Elastic Behaviors of Adsorbed Protein-like Chains

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CHINESE JOURNAL OF CHEMICAL PHYSICS VOLUME 23, NUMBER 1 FEBRUARY 27, 2010

ARTICLE

Elastic Behaviors of Adsorbed Protein-like Chains

Ting-ting Sun∗, Hai-zhu Ma

College of Information and Electronic Engineering, Zhejiang Gongshang University, Hangzhou 310018,China

(Dated: Received on May 26, 2009; Accepted on August 21, 2009)

Elastic behaviors of protein-like chains are investigated by Pruned-Enriched-Rosenbluthmethod and modified orientation-dependent monomer-monomer interactions model. Theprotein-like chain is pulled away from the attractive surface slowly with elastic force actingon it. Strong adsorption interaction and no adsorption interaction are both considered. Wecalculate the characteristic ratio and shape factor of protein-like chains in the process ofelongation. The conformation change of the protein-like chain is well depicted. The shapeof chain changes from “rod” to “sphere” at the beginning of elongation. Then, the shapechanges from “sphere” to “rod”. In the end, the shape becomes a “sphere” as the chainleaves away from the surface. In the meantime, we discuss average Helmoholtz free energyper bond, average energy per bond, average adsorbed energy per bond, average α-helicalenergy per bond, average β-sheet energy per bond and average contact energy per bond.On the other hand, elastic force is also studied. It is found that elastic force has a longplateau during the tensile elongation when there exists adsorption interaction. This result isconsistent with SMFS experiment of general polymers. Energy contribution to elastic forceand contact energy contribution to elastic force are both discussed. These investigations canprovide some insights into the elastic behaviors of adsorbed protein chains.

Key words: Elastic behavior, Adsorbed protein-like chain, Pruned-Enriched-Rosenbluthmethod, Orientation-dependent monomer-monomer interactions model

I. INTRODUCTION

Adsorption phenomenon plays an important role formany research areas including colloidal dispersions [1,2], polymer adhesion [3], biocompatibility [4], and chro-matographic separation [5]. The adsorption of proteinmolecules on solid interfaces is much more significantbecause many fields such as biomedical materials en-gineering [6], chromatography [7], and nanotechnology[8, 9] are all related to interface adsorption. The in-teractions between protein and surface will drive thechange of protein conformation. Therefore, there aremany researches in the adsorption-induced conforma-tional change of proteins [10, 11]. It is quite essential tounderstand protein adsorption. And the investigationof the protein conformation in the process of elongationfrom an adsorbed surface is significant.

There are numerous experimental and theoreticalstudies devoted to the phenomenon of adsorption ontosolid surfaces. Various parameters are paid attentionto, which describe the conformations of polymer chainsadsorbed on a surface, such as the adsorbed amount,surface coverage, layer thickness, average bound frac-

∗Author to whom correspondence should be addressed. E-mail:Tingtingsun@mail.zjgsu.edu.cn

tion of polymer monomers, and polymer volume frac-tion profile normal [12−21]. With the development ofexperimental techniques, the parameters can be moreand more exactly measured to study the adsorptionphenomenon. Atomic force microscopy (AFM) [22] hasbeen used to study the adsorption phenomenon. It isimportant that the single-molecule force spectroscopy(SMFS), based on AFM can be used to study inter-molecular and intra-molecular interactions with its ex-tremely high force sensitivity [23]. One of the classi-cal example is done by Hugel et al., in which the singlepoly-vinyl-amine chains of varied line charge density areinvestigated by using AFM based on SMFS [24]. Cui etal. also measured the force of a single polyelectrolytechain from a substrate [25]. The interesting finding isthat typical force curves has a long plateau and thendrops to zero [24, 25].

