DISORDERED HETEROPOLYMERS: MODELS FOR ... papers/69.pdfPolymers are good candidates for materials...

DISORDERED HETEROPOLYMERS:MODELS FOR BIOMIMETIC POLYMERSAND POLYMERS WITH FRUSTRATING

QUENCHED DISORDER

Arup K. CHAKRABORTY

Department of Chemical Engineering, and Department of Chemistry, University of California,Berkeley, CA 94720, USA

AMSTERDAM } LONDON } NEW YORK } OXFORD } PARIS } SHANNON } TOKYO

A.K. Chakraborty / Physics Reports 342 (2001) 1}61 1

E-mail address: [email protected] (A.K. Chakraborty).

Contents

1. Introduction 42. Biomimetic recognition between DHPs and

multifunctional surfaces 52.1. Theory of thermodynamic properties 82.2. Monte-Carlo simulations of

thermodynamic properties 232.3. Kinetics of recognition due to statistical

pattern matching 30

2.4. Connection to experiments and issuespertinent to evolution 41

3. Branched DHPs in the molten state } modelsystem for studying microphase ordering insystems with quenched disorder 44

Acknowledgements 52Appendix 52References 59

Physics Reports 342 (2001) 1}61

Disordered heteropolymers: models for biomimetic polymersand polymers with frustrating quenched disorder

Arup K. Chakraborty

Department of Chemical Engineering, and Department of Chemistry, University of California, Berkeley,CA 94720, USA

Received December 1999; editor: M.L. Klein

Abstract

The ability to design and synthesize polymers that can perform functions with great speci"city wouldimpact advanced technologies in important ways. Biological macromolecules can self-assemble into motifsthat allow them to perform very speci"c functions. Thus, in recent years, attention has been directed towardelucidating strategies that would allow synthetic polymers to perform biomimetic functions. In this article,we review recent research e!orts exploring the possibility that heteropolymers with disordered sequencedistributions (disordered heteropolymers) can mimic the ability of biological macromolecules to recognizepatterns. Results of this body of work suggests that frustration due to competing interactions and quencheddisorder may be the essential physics that can enable such biomimetic behavior. These results also show thatrecognition between disordered heteropolymers and multifunctional surfaces due to statistical patternmatching may be a good model to study kinetics in frustrated systems with quenched disorder. We alsoreview work which demonstrates that disordered heteropolymers with branched architectures are goodmodel systems to study the e!ects of quenched sequence disorder on microphase ordering of molten

0370-1573/01/$ - see front matter ( 2001 Elsevier Science B.V. All rights reserved.PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 0 0 6 - 5

copolymers. The results we describe show that frustrating quenched disorder a!ects the way in whichthese materials form ordered nanostructures in ways which might be pro"tably exploited in applications.Although the focus of this review is on theoretical and computational research, we discuss connectionswith existing experimental work and suggest future experiments that are expected to yield further in-sights. ( 2001 Elsevier Science B.V. All rights reserved.

PACS: 87.15.Aa; 82.35.#t

3A.K. Chakraborty / Physics Reports 342 (2001) 1}61

1. Introduction

Synthetic polymers have enormously impacted societal and economic conditions because theyare commonly used to manufacture a plethora of commodity products. This is one of the drivingforces that continues to spur fundamental research aimed toward understanding the physics ofmacromolecules and learning how to chemically synthesize them. Another motivation for suchresearch is, of course, intrinsic interest in the fascinating behavior of macromolecules. Researchconducted by several physical and chemical scientists has led to substantial advances in our abilityto synthesize macromolecules and understand their physical behavior.

In recent years, technological advances have begun to demand materials which exhibit veryspeci"c properties. If polymers are to continue to impact society in important ways, they mustmeet this need. Polymers are good candidates for materials which can perform functionswith a high degree of speci"city. We can make this claim because it is well-established thatbiological macromolecules are able to carry out very speci"c functions. One feature thatallows biological macromolecules to perform speci"c functions is their ability to self-assembleinto particular motifs. Polymeric materials could impact advanced technologies in importantways if we could learn how to design and synthesize macromolecular systems that can self-assemble into functionally interesting structures and phases. One way to confront this challengeis to take lessons from nature since millenia of evolution have allowed biological systemsto learn how to create functionally useful self-assembled structures from polymeric buildingblocks. By suggesting that we take lessons from nature, we do not imply copying thedetailed chemistry which allows a biological system to carry out a speci"c function thatwe seek. This would be impractical in many contexts. Rather, we suggest asking the followingquestions: are there underlying universalities in the design strategies that nature employs inorder to mediate a certain class of functions? If so, can we exploit similar strategies todesign synthetic materials that can perform the same class of functions with biomimetic speci"city?The reason for the interest in universal strategies is that these may be easier to implementin synthetic systems than the detailed chemistries of natural systems, and may illuminate theessential physics. However, it is also important to realize that universal strategies will also leadto lower degrees of speci"city compared to situations where the detailed chemistry has been"ne-tuned.

Recent work suggests that a possible design strategy employed by natural polymers to a!ectassembly into functionally interesting materials is to exploit multifunctionality and disorderedsequence distributions (e.g., [1}3]). Disordered heteropolymers (DHPs) constitute a class ofsynthetic polymers that embodies these features. These are copolymers containing more than onetype of monomer unit, with the monomers connected together in a disordered sequence. Themonomers may also be connected with di!erent architectures; e.g., branched versus linear connect-ivity. An important point is that once synthesis is complete, the sequence and the architecturecannot change in response to the environment. Since DHPs embody multifunctionality andquenched disorder, they serve as excellent vehicles to explore the suggestion that these features maybe essential elements for mediating certain types of biomimetic function in synthetic systemsrelevant to applications. Competing interactions (due to the presence of di!erent types of monomerunits), connectivity, and the quenched character of the disordered sequence also make DHPsquintessential examples of frustrated systems.

A.K. Chakraborty / Physics Reports 342 (2001) 1}614

In the latter half of this century, physical scientists interested in the condensed phase havedirected considerable attention to two broad classes of problems: those involving biologicalphenomena (e.g., protein folding) and frustrated systems (e.g., the e!ects of frustrating quenchedrandomness in spin glass physics). In addition to being a way of exploring how biomimeticsynthetic soft materials can be designed, studying DHPs also allows us to study those aspects ofbiopolymer behavior that may be termed physics (as distinguished from detailed chemistry) (e.g.,[1}3]). DHPs of particular types (vide infra) also o!er the potential for being excellent vehicles forcareful experimental studies of the manifestations of frustrating quenched disorder.

In this article, we try to illustrate (via examples) how DHPs can serve as biomimetic polymersand/or as model systems to study the physics of frustrated systems with quenched randomness. Webegin (Section 2) by discussing the adsorption of DHPs from solution, and these considerationsshow that such molecules can exhibit a phenomenon akin to recognition in biological systems.These studies also suggest that the systems that we discuss may be good (and simple) model systemsfor experimental studies of kinetic phenomena in frustrated systems with quenched disorder.Section 2 also includes a discussion of the connections of the work we describe with experimentsand some provocative ideas being considered in evolutionary biology. In Section 3, we discusstheoretical and experimental work which demonstrates that DHPs with branched architectures aregood model systems to study e!ects of frustrating quenched disorder on microphase ordering.

This article is not a comprehensive or encyclopedic review of DHP physics. However, this article,some recent reviews of the use of these macromolecules as minimalist models to study proteinfolding (e.g., [1}5]), and a recent review in this journal on theoretical considerations of microphaseordering in molten DHPs with linear architectures [6] provide a glimpse of much of what is knownabout the physical behavior of these macromolecules.

2. Biomimetic recognition between DHPs and multifunctional surfaces

Many vital biological processes, such as transmembrane signaling, are initiated by a biopolymer(e.g., a protein) recognizing a speci"c pattern of binding sites that constitutes a receptor located ina certain part of the surface of a cell membrane. By recognition we imply that the protein adsorbsstrongly on the pattern-matched region, and not on other parts of the surface; furthermore, itevolves to the pattern-matched region and binds strongly to it in relatively fast time scales withoutgetting trapped in long-lived metastable adsorbed states in the wrong parts of the cell surface. Ifsynthetic polymers were able to mimic such recognition, it would indeed be useful for manyadvanced applications. Examples of such applications include sequence selective separation pro-cesses [7,8], the development of viral inhibition agents [9}11], and sensors.

Polymer adsorption from solution has been studied extensively in recent years (see [12,13] forrecent reviews). Most studies have been concerned with the adsorption of polymers with orderedsequence distributions (e.g., homopolymers and diblock copolymers). These studies have taught usmany important lessons. One lesson pertinent to our concerns can be illustrated by considering theexample of a homopolymer interacting with a chemically homogeneous surface. In this case, oncewe have chosen the chemical identity of the polymer segments, di!erent surfaces are characterizedby the attractive energy per segment between the surface in question and the polymer segments(E/k¹). Thus, if we plot the polymer adsorbed fraction (at equilibrium) as a function of E/k¹, points


Fig. 1. Schematic representation of an adsorbed polymer chain. The sketch provides the de"nitions of loops, trains,and tails.

on the abcissa correspond to di!erent surfaces. Theoretical and experimental studies have "rmlyestablished the nature of this plot. For small values of E/k¹, there is no adsorption because theenergetic advantage associated with segmental binding is not su$cient to beat the entropic penaltyassociated with chain adsorption. For su$ciently large values of E/k¹, adsorption does occur. Thetransition from desorbed states to adsorption is a second order phase transition for #exible chains[14]. Adsorbed polymer conformations can be characterized by loops, tails, and trains (see Fig. 1).Fluctuating the distribution of loops while maintaining the same number of contacts is favoredbecause this increases the entropy. These loop #uctuations cause the adsorption transition to becontinuous. The practical consequence is that thermodynamic discrimination between surfaces isnot sharp } a requisite feature for recognition.

The adsorption characteristics of diblock copolymers on striped surfaces have also beenunderstood (e.g., [15,16]). At equilibrium, they adsorb at the interface of the stripes with each blockadsorbed on the stripe that is energetically favored. This phenomenon is di!erent from what ismeant by recognition in important ways. Firstly, the chain is not localized in a region commensur-ate with chain dimensions. This is so because it is entropically favorable for the chain to sample theentire interface. Secondly, it seems highly likely that the diblock copolymer chains would bekinetically trapped in regions away from the interface. This is so because adsorbing one block on anenergetically favorable stripe while allowing the non-adsorbed block to sample many conforma-tions appears to be a deep free energy minimum. Thus, this system does not seem to exhibit thehallmarks of recognition either. (Much work has also been done on the behavior of molten diblockcopolymer layers on patterned surfaces. We do not discuss these studies here because the focus ofthis section is on adsorption from solution. Readers interested in this topic are directed to a recentreview [17] and references therein.)

In short, recognition implies a sharp discrimination between di!erent regions of a surface andlocalization of the chain to a relatively small pattern matched region without getting kineticallytrapped in the `wronga parts of the surface. In biological systems it also usually entails adsorptionin a particular conformation or shape. Synthetic polymers with ordered sequence distributions donot seem to exhibit these characteristics. Similar conclusions can be reached by perusing interestingstudies of DHP adsorption on homogeneous surfaces [18}21] and homopolymer adsorption onchemically disordered surfaces [22].

One way to make synthetic systems mimic recognition is to copy the detailed chemistries whichallow natural systems to a!ect recognition. This is not a practical solution in most cases. Recently,


some work has been done to explore whether there are any universal strategies that may allowsynthetic polymers and surfaces to mimic recognition [23}31]. (Such universal strategies may besimpler to implement in practical situations, and may also shed light on the minimal ingredients, orprinciples, required for synthetic systems to mimic recognition.) This body of work is the primaryfocus of this section. The purpose of this work has not been to explain the physical and chemicalmechanisms that allow recognition in biological systems. However, in order to deduce possibleuniversal strategies, some coarse-grained observations about biological systems have provided theinspiration.

