice model as well as that the problem can be solved exactly for the finite chainlength [29]. Hence, despite their simplicity, it may enhance our understanding about themechanism involved in the force-induced transitions of biopolymers [9], [19]–[21].

In this paper, we consider self-avoiding walks (SAWs) to model a biopolymer chain ona triangular lattice. We used this lattice because all nearest neighbors are equivalent andthe respective coordination number (z = 6) ensures that the ground state conformation isclose to the one obtained on the counter off-lattice. We show that, by including suitablephysical parameters in the model, one can study different force–extension curves obtainedfor different biopolymers, namely dextran (polysaccharides), DNA and proteins. Thispaper is organized as follows: in section 2, we define the model and briefly discuss theexact enumeration method to calculate relevant thermodynamic quantities associated with

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(a)

(b)60o

120o

Click–I

Click–II

Figure 1. (a) Schematic representation of a polymer on a triangular lattice.(b) Representations of two kinds of bending on the lattice are shown: (i) 60◦ and(ii) 120◦.

the force-induced transitions. Section 3 reviews the stretching of polysaccharides. Inthis section, we also show that, by including the bending rigidity in the model, one canobtain the force–extension curve, which is qualitatively similar to the one obtained forthe polysaccharides [16]. Section 4 dwells on the issues related to the solvent and itsimpact on the statistics of polymers. The force–extension behavior of a homopolymeris calculated by adding an attractive interaction between non-bonded nearest-neighbormonomers. Unlike the square or cubic lattice, where some of the pairwise interactions arenot possible because of the lattice constraint, on the triangular lattice, all such interactionsmay be included in the description of the model [30]. In section 5, we extend the methoddeveloped in section 4 to study the protein unfolding. Following the model developed byDill et al [31], we model a protein chain by a sequence of hydrophobic (H) and hydrophilic(P) monomers. We compare the force–extension curve for this model with and withoutthe solvent. The paper ends with a brief discussion in section 6.

2. Model and method

In general, the choice of lattice depends very much on numerical convenience as wellas the interaction involved among the monomers. We consider here a biopolymer chainconsisting of N monomers connected through the covalent bonds on a triangular lattice(figure 1). In the numerical simulation (Monte Carlo or molecular dynamics) [32], a largenumber of conformations are generated by repositioning of the monomers, while keepingthe covalent bonds intact. Here, we generate all possible conformations of a chain so that

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the partition functions can be calculated exactly. The canonical partition function of apolymer chain of length N can be written as

ZN =∑

N1,N2,m,x

C(N1, N2, m, x)bN11 bN2

2 ωmux. (1)

On a triangular lattice, the polymer chain can have two types of bends, say click-I (60◦) and click-II (120◦) as shown in figure 1(b). We associate energies εb1 and εb2

with click-I and click-II, respectively. N1 and N2 are the numbers of click-I and click-II in a given conformation of the chain. The Boltzmann weights for click-I and click-IIare b1(= exp(βεb1)) and b2(= exp(βεb2)), respectively. ω (= exp(−βε)) is the Boltzmannweight for the attractive interaction among non-bonded nearest-neighbor monomers and

m is the total number of such pairs in that conformation. u (= exp[β(�f · x)]) (x = unitvector along the x axis) is the Boltzmann weight associated with the force which is appliedat one end, keeping the other end fixed. Here, β = 1/kBT , kB is the Boltzmann constantand T is the temperature of the system. C(N1, N2, m, x) is the number of distinct walksof length N having m numbers of pairs, whose end points are at a distance x apart. Itmay be noted that, for the finite length of a polymer chain, the density of states (C) canbe evaluated exactly and hence all the results presented here are exact. We have obtainedC(N1, N2, m, x) for N ≤ 25 monomers (almost double that in the previous studies [33] inthis context) and analyzed the partition function.