However, these researches all focus on general poly-mer chains, such as single polyvinylamine chains or sin-gle polyelectrolyte chains etc. Interactions between sin-gle polymer chains and the substrate are strong. It isdifficult to measure elastic force directly based on SMFSif the interactions between single polymer chains andthe substrate are weak. However, this disadvantage canbe solved by simulation investigation. Steered molec-ular dynamic (SMD) as a new atomic scale simulationmethod developed in recent years [26−28], can simu-late AFM or SMFS considering a suppositional force to

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12 Chin. J. Chem. Phys., Vol. 23, No. 1 Ting-ting Sun et al.

operate polymers or biopolymers. However, SMD sim-ulation should be used in disequilibrium stretching forgeneral polymers [29−31]. In these studies, it is foundthat force curve has a long plateau and then drops tozero. It agrees well with experimental results. How-ever, the average Helmoholtz free energy and averageinner energy can not be obtained in detail. There-fore, the enumeration calculation method and MonteCarlo method are used to study the elongation underequilibrium, and the average conformations and aver-age energies can be well investigated [32−34]. For longchains, since the number of conformations is very large,Pruned-Enriched-Rosenbluth method (PERM) is usedto stead the enumeration calculation method [35, 36].At the same time, the elasticity of protein chains hasnot been investigated by SMFS yet. Simulation inves-tigation is quite needed to study the elastic behaviorof proteins. Because most of proteins are copolymersand polyelectrolytes, protein adsorption bears some re-semblance to the adsorption of general polymer chains.Also through the simulation research, there maybe alsoexist some different elasticity behaviors between gen-eral polymer chains and proteins, especially for weakadsorption interaction.

In this work, we investigate the elastic behaviors ofadsorbed protein-like chains by using the PERM [35,36] and the modified orientation-dependent monomer-monomer interactions (ODI) model [37, 38]. It can sim-ulate the force curves, which can also been obtained bySMD simulation. However, we can also obtain shapeproperties and thermodynamic properties during theelongation.

II. METHOD OF CALCULATION

In Fig.1, we draw the schematics of a protein-likechain in the process of tensile elongation along Z-axisdirection. The force is acted on the first atom.

FIG. 1 Schematics of a protein-like chain in the process oftensile elongation along Z-axis direction. The force is actedon the first atom.

In the simulation, we used the modified ODI model[37, 38] based on original ODI model [39]. In thismodel, a protein-like chain is schematically viewed asa linear sequence and the chains are constrained to bethe nearest-neighbors on a three-dimensional cubic lat-tice. And each lattice site can only be occupied byone monomer. The merit of our model is that α-helicaland β-sheet structures can be taken into account si-multaneously. Therefore, the model is much closer tothe real protein. Considering the adsorption of surface,the Hamiltonian of the adsorption system is assumed tocontain five terms:

H =∑

|i−j|≥3

εcδ(rij − a) +∑

|i−j|≥3~ri+~si=~rj

εbδ(rij − a) +

N−1∑

i=1

ω(~si~si+1) + εhnh + εana (1)

where rij is the distance between monomers i and j. ais the lattice spacing (in fact, a=1 in our calculation).Here δ(x)=1 for x=0, and δ(x)=0 for x6=0. The unitvector ~si represents the orientation of monomer, and(~si~si+1) is a scalar product of the orientation vectors.εc is the energy of one contact interaction. Monomers iand j can form a contact if the two monomers are ad-joining lattice sites while not adjacent along the chain.The first term of Eq.(1) represents the total contact en-ergy of the chain. εb corresponds to the hydrogen bondsin topological contacts. Bearing in mind the conditionnecessarily for formation of hydrogen bonds, we con-sider that the value of εb is negative if (i) one of themonomers is directed toward the other monomer (e.g.,~ri+~si=~rj), and (ii) simultaneously the vectors ~si and~sj are parallel. In all the other cases, εb=0. In fact,we consider the second term of Eq.(1) to be β-sheet en-ergy. The third term (with ω>0) of Eq.(1), constructedin analogy with the Ising antiferromagnetic spin-spininteraction, describes the orientational dependence ofthe interaction between the nearest-neighbor monomerslinked via the peptide bonds. If the scalar product ofthe orientation vectors (~si~si+1) is negative, the thirdterm is negative [37−39]. Therefore its goal is to re-produce an antiparallel orientation of nearest-neighbormonomers. In the fourth term of Eq.(1), εh is the energyof one helix interaction, nh is the number of helices inthe chain, and εhnh represents the total energy of he-lices. Therefore, the α-helical and β-sheet energy areboth considered here. The last term, εa is the energyof adsorption, na is the number of monomers which areadsorbed on the surface.