Each protein carries a speci"c pattern encoded in its sequence of amino acids. In recent years,great interest in elucidating the physics of protein folding has led to many coarse-grained modelsfor amino acid sequences in proteins. All models exhibit a common feature. In order to illustratethis feature, consider the H/P model [32] wherein amino acids are considered to belong to twoclasses } hydrophobic and polar. This (and other) models have been used to characterize proteinsequences, and it has been found [33}35] that the pattern of H- and P-type moieties is usually notperiodically repeating. Similarly, examination of cell and virus surfaces reveals that the chemicallydi!erent binding sites that constitute receptors (which are recognized by proteins) are also notarranged in a periodically repeating pattern. These observations suggest that disorder and compet-ing interactions (due to preferential interactions between polymer segments and surface sites) maybe key ingredients for recognition between synthetic polymers and surfaces.

Heteropolymers with disordered sequences carry a pattern encoded in their sequence distribu-tion. The information content is statistical, however, since the sequences are characterized statist-ically. For example, for a 2-letter DHP (say, A- and B-type segments), the simplest way to describethe disordered sequence distribution is by specifying the average fraction of segments of one type( f ), and a quantity j that measures the strength of two-point correlations in the chemical identity ofsegments along the chain [36]. j is directly related to the synthetic conditions and the matrix ofreaction probabilities, P. Elements of this matrix, P

ijare the conditional probabilities that

a segment of type j directly follows a segment of type i. Clearly, j depends upon the choice of thechemical identity of the segments and synthesis conditions. Consequently, synthesis conditions andthe choice of chemistry determines the statistical pattern carried by DHPs. If j'0, withina correlation length measured along the chain, there is a high probability of "nding segments of thesame type. We shall refer to such an ensemble of sequences as statistically blocky. If j(0, withina certain correlation length measured along the chain, there is a high probability of "nding analternating pattern of segments. The absolute magnitude of j measures the correlation length. Forexample, j"0 corresponds to perfectly random sequences, and j"1 implies the correlationlength is the entire chain length. Characterization of sequence statistics using f and j implies thatwe are only looking at two-point correlations to describe the statistical patterns. More elaboratestatistical patterns can be described by considering higher order correlations and/or more than twotypes of segments.

Consider the interaction of DHPs with surfaces bearing more than one type of site, with the sitesbeing distributed in a disordered manner. Examples of such surfaces with two kinds of sitesdistributed in a disordered fashion are shown schematically in Fig. 2. The distribution of thesesites on the neutral surface can be characterized statistically. For example, a simple way would be tospecify the total number density of both kinds of sites per unit area, the fraction of sites of one type,and the two point correlation function describing how the probability of having a site of type A at


Fig. 2. Examples of statistically patterned surfaces. White represents a neutral background. The two types of `activeasurface sites are depicted using light and dark grey dots. In the panel on the left, within some length scale, there is a highprobability of "nding sites of opposite types adjacent to each other. Such surfaces are referred to in the text as statisticallyalternating. In the panel on the right, within some length scale, there is a high probability of "nding sites of the same typeadjacent to each other. Such surfaces are referred to in the text as statistically patchy.

position r is related to the probability of having a site of the same type at position r@. Fig. 2 showsspeci"c realizations of two surfaces bearing simple statistical patterns.

In nature, recognition (with all its hallmarks noted earlier) occurs when the speci"c patternencoded in its sequence distribution and that carried by the binding sites is matched (i.e., related ina special way). DHP sequences and the surfaces we have described in the preceding paragraphcarry statistical patterns. The question we now ask is: will statistically patterned surfaces be able torecognize the statistical information contained in an ensemble of DHP sequences when thestatistics characterizing the DHP sequence and surface site distributions are related in a specialway? In other words, is statistical pattern matching su$cient for recognition to occur?

This question is interesting for three reasons: (1) the answer may tell us what the minimalingredients are for the occurrence of a phenomenon akin to recognition; (2) if recognition canoccur via statistical pattern matching, the phenomenon might be pro"tably exploited inapplications; (3) DHPs interacting with functionalized surfaces bearing statistical patterns may begood model systems to study the physics of frustrated systems with quenched disorder.

In order to answer this question, we have to address several issues. We must determine whethercompeting interactions and disorder are su$cient for sharp discrimination between di!erentstatistical patterns, and whether the inherent frustration allows localization (in reasonable timescales) on a relatively small part of the surface which is statistically pattern matched. Addressingthese issues requires that we study both thermodynamic and kinetic behavior. Let us begin bydescribing the thermodynamics.

2.1. Theory of thermodynamic properties

Srebnik et al. [23] analyzed a bare bones version of the problem in order to examine whetherfrustration due to competing interactions and quenched disorder are su$cient to obtain sharpdiscrimination between surfaces with di!erent statistical patterns. In this model, the 2-letter DHPs


are considered to be Gaussian chains in solution. The surface is comprised of two di!erent types ofsites on a neutral background, and each type of site interacts di!erently with the two types of DHPsegments. In the in"nitely dilute limit, the physical situation described above corresponds to thefollowing Hamiltonian:

!bH"!

32lP

N

0

dnAdrdnB

2!P

N

0

dnPdr]k(r)d(r(n)!r)h(n)d(z) (1)

where r(n) represents chain conformation, k(r) is the interaction strength with a surface site locatedat r, the factor of d(z) ensures that these sites live on a 2-D plane, l is the usual statistical segmentlength, and h(n) represents the chemical identity of the nth segment. For a two-letter DHP, thisquantity is $1 depending upon whether we have an A- or a B-type segment. Eq. (1) implies thata surface site which exhibits attractive interactions with one type of DHP segment has an equallyrepulsive interaction with segments of the other type. This is assumed for simplicity. What isimportant is that interactions between a surface site and the two di!erent types of segments aredi!erent.

Since the DHP sequence and the surface site distribution are disordered, k(r) and h(n) are#uctuating variables. Later, we shall have much to say about how di!erent types of correlated#uctuations of these variables a!ects the physical behavior. For now, in order to explore theessential physics, let us consider the #uctuations in k(r) and h(n) to be uncorrelated. Further, inorder to simplify the analysis, these #uctuations are described by Gaussian processes. This latterapproximation should not a!ect the qualitative physics, and generalizes the results to a physicalsituation with many types of DHP segments and surface sites.

Speci"cally, Srebnik et al. [23] take the surface to be neutral on average (i.e., k has a mean valueof zero), and the variance of the #uctuations in k is p2

1. This implies that p2

1is the only variable

which measures the statistical pattern carried by the surface. Di!erent values of this quantityrepresent di!erent surfaces. Physically, p2

1is proportional to the total number density of both types

of sites on the neutral surface.The uncorrelated sequence distribution is described by h(n) having a mean value of (2f!1),

where f is the average composition of one type of segment. The variance is p22, and is also related to

the average composition (it equals 4f (1!f )). Thus, in this simple case the statistical pattern carriedby the DHP sequence is measured by the average composition.

In order to proceed, we must average over the quenched sequence distribution and the#uctuations in k that characterize the surface site distribution. Consider the latter issue "rst. If thesites on the surface can anneal in response to the adsorbing chain molecule, then the partitionfunction is self-averaging with respect to the #uctuations in k. This could be the physical situation ifthe functional groups that represent the sites on the surface were weakly bonded to the surface. If,however, the sites on the surface cannot respond to the presence of the DHP, then the partitionfunction is self-averaging with respect to the #uctuations in k only under restricted circumstances.As has been explicated in many contexts (e.g., [37}39]), the quenched and annealed averages overdisorders external to the #uid of interest are equivalent when the medium is su$ciently large, andthe time of observation is long enough for the #uid (in our case, the polymer) to sample the medium.Later, we shall quantify these statements by examining results of Monte-Carlo simulations. For themoment, we carry out an analysis that holds for annealed disorders under all circumstances, and


is appropriate for quenched surface disorders under the restrictions noted above. Therefore,following Feynman [40], we calculate the in#uence functional by averaging over the Gaussian#uctuations in k(r). This obtains the following e!ective Boltzmann factor:

exp[!bH$*4

]"PPDk(r) expC!1

2p21PdrPdr@ k(r)d(r!r@)k(r@)d(z)d(z@)D

]expC!32lP

N

0

dnAdrdnB

2!P dnP dr k(r)d(r(n)!r)h(n)d(z)D . (2)

The partition function is not self-averaging with respect to the quenched sequence #uctuationsunder any circumstances. Replica methods [41,42] provide one way to carry out the quenchedaverage. We will consider f"0.5, thereby "xing the statistics of the DHP sequences. We then studyadsorption as a function of the surface statistics (i.e., the variance of the distribution thatcharacterizes the #uctuations in k(r)). Replicating the e!ective Hamiltonian in Eq. (2), and carryingout the functional integral corresponding to the average over the distribution of h obtains thefollowing m-replica partition function:

SGmT"m<a/1PPDra (n)PPDka (r)

]expC!32l

m+a/1PdnA

dradn B

2

D]expC!

12p2

1

+a,b PdrP dr@ ka (r)da,bd(r!r@)kb(r@)d(z)d(z@)D

]expCp222

+a,b PdnP drP dr@ ka (r)kb (r@)d(ra (n)!r)d(rb(n)!r@)d(z)d(z@)D . (3)

This replicated partition function can be written in a form that is more convenient both forthinking and computing. De"ne the following order parameter, Qab (r!r@), which measures theconformational overlap on the surface between the replicas:

Qab(r, r@)"P dn d(ra (n)!r)]d(rb (n)!r@)d(z)d(z@) . (4)

This de"nition allows us to rewrite Eq. (3) as a functional integral over this overlap orderparameter in the following way:

SGmT"PPDQab exp(!E[Qab]#S[Qab]) ,


where

E"!lnm<a~1PPDka (r) expC!

12PdrP dr@ ka (r)Pab (r, r@)kb(r@)D ,

S"lnm<a~1P PDra (n) expC!

32l

m+a~1P dnA

dradn B

2

D]d[Qab(r, r@)!P dn d(ra(n)!r)]d(rb(n)!r@)] (5)

and Pab(r, r@) is

Pab(r, r@)"d(r!r@)dab

p21

d(z)d(z@)!p22Qab(r, r@) . (6)

The quantity S is clearly the entropy associated with a given overlap order parameter "eld since it isthe logarithm of the number of ways in which the DHP can organize itself in 3D space with theconstraint that the overlap between replicas on the 2D surface is Qab . Then, E is the associatedenergy.

As has been demonstrated in the context of protein folding (e.g., [1}5]) and the behavior ofDHPs in 3D disordered media [43], these polymers with quenched sequence distributions canexhibit behavior akin to the REM model, the Potts glass with many states, or p-spin models. Oneconsequence of this is that, under certain circumstances, the thermodynamics is determined bya few dominant conformations. This is because frustration due to competing interactions andquenched disorder makes these few conformations energetically much more favorable compared toall others. Since the physical situation that we are considering also embodies the frustrating e!ectsof competing interactions and quenched disorder, we must allow for the possibility of sucha phenomenon (we will refer to it as freezing for convenience). In fact, since the competinginteractions in our case occur on a 2D plane, this e!ect might be enhanced. It is very important tounderstand that the preceding sentence does not imply that the problem we are considering is onewherein the dimensionality of space is 2. The polymer conformations can (and do) #uctuate in 3Dspace by forming loops and tails; only the competing interactions in this simple scenario aremanifested in 2D space. We shall return to the importance of loop #uctuations later in this section.

Mathematically, allowing for the possibility of a few dominant conformations implies that wemust allow for broken replica symmetry. Parisi [44] pioneered the way in which to compute andthink about broken replica symmetry in the context of spin glasses. For SK spin glasses, replicasymmetry is broken in a hierarchical manner [42,44]. For the REM and p-spin models with p'2,one stage of the symmetry breaking process is su$cient for sensible calculations [42]. As notedearlier, since DHPs share some features with these models, a one-step replica symmetry breaking(RSB) scheme is a reasonable approximation (e.g., [1}5,42,43]).