Quantities of interest like the average reaction coordinate (〈x〉) and average number ofnon-bonded nearest-neighbor pairs (〈m〉) can be calculated from the following expressionsby setting moment, k = 1:

〈x〉 =

∑N1,N2,m,x xC(N1, N2, m, x)ωmux

∑N1,N2,m,x C(N1, N2, m, x)ωmux

(2)

and

〈mk〉 =

∑N1,N2,m,x mkC(N1, N2, m, x)ωmux

∑N1,N2,m,x C(N1, N2, m, x)ωmux

. (3)

Since in a system of finite length no true phase transition takes place, we calculatea sudden change in the energy to identify the transition [10]. For this, we monitor thefluctuation in the number of nearest-neighbor contacts, which can be obtained by settingk = 2 in the following equation:

χ = 〈mk〉 − 〈m〉k. (4)

If the peak height in the fluctuation curve increases with N , it may be treated as asignature of the phase transition. This quantity is also related to the specific heat of thesystem as Cv = χ/T 2. In the following, we set kB = 1 and work in the reduced units.

3. Pulling polysaccharides

Polysaccharides are one of the key biomolecules and have widespread mechanical functionsin biology. Since polysaccharides do not fold or form a double helix, it was earlier thoughtthat polysaccharides may not offer the same exciting opportunities as nucleotides, whose

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(b)

(a)

(c)

Figure 2. (a) Schematic representation of polysaccharide structure, (b) two formsof chair and (c) boat form (causes steric crowding).

duplex can be unzipped, or proteins, which can be mechanically unfolded. However, theyexhibit their own richness in terms of the force-induced conformational transitions asdiscussed above. It was noted that bonds flip on a faster time scale than the experiment,and hence stretching is identified as an equilibrium process [16]. Although a large numberof polysaccharides have been stretched in recent years, the vast majority of studies werebased on homopolymers. There is a debate in recent years on whether the conformationalchanges that occur are a ‘chair–boat’ transition (figure 2) or a ‘helix–sheet’ transition [34].Moreover, it was also revealed in the earlier studies [16] that there may be one, two orno conformational transformations in polysaccharides. For example, amylose (1a-4e) anddermochondan sulfate (1a-4e) have one conformational transformation [35], whereas pectin(1a-4a) undergoes at least two conformational transformations [36]. However, cellulose(1e-4e) does not exhibit such transformations [35] under a force.

Based on the thermodynamics of an elastically coupled two-level system, Rief et al[18] have calculated the probability for a transition and a related change in the lengthof each segment. Using computer simulation, they reproduced the force–extension curve,which shows a kink around 700 pN. Haverkamp et al [35]–[37] developed a model toinclude another conformational transformation of the glycan rings, which is based on theconcept of equilibrium between the two (referred to as ‘clicked’ and ‘unclicked’) states.The equilibration is determined by the Gibbs energy difference between the states and it isperturbed in favor of the clicked states by the force applied to the molecules. The partitionfunction defined in equation (1) may be able to describe the ‘chair–boat’ transition, if weset an attractive interaction parameter equal to zero, i.e. ε = 0. The partition function

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f

f

f

f

(c)

(b)

(a)

(d)

Figure 3. Schematic representation of a polymer on a triangular lattice underthe application of force: (a) the coil form, (b) the stretched form with 60◦ bend,(c) the stretched form with 120◦ bend and (d) the fully stretched conformation.

(equation (1)) takes the form

ZN =∑

N1,N2,x

C(N1, N2, x)bN11 bN2

2 ux. (5)

A schematic representation of some of the conformations in the presence of the appliedforce, which a polymer chain on a triangular lattice may exhibit, is shown in figure 3. Athigh temperature or in the absence of the bending interactions (i.e. for b1 = b2 = 1),the chain will be in the high entropic state and it can be well represented by SAWs(figure 3(a)). The force–extension curve shown in figure 4(a) represents the entropicresponse. In the presence of the bending interaction at low temperature, the systemattains the low entropy state. We consider two limiting cases of the bending energy:(i) εb1 �= 0 and εb2 = 0, and (ii) εb1 = 0 and εb2 �= 0. The corresponding conformations areshown in figures 3(b) and (c). If b1 > b2, the polymer chain will acquire the conformationshown in figure 3(b), while for b2 > b1 it will acquire the conformation shown in figure 3(c).

