Heteropolymers in a Solvent at an Interface

4 Pages·0.14 MB·English

by  A. Maritan

Heteropolymers in a Solvent at an Interface
A. Maritan, M. P. Riva, and A. Trovato
International School for Advanced Studies (SISSA),
and Istituto Nazionale di Fisica della Materia,
Via Beirut 2-4, 34014 Trieste, Italy
9
9 The Abdus Salam International Center for Theoretical Physics,
9 Strada Costiera 11, 34100 Trieste (Italy)
1
Exactboundsareobtainedforthequenchedfreeenergyofapolymerwithrandomhydrophobicities

n in the presence of an interface separating a polar from a non polar solvent. The polymer may be
a idealorhavestericself-interactions. Theboundsallowtoprovethata“neutral”randompolymeris
J
localized neartheinterfaceatanytemperature,whereasa“non-neutral”chainisshowntoundergo
6 adelocalizationtransitionatafinitetemperature. Theseresultsarevalidforaquitegeneralapriori
2
probability distribution for both independent and correlated hydrophobic charges. As a particular

case we consider random AB-copolymers and confirm recent numerical studies.
]
h
c PACS numbers: 61.41.+e, 05.40.+j, 36.20.-r
e
m Thestatisticalbehaviorofheteropolymershasbeenin- ~r = (x ,...,x ), i = 0,...,N of a d-dimensional hy-
i i1 id
- tensively studied in recent years [1–7]. They model ran- percubic lattice. A flat interface passing through the
t
a dom copolymers [2,3] and to some extent protein fold- origin and perpendicular to the ~u = (1,...,1) direction
t ing [1]. For example, a chain composed by hydrophobic separates a polar (e.g. water), on the ~u ~r > 0 side,
s ·
. and hydrophilic (polar or charged) components in a po- from a nonpolar (e.g. oil or air), on the ~u ~r < 0 side,
t ·
a lar(aqueous)solventevolvestowardconformationswhere solvent. The i-th monomer interacts with the solvent
m
the hydrophobic partare buried in orderto avoidwater, throughits hydrophilic chargeq >0 (q <0 means that
i i
- wheras the polar part is mainly exposed to the solvent it is hydrophobic) and contributes to the energy with a
d
[8]. This is what commonly happens to proteins and term q sgn(~u ~r ) [12]. For simplicity we can associate
n i i
− ·
o makesthem soluble in aqueoussolutions. Howeverother the charge to the link between adjacent positions on the
c proteins (e.g. structural proteins) are almost insoluble chain instead that to the single monomer.
[
under physiologicalconditions and prefer to form aggre- The partition function of the model for a chain W

1 gates [8]. Many of the proteins which are insoluble in starting at position~r is
v water are segregated into membranes which have a lipid
2 bilayer structure [8]. Membrane proteins have a biologi- N
9 (~r, q )= exp β q sgn(~u ~r ) , (1)
2 calimportanceatleastasgreatasthosewhicharewater Z { i} ( i · i )
1 soluble. Usually one distinguishes integral and non in- WX:~r→. Xi=1
0 tegral membrane proteins according to whether the pro- where β 1 = k T. If one sums over non-interacting
− B
9 tein is most immersed in the lipid bilayers or simply an-
(ideal) chains, then the lattice version of the model in-
9 chored to the membrane respectively (in the latter case
troduced in Ref. [2] is recovered. We will consider also
/
t theproteinisessentiallywatersoluble)[8]. Ananalogous the more physical case where steric interaction among
a
situation occurs for copolymers at interfaces separating
m monomers does not allow for multiple occupancy of lat-
two immiscible fluids. If the solvents are selective (i.e.
tice nodes, studied for a particular case in Ref. [3].
-
d poor for one of the two species and good for another), The free energy density of the system reads
n AB-copolymers are found to stabilize the interface [9]. f(~r,β) = lim 1 ln (~r, q ), where denotes
o Moreover, random copolymers have been claimed to be − N→∞ βN Z { i} ···
thequenchedaverageoverthedistributionofthecharges
c
more effective in carrying out this re-enforcement action
: qi . In the following we will assume that qi are inde-
v [10]. { } { }
pendentrandomvariableshavingaGaussiandistribution
i
X of the form
r The simplest theoretical approach to the above prob-
a lems has been proposed by Garel et al. [2]. In the case 1 (qi q0)2
P(q )= exp − . (2)
of membrane proteins the finite layer of lipidic environ- i √2π∆2 − 2∆2
(cid:20) (cid:21)
ment is modeled as an infinite semi-space. Though a
More general cases will be treated at the end. In par-
quite rough approximation, this is the simplest attempt
ticularconsideringchargesnotindependentlydistributed
incapturingthe relevantfeaturesduetothecompetition
is of interest for designed sequences as happens for real
of different selective effects [11].
proteins.