Replicas are divided into groups. Replicas within a group have perfect overlap on the surface,and those in di!erent groups do not overlap at all on the surface. The energy can be computed byevaluating the logarithm of the determinant of the matrix Pab . Mezard and Parisi [44] have


provided formulas for this quantity when there is broken replica symmetry. Using their formulaand a 1-step RSB scheme the energy is computed to be:

E"

12C!lnp2

1#

1x0

ln(1!C1px

0)D ,

p6 "p/N ,

C1"p2

1p22N/A , (7)

where x0

is the number of replicas in a group, p is the number of contacts with the surface, and A isthe surface area of the solid. In writing Eq. (7), the density of adsorbed segments on the surface hasbeen approximated to be uniform.

In order to compute the entropy a physically transparent method can be employed. The "rstquantity that we need to compute is the number of ways in which x

0replicas can be arranged such

that their surface conformations overlap perfectly. When polymers adsorb, the conformations arecharacterized by loops, trains, and tails (see Fig. 1). In the long chain limit, we may ignore tails. Letf/*

(r,!r@

,) be the probability that a loop of length ni starts at r

,on the surface and ends at r@

,. The

loop length ni ranges from 1 to the chain length, N. Including loops of length 1 implies that trainsare incorporated in the computation of the entropy. With this de"nition, the restricted partitionfunction for x

0replicas in a group can be written as:

Z(r,1)" +

n1 ,n2 ,2,npPdr,

22Pdr,

pf x0n1

(r,2!r,

1)2f x0

np(r,p!r,

p~1)d(n

1#n

2#2#n

p!N) . (8)

The delta function conserves total chain length while allowing loop length #uctuations. We do notintegrate over the position of the "rst adsorbed segment, r

1,, for later mathematical convenience.

The entropy for x0

replicas in a group is obtained by integrating Z over r1,

and then taking thenegative logarithm.

In order to compute this partition function, let us "rst introduce a Laplace transform conjugateto N; i.e.,

z(k)"PdNz(N)e6 kN (9)

where k is the Laplace variable conjugate to N. The product of the functions that describe the loopprobabilities exhibits a convolution structure, and so it is convenient to introduce a 2D Fouriertransform conjugate to the 2D spatial coordinate that de"nes position on the surface. TheFourier}Laplace transform of the restricted partition function is

Z(k, j)"CN+n/1

f x0n

(k)e~jnDp

. (10)

The loop probability factors must be of two types. Following Hoeve et al. [45], the factor f1

fora loop of unit length is taken to be

f1(r)"ud(r!l) (11)


where u is the partition function for one adsorbed segment, and depends upon the chemical detailsof chain constitution.

In the simple model that we are considering now, the DHP segments do not interact with eachother. So, the loop probability factor for longer loops is Gaussian:

fn(r)"

Cn3@2

expC!r2

2nl2D (12)

where C is a normalization constant that depends upon chain sti!ness, n is the loop length, and r isthe distance between loop ends.

Substituting the Fourier transforms of the loop probability factors into Eq. (10), integrating overr1,

, and inverting the Fourier and Laplace transforms yields the partition function that we seek.Now, noting that there are m/x

0groups of replicas, the entropy in Eq. (5) is calculated as the

product of m/x0

and the negative logarithm of this partition function. In carrying out thesemanipulations, the sum over loop lengths in Eq. (10) can be taken to be an integral since we areconcerned with the long chain limit.

Combining the result of the entropy calculation with Eq. (7) for the energy obtains the followingfree-energy functional:

F"

12C!lnp2

1#

1x0

ln(1!C1p6 x

0)D!

1x0N

]lnCp6 N+q/0Ap6 N

q BA2pl23x

0B

qCqx0u(PM N~q)x0

Cq((4!3x0)/2)

C(q(4!3x0)/2)

][N!(p6 N!q)]*(4~3x0 )@2+q~1D .

(13)

Srebnik et al. [23] also added a term that represents contributions from non-speci"c three-bodyrepulsions to the energy. It is not essential to add this term, but at high values of the adsorbedfraction it may be necessary for stability.

A mean-"eld solution for the order parameters, p6 and x0, is obtained by extremizing the free

energy functional with respect to them. It is worth noting that the free energy functional must beminimized with respect to p6 and maximized with respect to x

0. The reason that the free energy

functional has to be maximized with respect to x0

is that when the mP0 limit is taken in thereplica calculation the lowest-order correction to the free energy evaluated at the saddle pointvalue is negative; this is because the dimensionality of the integral is m(m!1), which is negativewhen the replica limit is taken [42]. A simple computer code allowed Srebnik et al. [23] to obtainthe saddle point values of the two order parameters pN and x

0.

By following p6 and x0

we can learn about the adsorption characteristics of DHPs ontodisordered multifunctional surfaces. The order parameter p6 is simply the fraction of adsorbedsegments. It acquires values greater than zero when adsorption occurs. The order parameter x

0has

been interpreted to be 1!+iP2

i, where P

iis the probability with which conformation i occurs.

When a multitude of conformations are sampled, each of these probabilities is very small andx0

acquires the asymptotic value of unity. This is the usual situation in polymer physics becauseentropic considerations lead to large conformational #uctuations. Natural polymers seem to be


Fig. 3. The order parameters p6 and x0

plotted as a function of C1. For the calculation described in the text, each point on

the abcissa represents a di!erent statistically patterned surface.

designed such that, under appropriate circumstances, conformational #uctuations are suppressedand a few dominant conformations determine the thermodynamics. DHPs can also exhibit similarphysics (e.g., [1}5,43]). In fact, our quest for biomimetic recognition can only be successful ifadsorption occurs in a few dominant conformations.

Let us "x all the parameters that determine the nature of the chains. Then, let us study thevariation of the order parameters p6 and x

0with C

1. Di!erent values of this parameter correspond

to di!erent surfaces. Speci"cally, each value of C1

corresponds to a di!erent total number densityof sites on the surface. The number of sites per unit area on the surface can be adjustedexperimentally by a number of means, the simplest being the adsorption of functional groups ontoa surface from solutions of varying concentrations [7].

Fig. 3 shows how the two relevant order parameters vary with C1. A uniform neutral surface

corresponds to C1"0. Therefore, when this parameter is small no adsorption occurs. The order

parameter x0

is unity since in the absence of intersegment interactions and adsorption allconformations are energetically equally likely. Fig. 2 shows that above some value of C

1weak

adsorption occurs with a multitude of conformations being sampled. The transition from noadsorption to weak adsorption is continuous. The theory also predicts that at a higher thresholdvalue of C

1a sharp transition from weak to strong adsorption occurs. This adsorption transition is

accompanied by x0

becoming less than unity. This signals that the polymer adsorbs in only a fewdominant conformations (at least as far as the adsorbed segments, and hence, the loop structure isconcerned). As noted earlier, in the simple model that we have been considering, each point onthe abcissa corresponds to a di!erent surface. Thus, the sharp transition from weak to strong


Fig. 4. Schematic representation of DHPs interacting with a surface bearing just a few sites. The two panels depictdi!erent conformations with the same energy.

adsorption implies sharp thermodynamic discrimination between surfaces bearing di!erent statist-ical patterns } one of the hallmarks of recognition.

The physical reason for the sharp transition from weak to strong adsorption (and the freezinginto few dominant conformations that accompanies it) can be understood by "rst discussing whathappens when the continuous transition to weak adsorption occurs. When there are only a few siteson the surface (small C

1), the energetic advantage associated with chain segments binding to

preferred sites is not su$cient to overcome the entropic penalty for chain adsorption. For highervalues of C

1, adsorption occurs because now the number of sites is su$cient for the favorable

energetics of preferential segmental binding to overcome the entropic penalty. At the same time,since the number of surface sites per unit area is small, it is very easy for the chain to avoidunfavorable interactions. Furthermore, as shown schematically in Fig. 4, because it is easy to avoidunfavorable interactions, the chain can obtain the same energetic advantage in many di!erentconformations. Thus, the system minimizes free energy by sampling a multitude of conformationswhich have roughly the same energy. As the loading of surface sites increases, however, it isintuitively obvious that it becomes increasingly di$cult to avoid unfavorable interactions. In fact,it is clear that above some threshold loading of surface sites, most arbitrary adsorbed conforma-tions will be subjected to many unfavorable interactions. Thus, for a su$ciently high loading (andhence, adsorbed fraction), most adsorbed conformations constitute a continuous spectrum of highenergy states. However, there will be a small ensemble of conformations that are signi"cantly lowerin energy. These conformations are the few that carefully avoid unfavorable interactions (as bestas possible given the disorder in the sequence and the surface site distributions). These few


conformations are energetically more favorable than all others. Thus, in a manner reminiscent ofthe random energy model of spin glasses, the energy spectrum develops a gap between the smallensemble of conformations that adsorb in a pattern-matched way and the multitude of others. Bypattern matching we mean registry between adsorbed segments and the preferred sites. As theloading of surface sites increases, the system becomes more frustrated, and the energy gap betweenthe pattern-matched adsorbed conformations and all others increases. Beyond a threshold value ofthe loading, the energy gap becomes much greater than the thermal energy, k¹. This causes thepolymer chain to sacri"ce the entropic advantage of sampling a multitude of conformations, and itadsorbs in the few pattern-matched conformations. Of course, these pattern-matched conforma-tions are strongly adsorbed. Thus, we get a phase transition with a sharp increase in the adsorbedfraction and passage to a thermodynamic state where only a small ensemble of conformations aresampled. Mean-"eld theory predicts that this transition is "rst order. There are two free-energyminima, one corresponding to a replica symmetric solution of the equations and the other toa solution with broken replica symmetry [46]. Before the transition from weak to strong adsorp-tion, the minimum corresponding to the replica symmetric solution is the global minimum. Whenthe transition occurs, the solution with broken replica symmetry becomes the global minimum.

It is worth remarking that, while a two-letter DHP in solution does exhibit REM like behavior,the energy gap between the low-lying conformations and the continuous part of the spectrum is notlarge [5]. Signi"cantly larger gaps are obtained for designed sequences. In the situation we havebeen considering, interaction with the disordered distribution of surface sites adds another sourceof frustration which makes the energy gap in a REM-like picture quite large when the surfaceloading exceeds a threshold value even for random sequences and surface site distributions. At leastthis appears to be true for the thermodynamic behavior. We shall see later that kinetic consider-ations require delicate design of the statistics of heteropolymer sequence and surface site distribu-tion statistics.

The preceding discussion provides a compelling physical argument for the existence ofa transition from weak to strong adsorption accompanied by the adoption of a few dominantconformations when the statistics of the DHP sequence and surface site distributions are related ina special way. However, we have provided no physical reason for the transition to be sharp (or "rstorder as predicted by the mean-"eld calculation). The order of the transition can be establishedrigorously only by carrying out a renormalization group calculation. A proper calculation of thissort has not yet been performed. From a fundamental standpoint, it is important that the order ofthe transition be established in a rigorous way. From a practical standpoint what is important isthat the transition is sharp and hence can display one of the hallmarks of recognition } a sharpdiscrimination between surfaces to which the chains bind weakly and others to which it binds verystrongly. The physical reason for the sharpness of the transition is suggested by a simple model ofthe phenomenon under consideration; Monte-Carlo simulations for "nite size systems is alsoindicative of a "rst-order transition.

First, let us discuss a model [26] which complements the replica "eld theory that we havediscussed and is motivated by simple physical considerations. Again, consider DHP chainscomprised of two types of segments (A and B) interacting with a surface functionalized by twodi!erent types of sites. The segments of type A prefer to interact with one type of surface site, andthose of type B exhibit the opposite preference. Thus, from an energetic standpoint, there aretwo types of segment-surface contacts: good and bad contacts. Good contacts are those that


involve preferred segment-surface site interactions. Let us try to develop a free-energy functionalwith the order parameters being the total number of adsorbed segments (p) and the numberof good contacts (q).