In this model, it is possible to go from click-I to click-II under the application of aforce, but the converse is not true. It is interesting to see that the simple model presentedhere captures the features of the force–extension curve of the polysaccharides quite nicely.Moreover, by varying the bending energies εb1 and εb2 , it is possible to generate the force–extension curves for a range of model polysaccharides [16] in which there may be one,two or no conformational transformations. In figure 4(b), we show the variation of theforce–extension curves for some values of εb2 , keeping εb1 constant. One can notice that,up to a certain force, the force–extension curve and position of the kink remain the same.

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0.2 0.4 0.6 0.8 1.0<X>

0.0

1.0

2.0

3.0

f

εb1

= 0.50, εb2

= 0.00

εb1

= 0.00, εb2

= 0.50

εb1

= 0.00, εb2

= 0.00

0.2 0.4 0.6 0.8 1.0<X>

0.0

2.0

4.0

6.0

f

εb1

= 0.55, εb2

= 0.388

εb1

= 0.55, εb2

= 0.488

εb1

= 0.55, εb2

= 0.588

(a)

(b)

T = 0.20

T = 0.20

Figure 4. (a) Force versus average extension diagram at low temperature(T = 0.2) for different combinations of bending energies: (i) dotted line showsthe entropic response, while (ii) dashed and (iii) solid lines show the kinkcorresponding to the conformational transition from click-I to click-II. (b) Forceversus average extension diagram at low temperature (T = 0.2) at fixed εb1 =0.55. Different values of εb2 show the variation in the force–extension curve,which is qualitatively similar to the one obtained for different polysaccharides.In the constant force ensemble, the x axis is for the force and the y axis is forthe extension. However, in experiments, it is usually reported the other way. Forthe sake of comparison with the experiment [16], we plot extension on the x axisand corresponding force on the y axis.

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The force–extension curves thus obtained are qualitatively similar to the ones reportedfor different polysaccharides [16].

4. Modeling of unfolding of homopolymer

A long flexible polymer chain can collapse to a compact globule state by lowering thetemperature. Such a transition occurs at the θ temperature and is referred to as a coil–globule transition [38, 39]. Away from the θ temperature, a polymer chain can exist eitherin a collapsed or a swollen state. The mean square radius of gyration 〈R2

g〉 scales with

the chain length N as 〈R2g〉 ∼ bN2ν , where b and ν are the bond length and the critical

exponent, respectively. At low temperatures, when the polymer is in the collapsed state,ν = 1/d, while at high temperature ν is given by the Flory formula [38]: ν = 3/(2 + d).These values are believed to be exact for d = 1, 2 and 4 with a logarithmic correction. Ford = 3 the (approximate) numerical estimate is obtained from a variety of series expansion,Monte Carlo and field theory methods [38]–[41], which are in agreement with the Floryformula within the error bars. It should be emphasized here that a polymer chain cannotacquire the conformation of a fully stretched state (characterized by ν = 1) by varying thetemperature alone. However, application of the force may drive the polymer to achievethis stretched state. The applied force not only tilts the free energy barrier, but alsointroduces a new stretched state, which is not otherwise accessible [10].

Several attempts have been made to understand the force-induced unfolding ofhomopolymers in recent years [7, 10], [42]–[46]. However, most of the studies were eitherconfined to the simple models, e.g. WLC model and FJC model, or the exact resultsbased on the square and cubic lattices [9, 10, 19] for a chain of small length. Effortshave also been made to understand the unfolding of homopolymer from the Monte Carloand molecular dynamics simulations, etc [26]–[28, 46]. These approaches can be, andhave been, modified to include the excluded-volume effects, polymer-specific energeticsand higher order energetics, e.g. rotational energy, etc [47]. However, we focus ourattention particularly on the low-temperature regime, which is difficult to access eitherexperimentally or by other numerical techniques, such as Monte Carlo or moleculardynamics methods.

