Table Of ContentA Model Ground State of Polyampholytes
Shay Wolfling and Yacov Kantor
School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel
Thegroundstateofrandomlychargedpolyampholytesisconjecturedtohaveastructuresimilar
to a necklace, made of weakly charged parts of the chain, compacting into globules, connected by
highlychargedstretched‘strings’. Wesuggestaspecificstructure,withinthenecklacemodel,where
alltheneutralpartsofthechaincompactintoglobules: Thelongest neutralsegmentcompactsinto
8
a globule; in the remaining part of the chain, the longest neutral segment (the 2nd longest neutral
9
9 segment) compacts into a globule, then the3rd, and so on. We investigate the size distributions of
1 the longest neutral segments in random charge sequences, using analytical and Monte Carlo meth-
ods. We show that the length of the nth longest neutral segment in a sequence of N monomers is
n proportional to N/n2, while the mean number of neutral segments increases as √N. The polyam-
a
J pholyteinthegroundstatewithinourmodelisfoundtohaveanaveragelinearsizeproportionalto
√N, and an average surface area proportional to N2/3.
1
36.20.-r,02.50.-r,05.40.+j,33.15.Bh
]
t
f
o
s I. INTRODUCTION imation. The extreme sensitivity of the ground state
. structure to the excess charge, noted by Kantor et al.
t
a The desire to understand long chain biological macro- [10,11],andsupportedbyMonteCarlo(MC)simulations
m
molecules, and especially proteins, stimulates extensive aswellasvariationalmeanfieldcalculations[12],resolves
- studies of polymers [1–3]. An important class of poly- the apparent contradiction between the scaling and the
d
n mers are those with an electric charge along their back- Debye-Hu¨ckel motivated arguments.
o bone. This work deals with heteropolymers,which carry To gain some insight into the behavior of PA’s, analo-
c positive and negative charges, known as polyampholytes gies to charged drops were explored [13–15]: A spher-
[ (PA’s) [4]. Models of PA’s are important to the study of ical drop charged beyond a certain charge, called the
1 proteins, since under normal physiological conditions, 5 Rayleigh charge QR, which depends on the surface ten-
v of the 20 naturally occurring amino acids have anexcess sion and volume of the drop, becomes locally unstable
2 charge[1]. We considera polymerofchargedmonomers, to elongation, since the pressure difference between the
0 interactingviaunscreenedCoulombinteractions,andwe inside and outside of the drop vanishes. Even before the
0
investigate its ground state structure. Throughout this total charge reaches QR, the drop becomes unstable to
1
work,wediscussPA’sthatconsistofarandommixtureof splitting into two equal drops. Additional splittings of
0
8 positive and negative charges, which cannot move along the drop occur for larger charges. Similar behavior is
9 the chain. expected in PA’s charged to QR √N. Although a
∼
/ We are interested in the geometrical features of the PA cannot split, the analogy to charged drops can still
t
a ground state of a PA, and in particular in the depen- be exploited: Constraining the structure to maintain its
m dence of its radius of gyration (r.m.s. size) R on the connectivitybyattachingdropletswithnarrowtubes,re-
g
- numberofmonomersN: Rg Nν. HiggsandJoanny[5] sults in a necklace-type structure of droplets connected
d and later Wittmer et al. [6],∼elaborated on arguments of by strings.