The energy corresponding to given values of p and q is:

Ek¹

"[qdE#pE1] (14)

where dE is the energy di!erence between good and bad contacts, and E1

is the energy of a badcontact.

We now need to compute the entropy for a chain of length N and p adsorbed segments of whichq are good contacts. This entropy can be partitioned into three separate contributions. Firstly,there is a `mixinga entropy (S

.*9) associated with the number of ways to choose p adsorbed

segments out of N. There is also the entropy loss (S!$4

) associated with segmental binding, and theentropy (S

-001) associated with loop #uctuations in the adsorbed conformations. As we shall see, the

last contribution is of crucial importance.The simplest possible approximation for S

.*9yields

S.*9N

"!p6 ln p6 !(1!p6 ) ln(1!p6 ) (15)

where p6 "p/N is the fraction of adsorbed segments.As is usual, the loss in entropy upon segmental binding is given by

S!$4

/N"!w@p6 (16)

where u@ is a constant related to chain #exibility and solution conditions.Now consider the computation of the loop entropy. When a homopolymer adsorbs on a chemic-

ally homogeneous surface, the energetic advantage associated with every segment-surface contactis the same. Thus, the adsorbed chain exhibits large #uctuations in the loop structure to maximizeentropy. It is important to note that the loops live in three-dimensional space, and any descriptionof the physics must account for the loop #uctuations properly.

The problem that we are considering is distinctly di!erent from homopolymer adsorptionbecause the segment}surface contacts are of two types. The existence of good and bad contactsimplies that there are two types of loops. There are loops associated with forming good contacts atboth ends, and those that are associated with forming the other contacts. These two types of loopsare fundamentally di!erent in character. Each loop is characterized by the loop length and thedistance between loop ends on the surface. For the loops associated with forming good contacts atboth ends, only certain values of these quantities are allowed. This is because the two segments thatcorrespond to the loop ends must be bound to surface sites with which they prefer to interact. Theallowed loop lengths and distance between the loop ends on the surface are intimately related to theprobabilities of "nding certain types of sites and segments at di!erent locations along the chain andon the surface. Thus, the allowed #uctuations of loops associated with good contacts are deter-mined by the statistics that characterize the chain sequence and the surface site distribution. Loopsassociated with contacts that are not good are not restricted in this manner, and the usual


#uctuations in loop length and distance between loop ends are allowed. The above argumentsuggests that competing interactions and disorder cause loop #uctuations to be suppressed by theformation of good contacts. It is reasonable to suspect that if the statistics of the chain sequenceand the surface site distribution are such that there is a high probability for the formation of goodcontacts, loop #uctuations are strongly suppressed. Suppression of loop #uctuations makes thechain e!ectively sti!er. Su$ciently sti! chains are known to undergo sharp ("rst-order) adsorptiontransitions [14]. These arguments suggest that strong suppression of loop #uctuations resultingfrom frustration due to competing interactions and quenched disorder, and statistical patternmatching are the origins of the sharp adsorption transition. The meaning of statistical patternmatching also is made more clear; we mean that the statistics that describe the DHP sequence andthe surface site distribution are such that the probability of making good contacts in certainadsorbed conformations becomes su$ciently high.

In order to explore the veracity of these arguments, we need to write down a mathematicalformula for the entropy corresponding to loops associated with good and bad contacts when theycoexist. (For ease of reference, we shall refer to these loops as quenched and annealed, respectively.)In order to develop such a formula, it is instructive to "rst consider the entropy associated witheach type of loop when it is the only type of loop that exists. Once these formulas are available, it isrelatively straightforward to combine them properly to obtain what we seek.

Let us begin with the quenched loops. Since the bare chains are non-interacting (and henceGaussian) in our model, the loop factor for a loop of length n returning to the plane with the twoends separated by a distance d is again (see Eq. (12)) given by

P(n, d)"C

n3@2expC!

d2

2nb2D . (17)

The quantities n and d depend upon the statistics that describe the chain sequence and surface sitedistributions. For example, in the case of uncorrelated #uctuations in the surface site distribution,the most probable value of d2&1/p2

1, where p2

1is the width of the distribution and is proportional

to the surface loading of both types of sites. Since we are considering a situation where onlyquenched loops exist, if there are q adsorbed segments, there are q quenched loops with the averageloop length being N/q. Shortly, we shall see how the average loop length (and hence n) is closelyrelated to the statistics which describe the sequence and surface site distributions. In view of theconsiderations noted above and Eq. (17), it is easy to write down an expression for the entropycorresponding to q quenched loops. Speci"cally, taking n and d to equal their most probablevalues, obtains

S26%/ -0014

N"q6 ln(Cq3@2)!C

aq2

2b2p21D (18)

where q6 "q/N. Now consider the entropy associated with forming annealed loops only. Towardthis end, consider the well-known problem of a homopolymer adsorbing on a chemically homo-geneous surface since herein the loops are annealed. Let the energy bonus for segmental adsorptionbe represented by a potential /(z) which is zero everywhere except at the surface; z is the coordinatenormal to the surface. The e!ective energy bonus for segmental adsorption, lnb is then given by the


following formula:

lnb"lnP=

0

dz[e~((z)@kT!1] . (19)

As noted earlier, in this case, the loop lengths and the distance between loop ends can #uctuate withthe only constraint being the "xed chain length. This problem is similar to the adsorption ofa homopolymer chain (which lives in three-dimensions) to a point, and its solution is presented in[14]. This method can be adapted to solve the problem we are considering. The main di!erencebetween the problem considered in [14] and our concern is that the potential is imposed by animpenetrable two-dimensional manifold rather than a point. This imposes certain additionalsymmetries. Exploiting these symmetries, it is easy to show [26] that the following Schrodinger-likeequation describes the problem under consideration:

[1#bd(z)]g( t(z)"lt(z) (20)

where g is the standard connectivity operator, and the eigenfunction t and the eigenvalue l havetheir usual meaning. Very simple manipulations (described in [14]) lead to the following relation-ship between b and l:

1b"

12pPdk

g(k)[l!g(k)]

, (21)

where k is the Fourier variable conjugate to z.We seek the entropy corresponding to the formation of P annealed loops. This can be obtained

by "rst deriving a relationship between b and P. In order to "nd this relationship, it is convenient tode"ne a generating function z(N,b) as follows:

z(N,b)"+P

bPZ(N,P) (22)

where Z(N,P) is the partition function for a chain of length N with P annealed loops. Of course,this generating function is related to the eigenvalue in Eq. (20) in the usual way; i.e., lN"z(N,b).For adsorption problems such as this, the ground state dominance approximation is appropriate[14,47]. This implies that we may evaluate the sum in Eq. (22) using the saddle point approxima-tion. In other words,

P"bR ln zRb . (23)

With this approximation, making use of the relationships between b and l (Eq. (21)) and thatbetween l and the generating function allows us to obtain the equilibrium value for P/N.Speci"cally, we "nd that

PN

"

1l

:dk g(k)/[l!g(k)]:dk g(k)/[l!g(k)]2

. (24)


Fig. 5. Schematic depicting quenched and annealed loops. The darkly shaded loops correspond to loops with both endsbeing good contacts. The lightly shaded loops do not have good contacts at the end and exhibit signi"cant #uctuations.

The entropy is now easy to calculate as the free energy equals !N ln l and the energy bonusassociated with segmental adsorption is lnb. We "nd that

SN

"!

FN¹

#

EN¹

"ln l!PN

lnC12pPdk

g(k)[l!g(k)]D . (25)

Eqs. (21) and (24) can be solved simultaneously to obtain P as a function of b, and Eqs. (21) and (25)yield the relationship between the entropy and b. Thus, we obtain the entropy as a function of P.

Now, we need to compute the entropy when quenched and annealed loops co-exist. On physicalgrounds, it is clear that the total number of segment}surface contacts is greater than or equal to thenumber of good contacts. This implies that the annealed loops live within the quenched ones. Thisis illustrated schematically in Fig. 5. The annealed loops can redistribute among the quenched onesin an unconstrained manner. It is most convenient to consider this physics in the grand canonicalensemble, whence we can say that the chemical potential of the annealed loops is the same in all thequenched loops. The chemical potential can be easily calculated from the equations we havederived so far since it equals !RS/RP. One "nds that it is a monotonic function of P. This fact,when combined with the observation that the chemical potential of annealed loops must be equalin the quenched loops, leads to the conclusion that the concentration of annealed loops is the samein each quenched loop. By concentration, we mean the quantity p6

!"number of annealed

loops/length of quenched loop. These remarks allow us to properly combine the formulas for theentropies of quenched and annealed loops (Eqs. (18), (21), (24) and (25)). Noting that there areq quenched loops (q good contacts), that p6

!"p6 !q6 , and that p6

!is the same in each quenched loop

lead us to the following expression for the loop entropy corresponding to p adsorbed segments ofwhich q are good contacts:

S-0014N

"q6 ClnP(1/q6 , d)#1q6s(pN

!)D (26)

where P@ is the probability written down in Eq. (17), and s(p6!) is the entropy of annealed loops with

concentration p6!divided by n. The latter quantity is obtained from Eqs. (21), (24), and (25) with one


modi"cation. The calculation leading to these equations considered a uniform surface. We areconcerned with a situation where adsorption can only occur on particular surface sites which donot uniformly cover the surface. The concomitant entropy loss is proportional to lnp2

1, and is

accounted for by Chakraborty and Bratko [26].Combining Eqs. (14) and (26), we obtain the free energy density f as a function of the order

parameters, p and q to be:

f"q6 dE#(u#E1)p6 #p6 ln p6 #(1!p6 ) ln(1!p6 )

!q6 ln(Cq3@2)#aq2

2p21b2

!s(p6 !q6 )!(p6 !q6 ) lnCp21

b2

a D . (27)

These two order parameters can be further related by noting that

q6 "p6 P'

(28)

where P'

is the probability of making a good contact. At in"nite temperature, when entropy isirrelevant, P

'is simply related to the statistics of the sequence and surface site distributions. Let us

denote this intrinsic probability for making good contacts by P@'. It has been conjectured that [26]

the relationship between P@'

and the sequences and surface site statistics is the following:P@'"+

mP

4(m)P

#(m). Here P

#(m) is the probability of "nding a block of length m of like segments on

the chain sequence, and P4(m) is the probability of "nding a patch of size m of like sites on the

surface. This conjecture has been found to be consistent with Monte-Carlo simulation results[24,25]. Note that given the statistics that describe the DHP chain sequence and surface sitedistributions, P

#(m) and P

4(m) can be easily computed. At "nite temperatures, the probability of

making good contacts is modi"ed by entropic considerations, and P@'must be weighted by the free

energy for making good contacts in the following manner:

P'"P@

'

e~bF2

P@'e~bF2#(1!P@

')e~bF!

(29)

where F2

and F!are the free energies associated with quenched and annealed contacts (loops), and

can be calculated from the equations derived earlier.Eqs. (27)}(29) need to be solved numerically to obtain the values of the order parameters pN and

qN which minimize the free energy. Chakraborty and Bratko [26] have obtained this mean-"eldsolution. The purpose of this exercise is to obtain some insight into the origin of the sharptransition. Thus, let us consider results for the simplest possible scenario } uncorrelated sequenceand surface site #uctuations. In this case, P@

'is simply proportional to the product of the widths of

the two statistical distributions. Fig. 6 shows the variation of p6 and q6 with p1, the width of the

distribution that characterizes the surface site #uctuations. We have taken the statistics of the DHPsequence to be "xed in constructing Fig. 6. Thus, points on the abcissa in this "gure correspond todi!erent surfaces.