The force–extension behavior of the homopolymer is calculated by adding anattractive interaction between the non-bonded nearest-neighbor monomers. Unlike thesquare or cubic lattice, where some of the pairwise interactions are not possible becauseof the lattice constraints, it is possible to include all such interactions in the description ofthe triangular lattice model. A complete phase diagram can be probed by using an exactenumerations analysis of finite chains. We consider a flexible polymer chain by settingb1 = b2 = 1 in equation (1) and calculate the thermodynamic variables using equations (2)and (3).

In figure 5, we show the variation of the reaction coordinate with force at differenttemperatures. For the sake of comparison, we also show the response of the applied forceon a polymer chain of the same length on the square lattice. It is evident from figure 5that the 〈x〉 versus f curves for these two cases are distinctly different. One can noticea shift in the magnitude of the force towards the higher side for the triangular latticein comparison to that of the square lattice. This shift arises because of a large numberof nearest-neighbor contacts, which exist on a triangular lattice as compared to a square

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0.0 1.0 2.0 3.0f

0.2

0.4

0.6

0.8

1.0E

xten

sion

(<x

>)

TriangularSquare

0.0 1.0 2.0 3.0f

0.0

0.2

0.4

0.6

0.8

1.0

Ext

ensi

on (

<x>)

TriangularSquare

T = 0.1

T = 1.0

(a)

(b)

Figure 5. Response of the applied force on a flexible homopolymer chain on atriangular lattice: (a) low (T = 0.1) and (b) high (T = 1.0) temperature. Wealso plot the force–extension curve for the square lattice for comparison.

lattice. Moreover, there are more numbers of plateaus in the case of triangular latticesat low temperature than in the case of square lattices. This indicates that the existenceof an intermediate state induced by the force will get more prominence in the off-latticemodel [9, 25]. A minimal model adopted here on a triangular lattice exhibits the plateausin the force–extension curve, which correspond to the ball transforms to an ellipse andthen finally to the string under the applied force [15].

In figures 6(a) and (b), we plot the average number of non-bonded nearest neighbors(〈m〉) with temperature for different values of the force. We find that 〈m〉 decreases muchfaster on the square lattice than on the triangular lattice with temperature. This is becauseof the fact that the probability of the formation of contacts on the triangular lattice ismuch higher than on the square lattice. In fact, on the triangular lattice, every step givestwo contacts out of five (or less) possibilities, whereas on a square lattice one requires atleast two steps to make two contacts out of nine (or less) possibilities. Figures 6(c) and (d)show the variation of χ with temperature for triangular and square lattices, respectively.

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0.0 1.0 2.0 3.0T

0.2

0.4

0.6

0.8

1.0

<m>

f = 0.00f = 0.20f = 0.50

0.0 1.0 2.0 3.0T

0.2

0.4

0.6

0.8

1.0

<m>

f = 0.00f = 0.20f = 0.50

0.0 2.5 5.0 7.5 10.0T

0

5

10

15

20

25

30

χf = 0.00f = 0.20f = 0.50

0.0 2.5 5.0 7.5 10.0T

0

2

4

6

8

χ

f = 0.00f = 0.20f = 0.50

(a)

(b)

Triangular

Square

Triangular

Square

(c)

(d)

Figure 6. Variation of the average number of non-bonded nearest-neighborcontacts (〈m〉) with temperature at different forces. This is different from thebehavior of 〈m〉 on the square lattice which can be seen from (b). Correspondingfluctuation curves for the number of nearest-neighbor contacts (χ = 〈m2〉−〈m〉2)are shown in ((c) and (d)). The phase boundary in the force–temperature diagramshown in figure 7 has been obtained from the peak position of the fluctuationcurve for a given f and T .

The phase boundary in the force–temperature diagram of the triangular lattice has beenobtained from the peak of the fluctuation curves. The force–temperature diagram for thefolding and unfolding of the polymer is shown in figure 7. The critical force, which is ameasure of the stability of the polymer globule compared to the extended state, has beendetermined by using equation (4) as a function of temperature. The phase boundary inthe force–temperature diagram (figure 7) separates the regions where the polymer existsin a globule state from the region where it exists in the extended conformation. The model

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0.0 0.5 1.0 1.5 2.0 2.5 3.0T