We will study a lattice discretized version of their
Wewillshowthatforaneutralchain(q =0)f(0,β)<
model. The nodes of an N links chain occupy the sites 0
f(~r,β) with ~r N, in the large N limit, and for all β.
| |≥
1
wTihtehsβaumppeerh→old0s iafls|qo∆0|fo→r q00. T6=hi0s iimf βpli>es tβhuaptpetrh(|eq0c|h,a∆in) G(x)= √1πx1e−x2 −[1−erf(x)]. (6)
is localized around the interface at any temperature if
G(x)isapositivedecreasingmonotonicfunctionforpos-
q =0,andatsufficently lowtemperatureif q =0. The
0 0 6 itive arguments.
proofis rigorousfor the idealchain, whereasfor the self-
We consider separately the neutral and the q = 0
0
avoidingcaseonlyamildandwellacceptedhypothesison 6
cases. In the neutral case it turns out that the chain is
the asymptotic behavior ofthe entropy is needed. When
always localized at the interface. In fact, if q = 0 we
0
q =0arigorouslowerboundonthefreeenergyforboth
0 have
6
the ideal and the self-avoiding chain allows to determine
a β (q ,∆) below which the chain is delocalized. 1 lnC 2 1
lower | 0| ∆f h (k,β)= ln2 k ∆ . (7)
We will first consider the ideal chain case and then ≤ 0 β − k − π √k
(cid:20) (cid:21) r
explainthe modifications necessaryto extendthe results
to self-avoiding chains. It is easy to see that for any β there exists a value k(β)
such that h (k,β) < 0 for k > k(β). For example at
0
high temperature k(β) [ln(β∆)]2(β∆) 2. This shows
Ideal chain. For clarity we derive the bounds in the −
∼
thatatanytemperatureaneutralrandomchainisalways
d=1case. Thegeneralcasedoesnotcontainanyfurther
adsorbed by the interface.
difficulty [13]. Let us first consider initial positions far
In the non-neutral case with k =2, one has ∆f <0 if
from the interface in the favorable solvent, x N if
≥
q >0orx N ifq <0. Undertheseassumptionsall
0 0 ln2
≤−
1ch(aoirnssgrne(mxai)in=in t1hreessapmecetisviedley,)ifmorpalyllini.gUthpaotnsagvne(rxagi)in=g β >βupper = |q0|G(|q∆0|). (8)
−
over the charge distribution, we obtain the free energy
The limit lim h (k,β) = 0 does not allow to de-
density of a walk in the favorable solvent: k→∞ q0 +
duce the existence of a negative minimum in k, so that
1 the previous argument, showing that the neutral chain
f∗ = ln2 q0 . (3) is always localized, does not hold for q = 0. Equation
−β −| | 0
6
(8) proves localization at sufficiently low temperatures.
We give an upper bound to the free energy as fol- For q ∆, it yields β = √πln2, and in the op-
lows. Consider only chains made up of blobs of k | 0| ≪ upper ∆
steps, with k even. Bringing a blob in its globally fa- posite regime ∆ ≪ |q0|, βupper = 2√πln2e|q∆02|2 |q∆03|2. In
vored side leads to an energy contribution of the form the limit q0 /∆ 1 we can give a better estimate for
| | ≪
H = k q , so that βupper, such that βupper 0 as q0 0, by considering
j−thblob − i=1 k(j−1)+i a larger blob size k = 2x→2(∆/q| )2|,→where x q /∆
(cid:12)(cid:12)(cid:12)P (cid:12)(cid:12)(cid:12) Nk k is fixed: 0 | 0| 0 ≫ | 0|
N
Z(x=0,{qi})≥(Ck)k expβ (cid:12) qk(j−1)+i(cid:12), β = 3 ln √2x ∆/q |q0|. (9)
 Xj=1(cid:12)(cid:12)Xi=1 (cid:12)(cid:12) upper 2x20G(x0) 0 | 0| ∆2
(cid:12) (cid:12) (cid:16) (cid:17)
where Ck is the number of chains st(cid:12)arting and e(cid:12)nding Wenowlookforalowerboundonf forallchaininitial
in the origin and remaining in the same side. In the
positions. If,forexample,q >0,thepreferredsideisthe
0
one-dimensional case it is easy to exactly determine C .
