Collapse Transition of Two-Dimensional Flexible and Semiflexible Polymers

0.15 MB·English

by  Haijun Zhou

Collapse Transition of Two-Dimensional Flexible and Semiflexible Polymers ∗
Haijun Zhou1, Jie Zhou1, Zhong-Can Ou-Yang1, and Sanjay Kumar2
1Institute of Theoretical Physics, the Chinese Academy of Sciences, Beijing 100080, China and
2Department of Physics, Banaras Hindu University, Varanasi 221 005, India
(Dated: February 6, 2008)
Thenatureoftheglobule-coiltransitionofsurface-confinedpolymershasbeenanissueofdebate.
7 Herethis2Dcollapsetransitionisstudiedthroughapartiallydirectedlatticemodel. Inthegeneral
0 case of polymers with positive bending stiffness (∆ > 0), the collapse transition is first-order; it
0 becomes second-order only in the limiting case of zero bending stiffness (∆ ≡0). These analytical
2 resultsareconfirmedbyMonteCarlosimulations. Wealsosuggestsomepossiblefutureexperiments.

n
PACSnumbers: 82.35.Lr,05.50.+q,64.60.Cn,82.35.Gh
a
Keywords: collapse,globule-coiltransition,semiflexible,latticepolymer
J

8
1
The collapse (globule-coil) transition of a self- firmed by MC simulation in 3D [16], while in 2D the

attracting chain is one of the fundamental problems in transition is argued to be second-order by MC simula-
]
t polymer physics [1, 2, 3]. Being deeply connected with tion [16] and scaling analysis [17, 18].
f
o biophysical problems such as DNA condensation, chro- In order to understand the collapse transition more
s matin organization, and protein folding, the collapse deeply, here we investigate a 2D partially directed lat-
.
at transitionhasalsobiologicalrelevance. Generallyspeak- tice model of a polymer chain which is exactly solvable
m ing, this transition is caused by the competition be- (Fig.1). As the free energydensity ofthis modelsystem
tweenmonomer-monomerattractionandconfigurational can be calculated precisely, we are able to draw definite
-
d entropy: Formation of contacts lowers the energy, but it conclusions concerning the nature of the collapse transi-
n requiresmonomersto bealignedandclosetoeachother, tion. We findthatthe collapsetransitionis second-order
o
thereby decreasing the polymer’s degrees of configura- for a polymer with exactly zero bending stiffness, while
c
[ tionalfreedom. Whenthe monomer-monomerattraction it changes to be first-orderwhen the bending stiffness ∆