0.0

0.5

1.0

1.5

2.0

fFolded

Unfolded

Figure 7. Force–temperature phase diagram for the flexible polymer chain.The phase boundary separates the globule and the extended state. At lowtemperature, re-entrance is visible.

also shows the existence of re-entrance at the low temperature. The origin of re-entrancecan be explained by using a phenomenological argument [9, 10] near T = 0. By equatingthe free energy of the folded state and the free energy of the stretched state due to theforce, one can calculate the critical force for the unfolding as fc(T ) ∼ −ε+(1/

√N)ε+TS,

where S is the entropy associated with each monomer. The dominant contribution to thefree energy is due to the non-bonded nearest-neighbor interactions. The second term isa surface correction term which vanishes in the thermodynamic limit. It should be notedthat for a flexible polymer chain the entropy associated with a globule is finite and hencea positive slope dfc/dT , at T → 0, confirms the existence of re-entrance. If one increasesthe length of the polymer, the re-entrance seen in the force–temperature diagram will bemore prominent [48].

5. Protein unfolding

The lattice-based models have been widely used to study the thermodynamic properties ofthe protein unfolding [31, 49, 50]. Although the models simplify the molecular complexityof the real proteins, it includes the basic features of the protein unfolding, e.g. nativeconformation (only one conformation with the minimum global energy) and a large numberof conformations for a given sequence (Levinthal paradox) [51]. Moreover, as discussed inthe preceding sections, the model permits us to extract the thermodynamic features of theunfolding. In this approach, it is possible to obtain the exact density of states, which isnot possible in the full atomistic or off-lattice simulations. A protein molecule is modeledon a triangular lattice, where each monomer corresponds to an amino acid residue. In thesimplified notation, the amino acids may be classified as hydrophobic (H) or hydrophilic(P). The values of different interactions of these molecules are shown in table 1 [52].

The major assumption behind the HP model is that the driving force behind theprotein folding is the hydrophobicity [31]. The hydrophobic acids are nonpolar and thusprefer the other neutral molecules and nonpolar solvents in their surroundings, whereasthe hydrophilic amino acids are polar in nature. When a protein is subjected to a solvent

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(b)

(a)

Figure 8. Two possible degenerate ground state structures of a given sequenceon a triangular lattice. The black and white circles correspond to hydrophobicand hydrophilic amino acids, respectively.

(S), like water, the hydrophobic monomers tend to form clusters to minimize their contactwith the water surface, while most of the polar (hydrophilic) monomers are exposed tothe water surface. The free energy is minimized when the maximum number of contactsbetween the non-bonded hydrophobic monomers is formed. Thus, in addition to HH,HP and PP pairs, there is a possibility of the formation of pairs of hydrophobic andhydrophilic amino acids with the solvent molecules as PS and HS, respectively. In orderto study this, we use a lattice Flory–Huggins model like [38], where lattice sites, whichare not occupied by the chain, are filled by the solvent molecules and these are labeledas solvent sites. In most of the studies, HP, PP, PS and HS interactions are set tozero [53]–[55]. However, there are recent studies which revealed that the solvents alsoplay a crucial role [56]. Here, we shall use the interaction matrix shown in table 1 tostudy the two cases, namely the HP model with and without a solvent, and compare theircorresponding force–extension curves. In order to avoid whether a sequence folds or not,we choose the following sequence: PPHPPHHPPPPHHPPPPHHPPPPHH [57], which isknown to fold. Two possible degenerate ground states of the chain in the presence ofsolvent molecules are shown in figure 8.

The force–extension curves for three different cases (with and without solvent) areshown in figure 9. For a given sequence, if the interaction between hydrophobic andpolar (HP) amino acids is set equal to zero, then the HP(1,0) model without solvent

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0.0 0.5 1.0 1.5 2.0 2.5 3.0f

0

5

10

15

20

25

30

Ext

ensi

on (

<x>)

εHH

= -4.0, εHP

= 0.5, εPP

= 0.0, εHS

= 1.0, εPS

= -0.5

εHH

= -4.0, εHP

= 0.5, εPP

= 0.0, εHS

= 0.0, εPS

= 0.0

εHH

= -4.0, εHP

= 0.0, εPP

= 0.0, εHS

= 0.0, εPS

= 0.0

T=0.1

Figure 9. Extension versus force curves with and without the solvent. Multi-step plateaus correspond to different intermediate states, which are visible at lowtemperature.