k rightone. Considerthen the startingpointatx=N k
1 k −
It turns out Ck = k2 +1(cid:18) k2 (cid:19), so that, by using the (k0s<tepks≤ofNth)eawndalklet(0gE<, wEith Ek wsoimthe Esubtsheet onfumthbeelraostf
Stirling’s formula, the asymptotic result Ck 2kk−23 elements in E), be the nu|m|b≤er of walk|s h|aving the steps
is found (in d dimensions Ck (2d)kk−d+22 [1∼4]). The belonging to E in the unfavorable side. Upon defining
upper bound on the free energy∼is then: Gk = EgE (Gk <2k) one has
1 lnCk 1 k (xP=N k, qi )= 2N Gk eβ Ni=1qi+
f(0,β) q , (4) Z − { } −
i
≤−β k − k(cid:12)(cid:12)(cid:12)Xi=1 (cid:12)(cid:12)(cid:12) +eβ Ni=1qi (cid:0)gEe−2β (cid:1)i∈EPqi, (10)
and, by using Eq. (2), we obtain: (cid:12) (cid:12)
(cid:12) (cid:12) P XE P
∆f =f(0,β)−f∗ ≤hq0(k,β)≡ sothat,byusingtheinequalitylnx≥lnxandaveraging
1 lnC √k q over the charge distribution, one obtains
k 0
ln2 q G | | , (5)
0
≡ β (cid:20) − k (cid:21)−| | √2∆ ! 1 a(∆,q ,β)E 1
o | |
where the scaling function G is given by f(N −k)≥f∗− βN ln"1+ gE 2N − #,
E
X
2
with a(∆,q ,β) = exp 2∆2β2 2βq . This equation These considerations are sufficient to generalize the
o 0
−
and its analogous in the q <0 case show that there is a previousresultstotheself-avoidingcase,yieldingthefol-
0
(cid:2) (cid:3)
delocalization temperature lowing bounds for the critical temperature:
such that, if β < ββlowe,rf=(N|∆q02| k ) f and the ch(1a1in) |q0|Gl(n√κ2∆|q0|) ≤βc ≤ |∆q02| , (13)
lower ∗
| − | ≥
delocalizes. This argument can be extended to the d- whichdonotdependontheassumption(12)andisthere-
d(2imd)eNnesβionaN=l1cqaisaen,dinfd∗w=hic−hβ11ln≤2dG−k |<q0|(.2d)k, Zd∗(qi) = feosrteimraigteorβouupsp.erAcgaaninb,eindetrhiveedlimbyitu|sqi0n|g/∆Eq≪. (112a):better
The bounds we have proved above allow to conclude
P 1 γ q
that there is a critical value βc such that for values of β βupper = x2G−(xs)ln √2x0∆/|q0| |∆02|. (14)
smaller than β the chain is delocalized in the favorable 0 0
c (cid:16) (cid:17)
solvent,whileforlargervaluesitisadsorbedbytheinter-
face, with the estimates β < β < β (it is easy
lower c upper Generic probability distribution. Up to now we have
to verify that β < β ). The lower bound (11)
lower upper consideredthehydrophobicchargesasindependentlydis-
and the upper bound (9), in the limit q /∆ 1, show
| 0| ≪ tributed Gaussian random variables. Actually, the re-
thesamebehaviorfoundbyusingbothanImry-Matype
sults we have proved do not depend on this assumption.
argument [2] and variational approaches [5,7].
We will briefly sketch this in a few cases [18].