is muchstrongerthanthermalenergy,the polymertakes is non-zero. Therefore bending energies have a dramatic
1
globular conformations to maximize the number of con- effect on the cooperativity of the 2D globule-coil transi-
v
4 tacts. At the other limit of high temperatures, the poly- tion. We alsosubstantiateourpredictionsbyperforming
3 mer is in completely disorderedcoil states with maximal extensive MC simulations. It becomes experimentally
4 entropy. At zero external force, the collapse transition feasible to confine polymers to 2D mobile surfaces (see,
1 occurs at a so-called θ-temperature [1]. e.g., [19]), therefore, future experiments will be able to
0
The exact nature of the collapse transition, however, verify the theoretical predictions. The present work is
7
0 is not yet completely settled, despite decades of exten- alsodirectlyapplicable to the biology-relatedproblemof
/ sive efforts [1, 2, 3]. This transition can be studied by protein β-sheet unfolding.
t
a mapping to the tricritical point of the φ4-φ6 O(n) field The model.—The 2D partially directed polymer of N
m
theory at the n→0 limit [4, 5, 6], and is expected to be identicalunitsonasquarelatticeisshowninFig.1. The
- second-order in 2D and beyond (see, e.g., [7]); further- lengthofthebondconnectingtwoconsecutivemonomers
d
n more, exponents of the temperature-driven collapse and iandi+1isfixedtoa0,whilethedirectionofbondin−z0
o adsorption of a 2D polymer grafted on a linear bound- is prohibited. If any two monomers i and i+m (m≥3)
c ary was obtained by analytical calculations [8, 9]. How- occupynearestneighboringlatticesites,anattractiveen-
: ever,experimentallyobservedcollapsetransitionsofboth
v
i relatively flexible [10] and semiflexible [11] 3D polymers
X
aremorelikefirst-ordertransitions. RecentMonteCarlo
r (MC) simulation [12] is also in favor of the latter inter-
a
pretation.
Withthe advancementinsingle-moleculeforcemanip-
ulationmethods, nowtheglobule-coiltransitioncanalso
be induced by external stretching [13, 14]. According to .
mean-field theory [15] the force-induced collapse transi-
tion is first-order in all dimensions. This claim is con-
FIG.1: The2D latticepolymermodelwith contacting inter-
∗Citation information: H. Zhou, J. Zhou, Z.-C. Ou-Yang, and S. action, bending stiffness, and external stretching. The arrow
Kumar,Phys. Rev. Lett.97,158302 (2006). shows the z0 direction.
2
ergy of magnitudeǫ is gained. Usually realpolymers are calculated by Eq. (4). The point ζ = 1 is therefore a
semiflexible,weassociateanenergypenaltyofmagnitude singularpoint ofG (ζ), where it attains a finite value as
β
∆toeachlocaldirectionchangeofthechain[20,21]. For longasλ (1)<a/(ps2). Atthispoint,thefirstderivative
1
thismodel,onlythespecialcaseofzerobendingstiffness of G (ζ) with respect to ζ is [25]
β
has been studied analytically [18, 22, 23]. The results
L
of the present paper suggest that, in the general case of
∆>0 the properties of the polymer will be dramatically G′β(1)≡dGβ(ζ)/dζ|ζ=1 = lim Xny(n)+O(1) , (5)
L→∞
n=1
different from these earlier predictions [18, 22, 23].
Partition functions.—Tocalculatethefreeenergyden- where y(n)= L A A x λ /(1−ps2λ /a) 2.
sityofthepolymer,agivenconfigurationofthe2Dchain (cid:0)Pj,k=1 nj kj k j j (cid:1)
The configurational energy of a coil segment is
is divided into a linear sequenceof β-sheetsegments and
coil segments [24]. A β-sheet segment is defined as a E =−n fa +m ∆ , (6)
c c 0 c
folded segment of n ≥2 consecutive columns, in which
β
contacting interactions exist between any two adjacent where nc and mc are, respectively, the total number of
ones. Two consecutive β-sheet segments are separated columns andthe total number ofbends in the configura-
by a coil segment, which is a segment of nc≥0 consecu- tion. Inacoilsegmentthevalueofmc stronglydepdends
tive columns in which all monomers are free of contacts. ontheconfigurations. Tocalculatethepartitionfunction
For example, the configuration shown in Fig. 1 has two Zc(n)ofa coilsegmentofn monomers,oneneeds to dis-