n
Edwardsetal.[7]andsuggested,onthebasisofaDebye- For homogeneously charged polymers (polyelec-
o
Hu¨ckel type theory [8], a collapsed structure (ν = 1/3) trolytes), the charged drop analogy was used [16] to
c
: forneutralPA’sinthegroundstate: Thechaintakesad- characterize the structure completely within the neck-
v
vantage of the presence of two types of charges along its lace model (including the number of ‘beads’ and strings,
i
X backbone, and assumes a spatial conformation in which their sizes, and the number of monomers in them, for a
r every charge is predominantly surrounded by charges of given temperature and charge density). However, trying
a an opposite sign. to apply the necklace model to quenched PA’s having
A different approach to the study of the ground state random charges [13,14], several difficulties occur due to
ofPA’s[9],isbyscalingarguments,requiringthatthein- the randomness. It wasnoted, for instance,that a situa-
teraction energy be the same on all length scales. These tionoccurs,inwhichmostsphericalshapesareunstable,
arguments lead to a stretched structure of the ground while there is on average no energetic gain in splitting
state, where R N. In this approach R is averaged the sphere into two equalparts. A consistent theoretical
g g
∼
overacompleteensembleofallquenches,andthetypical picture for random PA’s beyond the instability thresh-
overall(excess)chargeQofthePAis √N. Whereasin old was not found, but a typical PA is conjectured to be
∼
the Debye-Hu¨ckel approximation,all the chains are neu- composed of rather compact globules connected by long
tral, since the overallneutrality of the PA is an essential strings. In orderto reduce the electrostaticalenergy,the
condition for the validity of the Debye-Hu¨ckel approx- globulesconsistofsegmentsofthechainthatareapproxi-
1
matelyneutral(collapsingaccordingtotheDebye-Hu¨ckel Kantor and Erta¸s [17–19] attempted to quantify the
picture), while the strings are formed by highly charged qualitative necklace model, by postulating that the
segments. ground state of a PA will consist of a single globule,
Since the structure of a randomly charged PA cannot formed by the longest neutral segment of the PA, while
be characterized completely analytically, we turn to nu- theremainingpartwillformatail. Theyinvestigatedthe
mericalMC methods in orderto characterizethe ground probability that the longest neutral segment in a chain
state of such PA’s within the necklace model. A key of N monomers has length L. A probability density was
role in the structure of randomly charged PA’s is played defined and investigated in the limit where N,L ,
→ ∞
by the neutral segments in the chain (forming the beads while the reduced length l L/N is fixed. The proba-
≡
in the necklace). We therefore apply MC methods to bility density was obtained numerically and investigated
study the neutralsegments sizedistributionofrandomly analytically, but a complete analytical solution to the
charged PA’s. The rest of the paper is organized as fol- problem was not found. We follow [17], and investigate
lows: InsectionIIwedescribetheprocessofconstruction the problem of the size distribution of neutral segments
ofagroundstateforarandomlychargedPA,bydividing in randomly charged PA’s by mapping the charge se-
it into neutral segments. We discuss the motivation for quence into a one-dimensional random walk (1-d RW).
thisprocess,definetheimportantparametersoftheprob- The charge sequence ω = q (i = 1,...,N; q = 1)
i i
{ } ±
lem and compare the process to similar existing models. is mapped into a sequence of positions S (ω) = i q
i j=1 j
In section III we investigate the sizes of the globules in (S =0)ofarandomwalker. (Fromnowwewillmeasure
0
the suggested ground state, including their dependence charges in units of the basic charge q , and therPefore q
0 j
on the total number of monomers and on other param- will be dimensionless.) The random sequence of charges
eters. In section IV we discuss effects of finite chains, is thus equivalent to a RW, a chain segment with an ex-
investigate the dependence of the total number of neu- cess charge Q corresponds to a RW segment with total
tral segments in such finite chains on N, and construct displacementof Q steps, anda neutralsegmentis equiv-
a self consistent picture of the structure of the ground alent to a loop inside the RW. Throughoutthe paper we
state. In section V we obtain some of the physical char- will use the terminologies of randomly chargedPA’s and
acteristics of the constructed ground state, such as its of RW’s interchangeably.
linear size and surface area, and in the last section we
discuss our results and compare them to other studies.