The argument showing localization at any tempera-
Self-avoiding chain. All the results shown for a ran-
ture for a neutral chain holds true, both for randomand
dom chain can be readily generalized for a self-avoiding
self-avoiding chains, if k q √k as k . The
chain. Namely,aneutralchainislocalizedatalltemper- i=1 i ≃ → ∞
atures,whereasanon-neutralchainundergoesalocaliza- central limit theorem e(cid:12)nsures t(cid:12)his for independent ran-
(cid:12)P (cid:12)
tion transition at some critical temperature β . dom variables having a(cid:12) generic(cid:12)probability distribution
c
The delocalization temperature β can be derived with finite variance and null mean. In the non-neutral
lower
exactly in the same way, since the division of walks into case, the existence of a delocalization transition can be
classesaccordingto the number ofsteps made inthe un- provede.g. for a bimodal distribution. This corresponds
favorable solvent does not depend on the self-avoidance to the more realistic case of two kinds of monomers, one
constraint. hydrophilicandtheotherhydrophobic. Wethusconsider
The upper bounds on the free energy, which allow to the generic bimodal distribution [19]:
prove chain localization, requires instead some refine-
ments with respect to the previous case. While the en- P(qi)=αδ(qi q+)+(1 α)δ(qi+q ) , (15)
− − −
ergy term is computed in the same way as before, the
with q ,q > 0. The probability distribution (15)
entropy term is different. Firstly, the connective con- +
−
has three independent parameters, and fixing the av-
stant (κ = 2d for a random walk in d dimensions) is
erage charge q = α(q +q ) q and the variance
different. We recall that the existence of the connective 0 + − − −
∆ = α(1 α)(q +q ) we are left with one free pa-
constant, κ = limN lnSN /N, for self-avoiding walks − + −
→∞ rameter. Itisinterestingtoreportthedelocalizationtem-
(SAW)hasbeenrigorouslyestablished[15](SN istheto- p
perature β , which provides a good estimate for the
talnumberofN-stepsSAWstartingfromthesamesite). lower
critical temperature in the previous cases:
The subleading correction of the form S κNNγ 1
N −
≃
is widely agreed upon, although not rigorously proved
α(1 α) q
[14]. Secondly we introduce the notion of loop, follow- βbim = − ln 1+ 0
ing e.g. [16], and consider only walks made up of N/k lower p 2∆ (cid:12) " α(1 α)∆ (1 α)q0#(cid:12)
(cid:12) − − − (cid:12)
blobs, each blob being a k-loop, in such a way that (cid:12) (cid:12)
(cid:12) p (cid:12)
different blobs can be embedded independently, as well Noticethatinthe li(cid:12)mitofnearlyneutralchain(q0 (cid:12)
| |≪
as for a random chain. A N-loop is a N-steps SAW, ∆) we get βcd ≃ 2|q∆02|, which is the same function of |q0|
starting and ending on the interface, which always re- and∆asintheGaussiancase,suggestingtheexistenceof
mains in the same half-space, with the further condition auniversalbehavior. Inthe limit ofnearlyhomogeneous
x01−x02 ≤xi1−xi2 <xN1−xN2 ∀i. Ithasbeenproved chain (∆ ≪ |q0| which implies α ≃ 0 or α ≃ 1) instead
[17] that the free energy density of loops is the same as βbim = 1 ln α q+ diverges logarithmically
for SAW, κ lim lnL /N = κ, where L is the lower 2(q++q−) 1−αq−
number ofNl -≡loopsN. →Th∞e subNleading correctionisNusually in contrast with th(cid:12)(cid:12)e Ghaussianic(cid:12)(cid:12)ase.
We consider now(cid:12) the case(cid:12)in which the hydropho-
assumed in the same form as for the number of SAW:
bic charges q are not independent random variables,
i
{ }
LN κNNγs−1 . (12) but are Gaussianly distributed with qi = q0 ∀i, qiqj −
≃
3
qi qj = Mi−j1. We assume Mi−i1 = ∆2 ∀i, in anal- We would like to thank Jayanth Banavar, Cristian
ogy with the non-correlated case, and also translational Micheletti and Flavio Seno for useful discussion and on-
invariance along the chain for the correlation matrix: goingcollaboration,andEnzoOrlandiniforbringingref-
M = b(i j ). One can prove that if long range cor- erences [19] to our attention.
ij
| − |
relations decay exponentially or even algebraically the
neutral chain is again localized at all temperatures. In
fact,byassuminganalgebraicdecay,b(r) r η,itturns
−
≃
out that k q kδ/2 with δ = min(η,1). Only if
i=1 i ≃
correlatio(cid:12)ns are so(cid:12)strong that they do not vanish along
(cid:12)P (cid:12)
the chain(cid:12)(η =0),(cid:12)the chain does not localize at all tem- [1] T.Garel,H.Orland,andD.Thirumalai, AnalyticalThe-
peratures. ories of Protein Folding, in New developments in the-
In the non neutral case the existence of the transition oretical studies of proteins, edited by R. Elber (World
can be proved. For example the estimate of the delocal- Scientific), and references therein.