β-sheets and two coils. After having made such a dis- tinguish among four different boundary conditions [25].
tinction between β-sheets and coils, we proceed by first The generating function Gc(ζ) of Zc(n) in this case is
calculating the partition functions of β-sheets and coils expressed as
separately.
∞
Under the action of an external force f, the energy of Gc(ζ)≡ X(cid:0)ζ/a(cid:1)nZc(n) (7)
a β-sheet of n columns is
β n=0
as2(a−ζ)(a+ζp)
nβ−1 = a3−ζa2(1+p)+ζ2ap(1−s2)−ζ3p2s2 . (8)
Eβ =−ǫ X v(lj,lj+1)−nβfa0+2(nβ −1)∆ , (1)
j=1 The divergence radius of G (ζ) is easy to obtain from
c
Eq. (8). G (ζ) approaches +∞ as ζ approaches this di-
where l is the number of monomers in the j-th column c
j vergence radius.
of the β-sheet, and v(l ,l ) = min(l ,l )−1. The
j j+1 j j+1 The collapse transition.—Since every configuration of
partition function of a β-sheet with n≥4 monomers is
the polymer is of the form ...-β-c-β-c..., the generating
function G(ζ) of the whole polymer’s partition function
[n/2] m−1
Zβ(n)= X Xδln1+···+lnβpms2(m−1) Y av(lj,lj+1) , Z(N) can readily be written down [24, 25]:
nβ=2lj≥2 j=1 (2) G(ζ)≡ X∞ (cid:0)ζ/a(cid:1)NZ(N)= [1+Gβ(ζ)]Gc(ζ) . (9)
wherep=efa0/T,s=e−∆/T,a=eǫ/T,andT isthetem- 1−Gβ(ζ)Gc(ζ)
N=0
perature. Itiseasiertocalculatethe generatingfunction
Z(N) is related to the free energy density g(f,T) of the
G (ζ) of the partition function Z (n) than to calculate
β β
polymerbyZ(N)=exp −Nβg(f,T) . Therefore,inthe
Z (n) directly. After some simple matrixoperations[25] (cid:0) (cid:1)
β
thermodynamic limit N →∞, the free energy density is
we find that
∞ g(f,T)=−ǫ+T lnζ0 , (10)
n
Gβ(ζ) ≡ X(cid:0)ζ/a(cid:1) Zβ(n) (3)
where ζ is the smallest positive root in the range 0 <
n=4 0
p2s2 x x A (ζ)A (ζ)λ (ζ) ζ0 <1 of the following equation
i j ik jk k
= X . (4)
a 1−(ps2/a)λ (ζ)
k G (ζ )G (ζ )=1 . (11)
i,j,k=1 β 0 c 0
InEq.(4),x =(ζ/a)(j+1)/2;λ (ζ)≥λ (ζ)≥...arethe If Eq. (11) has no root in the range of 0 < ζ0 < 1, then
j 1 2 ζ =1. The polymer’s relative extension ℓ along z is
eigenvalues of a L×L real-symmetric matrix Λ(ζ) with 0 0
elements Λ (ζ) = ζ1+(i+j)/2/a|i−j|/2 (i,j = 1,2,...,L);
ij ∂lnζ 1 ∂ζ
0 0
and the orthogonal matrix A(ζ) contains the eigenvec- ℓ≡− =− . (12)
∂lnp βa ζ ∂f
0 0
tors ofmatrix Λ(ζ). The parameterL should be infinity.
When ζ > 1, λ (ζ) = +∞ and consequently G (ζ) is A similar expression can also be obtained for the den-
1 β
not properly defined. When ζ ≤ 1, all the eigenvalues sity of contacts. For the simplicity we take the relative
of matrix Λ(ζ) are finite, and the value of G (ζ) can be extension ℓ as our order parameter.
β
3
When both temperature T and external force f are Results for ∆ ≡ 0. We perform analytical calcula-
sufficientlylow,Eq.(11)hasnorootin0<ζ <1. Then tions as well as Monte Carlo simulations on the model
0
g(f,T) = −ǫ. The polymer is in the β-sheet phase, and system. The ℓ-T relation at f = 0 is shown in Fig. 2a,
ℓ ≡ 0. There is no significant entropic contribution to and the ℓ-f relation at T = 0.591ǫ is shown in Fig. 2b.
thefreeenergydensity. Asthetemperatureortheexter- When there is no external force, a second-order globule-
nalforceiselevatedtocertainpointsuchthatEq.(11)is coil phase transition occurs at T (0) = 0.8205ǫ. The
gc
satisfied exactly at ζ =1, a phase transition occurs. At force-induced globule-coiltransition at constant temper-
0
this point, the β-sheetphase changeto the extended coil atureisalsosecond-order. Ourfindingsareinaccordance
phase. AccordingtotheworkofLifson[24],thisglobule- with Refs. [16, 18]. In the vicinity of the collapse transi-
coiltransitionisfirst-orderifthevalueofG′ (1)inEq.(5) tion,therelativeextensionoftheextendedcoilphasecan
β
is finite at the phase transition point; otherwise, it is a be expressed in a scaling form in terms of temperature
continuous phase transition. At the collapse phase tran- change δT or force change δf:
sition point, high-precision numerical calculations reveal
that y(n) of Eq. (5) decays exponencially with n as long ℓ(cid:0)Tgc(f)+δT(cid:1) ≈ ℓ(cid:0)Tgc(f)(cid:1)+c1 (δT)γ1 , (14)