w(cid:13) = {+ - - - + + - + + + + + - + - + + + + - - + - + + + - + }(cid:13)
L(cid:13) L(cid:13) L(cid:13) L(cid:13)
1(cid:13) 3(cid:13) 2(cid:13) 4(cid:13)
II. THE MODEL - MOTIVATION AND
DEFINITIONS 8(cid:13)
6(cid:13)
Empirical observations of the energy of randomly
4(cid:13)
chargedPA’ssuggestedthatthe quench-averagedenergy
can be presented as a sum of condensation, surface and S(cid:13)((cid:13)w(cid:13) )(cid:13) 2(cid:13)
i(cid:13)
electrostatic energies [13,14]: 0(cid:13) i(cid:13)
q2 Q2 -2(cid:13)
E = 0N +γS+ , (1)
− a R
L(cid:13) L(cid:13) L(cid:13) L(cid:13)
1(cid:13) 3(cid:13) 2(cid:13) 4(cid:13)
where q2/a is the condensation energy gain per parti-
0
cle (q is the typical charge of a monomer and a is a
0
microscopic distance such as diameter of the monomer), L(cid:13)1/3(cid:13)
1(cid:13)
γ ∼q02/a3 is the surface tension, and R is the linear size L(cid:13)12(cid:13)/3(cid:13)
of the chain. (In Eq. (1) we omitted dimensionless pref- FIG.1. Anexampleofachargesequenceω,mappedintoa1-d
actors of orderunity.) When the excess chargeQ is very
RWSi(ω),andatypicalloopsstructure. Filledcirclesindicatethe
small, the PA will form a single globule. When Q in- starting and ending points of loops. The longest loop in the RW
creases,the electrostaticforceswill overcomethe surface has 8 steps (L1 = 8), the 2nd longest loop has 6 steps (L2 = 6),
tension and the globule will split (similarly to the split- L3=4andL4=2. Theexcess charge(whichisequivalenttothe
tingofadropchargedbeyondthe Rayleighcharge). Nu- total displacement of the RW) is Q = +8 and the total length is
mericalstudiessuggest[13–15]thataPAformsanecklace N =28.
of weakly chargedglobules, connected by highly charged
strings. This structureis acompromisebetweenthe ten- When the longest neutral segment forms a globule (as
dency to reduce the surface area (i.e. to form globules) assumed in [17–19]), the remaining part of the chain is
due to surface tension, and the tendency to expand, in very large ( N). It is natural to assume that neutral
∼
order to reduce the Coulomb interaction caused by the segments on that tail will further reduce the total en-
excess charge. ergy by folding into globules. Eventually, the necklace
2
willconsistofmanyneutralglobules. However,thereare ferent.) We will return to this absence of self-similarity
many ways in which the chain can be divided into neu- in section III.
tral segments, and we are interested in a simple unique
structure. We therefore suggest a specific necklace-type
structure,andconstructthegroundstateforarandomly III. SIZE DISTRIBUTION OF NEUTRAL
chargedPA in the following way, depicted in Fig.1: The SEGMENTS
longest neutral segment contains L monomers; it com-
1
pactsintoaglobuleoflinearsizeproportionaltoL1/3;in In order to characterizea PA in the necklace-type low
1
the remaining part of the chain the longest neutral seg- energystate,madeofneutralglobulesandhighlycharged
ment (the 2nd longest neutral segment) of size L also strings,we areinterestednotjust inthe sizedistribution
2
compactsintoaglobule,thenthe3rdandsoon,untilthe of the longest neutral segment, but also in the size dis-
segments become very small (of only a few monomers). tribution of the 2nd longest segment, 3rd longest, and
Generally, L denotes the number of monomers in the so on. We examined the statistics of loops in a 1-d
n
nthlongestneutralsegment,whichcompactsintoaglob- RW of N steps, using MC method for several N’s, up
ule of radius L1/3. Eventually, all the neutral segments to N = 104. For each N we randomly selected 106 se-
n
quences. For each sequence we found the lengths L of
are exhausted and we are left only with strings which n
the (non-overlapping) loops, and calculated their distri-
carry the PA’s excess charge Q, and connect the glob-
butions and averages.
ules. The total number of monomers in the chain is the
We denote by P (L ) the probability of the nth
number of monomers in all the neutral segments, plus N,n n
longestloop ina RW ofN steps to be of lengthL . The
the excess charge. n
averagelengthofthelongestloopwasfound[17,18]tobe
The‘groundstate’,generatedbythisprocessoflongest
proportional to N. Expecting the same behavior for the
neutral segments compacting into globules, does not
length of the nth longest loop, we define the probability
necessarily minimize the total energy of the PA. First
density of the nth longest loop:
of all, the process does not consider the possibility of
weakly charged globules, which can include much more N
p (l ) [P (L )+P (L +1)] , (2)
monomers than the neutral globules, thus compensating n n ≡ 2 N,n n N,n n
by surface energy for the additional electrostatic energy.