In this work, we adopt two adsorption energy:εa=0 (no adsorption), εa=−1.5 (strong adsorption)(in units of kT ). We also select three typical en-ergy groups: εh=−0.5, εb=−0.5; εh=−0.5, εb=0; andεh=0, εb=−0.5. In the meantime, εc=−0.5 and ω=0.2[37−39].

An effective simulation method is used here, which is

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Chin. J. Chem. Phys., Vol. 23, No. 1 Elastic Behaviors of Adsorbed Protein-like Chains 13

called the PERM [35, 36]. Grassberger had used thisalgorithm for simulating flexible chain polymers and hisresults showed this method to be the most efficient forthree-dimensional polymers on the simple-cubic lattice[36]. Also PERM can be used instead of the enumera-tion calculation method, and the partition function canbe calculated [40, 41]. In our calculation, the lowerthreshold is 0.2, and the upper threshold is 5.0.

The partition function Z of the system is

Z =∑

i

exp(− Ei

kT

)(2)

where∑

i

is the sum of all conformations whose Z-axis

component of the first monomers is at the position Z.The Helmholtz free energy of the protein-like chains

per lattice can be derived from the partition function:

A = −kT lnZ (3)

At the same time, the mechanical force f can be ob-tained from the dependence of A on the elongated dis-tance along the force direction [42−45]:

f =∂A

∂Z(4)

In the meantime, energy contribution to elastic force fu

is defined by

fU =∂〈U〉∂Z

(5)

And contact energy contribution to elastic force fUcis

defined by

fUc=

∂〈Uc〉∂Z

(6)

Here we define Z0 as

Z0 =Z

N(7)

Thus, the influence of chain length N will be ignored.

III. RESULTS AND DISCUSSION

A. Chain size and shape

〈R2〉 is the mean-square end-to-end distance. Fig-ure 2 shows characteristic ratio 〈R2〉/N as a function ofZ0 during the elongation process with different adsorp-tion energy εa and different secondary structure. Here,N=30. For εa=0, when Z0 increases, 〈R2〉/N decreasesa little, then hold the line. The chain size of protein-likechains is nearly unchangeable in the process of elonga-tion from the adsorption surface. It can be found forεh=−0.5, εb=−0.5, 〈R2〉/N decreases from 0.99 to 0.85.

FIG. 2 Characteristic ratio 〈R2〉/N vs. Z0 during the elon-gation process with different secondary structure and ad-sorption energy for N=30, Z0=Z/N .

However, for strong attractivity (εa=−1.5), 〈R2〉/N de-creases first, then increases to a peak. At last it dropsto a fix value. At the beginning of elongation, adsorbedprotein-like chains move away from the attractive sur-face, the shape of the chains changes from a rod to asphere slowly. It causes 〈R2〉/N decreases. When thechain is elongated more, the shape is from a sphereto a rod. This leads 〈R2〉/N to increase. At last,〈R2〉/N drops which means the shape of chain changesto a sphere again. For exampe, for the case εh=−0.5,εb=−0.5, the ratio of 〈R2〉/N decreases from 1.88 to1.66 in the region of Z0=0−0.10, then it changes from1.66 to 2.46 in the region of Z0=0.10−0.27, and at lastdecreases from 2.46 to 0.87 of Z0=0.27−0.40.