Fig. 6 shows that when p1

is small, both pN and qN are zero. This is simply a re#ection of the factthat the energetic advantage associated with segmental binding is not su$cient to overcome theconcomitant entropic penalty. This is because the number of binding sites available on thesesurfaces is not su$cient. When p

1becomes su$ciently large, adsorption does occur. However, it is


Fig. 6. The order parameters p6 (solid line) and q6 (dotted line) plotted as a function of p1. For the calculation described in

the text, each point on the abcissa corresponds to a di!erent surface.

important to note that the number of good contacts is very small in this weak adsorption limit.Above a threshold value of p

1, our theory predicts a sharp transition from weak to strong

adsorption. This transition coincides with a jump in q6 , signifying that now the preponderance ofcontacts are good ones. The entropy is now dominated by that associated with the quenched loops,and is low because loop #uctuations are suppressed. Notice that after the sharp transition q6 growsfaster than p6 , and ultimately approaches p6 .

These results provide evidence for the argument made earlier that the suppression of loop#uctuations is the origin of the sharp transition from weak to strong adsorption. This is evidentfrom the result that the number of good contacts (quenched loops) jumps at this transition.A preponderance of good contacts implies a strong suppression of loop #uctuations; i.e., we havea situation resembling the adsorption of a sti! chain, a case for which the adsorption transition isknown to be "rst order [14].

In some ways, the phenomenon we are considering resembles protein folding (or hetero-polymeric models of folding). In the latter situation, a "rst-order transition called the coil}globuletransition occurs wherein the preponderance of contacts become native ones. This is followed bya continuous transition to the "nal low entropy folded state. The sharp transition we see may beconsidered the analog of the coil}globule transition.

The fact that in the strongly adsorbed state the quenched loops dominate is very signi"cant. Thedominance of quenched loops implies that the chain adopts a small number of conformationscharacterized by a certain distribution of loops speci"c to the sequence and surface site distribu-tions. Only small #uctuations around these conformations (shapes) of the adsorbed chain occurafter the transition. This adoption of a small class of shapes makes the phenomenon we areconsidering richer than other successful e!orts to elucidate strategies that can localize chains tocertain regions of a surface [29}31]. This feature was also signaled in the replica "eld theory bybroken replica symmetry coinciding with the transition from weak to strong adsorption. In thesimple model that we have just discussed, the structure of the quenched loops is measured only bythe average length, q. Notice that even this quantity is determined by the probability of goodcontacts; i.e., the statistics of the sequence and surface site distributions. This suggests that the class


of shapes that are adopted upon strong adsorption is determined by the statistical patterns on thechain and the surface. This issue of the emergence of a particular class of shapes (conformations)upon recognition will be explored in great detail later when we consider the kinetics of patternrecognition, and the suggestions of the thermodynamic model we have been considering willbecome vivid.

2.2. Monte-Carlo simulations of thermodynamic properties

The predictions of the models we have been considering are consistent with a series ofMonte-Carlo simulations designed to compute thermodynamic properties [24,25,27]. These stud-ies were carried out using an adaptation of the non-dynamic ensemble growth method pioneeredby Higgs and Orland [48,49].

The simulations were carried out on a cubic lattice. The introduction of the lattice does a!ectquantitative predictions. However, the phenomenology is expected to be the same because ofreasons that have been explained in detail in [48]. A particular sequence is "rst drawn from thestatistical distribution under consideration. A particular realization of the surface site distributionis also drawn from the statistical distribution under consideration. M monomers are then placedrandomly with Boltzmann probabilities dictating the positional probabilities. These positions areallowed to vary between 0 and 2N where N is the length of the polymer we wish to simulate. This isequivalent to studying isolated chains con"ned between identical surfaces separated by a distance,4N. One then attempts to add a second segment of type A or B (as speci"ed by the particularrealization of the sequence) at the end of each monomer. In other words, 6M trials are made.M dimers are then chosen with Boltzmann probabilities. The potential energy is determined byintersegment interactions and interactions with the surface sites; excluded volume interactions areenforced. This process is continued until chains with the desired length have been grown. ForM<N, a canonical distribution results [48}50], and properties are computed as non-weightedaverages over the ensemble. If necessary for statistical accuracy, many populations of the samesequence are simulated. The same procedure is repeated for several realizations of the quenchedsequence distribution. Averages over di!erent realizations of the surface site distribution arecomputed in two di!erent ways. In principle, if the DHP samples a su$ciently large surface overa su$ciently long observation time, then treating the external disorder that constitutes the surfaceas quenched or annealed yields equivalent results (e.g., [37}39]). The annealed case is easier tosimulate because the e!ects of the #uctuating surfaces sites can be integrated out to obtain ane!ective Hamiltonian which can then be simulated. However, the conditions for this equivalence tohold are di$cult to establish in a quantitative manner. Thus, explicit quenched averages over thedisordered surface sites have been computed and compared to results obtained by simulatinga pre-averaged Hamiltonian [25]. It was found that, for N"32, once the surface sites live ona lattice larger than 200]200, both procedures yield equivalent results. The results we discuss havetypically been obtained by taking M to be 64 000 for N"32, 64 and 128.

In order to compare the simulation results to the basic phenomenology predicted by theanalytical models, the "rst simulations [25] were carried out for situations where both the DHPsequence and surface site distributions exhibit uncorrelated #uctuations. Thus, two-letter DHPswith the sequence statistics characterized by the average composition f were examined. Considersurfaces with two types of sites with the average composition always being equal to half. As in the


Fig. 7. Results of Ensemble Growth Monte-Carlo simulations. Adsorbed fraction p6 and the parameter x as a function ofp. Each point on the abcissa corresponds to a di!erent surface.

analytical models, the statistics of the surface site distribution is then varied by changing the totalnumber of sites per unit area (i.e., loading).

Fig. 7 depicts the results of the ensemble growth MC simulations for these situations by plottingthe adsorbed fraction p6 as a function of p; p is the width of the distribution characterizing thesurface site distribution, and is proportional to the loading. The absorbed fraction p6 is vanishinglysmall for small loadings of the surface. For su$ciently large values of p, we "nd that a gradualtransition to weakly adsorbed states occurs. We also observe that beyond a higher threshold valueof p a sharp transition occurs from weak to strong adsorption. In Fig. 7, we also plot the variationof a parameter x with p. This quantity is de"ned in a manner analogous to the way in which x

0was

de"ned in the replica "eld theory. The quantity x"1!+iP2

i, where P

iis the probability of "nding

a conformation with energy Ei. Due to degeneracy, x and x

0are not identical. However, simulation

results [25] show that x faithfully reproduces the qualitative trends obtained by computing x0. It is

much easier to compute x from the simulation data as Eiis easily obtained from the simulation

results. Fig. 7 shows that x equals unity for values of p for which we have no adsorption or weakadsorption. However, the transition from weak to strong adsorption is accompanied by x becom-ing less than unity. (The simulation results suggest that x become less than unity for a value ofp that is slightly larger than that corresponding to the weak to strong adsorption transition, andsome reasons have been discussed to explain this [26]. We shall not focus on this minor detail here.)

The simulation results shown in Fig. 7, therefore, reveal the same phenomenology as thetheoretical models. For uncorrelated sequence and surface site #uctuations, and for "xed DHPsequence statistics, when the surface loading exceeds a threshold value a sharp change from weakto strong adsorption occurs with the thermodynamics being determined by a few dominantconformations when strong adsorption occurs. The transition is rounded in the Monte-Carlosimulations because of "nite size e!ects. Unlike the theoretical predictions, however, it is di$cult toascertain the order of the transition from the simulation results. This question can be explored byexamining "nite size e!ects. Thus, the quantity S(dp)2T has been computed. This quantity exhibitsa peak when the sharp adsorption transition occurs. Furthermore, the width of the peak narrows


and shifts to the left as the chain size increases. Unfortunately, no "rm conclusions can be drawnabout the order of the transition as simulations have been carried out only for three chain lengths,a number insu$cient to extract "nite size scaling exponents with con"dence [25]. One reason forthe lack of simulation data is that the important result is that a sharp transition occurs from weakto strong adsorption when the statistics of the sequence and surface sites are related in a specialway. The order of the transition is fundamentally very important, but not crucial for examining thephysics of discriminatory pattern recognition exploiting the notion of self-assembly driven bystatistical pattern matching. We now turn to these exciting questions, for which the precedingdiscussions serve as a prelude.

We begin our considerations of how the basic physics described above can be exploited to causeDHPs bearing certain statistical patterns to recognize surfaces that bear complementary statisticalpatterns by discussing some simulation results that pertain to thermodynamics. This will befollowed by detailed considerations of the kinetics of the self-assembly process that leads torecognition driven by statistical pattern matching.

Consider DHPs with symmetric average compositions that carry two di!erent types of statisticalpatterns encoded in their sequence. The two types of statistical patterns are characterized withtwo-point correlations only. As we have noted earlier, these two-point correlations are speci"ed bya parameter j. Positive values of j imply that within a certain length along the chain, there is a highprobability of "nding segments of the same type. We shall call the ensemble of DHPs with suchsequences statistically blocky. Negative values of j imply that within a certain length along thechain there is a high probability for "nding the two types of segments arranged in an alternatingfashion. We shall refer to these types of sequences as statistically alternating.

We shall also consider two types of statistically patterned surfaces. Let us restrict attention tosituations where the composition specifying the relative amounts of the two types of sites issymmetric. Surfaces that we call statistically patchy have a high probability for sites of the sametype to be adjacent to each other within some correlation length, i, measured on the two-dimensional surface. Those that we call statistically striated have a high probability for sites of theopposite type to be adjacent to each other within a correlation length i. Speci"cally, thedensity}density correlation function specifying the distribution of sites on the surface for the twosituations are: p2 exp(!ir) and p2(!1)*x`*y exp(!ir) for statistically blocky and striatedsurfaces, respectively. r is distance measured on the two-dimensional surface, and *x and *y aredisplacements in the two orthogonal cartesian coordinates used to specify position on the surface.p2 measures the strength of the surface disorder, and is proportional to the total surface loading,a physical parameter with which we are already familiar. As we shall see later, the total loading canbe varied conveniently in experiments.

Ensemble growth MC simulations have been performed to study whether statistically blockyDHPs can recognize statistically patchy surfaces more easily than statistically striated surfaces, andvice versa [24]. The two types of statistically patterned DHPs were characterized by j"0.4 andj"!0.4. The statistically patterned surfaces are characterized by a value of i"0.7/l, where l isthe statistical segment length. The way the statistics of surfaces within a class (statistically patchy orstriated) is varied is by changing the total loading, i.e., p.

Figs. 8a and b show the results of the ensemble growth MC simulations for statisticallyalternating and patchy surfaces. The adsorbed fraction pN is plotted as a function of p for statisticallyblocky and alternating DHPs in each panel. In all four cases, we "nd that a sharp adsorption


Fig. 8. Results of Ensemble Growth Monte-Carlo simulations for the interactions of statistically alternating (opencircles) and blocky ("lled circles) DHPs with (a) statistically patchy surfaces and (b) statistically striated surfaces.

transition occurs on the background of weak adsorption when the loading acquires a thresholdvalue. For the statistically patchy surfaces, we see that the sharp adsorption transition occurs ata smaller value of p for the statistically blocky chains. The opposite is true for the statisticallystriated surface. Thus, a statistically patchy (striated) surface with a loading that will stronglyadsorb the statistically blocky (alternating) ensemble of DHPs will not strongly bind the statist-ically alternating (blocky) chains. These results show that, at least from a thermodynamic stand-point, DHPs can discriminate between surfaces bearing di!erent statistical patterns, and vice versa.In other words, they can recognize each others statistics.