where l L /N. (Note that at least one term in
Secondly, even when considering only neutral globules, n ≡ n
Eq. (2) vanishes, since loops can be only of even length.
it is not necessary that the procedure of compacting the
Therefore,definition(2)includesaverageofprobabilities
longest neutral segment at each step generates the low-
for L and L +1 as in the definitions used in contin-
est energy state: It is possible that two ‘medium sized’ n n
uum limits for discrete RW’s, in order to prevent even-
globules will have lower surface energy than a long one
odd oscillations.) For small N’s this definition of p (l )
and a very short one (which remains after the first long n n
still depends on N, but in the N,L limit it be-
segment was already chosen). n → ∞
comes a function of only l . Numerical evidence for the
n
A similar model, describing a breaking of a RW into N independence of p (l ) for N was presented in
1 1
loops, is the Loop Erased Self-Avoiding Walk (LESAW) [18−]. MC results for p (l ) for se→ver∞al values of n indi-
n n
model [20]. A LESAW is constructed from a RW, cate that the probability densities are virtually indepen-
in which any loop generated by a self intersection is dent of N [28]. For small N and large n the probability
erased. The LESAW model was subject for intensive density p (l ) depends on N, since L ’s are short, and
n n n
study [21–25], and many properties of the lengths of the continuum limit is expected only when L 1. In or-
n
erased loops in it were found. However, these results do der to overcome effects of finite N, we dete≫rmined the
not consider issues of the longest erased loops, and are N dependence of L , through an extrapolationof the
n
therefore of little help to us. slo−pes of the linearhfitsiof log L vs. logN to N .
n
h i → ∞
Other related models are those of randomly broken For any given n we calculated, through a local linear fit,
objects [26,27], in which an object, such as a segment slopes for several values of N, and estimated them at
of unit length, is divided into mutually exclusive parts N limit. Furthermore, it can be proven analyti-
→ ∞
by a self-similar random process. The division of a seg- cally [28]thatthe probability density of the longestloop
ment into parts resembles the ‘breaking’ of a RW into is ‘universal’, i.e. for long RW’s, it does not depend on
loops: The probability densities of the longest segments the number of steps, or on details of a single step.
insome models ofrandomlybrokenobjects resemblethe Since p (l ) are N independent, we can investigate
n n
−
probability density of the longest loop in [18]. However, them for any given (large enough) N. The probability
one main difference between models of randomly broken densities of the five longest loops in a chain are depicted
objects and the longest loop problem, is that the prob- in Fig.2 for N = 1000. Several properties of p (l ) are
n n
abilities of the longest loops are not self-similar. (The evident from this figure. The probability density p (l )
1 1
probability of a certain fraction of the chain to form the wasshown[18]tohaveasquarerootdivergenceatl =1,
1
longest loop and the probability of the same fraction of and a discontinuous derivative at l = 1. When the
1 2
theremainingchaintoformasecondlongestlooparedif- length of the first loop L in a specific chain is long,
1
3
then the lengths of the other loops L (for n > 1) in at1/n′,since alltheprobabilitydensitiesaredependent.
n
thatchainmustbe short,sincethe totallengthofallthe Apparently,thesesingularitiesdonotcausediscontinuity
loops cannot exceed N. Therefore, since in many chains ofthefirstderivative,and,thereforearenotvisibleinthe
the length of the longest loop is almost equal to N (as numeric data. In severalmodels of randomly broken ob-
indicated by the divergence of p (l ) when l 1), the jects[26,27],theprobabilitydensitiesofthelengthofthe
1 1 1
lengths ofthe other loops approachzero. This→is evident nth segment have singularities at l = 1/n′ (for n n′).