In order to investigate the shape of adsorbed protein-like chains in more detail, we consider the shape-factors〈δ〉 [46, 47]. It origins from the radius of gyration tensorS, which is defined as:

S =1

N + 1

N∑

i=0

SiSTi

=

Sxx Sxy Sxz

Syx Syy Syz

Szx Szy Szz

(8)

here, Si=col(xi, yi, zi) is the position of monomer i ina frame of reference with its origin at the center of achain. The tensor S can be diagonalized to form a di-agonal matrix with three eigenvalues L1

2, L22, and L3

2

(L12≤L2

2≤L32). Then the shape-factors 〈δ〉 could be

defined as:

〈δ〉 = 1− 3⟨

L12L2

2 + L22L3

2 + L12L3

2

(L12 + L2

2 + L32)2

⟩(9)

〈δ〉 varies between 0 (sphere) and 1 (rod).In Fig.3, The shape factor 〈δ〉 as a function of Z0 dur-

ing the elongation process with different adsorption en-ergy and different secondary structure is shown. When

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14 Chin. J. Chem. Phys., Vol. 23, No. 1 Ting-ting Sun et al.

FIG. 3 Shape factor 〈δ〉 vs. Z0 during the elongation pro-cess with different secondary structure and adsorption en-ergy for N=30, Z0=Z/N .

the adsorption interaction energy is zero, 〈δ〉 is around0.30−0.35 for the three energy parameter at Z0=0. Inthe process of elongation, 〈δ〉 changes little. Becausein this work, the chain are not allowed to pass throughthe adsorption surface. The value of 〈δ〉 can not reachzero. However, it can be certified that the chain is likea sphere. In the other hand, for large adsorption inter-action, the curves decrease firstly, and then increase, atlast decrease again. The changement of 〈δ〉 is similarto 〈R2〉/N . In order to illuminate these change visu-ally, six sketch maps are added on Fig.3. We selectεh=−0.5, εb=−0.5, and εa=−1.5 to discuss in detail.At the beginning of the elongation (Z0=0), the value of〈δ〉 is 0.53. Due to the adsorption of surface and the sec-ondary structure in the protein, the chain is like a rod.Secondly, with Z0 increasing, the value of 〈δ〉 drops to0.37 (Z0=0.16). When the chain is elongated, the shapetrends to be a sphere. Thirdly, if Z0 increases again, thevalue of 〈δ〉 reaches 0.55 (Z0=0.27). The chain extendto a rod shape along the direction of force, and a fewmonomers are adsorbed on the attractive surface yet.At last, the whole chains are pulled away from the sur-face. There is no effect of the adsorption on compactchains, therefore the adsorbed protein-like chains be-come the general compact chains. The value of 〈δ〉 is0.30 (Z0=0.40).

B. Thermodynamics properties

The average helmholtz free energy per bond A/N as afunction of Z0 during the elongation process with differ-ent adsorption energy and different secondary structureis shown in Fig.4. For εa=0, A/N has a little decrease-ment in small Z0, then keeps unchanged. For strongattraction (εa=−1.5), it increases to a certain valueat about Z0=0.4. The changement of thermodynamicsproperties for adsorbed protein-like chains in this pro-cess is due to the change of conformations. For exam-

FIG. 4 Average helmholtz free energy per bond A/N vs.Z0 during the elongation process with different secondarystructure and adsorption energy for N=30, Z0=Z/N .

ple, for εh=−0.5, εb=−0.5, the value of A/N increasesfrom −4.2 to −3.43, for εh=−0.5, εb=0, it changesfrom −4.17 to −3.39, and for εh=0, εb=−0.5, the valueis from −4.17 to −3.26. This result also illuminatesthe different elastic behaviors of adsorbed protein-likechains from protein-like chains with no surface adsorp-tion.