as ∆>0 [25]. In the vicinity of ∆=0 we have ℓ f (T)+δf ≈ ℓ f (T) +c (δf)γ2 , (15)
(cid:0) gc (cid:1) (cid:0) gc (cid:1) 2
y(n)∼e−0.88n∆/ǫ (∆≪ǫ) . (13) where T (f) and f (T) are, respectively, the transition
gc gc
temperature at fixed force and the transition force at
Therefore,atthephasetransitionpointG′ (1)isfiniteas fixed temperature; γ and γ are two scaling exponents;
β 1 2
long as ∆ > 0. In other words, the collapse phase tran- and c ,c are two numerical constants. In the case of
1 2
sition is first-order for polymers with bending stiffness ∆ ≡ 0, we have ℓ T (f) = ℓ f (T) = 0; and we
(cid:0) gc (cid:1) (cid:0) gc (cid:1)
∆>0; it is second-order for polymers with exactly zero findthatthe scalingexponentinEq.(14)isindependent
bending stiffness. Itisremarkablethatthe cooperativity of force with γ = 1/2 (see Fig. 3), which is consistent
1
of the collapse transition can be changed by adding just with a second-order phase transition. The scaling rela-
a small bending stiffness. tion Eq. (15) with γ =1/2 is also confirmed by numer-
2
ical calculation (data not shown).
Results for ∆ > 0. In this case, the collapse transi-
A Theory tion becomes first-order. For example at ∆ = 0.5ǫ and
0.3 MC (N=1024)
MC (N=3600) f =0,therelativeextensionjumpsfromzerotoℓ=0.193
n MC (N=4900)
sio MC (N=10000) at the transition temperature Tgc(0) = 1.209ǫ (Fig. 2a).
n Such a large jump is also observed in the force-induced
exte0.2 transition (Fig. 2b). A non-zero bending stiffness there-
ve ∆=0 ∆=0.5 ε fore is able to dramatically enhance the cooperativity of
ati
el0.1 the globule-coil phase transition. This may be partially
r
understood in the following way. A positive bending en-
ergy significantly decreases the configurational entropy
0 of a coil segment. Consequently the globule-coil transi-
0.8 0.9 1 1.1 1.2 1.3
temperature (ε) tion will occur at higher temperature and higher force,
and once the polymer is unfolded it favors those highly
B Theory elongated configurations which have fewer bends.
MC (N=1024) As is expectedfor afirst-orderphase transition,inthe
0.6 MC (N=1600)
n MC (N=3600) caseof ∆>0, the scalingexponents in the scaling forms
nsio MMCC ((NN==41900000)0) (14) and (15) are γ1 =γ2 =1 (Fig. 3).
e
xt0.4 Conclusion.—Inthisworkwestudiedthecollapsetran-
e
e sition of a 2D partially directed lattice polymer model
v
ati ∆=0 ∆=0.25 ε using the Lifson approach [24]. We devided each con-
rel0.2 figuration of the polymer into β-sheet segments and coil
segments and then calculated their partition functions
separately. This approach enables us to precisely cal-
0
0.4 0.5 0.6 force 0(.ε7/a 0) 0.8 0.9 culate the free energy density of the system. Previous
tricriticalfieldtheory[22,23]andnumericalscalinganal-
ysis [18] predicted that the collapse transition of a flex-
FIG. 2: (Color online) Response behavior of the polymer.
ible (∆≡0) partially directed lattice polymer is second-
(A) temperature–extension curve at force f = 0; (B) Force–
extensioncurveattemperatureT =0.590928ǫ. Solidlinesare order. We have confirmed this prediction. Our calcula-
analyticalpredictionsandsymbolsareMCsimulation results tions further revealed that, the susceptibility χ=∂ℓ/∂f
for different system sizes. of the polymer diverges as χ ∝ f − f (T) −1/2 or
(cid:0) gc (cid:1)
4
-1 a mobile lipid bilayer [19]. Polymer configurations can
be recorded in real-time and manipulated by controlling
-2 temperature or external force.
δ0(l)-3 We thank X.-S. Chen, D. Giri, P. Grassberger,
1
g D. Marenduzzo, H. Orland for critical comments, and
o
l
-4 the StateKeyLab.ofSci.andEngn.ComputingofCAS
∆=0, f=0
∆=0, f=0.5 ε/a0 for computational facilities. HZ acknowledges the initial
∆=0.25 ε, f=0
-5 ∆=0.25 ε, f=0.5 ε/a0 supportofR.Lipowskyandthe AvHfoundation,SKac-
-6 -5 -4 -3 knowledges the hospitality of the MPIPKS-Dresden.
log10(δT/ε)
FIG. 3: The scaling relation between extension increment δℓ
[≡ℓ(cid:0)Tgc(f)+δT(cid:1)−ℓ(cid:0)Tgc(f)(cid:1)]andδT. Theupperandlower
dotted line has slope 1/2 and 1, respectively. The dimension
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