The physical reason that enables such recognition driven by statistical pattern matching is thefollowing. For pseudo patchy surfaces, arbitrary adsorbed conformations of blocky DHPs willexperience more unfavorable interactions at smaller surface loadings when compared to sequenceswhich are statistically alternating. Further, pattern matched adsorbed conformations with lowenergies are statistically far more probable for blocky DHPs interacting with pseudo patchysurfaces than for statistically alternating DHPs. These reasons cause the energetic driving force to


Fig. 9. Results of Ensemble Growth Monte-Carlo simulations. Adsorbed fraction p6 (full line) and x (dotted line) plottedagainst the surface disorder strength p. The results are for a small value of DbD.

adsorb in a few pattern-matched conformations (D) to be larger for statistically blocky chainsinteracting with pseudo patchy surfaces (at the same value of loading). The larger value of D whencombined with the fact that it is entropically less costly for the statistically blocky chain to organizeitself in a pattern matched conformation when interacting with a pseudo patchy surface leads suchstatistical sequences to adsorb strongly to pseudo patchy surfaces at smaller loadings compared tostatistically alternating sequences. Similar arguments explain why statistically alternating chainsrecognize pseudo striated surfaces at smaller loadings compared to statistically blocky chains.

The simulation results we have just described demonstrate that, from the standpoint of thermo-dynamics, statistically patterned chains can recognize statistically patterned surfaces when thestatistics characterizing the sequence and surface sites are related in a special way. The simulationresults have been found to be qualitatively captured by the simple formula described earlier withP@'"+

mP

4(m)P

#(m).

Static ensemble growth MC simulations have also been carried out to investigate the competi-tion between intersegment interactions and interactions between segments and surface sites [26].The main results of these simulations are as follows. There are two types of intersegmentinteractions. The "rst is a non-speci"c excluded volume interaction. The strength of these non-speci"c interactions is (<

AA#<

BB#2<

AB)/2, where <

ijare the strengths of the short-range

intersegment interactions between segments of type i and j. The sequence speci"c intersegmentinteractions which encourage segregation of like-type segments and freezing into a few dominantconformations are also taken to be short-ranged. The strength of these interactions isb"(<

AA#<

BB!2<

AB)/2. In the following, we will increase the strength of the interactions by

increasing the magnitude of b.Consider again 2-letter DHPs and multifunctional surfaces with short-range correlations de-

scribing the sequence and surface site #uctuations. When DbD is small, the variation of p and x withp is not qualitatively di!erent from that we have discussed earlier for b"0. This is shown inFig. 9 for DHPs with symmetric composition and for surfaces wherein the average composition ofthe two types of surface sites is also symmetric. Fig. 10 shows results for a relatively large value ofDbD"4. Even before adsorption occurs x(1. This is because of the well-known phenomenon


Fig. 10. Results of Ensemble Growth Monte-Carlo simulations. Adsorbed fraction p6 (full line) and x (dotted line) plottedagainst the surface disorder strength p. The results are for a relatively value of DbD"4k¹.

(e.g., [1}5]) wherein, for large enough values of DbD, frustration due to connectivity and the quenchedsequence disorder causes a few energetically favorable conformations to determine the thermo-dynamics. The situation we are studying is thus analogous to the adsorption of a folded proteinfrom solution. The results displayed in Fig. 10 show that the variation of p with p is of the sameform as that when DbD is small. Comparison of Figs. 9 and 10 shows that the sharp transition fromweak to strong adsorption occurs at higher values of p when DbD is larger. Physically, this is sobecause stronger speci"c intersegment interactions imply that the DHP chains are in lower energystates in solution compared to situations where DbD is small. Thus, compared to situations where thestrength of the intersegment interactions are small, stronger (more favorable) segment-surfaceinteractions are required for strong adsorption to become favorable.

A signi"cantly more interesting e!ect of speci"c intersegment interactions is revealed by thevariation of x with p displayed in Fig. 10. The few conformations that are adopted in solution aredetermined by the nature of the intersegment interactions and the sequence statistics. As theloading increases beyond the point where a gradual transition to weak adsorption occurs,x decreases even further. This is because even though the polymers are still essentially in the samefrozen conformations as those in solution, adsorption to the surface eliminates a few moreconformations from being sampled. When the loading is increased beyond the point where a sharptransition from weak to strong adsorption occurs, the MC simulation results show a ratherunusual variation of x with p. Fig. 10 shows that x "rst increases and then decreases again. This isinterpreted as follows. As the interactions of the chain with the surface increase because of thehigher density of surface sites, these interactions compete favorably with the intersegment interac-tions. Thus, the few dominant conformations favored by the intersegment interactions alone are nolonger energetically far more favorable than all other conformations. Thus, the system minimizesfree energy by gaining the entropy associated with sampling conformations other than thelow-energy conformations determined by the intersegment interactions alone. In other words, thefew dominant conformations adopted by the chains due to intersegment interactions unravel uponstrong adsorption to the surface. As the loading, and hence interactions with the surface, are


Fig. 11. Schematic depiction of the adsorption of DHPs with di!erent values of DbD: (a) moderate values of DbD. Here forlarge enough surface disorder strength, the conformations preferred by the intersegment interactions can unravel, (b)large values of DbD. Now, the intersegment interactions are too strong for the segment}surface interactions to be dominant.Adsorption always occurs in conformations very similar to those preferred by the intersegment interactions.

increased even further, the segment}surface interactions dominate over the intersegment interac-tions. Thus, the physics should be similar to that observed when the intersegment interactions areweak. That is, the strong segment}surface interactions coupled with the frustration due to disorderin DHP sequence and surface site #uctuations should cause the DHPs to freeze into a fewdominant conformations determined by the statistics of the chain sequence and surface sitedistributions. Consistent with this picture (shown schematically in Fig. 11), we see that x decreasesagain in Fig. 10.

The analytical and simulation results that we have described so far suggest the occurrence ofa phenomenon akin to recognition in biological systems due to statistical pattern matchingbetween the sequences of DHPs and the distribution of sites on multifunctional surfaces. Speci"-cally, these thermodynamic studies show that frustration (due to competing interactions anddisorder) and statistical pattern matching lead to one of the hallmarks of recognition: a sharpdiscrimination between surfaces to which a given ensemble of DHP sequences binds strongly andthose to which they do not. These studies also suggest that when strong adsorption occurs, thechains adsorb in a small class of conformations or shapes. These suggestions notwithstanding, it isunclear as to whether statistical pattern matching is su$cient to realize the other hallmarks ofrecognition and hence be pragmatically useful and of further scienti"c interest. The basic questionthat the aforementioned studies do not address is: can DHPs bearing a statistical patterndiscriminate between various parts of a single surface which bears di!erent statistical patterns indi!erent regions? Furthermore, can this happen in time scales of practical interest? If the answer to


these questions is to be a$rmative, then DHPs must not get kinetically trapped in the `wrongaregions of the surface. By wrong we mean those regions of the surface which bear statisticalpatterns of sites that are not matched to the DHP sequence statistics. To address these issueskinetic Monte-Carlo simulations have been performed [28]. The results of these simulations showthat the answers to the questions posed above are `yesa provided the statistical patterns areproperly designed. Furthermore, they reveal that the dynamical behavior of this system is typicalfor frustrated systems characterized by rugged free energy landscapes. As we shall discuss, theresults of the simulations demonstrate that the system we are considering may be a good modelsystem for experimental studies of kinetics in frustrated systems. Intriguing connections can also bemade to Kau!man's ideas [51] regarding the interactions between selection and the propensity forself-organization in evolution.

2.3. Kinetics of recognition due to statistical pattern matching

In order to discuss the kinetics of the phenomenon under consideration, let us begin again byconsidering 2-letter DHPs which carry statistical patterns. As before, let us concern ourselves withstatistically alternating and blocky patterns with the values of j being #0.4 and !0.4. Thecompositions of the two ensembles of sequences are symmetric.

As discussed earlier, surfaces with two types of sites on a neutral background can also bearsimple statistical patterns. The simplest statistical measures of the patterns carried by statisticallyalternating and patchy surfaces are the correlation length, the total density of A and B type sites onthe surface (loading), and the ratio f

4of the number of sites of A and B types (always equal to unity

in this discussion). Golumbfskie et al. [28] have generated statistically patterned surfaces of thistype by obtaining equilibrium realizations of a two-dimensional Ising like system using a Monte-Carlo (MC) algorithm. They simulate a lattice Hamiltonian with only nearest neighbor interac-tions, and MC moves which are exchanges of the identity of two sites. Typically 100 million MCsteps were run, well after equilibrium is established. For generating statistically patchy (alternating)surfaces, interactions between sites of the same type are taken to be attractive (repulsive) and thosebetween sites of the opposite type are repulsive (attractive). Neutral sites are non-interacting. Thecorrelation length is determined by the temperature, ¹

$, at which the MC simulation is carried out

and the loading. Large values of ¹$

lead to essentially random surfaces and, for a "xed loading,statistical patterns with larger correlation lengths are obtained upon reducing ¹

$. It is worth

noting that patterned surfaces of this sort can be created in practice by self-assembly of mixedmolecular adsorbents [52}56].

Consider a surface comprised of four quarters, each of size 100]100 lattice units. Let twoquarters have essentially random distributions of A (red) and B (yellow) type sites. Let the othertwo quarters be realizations of statistically patchy and alternating surfaces generated by themethods described earlier. Such an arrangement is shown in Fig. 12, where all quarters of thesurface are characterized by an average total loading of 20%, and the correlation length is &1.4for the statistically patterned regions.

Golumbfskie et al. [28] have carried out MC simulations with the Verdier}Stockmayer algo-rithm for chain motion to study the dynamic behavior of DHPs in the vicinity of such surfaces.They simulate a Hamiltonian with nearest-neighbor interactions. Attention is restricted to situ-ations where the segment}surface site interactions are much stronger than the intersegment


Fig. 12. Surface bearing four types of statistical patterns. White denotes the neutral background, and light and dark greydots correspond to the two types of active surface sites. The right top corner is statistically alternating and the left bottomcorner is statistically patchy. The other two quarters have a random distribution of surface sites.

interactions, and hence the latter are taken to be non-speci"c and of the excluded volume type. Thepreferred segment}surface interaction strengths are taken to be !1 and the ones that are notpreferred equal #1 in units of k¹

3%&, where ¹

3%&is a reference temperature. DHP segments of type

A prefer to interact with red sites on the surface, and those of type B prefer the yellow surface sites(see Fig. 12). The simulations are carried out at a temperature ¹ for chains of length 32, 100, and128. The results show that for "xed f and surface loading the important design variables are j, ¹

$and ¹/¹

3%&(determined by the chemical identities of segments and surface sites, and preparation

conditions).As we have seen from the thermodynamic results, statistically blocky DHPs are statistically

better pattern matched with the statistically patchy part of the surface, and statistically alternatingDHPs are statistically pattern matched with the statistically alternating region of the surface. The"rst question that one may ask is: If the surface shown in Fig. 12 is exposed to a solution containinga mixture of statistically blocky and statistically alternating DHPs, will the chain moleculesselectively adsorb on those regions of the surface with which they are statistically pattern matched?Such recognition due to statistical pattern matching requires not only that the `correcta patch bestrongly favored for binding thermodynamically, but also that the `wronga regions of the surfacenot serve as kinetic traps.


Fig. 13. Projection of typical center of mass trajectories for statistically blocky and alternating DHPs. The starting(ending) points of the trajectories are labelled 1 and 2 (1@ and 2@) for the statistically alternating and blocky chains,respectively.

1Local motion on the scale of monomers occurs in 10~10}10~11 seconds depending upon solvent conditions andmonomer size (see [47]). The MC moves employed by Golumbfskie et al. [28] are for motions of a statistical segmentlength. The time for local MC moves scales as the square of the statistical segment length for Rouse-Zimm dynamics.Estimates of the statistical segment lengths of synthetic polymers leads to the conclusion that the time scale associatedwith the MC moves in [28] ranges between 10~8}10~9 seconds. This is in agreement with previous estimates. Thenumbers reported by Golumbfskie et al. [28] are based on using 10~8 seconds as the time scale for primitive MC moves.