Then we plot the average energy per bond as a func-tion of X0 for adsorbed protein-like chains with εa=0and εa=−1.5. Figure 5 shows the average total energyper bond 〈U〉/N , average adsorbed energy 〈Ua〉/N , av-erage α-helical energy 〈Uh〉/N , average β-sheet energy〈Ub〉/N , and average contact energy 〈Uc〉/N vs. Z0

during the elongation process with different adsorptionenergy and different secondary structure.

In Fig.5(a), the curves are similar to the results inFig.4. The curves for protein-like chains without ad-sorption interaction are almost the lines parallel to theZ-axis and only a little undulation exists. For εa=−1.5,the value of 〈U〉/N increases from −1.48 to −0.41,which are −0.35, and −0.21 for (εh=−0.5, εb=−0.5),(εh=−0.5, εb=0), and (εh=0, εb=−0.5) respectively.In the process of elongation, the number of adsorbedresidues decrease. Until the chain is away from the sur-face, there is almost no adsorbed residues. It will influ-ence the value of adsorbed energy of protein-like chain.Figure 5(b) shows 〈Ua〉/N vs. Z0 in the process ofelongation. For εa=−1.5, with Z0 increasing, the num-ber of adsorbed residues becomes small. So the value of〈Ua〉/N increases. After Z0=0.40, 〈Ua〉/N reaches zero.There is no residue adsorbed on the surface.

Figure 5(c) also shows the α-helical energy 〈Uh〉/N ofεh=−0.5, εb=−0.5 and εh=−0.5, εb=0 as the chain isbe elongated from the surface. When εa=0, in the pro-cess of elongation, 〈Uh〉/N increases a little, the numberof helixes is nearly not changed. Therefore, 〈Uh〉/N isalmost at −0.20. However, if adsorption is not zero,in the process of elongation, the number of helixes in-creases apparently. That is because the protein-like

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Chin. J. Chem. Phys., Vol. 23, No. 1 Elastic Behaviors of Adsorbed Protein-like Chains 15

FIG. 5 Average total energy per bond 〈U〉/N (a), average adosrbed energy 〈Ua〉/N (b), average α-helical energy 〈Uh〉/N(c), average β-sheet energy 〈Ub〉/N (d), and average contact energy 〈Uc〉/N (e) per bond vs. Z0 during the elongationprocess with different secondary structure and adsorption energy for N=30, Z0=Z/N.

chain is constrained in a surface with strong adsorp-tion. With elongation, helixes are easily formed. Thenumber of helixes increases, 〈Uh〉/N decreases. Whenεa=−1.5, the value of 〈Uh〉/N has a large decrease from−0.06 to −0.20 (Z0=0.40). The changed extent is rel-atively large. It demonstrates the number of helicesincreases exceed 200% compared to that of the protein-like chain before being elongated. And it is also foundthat the helix number for strong interaction (εa=0) issmaller than that for no adsorption (εa=−1.5). Av-erage β-sheet energy 〈Ub〉/N vs. Z0 is presented inFig.5(d). The value decreases from −0.03 to −0.05 forεa=0, almost decreases 66.7%, and for large adsorp-tion interaction (εa=−1.5) the value of 〈Ub〉/N variesfrom −0.005 to −0.05. The changed extent is 90%,which is smaller than 〈Uh〉/N . For the case of εa=0,the change of β-sheet secondary structure is more obvi-ous than that of α-helix secondary structure during thestretching process of protein-like chains from the sur-faces. However, for εa=−1.5, the appearance is inverse.That is to say, strong adsorption interaction affects α-helix more than β-sheets and the surface bound withno adsorption affects β-sheet even more. The last en-ergy is average contact energy 〈Uc〉/N . It is shown inFig.5(e). The changement of 〈Uc〉/N with Z0 increas-ing is greatly important. We find when there is no ad-