≤
from the divergence of p (l ) when l 0 for all n>1. Thesesingularitiesareduetoaself-similarprocess,which
n n n
→
Becauseinanyspecificchainthelengthofthenthlongest leads to a different analytical expression for the proba-
loop is shorter than the length of the kth longest loop, bilities on each interval 1/(n′+1)<l <1/n′. Since the
for n > k, the divergence of p (l ) near zero is stronger random process of generating the longest loops in our
n n
for large n. Since the probability densities p (l ) are process is not self-similar (as indicated in the previous
n n
normalized separately for each n, then any two of them section), the reason for the singularities in the models
must intersect (i.e. pn(ln) always intersects pn′(ln′) for of randomly broken objects does not hold in the case of
n = n′). The length of the second longest loop never the nth longest loop (as was suggested by Frachebourg
6
exceeds the length of the first longest loop, and the sum et al. [29]). Therefore, the probability of the nth longest
of their lengths never exceeds N. Therefore, the length loop does not necessarily have singularities at values of
of the second longest loop cannot exceed half the length l=1/n′.
opf(tlhe) cvhaaniinsh.eCsoindseenqtuiceanltlyly,fol2r≤l 12>fo1r.alSlitmhielacrhlyainwse,acnand eraSginecleeningtahnoyfstpheecinfitchRlWon,gLes1t≥loLop2 ≥L···d≥ecLrnea,stehse(afovr-
2 2 2 2 h ni
showthatl 1/nforallnandp (l )=0forl >1/n. fixed N) with increasing loop number n. There is no
n n n n
≤
typicalscale inthe problem,andwe mayexpect apower
law dependence L Nn−α, with α > 0. The total
n
h i ∼
number of steps in all the loops in any given RW cannot
8
exceedN (i.e. L N),andtherefore L N.
n n ≤ nh ni≤
7 The convergence of the sum means that α > 1. We de-
pict in Fig.3 tPhe average reduced length Pl vs. n on
n
h i
6 a logarithmic scale. To avoid systematic errors due to
finite N, each value of l in the graph was determined
n
h i
5 through an extrapolation: For each n, we plotted l
n
h i
vs. 1/N, and found the extrapolated value of l near
p (l )nn4 1/N =0. ThesevaluesofhlniaredepictedinFihg.n3i. The
linearfittothedatapointshasaslopeof 2.3 0.4. We
3 − ±
therefore conclude that as N :
→∞
2 N
L , where α=2.3 0.4 . (3)
h ni∼ nα ±
1
At the next section we will argue that α = 2, which is
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 within the error limits of Eq. (3).
ln
FIG. 2. Probability densities of the 5 longest loops from
right to left: p1(l1)-thick points, p2(l2)-solid line, p3(l3)-dashed 0
line, p4(l4)-dot-dashed line, p5(l5)-dotted line, as a function of
ln = Ln/N from MC results of 106 random sequences of length −0.2
N =1000. −0.4
−0.6
Wenotethatp (l )isqualitativelysimilartotheprob-
2 2
ability densities of the length of the second segment in −0.8
>)
different models of randomly broken objects [26,27] (see <l n −1
Figs. 1(b) and 3(b) in [26]). However, the probability g (
o
densities ofthe lengthofthe secondsegmentin[26]have l−1.2
strong singularities (of the first derivative) at l2 = 1/4, −1.4
and are shown to have singularities at l = 1/k for all
2
−1.6
integer k 2. All the probability densities p (l ) of the
n n
≥
longest loops have a singularity when they become iden- −1.8
tically zero (p (l ) vanishes for l > 1/n and therefore
n n n −2
is non-analytical at l = 1/n). It is possible that due to −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
n log (n)
the singularity in pn′(ln′) at 1/n′ all the other probabil- FIG. 3. Logarithm (base 10) of average reduced length of the
ity densities pn(ln) (for n < n′) also have singularities nthlongestloopasafunctionoflogarithmofloopnumber n.