Fig. 13 depicts typical trajectories at ¹/¹3%&

"0.6. All points on these trajectories do notcorrespond to adsorbed states. Both the statistically alternating and blocky DHPs begin onrandomly patterned parts of the surface and ultimately "nd their way to the region of the surfacethat is statistically pattern matched with its sequence statistics. The length of the simulationsroughly corresponds to 1 s,1 over which separation or recognition is achieved with over 90%e$ciency (out of 1000 trials). These results show that biomimetic recognition between polymersand surfaces is possible due to statistical pattern matching. It seems possible to exploit this notionto design inexpensive devices that can separate a large library of macromolecules into groups ofstatistically similar sequences.


Fig. 14. Free energy landscape experienced by the statistically alternating DHPs as a function of center of mass position.The free energy is in arbitrary units, and the two deep minima on the right (in the wrong parts of the surface) are artifactsof periodic boundary conditions (see [28]).

In the trajectories shown in Fig. 13, the DHPs do adsorb onto the wrong parts of the surface.However, these adsorbed states are short lived compared to the time it takes to evolve to thestrongly adsorbed state. Once a chain is strongly adsorbed onto a region that is statistically patternmatched with its sequence the chain center of mass essentially does not move on simulation timescales (vide infra). The reason for this is made clear by the results shown in Fig. 14. The `wrongaregions of the surface correspond to local free energy minima which are separated by relativelysmall barriers from each other. In contrast, in the statistically pattern matched region there exista few deep global minima; each of these minima, in turn, is very rugged. It is important to note that¹/¹

3%&is an important design variable determined by the chemical identity of segments and surface

sites. Large values of this parameter will not allow the chains to adsorb anywhere to appreciableextent because the entropic penalty associated with adsorption dominates the free energy. Atsu$ciently low values of this ratio, the `wronga regions of the surface serve as kinetic traps, andlong-lived metastable adsorbed states exist in these regions. Thus, it is clear that due to both kineticand thermodynamic considerations there should be an optimal value of ¹/¹

3%&. Simulations have

been carried out for ¹/¹3%&

"0.45, 0.6, and 0.75. Of these conditions, Golumbfskie et al. [28] "ndthat 0.6 leads to the best discrimination and recognition. More work is needed in order to providequantitative criteria for the optimal value of ¹/¹

3%&.

The importance of both kinetics and thermodynamics for statistical pattern matching is furtheremphasized by the following point. We have provided a detailed discussion of thermodynamicarguments which suggest that DHPs with random sequences (j"0) sharply discriminate betweensurfaces with random site distributions which have di!erent loadings. However, Golumbfskie et al.[28] "nd that the likelihood of kinetic trapping in the wrong regions of the surface is rather highwhen uncorrelated sequence and surface site distributions are statistically pattern matched byadjusting the loading. In particular, let us compare the following two situations: (a) a surface withrandom site distribution bearing regions with loadings of 20% and 40% being exposed to DHPs


Fig. 15. Statistically patterned surface with three types (depicted in light grey, dark grey and black) of sites. Four di!erentstatistical patterns (described in the text) exist along with regions having a random site distribution.

with random sequences that thermodynamically strongly favor the region with a loading of 40%;(b) the surface shown in Fig. 12 exposed to the statistically alternating sequences. We note that theequilibrium adsorbed fraction in the region with a loading of 40% in case (a) and that for thestatistically alternating DHPs in the statistically alternating region in case (b) are roughly the same.Simulations (over 40 trials in each case) show that the window of ¹/¹

3%&for successful recognition

('80% e$ciency) is 50% wider for the latter case (b). The reason for this is that in the formersituation the free energy landscape that the chains negotiate is virtually uncorrelated, and thus rifewith local optima that lead to kinetic trapping [51].

Having illustrated the basic phenomenology of biomimetic recognition due to statistical patternmatching, Golumbfskie et al. [28] have also examined a situation that is a step closer toapplications. If the notion of statistical pattern matching is to be used to a!ect molecularly selectiveseparations, we need to separate more than two types of statistical patterns. A more diverse set ofstatistical patterns can be obtained by increasing the number of letters that code the statisticalpatterns. Using a 3-letter code, Golumbfskie et al. [28] generate four di!erent statistical patterns toillustrate a more complex separation than that depicted in Fig. 13. The surfaces are generated usinga lattice where the identity of each site can be A, B, C or neutral. For all cases, in the MCsimulations that generate statistically patterned surfaces the interaction energies (<

ij, where i

and j denote type of site) are symmetric and equal in magnitude, with the neutral sites being


Fig. 16. Starting (1,2,3,4) and ending (1@,2@,3@,4@) points of typical trajectories of four di!erent types of statisticallypatterned DHPs on the surface shown in Fig. 15. For these trajectories statistical pattern matching leads to successfulrecognition of complementary surface patches.

non-interacting. For statistical pattern number 1, <ii"<

AB"!1, <

AC"<

BC"#1. For statist-

ical pattern number 2, <ii"<

BC"!1, <

AC"<

AB"#1. For statistical pattern number 3,

<ii"<

BC"#1, <

AC"<

AB"!1. For statistical pattern number 4, <

ii"<

AC"#1,

<AB

"<BC

"!1. The realizations of the statistical patterns shown in Fig. 15 correspond to a totalloading of 30%, a 1 : 1 : 1 ratio of di!erent types of sites, and a correlation length of &1.4. Thecorrelation length is calculated for the one unique type of site (e.g., C for pattern 1).

The interaction energies above are also used to generate four types of DHP sequences that arestatistically pattern matched with the surface patches. The DHP sequences employed by us belongto the "rst three excited states in terms of the energy spectrum. Just the energy is clearly nota complete speci"cation of the sequence statistics, and points to the need to develop better designcriteria for 3-letter codes. What happens when four types of statistically patterned DHP sequences,each statistically pattern matched with one of four surface patches, are placed near the surface?

Fig. 16 shows what happens when four types of DHP sequences, each statistically patternmatched with one of the four patches, are placed near the surface. For clarity, only the starting and"nal positions of the chain centers of mass are shown. As depicted in Fig. 16, the DHP chains all"nd and then bind strongly to the complementary statistically patterned region, i.e., biomimeticrecognition due to statistical pattern matching is possible in this case also. However, unlike the100% e$ciency reported for the two letter cases, over a simulation time of 5 s, Golumbfskie et al.


Fig. 17. Distribution of "rst passage times for the event described in the text.

[28] "nd the lowest success rate to be 60%. The success rates increase with simulation time, andfailures are the result of kinetic trapping in the wrong regions of the surface. Simulations have notbeen carried out for long enough times to report the time scale over which the success rate wouldapproach 100% for all situations. While Fig. 16 demonstrates the feasibility of carrying outmolecular scale separations using statistical pattern matching with many such patterns, thee$ciencies found thus far (without detailed design considerations) also illustrate that we have notyet learned how to design statistical patterns with large number of letters in the code such thatlong-lived metastable states are largely eliminated.

Fig. 14 suggests that, for the system we are considering, chain dynamics on various time scalesand di!erent spatial locations should be quite di!erent and interesting. Detailed analyses ofdynamic MC simulation results serve to elucidate these issues, and provide evidence for macro-molecular shape selection when recognition occurs due to statistical pattern matching. Considerthe "rst passage time, which is de"ned to be the time taken for a chain starting in the random partof a statistically patterned surface to adsorb in the pattern-matched region with the energy beingbelow a certain cut-o! value. Fig. 17 shows the distribution of "rst passage times for chains oflength 32, with the energy cut-o! being !20k¹

3%&. It is clear that, after the usual turn-on time, the

distribution is decidedly non-exponential; a stretched exponential with a stretching exponent equalto 0.43 "ts the simulation results. The non-exponential character of the distribution of "rst passagetimes is indicative of highly cooperative chain dynamics. The event we are considering is tanta-mount to two events that occur in succession. These are chain motion to the edge of the statisticallypattern matched region of the surface followed by strong adsorption in one of the deep free energyminima in the statistically pattern matched region. The "rst passage times for these two events havebeen computed separately [28]. Signi"cantly, the distribution of "rst passage times for the "rstevent is exponential, and that for the second is non-exponential. This implies that as the chaintraverses the wrong parts of the surface, the center of mass motion is di!usive. This is because thefree energy minima and barriers it encounters in these regions of the surface are relatively small. Incontrast, it appears that strong adsorption in the statistically pattern matched region is highly


Fig. 18. Distribution of Pn: (a) center of mass in the wrong parts of the surface, (b) center of mass has entered the

statistically pattern matched region, (c) strongly bound state.

cooperative with many important free energy barriers encountered enroute to the bottom of one ofthe deep free energy valleys that exist in this part of the surface. What is the physical origin of thisbehavior in the statistically pattern matched region?

It is interesting to observe [28] that single particle MC dynamics on the free energy landscapeshown in Fig. 14 is exponential. This indicates something very important for understanding thenature of the dynamics. Single particle dynamics on the free energy landscape in Fig. 14 wouldreproduce the "rst passage time distribution obtained from the full chain dynamics if motion of thecenter of mass was always the slowest dynamic mode. Our results suggest that this is not true.

As we have discussed, adsorbed macromolecules are characterized by loops. The distribution ofthese loops #uctuates in time, and hence loop #uctuations are dynamic modes. Prior to consideringthe dynamics of these modes, let us examine the distribution of these loops as trajectories evolvestarting from wrong parts of the surface. Extensive MC simulations have been carried out withchains of length 100. All trajectories show the qualitative features depicted in Fig. 18 where weshow the probability distribution P

nfor an arbitrary segment to be part of a loop of length n for

a statistically alternating DHP. Each panel shows Pn

averaged over di!erent time (MC step)windows. The "rst panel shows that when the center of mass of the chain is in the wrong parts of thesurface P

nis structureless. This implies that all possible macromolecular shapes are being adopted

as the chain samples the wrong parts of the surface. Remarkably, the other panels in Fig. 18demonstrate that as the chain center of mass enters the statistically pattern matched regionPn

begins to show structure. Ultimately, it exhibits a spectrum of peaks which correspond to


Fig. 19. Squared and averaged displacement of the center of mass as a function of time measured in Monte-Carlo steps.

preferred loop lengths and hence macromolecular shape in the adsorbed state. This observation ofadsorption in preferred shapes due to statistical pattern matching is akin to recognition in biology,and we shall make it vivid shortly. Loops with the preferred lengths appear to be quenched in time,and so as before, we will call them quenched loops.

The physical reason for the lack of quenched loops in the wrong regions of the surface, whilethese types of loops seem to dominate the behavior in the statistically pattern matched region hasbeen discussed earlier. In the wrong region of the surface arbitrary adsorbed conformations withthe same number of contacts have roughly the same energy. This is because there are no particulararrangements that lead to much higher degrees of registry between adsorbed segments and theirpreferred sites, compared to other conformations. Thus, the distribution of loops #uctuatesstrongly, thereby gaining entropy while maintaining roughly the same energy. In the statisticallypattern-matched region of the surface, however, the situation is di!erent. While most adsorbedconformations are of relatively high energy as in the wrong region of the surface, now there exista few conformations that can bind segments to their preferred sites with high probability in certainregions because the statistics of the sequence and surface site distributions are matched. These fewpattern-matched conformations are much lower in energy than all others, and thus the chainsacri"ces the entropic advantage associated with loop #uctuations and `freezesa into one of theseconformations. The better the statistical pattern matching, the greater the suppression of loop#uctuations. Dramatic evidence for this thermodynamic argument (and the associated model) isprovided by the simulation results displayed in Fig. 18. We shall also make this vivid shortly byshowing snapshots of animations of the dynamic simulation results.