[2] T. Garel, D. A. Huse, L. Leibler, and H. Orland, Euro-
ization temperature is
phys. Lett. 8, 9 (1989).
[3] J.-U.Sommer,G.Peng,andA.Blumen, J. Chem. Phys.
E q
βcorr =min | || 0| . (16) 105, 8376 (1996).
lower E ( i,j EMi−j1) [4] Y. Kantor and I. Golding, Phys. Rev. E 56, R1318
∈
(1997).
If the charges are positivPely correlated (Mi−j1 > 0 for [5] S. Stepanow, J.-U. Sommer, and I. Y. Erukhimovich,
i=j),chainlocalizationismorefavoredthaninthenon- Phys. Rev. Lett. 81, 4412 (1998).
co6 rrelated case, whereas if charges are anti-correlated [6] P. Monari and A. L. Stella, cond-mat/9807133 (1998).
(Mi−j1 <0 for i6=j) it is less favored. [[78]] AT.. TEr.ovCarteoigahntdonA,.PMraortietianns,:coSntdru-mctautr/e9s81a2n3d21M(o1l9e9c8u)la.r
Properties, (W.H. Freeman, New York,1992).
Asymmetric interface potentials. Finally we extend [9] H.R.Brown,V.R.Deline,andP.F.Green,Nature341,
our demonstrations to the random AB-copolymers stud- 221 (1989).
ied by Sommer et al. [3]. Their model corresponds to [10] C.-A. Dai, B. J. Dair, K. H. Dai, C. K. Ober,
consider the following Hamiltonian: E.J.Kramer, C.-Y.Hui,andL.W.Jelinsky,Phys. Rev.
Lett. 73, 2472 (1994).
= q [λθ(q )θ( ~u ~r )+θ( q )θ(~u ~r )], (17) [11] For a more refined model, see e.g. R. Bonaccini and
i i i i i
H | | − · − · F. Seno,pre-print (1998).
i
X [12] Thecasewheretheenergyis−λ+qi andλ−qi(λ+,− >0)
with the charges distributed according to Eq. (15) with when~u·~ri >0and~u·~ri <0respectivelyisreadilyreduced
α = 1/2 and q = q (λ 1 measures the poten- to the one treated here apart from an additive constant
+
tial asymmetry). Such−an|AB−-co|polymer (Eq. (17)) is and a redefinition of the“charges” qi′ = λ++2λ−qi.
equivalent to a non-neutral chain in symmetric poten- [13] Noticethatintheoff-latticemodelofRef.[2]theentropy
hasonlyaharmonicterm(Edwardsfunctional)andthus
tials (λ=1), a case that we have already discussed. We
thed-dimensionalcasereducestriviallytothed=1case.
have proved the existence of a delocalization transition
Inthelatticemodelconsideredherethisisnomoretrue.
for a neutral chain also in the Gaussian case. For both
[14] B. D. Hughes, Random Walks and Random Environ-
distributions, the delocalization temperature shows the
ments, Vol. I, Clarendon Press, Oxford, 1995.
behavior βlower ≃ |λ∆−1| in the limit of nearly symmetric [15] J. M. Hammersley, Proc. Camb. Phil. Soc. 53, 642
potentials(λ 1),inagreementwiththescalinglawand (1957).
≃
the numerical results found in [3]. On the contrary, in [16] S. G. Whittington, J. Phys. A: Math. Gen. 31, 3769
thehighlyasymmetriccases(smallandlargeλ)different (1998).
asymptotic behaviors for β occur [18]. [17] J.M.Hammersley,G.M.Torrie,andS.G.Whittington,
lower
J. Phys. A: Math. Gen. 15, 539 (1982).
[18] Details will be published elsewhere.
To conclude, in this Letter we have proved several ex-
[19] Some previous exact results for the bimodal distribu-
act results on random heteropolymers in the presence of
tion and only for ideal chains were also obtained by
an interface. Namely, a neutral chain is localized at all Ya. G. Sinai, Theory Probab. Appl. 38, 382 (1993), and
temperatures,whereasachargedchaindelocalizesatafi- E. Bolthausen and F. den Hollander, Ann. Probab. 25,
nite temperature. The results arequite generaland hold 1334 (1997).
for ideal and self-avoiding chains, Gaussian and bimodal
distribution with independent and correlated charges.
Furthermore, our lower bounds for the transition tem-
perature confirm previous estimates.
4