4
IV. NEUTRAL SEGMENTS IN FINITE CHAINS Attheendofthissectionwewillarguethatα=2and
y =0.5, values which are within the error limits of those
deduced from the numeric data. In order to confirm the
So far we have discussed only long chains and their n √N relation, we show in Fig.4 the probability
f
properties in the N limit. There are several dif- h i ∼
ferences between the→pro∞perties of infinitely long chains density of nf divided by √N for several chain lengths
(N = 102 104). The division by √N causes a reason-
andfinitesizechains. Firstofall,infinitesizechainsthe ÷
able collapse of the graphs for different values of N to a
monomers that do not belong to any neutral segment
single function, which is (almost) N independent. We
constitute afinite partofthe chain,asopposedtoavan- −
see that the probability density has a maximum when
ishingpartasN . Secondly,ininfinitelylongchains
→∞ n /√N is close to zero (the most probable value of n
thenumberofneutralsegmentsisinfinite,whileforfinite f f
is finite and independent of N), and it decreases to zero
N at some point there are no more neutral segments.
with increasing n /√N.
Let us consider the total number of monomers which f
do not belong to any neutral segment, for finite N. It
is easily shown that the number of monomers left out 3
from all the neutral segments in any specific random se-
quence is exactly the excess charge Q of the correspond-
2.5
ing PA: The excess charge cannot exceed the number of
monomers left out, because, by definition, the neutral
2
segments do not have excess charge. On the other hand,
ablelpthaertmoofnthomeeexrscelessftcohuatrgfreo:mIf,thfoernineusttarnalcese,gamPeAnthsamsuexst- 0.5n /N )f1.5
cess positive charge, then no negative charge can be left p(
out from all the neutral segments. (If there had been a
1
groupofnegativecharges,itwouldhavejoinedagroupof
positive charges to make a neutral segment, or together
0.5
with a group of positive charges, would have joined an
existing neutral segment and make it longer). Since the
r.m.s. excess charge of a randomly charged sequence of 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
N charges of 1 is equal to √N, the r.m.s. number of n f /N0.5
monomersnot±inanyneutralsegmentinthechainisalso FIG.4. Probabilitydensityofnf,thetotalnumberofloopsin
equalto √N, becoming a vanishing fractionof the chain achain, divided by√N, forN = 100, 300, 500, 1000, 3000, 5000,
10000(fromrighttoleft).
as N .
→∞
Atsomepointtheprocessofsearchforthenextlongest
Knowingthe statisticalproperties of the chain, we try
loop exhausts all the loops in the RW. We investigate
to construct a self-consistent complete picture, in which
this stage in the process, by analyzing the number n of
f our numerical results fit together. We know that the
loops in a RW. When N is infinite, the averagelengthof
total length of all the loops ( nf L ) equals to the en-
the nth longest loop is given by L Nn−α. Applica- n=1 n
h ni∼ tire length of the chain minus the steps not in any loop
tionofthisequalityforfiniteN’swouldpredict,forlarge P
(which are the excess charge, which are on the average
enoughn’s, L 1. Sincethisisnotpossible(themini-
h ni≤ √N). Dividing this equality by N, taking an average
mallengthofaloopistwosteps),wearguethatEq.(3)is
over the random sequences, we get:
valid, for finite N’s, only to describe the averagelengths
of the longest nf loops. There is no typical scale to the 1 nf
problem of the total number of loops, and we therefore 1 = l . (5)
eoxfpthecetlaasptolowoepr,lfaowradlelpcehnadinenlceengotfhhsn,fisiuosnuNall.yTvheerylesnhgotrht − N12 hnX=1 ni
(consisting of only few positive and negative charges), On the other hand, from Eqs.(3) and (4) we know that,
having length independent of N, i.e. L N0, which omitting constants of order unity:
h nfi∼
means that l N−1. Since l n−α (substitut-
hnfi ∼ hnfi ∼ f hnfi hnfi Ny
ing n=nf in Eq. 3), we can expect: l = l n−αdn 1 Ny(1−α) .