sorption of the surface, the value of 〈Uc〉/N drops from−0.07 to −0.11. The number of contacts raises in thetensile process for εa=0. However, the trend of 〈Uc〉/Nfor εa=−1.5 is quite different. When we raise the ad-sorbed chain, the number of contacts first increase thendecrease. It is included the value of 〈Uc〉/N decreasesfrom −0.01 to −0.16, and it increases from −0.16 to−0.11. The minimum of 〈Uc〉/N is at Z0=0.27, which isthe most contacts appeared in the tensile process. ThisZ0 point is corresponding to the second turning pointin Fig.3. It is shown that most contacts does not occurat the point of minimum value of 〈δ〉. That is also tosay in globule structure, the number of contacts is notthe most, which is because the other secondary struc-tures (α-helix and β-sheet) is affected simultaneously inthe protein-like chain. At Z0=0.40, the value of 〈Uc〉/Nreaches a certain value, which is also corresponding tothe third turning point of 〈δ〉.

We also calculate the elastic force f according toEq.(4), and the results are shown in Fig.6(a). InFig.6(a), we find that the curves of no adsorption in-crease with Z0 increasing. In the beginning of elon-gation, the value of f is negative. It indicates thatthe chain leaves the surface spontaneously. However,in the case of εa=−1.5, the curves will increase firstly,then keep constant value, and at last decrease with Z0

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16 Chin. J. Chem. Phys., Vol. 23, No. 1 Ting-ting Sun et al.

FIG. 6 Elastic force f (a), energy contribution to elastic force fU (b) and contact energy contribution to elastic force fUc

(c) vs. Z0 during the elongation process with different secondary structure and adsorption energy for N=30, Z0=Z/N .

increasing. This trend is all found for the three en-ergy group. First, the force acting at the chain endis relatively small, and it increases with Z0. The longplateau suggests that the desorption process of protein-like chains from the surface is smooth. At last, the valueof f drops to zero when the chain is left from the surfacecompletely. This finding is similar to SMD simulationresults for adsorbed polymer chains [29]. For the threeenergy group, the value of elastic force f is in the rangeof 0.05−0.10.

We also calculate energy contribution to elastic forcefU according to the Eq.(5), and fU as a functionof Z0 in the elongation is shown in Fig.6(b). Forthe case of εa=0, fU has a little increasement first,and later reaches zero. When the adsorption interac-tion εa=−1.5, fU increases first, then it occurrencesa plateau with Z0 increasing. However, before theprotein-like chains are entirely pulled away from theadsorbed surface, maximumu values of fU are all foundin the three energy groups. It means at the maximumpoint, the energy changes obviously.

Lastly, contact energy contribution to elastic forcefUc

according to the Eq.(6) is also discussed here. Forεa=0, fUc

is −0.01 first, then increase to zero and keepthe value unchanged in the process of elongation. Forεa=−1.5, there also exists a plateau which is similar tothe curve of fU . Then the peaks of the curves showsthe mutation of contact energy 〈Uc〉/N . All these in-vestigations will deepen the understanding of proteinadsorption and elasticity.

IV. CONCLUSION

In this work, we perform the PERM method andmodified ODI model to study the elastic behavior of ad-sorbed protein-like chains. We investigate the changesof mean-square end-to-end distance, shape factor. Thenwe also study the thermodynamics properties of ad-sorbed protein-like chains. Different elastic behaviors

are obtained for chains adsorbed on the surface andchains without adsorption with the surface. The elas-tic force increases first in the small region of elonga-tion, and exists a long plateau for strong adsorptioninteraction, and the plateau force f is in the range of0.05−0.10. When the surface has no adsorption in-teraction with the chain, the force is negative first, itshows the chain spontaneously leaves from the surface.Plateaus and peaks are found in the curves of fU andfUc as a function of Z0.

V. ACKNOWLEDGMENT

This work was supported by the National NaturalScience Foundation of China (No.20904047).

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DOI:10.1088/1674-0068/23/01/11-17 c©2010 Chinese Physical Society