Table 1. Interaction matrix for the HP model with solvent.H P S

H −4.0 0.5 1.0P 0.5 0 −0.5S 1.0 −0.5 0

shows a multi-step transition indicating many intermediate states. It may be noted thatthere are many studies available in the literature using this approach [53]–[55]. However,if the hydrophobic polar interaction is set to be repulsive, the unfolding force not onlydecreases, but also some of the intermediate states vanish. Interestingly, in the presenceof a solvent, the unfolding force, not only decreases drastically but many intermediatestates disappear. The force–extension curve (figure 9) shows only one intermediate statewith a greater stability against the applied force. This is because of the fact that the HHcontacts get buried inside the PS contacts. The required unfolding force is relativelysmall, because during stretching many PS contacts are formed (depending upon thecoordination number), which also favor the force to unfold the chain. In the absenceof the PS interaction, the applied force competes with the HH interaction only and, as aresult, relatively more force is required. This is also evident from figure 10.

It was pointed out in [9, 10] that the re-entrance shown in figure 7 is the result of theground state entropy. Since, in this case, the ground state entropy is zero, one would notexpect any re-entrance behavior in the force–temperature diagram at low temperature,which is also evident from figure 10.

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0.0 0.5 1.0 1.5 2.0 2.5 3.0T

0.0

0.5

1.0

1.5

2.0

2.5

3.0

f

εHH

= -4.0, εHP

= 0.5, εPP

= -0.0, εHS

= 1.0, εPS

= -0.5, εSS

= 0.0

εHH

= -4.0, εHP

= 0.5, εPP

= 0.0, εHS

= 0.0, εPS

= 0.0, εSS

= 0.0

Figure 10. Force–temperature phase diagrams for a given sequence with andwithout the solvent. The re-entrance seen in figure 7 disappears because of theabsence of the ground state.

6. Conclusions

We have studied the conformational properties of the biopolymers in the presence of theapplied force on a triangular lattice and a square lattice. A simple, but realistic, modelpresented here shows that, by changing the parameters in the partition function definedin equation (1), one can qualitatively understand the force–extension curves of the variousbiopolymers. The choice of the triangular lattice keeps the system closer to the real system(off-lattice). It may be noted that, at higher coordination number (>6), the finite volumeassociated with each monomer will start affecting the globule structure of the polymer.In the case of 2D, the ground state of triangular lattice mimics the ground state of theoff-lattice. By assigning a proper Boltzmann weight to the two bends, click-I and click-II,we could reproduce the force–extension curves for the different polysaccharides. It wouldbe interesting to repeat these experiments at different temperatures and see whether thedifferential scanning calorimetry (DSC) gives a signature of the phase transition or not,when the system goes from click-I to click-II.

The mechanical stretching of the self-interacting homopolymers is studied on thesquare and triangular lattices and the force–extension curves have been compared. Bothcurves show the existence of plateaus at temperatures below the θ temperature, whichcorrespond to the situation in which the polymer adopts the ‘ball–ellipse–string’-typetransformations. It is found that, when the coordination number increases, the numberof plateaus also increases, reflecting that many plateaus are possible in the continuumspace. The re-entrance in the force–temperature plane has also been seen here, which isin accordance with the previous studies [43, 58].

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Our model study shows that the solvent has an important role in determining thestructure of the protein. In the absence of the solvent, the force–extension curve shows themulti-step plateaus corresponding to the different intermediate states. These intermediatestates appear to be an artifact of the lattice models [10, 48]. Our results clearly show thatthe presence of the solvent not only decreases the magnitude of the applied force, butalso plateaus seen in the force–extension curve disappear. This is in accordance with therecent atomistic simulation, as seen in the case of the unfolding of ubiquitin [59].

Acknowledgments

Financial support from the DST, New Delhi, India and the generous computer supportfrom MPIPKS, Dresden, Germany are acknowledged.

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