n n
*n=1 + n=1h i≃Zn=1 ∼ −
1 X X
nf Ny , where y = . (4) (6)
h i∼ α
Comparisonof the powersof N in Eqs.(5) and (6) leads
Substituting α from Eq. (3), leads to a value of y =
to
0.43 0.09. Investigating numerically the dependence of
±
n onN,wefoundthaty =0.46 0.06,whichiswithin 1
h fi ± y = , (7)
theerrorlimitsofthevaluepredictedbyEqs.(3)and(4). 2(α 1)
−
5
which together with Eq. (4) is satisfied by:
0.5
1
α=2 , y = . (8)
2
0.4
These equalities are satisfied by the values of y =0.46
±
0tu.0t6e,aansdelfα-c=on2si.s3t±en0t.4piocbtutarein,eidnnwuhmicehrictahlelya,vaenrdagceoncostni-- 0.5R/N ) 0.3
formationalpropertiesoftheconstructedgroundstatefit p (
together. 0.2
0.1
V. PHYSICAL PROPERTIES OF THE GROUND
STATE
0
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
(R−R0 )/N0.5
Inthissectionweinvestigatesomeofthephysicalchar- FIG. 5. Probability density of R, the linear size of the en-
acteristics of the constructed ground state of randomly tire chain, for several chain lengths: N =100 (dotted line), 300
charged PA’s. We focus on the linear size R and on the (dot-dashedline),1000(dashedline),3000(solidline),10000(thick
surfaceareaS oftheproposedgroundstate,tryingtoex- points). The minimal possible value of R (R0 = N1/3) is sub-
plaintheirdependenciesonN throughtheselfconsistent tracted from R, and the result is divided by √N to collapse the
pictureconstructedintheprevioussection. WedefineR, data.
the linear size of the chain, according to the picture of
Fig.1: The neutral segments in the chain compact into We can explain the dependence in Eq. (10), by as-
globules (each with a linear size of Rsegment ∼L1se/g3ment). suming that nhLn1/3i and nhLni1/3 have the same
If all the globules are linearly packed then the total lin- N dependence,andbyusingthepowerlawsof n and
f
ear size is the sum of the linear sizes of all the globules L− (Eqs. 3Pand 4), with α=P2 and y = 1 (Eq.h8):i
h ni 2
(R = R ). The linear size of the chain must
segment
includePthe monomers not in any neutral segment (the nf hnfi Ny N 1/3
‘strings’inthenecklace,whicharethetotalexcesscharge L1/3 L 1/3 N0.5 .
Q). Wethereforegetameanstodescribethechain’ssize: n=1h n i∼ n=1h ni ∼Zn=1(cid:18)nα(cid:19) ∼
X X
nf (11)
R L1/3+Q . (9)
≡ n
n=1 In order to confirm this dependence, we investigated
X
nf L1/3 as a function of N, and found the power
(Here, and throughout this section, we omit prefactors h n=1 n i
lawdependencewithexponent0.49 0.02,inaccordance
of order unity.) It is evident that the generated state P ±
with the prediction of Eq. (11). We see that the average
captures some essentialfeatures of the groundstate sug-
linear chain size R in Eq. (9) is a sum of two terms,
gested by the necklace model: The necklace type struc- h i
ture is compact (i.e. R N1/3) when the PA is neu- each proportionalto √N.
∼ Similarly to the definition of R, we can define the sur-
tral (the longest neutral segment is the entire chain) or
faceareaS ofthe chain. Eachneutralsegmentcompacts
has very small excess charge, and begins to stretch as
the excess charge increases (the charged strings become into a globule of surface area L2/3 , allthe globules
∼ segment
longer). Finally, the PA becomes completely stretched are linearly packed, and the surface area of the ‘strings’
(i.e. R N) for the fully chargedpolymer. is proportional to the number of monomers not in any
Inves∼tigating the average chain’s linear size depen- neutral segment, which is the excess charge. We can
dence on N, we get the dependence: therefore define:
R Nν , where ν =0.50 0.01 . (10) nf
h i∼ ± S L2/3+Q . (12)
≡ n
Fig.5 depicts the probability density of R, divided by n=1
X
√N for several values of N. From the data collapse we
deduce that the N dependence in Eq. (10) is valid not S canhavevaluesrangingfromN2/3,foraneutralchain,
−
just for the average linear size, but represents a scaling to N, for a completely charged chain. These limits in-
of the entire probability density. The ν 1 power in deed correspond to the expected behavior of PA. Obvi-
Eq. (10) means that the chain is not comp≃act2(although ously S N2/3. We analyzed the N-dependence of S,
≥
thedistributionis‘peaked’nearthelowestpossiblevalue and found that:
of R), and is not completely stretched, but has a linear
size as an ideal RW with N steps. S N0.67±0.01 . (13)
h i∼
6
When we subtract from S its minimal value, and divide of ‘beads’ and ‘strings’ in the necklace, the spatial ex-
the result by N2/3, we get a distribution which is iden- tent, the surface area and energy). This structure is
tical for all N. This dependence means that the average compact when the chain is neutral or weakly charged,
surface energy of the generated structure has the same and stretches as the chain becomes charged. We have
N dependenceasthesurfaceenergyofasinglecompact found that the groundstate structure has a very low en-
−
globule (or several compact globules, each containing a ergy, which depends on the number of monomers as the
finite part of the chain). From the same arguments that energyofasinglecompactneutralglobule. Wehavealso
led to Eq. (11), we get: shown that the unrestricted average of the linear size of
the polymer in the ground state depends on the number
hnfi Ny N 2/3 of monomers as the linear size of an ideal chain, with
hSi=hQi+n=1hLni2/3 ∼√N +Zn=1(cid:18)nα(cid:19) ∼N2/3 . qaucarliittiactaivleexbpehoanveniotrooffoνur=di0st.5ri0bu±ti0o.n01fo.rAthlethlionuegahrstihzee
X
(14) is similar to this obtained in [15] (it is peaked near its
smallest possible value, and has a tail, which determines
This power of N is in accordance with the numerically the asymptotic behavior of ν), Kantor and Kardar [15]
obtained value of 0.67 0.01, and indicates that the N- concluded that the average linear size increases with N
±
dependence of the averagesurface area is determined by at least as fast as a self-avoiding walk (i.e. ν >0.6). We
the neutral segments (i.e. the ‘beads’ in the necklace), believe that it is worthwhile to slightly alter the way in
and is not affected by the excess charge (the ‘strings’ in whichcompactglobulesareformedwithinourmodel(by
the necklace). allowingforweaklychargedglobulesorbynotforcingall
We investigated the expression S+ Q2, which has the theneutralsegmentstocompletelycompact),inorderto
R
sameN-dependenceastheenergyofthegeneratedstruc- try to reproduce this ‘swelling’ of the average chain.
ture. (We considered the energy terms of Eq. (1), and Theinvestigationofthesizedistributionofneutralseg-
omittedthecondensationterm,sinceitisthesameforall ments through the analogy to 1-d RW’s, was limited to
structures of a given N). Exploring the N dependence a particular class of RW’s, in which a unit displacement
of S+ Q2 (denoted as E), we found that: − appears at each step (i.e. each monomer in the chain
R is charged 1). In a future publication [28], we define
±
Q2 the problem of longest loops in the limit where the RW
E S+ N0.66±0.01 . (15) becomesatrueGaussianwalk,andprovethroughascal-
h i≡h R i∼
ing process the universality of the probability densities
WhensubtractingfromE itsminimalvalueanddividing of longest loops. This universality means that the prob-
the result by N2/3, we get a probability density which ability densities of longest loops are independent of the
is identical for all N. This dependence means that the number of steps and of the nature of the single step of
total energy is very low: The surface energy term (S) the RW. Therefore, the results obtained for a specific
gets its minimal value ( N2/3), and R N1/2, thus caseofrandomlychargedPA’s arevalidforallrandomly
∼ ∼
bringing the electrostatic energy term to a point where charged polymers, in which (for long chains) the charge
it does not affect the N-dependence of the total energy. distribution is an unbiased Gaussian.
The total energy behaves very much like the surface en-
ergy, as if the chain is constructed of a single compact
neutral globule. ACKNOWLEDGMENTS
This work was supported by the Israel Science Foun-
VI. CONCLUSIONS AND DISCUSSION
dation under grant No. 246/96.
We havestudied the propertiesof a conjecturedstruc-
ture of randomly charged PA’s in the ground state. Ac-
cording to the necklace model [13,14], in the ground
state ofPA’s neutralsegments inthe chaincompactinto
globules. Following Kantor and Erta¸s [17,18], we have
mapped the problem of longest neutral segments to the
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8