Let us return to the puzzle of why single particle dynamics on the free energy landscapecalculated for the chain center of mass does not reproduce the correct dynamical behavior.Consider the time correlation functions SP

n(t)P

n(t#q)T that describe how loop (and hence, shape)

#uctuations decay for n"17, 26, and 30. Attention is focused on these values of n since Fig. 18shows that the dominant shape contains loops with these lengths. The simulation results show that,for motion in the wrong parts of the surface, these correlations decay much more rapidly than thetime scale on which the chain center of mass moves. Thus, the modes corresponding to loop


Fig. 20. Relaxation time for loops of length 26. Data shown only after the center of mass enters the statistically patternmatched region. The relaxation time is vanishingly small for this loop length in the wrong parts of the surface.

#uctuations are the fast dynamic modes. After some time in the statistically pattern matchedregion, the chain's center of mass "nds the location corresponding to one of the deep free energyminima in this region. Now, the center of mass essentially does not move during the simulation. Asshown in Fig. 19, for chains of length 32 the center of mass ceases to move, and for chains of length100 the di!usion coe$cient becomes two orders of magnitude smaller than that in the wrong partsof the surface. On much longer time scales, the center of mass of these "nite chains will be able toescape from these minima and "nd other deep minima in the statistically pattern matched region.However, these long times are not relevant to these simulations, and for very long chains, theseevents would occur over time scales that are not experimentally relevant. The point is thatlocalization of the center of mass implies that this dynamic mode has equilibrated from a practicalstandpoint. Simulations show that, after the chain center of mass has equilibrated in the statist-ically pattern matched region, the time correlation functions describing loop #uctuations decayover very long times. In fact, ultimately, once shape selective adsorption occurs, they do not decayto zero and simply oscillate around a "nite value. Thus, these modes, which were the fastestdynamic modes in the wrong part of the surface become the slowest modes once the center of massreaches a location corresponding to a deep free energy minimum. This explains why single particledynamics on the free energy landscape appropriate for the center of mass cannot describe thecorrect dynamics obtained from the detailed simulations. Fig. 20 shows how the relaxation time forloops of length 26 changes as a typical trajectory evolves in the statistically pattern matched region.

The observations noted above imply the following physical picture. Once the center of mass ofthe chain is located in the region corresponding to a deep free energy minimum, the chain has toacquire the shape (conformation) that is most favorable on energetic grounds. In order to do this, itmust arrange its loops in a speci"c way. The necessary conformational rearrangements occur ina highly cooperative way, and there are important entropic barriers associated with this reorgan-ization. These entropic barriers which correspond to arranging the chain to acquire a speci"cshape (a hallmark of recognition) lead to the non-exponential character of the "rst passage timedistribution.

In order to make some of the results discussed above vivid, Golumbfskie et al. [28] haveanimated trajectories of their simulation runs. Fig. 21 shows snapshots from such a movie for


Fig. 21. Snapshots of chain conformations at di!erent times during a typical trajectory for a 128 segment statisticallyalternating DHP. Dark and light shadings represent the two kinds of surface sites and two kinds of segments that makeup the chain. The top three panels correspond to portions of the trajectory in the wrong parts of the surface, and thebottom three panels are for the statistically pattern matched region. The numbers correspond to time in arbitrary units.

a 128-segment statistically alternating 2-letter DHP. The trajectory starts with the chain above thestatistically random part of the surface. The DHP does adsorb somewhat on this region of thesurface. The absorbed conformations are characterized, as usual, by loops, trains, and tails whichexhibit large #uctuations. For example, both the loop length distribution and the distance on thesurface between loop ends #uctuate rapidly. The "rst three frames of Fig. 21 depict this. With time,this DHP evolves toward the statistically alternating region of the surface, and adsorbs strongly.Now the chain dynamics are dramatically di!erent (last three frames in Fig. 21). Several chainsegments adsorb on preferred sites on the surface. The resulting loops are quenched in time in thatthere are essentially no #uctuations of these loops on the time scale of our simulations. Within thesequenched loops live annealed loops that exhibit small conformational #uctuations.

The problem of how macromolecules recognize patterns on surfaces has also been studied byMuthukumar and co-workers [29}31] in the context of polyelectrolytes interacting with a surfacepattern of opposite sign. These studies complement the work described earlier in this section, andare very important contributions to the conceptual advances that are being made in understandingpattern recognition by macromolecules. We do not, however, provide a detailed review of this workbecause Muthukumar has recently provided insightful reviews [17,57].

The surfaces considered by Muthukumar and co-workers [29,30] contain a single pattern oftypical size ¸ made up of charges imprinted on a neutral background (of size <¸). The questionthey ask is: can a polymer consisting of oppositely charged segments distributed on the backbonein a particular sequence recognize (adsorb on) this region of the surface? Muthukumar [29] useda variational method to derive conditions for occurrence of such selective adsorption. Furtherinsight on this issue has been obtained via MC simulations [30]. Recent experiments (see referencesquoted in [17]) concerning the binding of polyelectrolytes to proteins seem to be consistent with


Fig. 22. The initial slope (proportional to adsorbed fraction at in"nite dilution) for horse cytochrome C adsorbing ona hydrophobic support randomly functionalized with copper. Di!erent points on the abcissa correspond to di!erentsurfaces each with a di!erent loading of copper. Data taken from [7].

some of the "ndings reported in [29,30]. The studies of Muthukumar and co-workers [17] alsoshow the importance of entropic frustration (vide supra). For the system they study, adsorptionoccurs in two stages, with the slower second step being associated with the development of registrybetween the charged segments on the polyelectrolyte and the oppositely charged surface sites.

2.4. Connection to experiments and issues pertinent to evolution

The results described in the previous subsections suggest that the phenomenon of recognitiondue to statistical pattern matching might prove useful in applications, and the class of systems thathave been considered are also good model systems to study kinetic phenomena in frustratedsystems. These reasons motivate pertinent experimental studies. Some examples of applicationswhere this phenomenon may be pro"tably exploited are chromatrography, the development ofviral inhibition agents, high throughput screening of vast numbers of macromolecules intoensembles of statistically similar sequences, and sensors.

Some coarse grained aspects of the phenomenology that we have described may already havebeen observed in experiments carried out in the context of chromatrography and viral inhibition.Arnold and co-workers have done some interesting experiments aimed toward the development ofinexpensive chromatographic materials for protein separations. In one such experiment [7], theymeasure the isotherms that describe the adsorption of horse cytochrome C on hydrophobicpolymer supports which have been functionalized with copper. Copper is adsorbed onto thesurface from a solution of a copper salt. The copper atoms adsorb randomly, and the loading ofcopper sites on the surface can be adjusted by varying the salt concentration in solution. Thesurface density of copper sites can be measured by titration. Hystidine residues on horse cyto-chrome C preferentially interact with the copper sites, while the others do not. Arnold andco-workers [7] have measured the initial slope of the adsorption isotherm as a function of theloading of copper sites. The initial slope is directly proportional to the adsorbed fraction forin"nitely dilute conditions. Their data are shown in Fig. 22. The shape of this curve is remarkably


similar to the theoretical predictions [23,26] and simulation results [24,25] that we have discussed.This indicates that statistical pattern matching may be the origin of the experimental observations,and that further work exploring whether the phenomenon can be used in chromatographicapplications is warranted.

Another important application where similar phenomenology may have been observed concernsviral inhibition. Viruses can attach to receptors on cell surfaces, and this is an important step whichleads to viral infections. Whitesides and co-workers [9}11] have considered the possibility ofinhibiting this process by adsorbing synthetic polymers onto the virus' surface, thereby blockingout the sites that can attach to the binding sites of receptors on cell surfaces. The question is: whatpolymer to use? Whitesides and co-workers have used random copolymers with the monomer unitsbeing methacrylic acid and a sugar [10]. Speci"cally, they have studied how the binding of thein#uenza virus on mammalian erythrocyte cells is inhibited by adsorbing such polymers on thevirus. In one set of experiments they have measured this inhibition constant as a function ofthe average composition of the DHPs. They "nd that when the inhibition constant is plotted asa function of the average polymer composition, it exhibits a maximum. The average composition isa measure of the statistical pattern encoded in the sequence statistics of the polymers (vide supra).The data from the Whitesides group suggests that a certain statistical ensemble of sequences (i.e.,with a certain range of average compositions) is most e$cient at inhibiting the in#uenza virus fromattaching to the erythrocyte cells. In other words, this ensemble adsorbs onto the surface of thevirus and blocks out its binding sites most e$ciently. This is consistent with the notion ofrecognition due to statistical pattern matching we have described. It is important to note, however,that theory and simulation predict a sharper maximum than that observed experimentally.

More detailed experiments that aim to study the phenomenon of statistical pattern matchingmore carefully are currently being considered and initiated by various groups. Surfaces bearingstatistical patterns can be created [52}54]. The major di$culty is associated with careful character-ization of the statistical patterns. Attempts to address this issue using photoemission electronmicroscopy (PEEM) are currently underway [58]. DHPs with careful control of sequence statisticscan and are being prepared to carry out experimental studies of recognition due to statisticalpattern matching [59]. Both synthetic and genetic methods are being used. The latter method [60]provides more control of the ensemble of sequences that are created. An experiment that shouldserve to study the phenomenology depicted in Figs. 12 and 13 is being planned [58,59]. Here, twoensembles of sequences with di!erent statistics will be tagged with red and green #uorescent labels.Adsorption from solution onto a surface resembling that depicted in Fig. 12 will be carried out, andoptical microscopy will be used to study whether the molecules "nd their statistically patternmatched complementary regions on the surface.

The theoretical and computational results that we have described demonstrate that the systemwe have been considering exhibits dramatic di!erences in dynamics on di!erent scales of space andtime. These are classic signatures of dynamics in free energy landscapes characteristic of frustratedsystems. For example, these features are similar to the behavior of spin glasses [42] and proteinfolding phenomena (e.g., [1}5]). Detailed experimental studies of the system we have beenconsidering may, however, be more tractable as a simple polymeric system is easier to synthesize,manipulate, and characterize. Consider as an example systems such as the one considered in theprevious paragraph. In the future, such systems may be employed to study kinetic phenomena also.This may be especially true given the rapid advances being made in single molecule spectroscopy.


Fig. 23. Decay of correlations in the #uctuations of the end-to-end vector in the wrong and statistically pattern matched(SPM) regions.

Single molecule spectroscopy [61] of the DHP should be able to measure the correlation functionsthat have been computed [28]. The problem seems ideal for this tool which is most revealing(compared to other experimental probes) when the ergodic hypothesis is violated. A convenientcorrelation function from the standpoint of single molecule spectroscopic experiments isSR(t#q) ) R(t)T, where R is the chain end-to-end vector. This quantity re#ects correlationsbetween shape #uctuations, and thus measures how chain conformation evolves in time. In Fig. 23we show how SR(t#q) ) R(t)T decays with time in the wrong and statistically pattern matchedregions [28]. In the former region, the correlations decay as usual. In the statistically patternmatched region, conformational #uctuations are strongly correlated over long times.

Our results also suggest an intriguing connection between the experimentally realizable systemthat we study and some provocative ideas concerning the competition between self-organizationand selection in evolution. Populations can be considered to evolve on "tness landscapes due tomutations [51]. Kau!man has suggested the NK model (which characterizes the varying degrees ofruggedness of "tness landscapes) to study how the competition between the ability to self-organizeand selection in#uences the manner in which populations evolve [51]. Population evolution on"tness landscapes due to a "nite mutation rate is analogous to the "nite temperature dynamics ofphysical systems on free energy landscapes [49]. The model we study is a kind of NK model. Wehave described the motion of a statistically alternating DHP on a free energy landscape corre-sponding to a surface which has random regions and a region that is statistically pattern matchedwith the DHP. This type of chain (analog of genotype) which has the ability to self-organize isnaturally led to a free energy minimum in the statistically pattern matched region. Consequently,members of this ensemble of DHP sequences develop speci"c characteristics } viz., a speci"cdistribution of loop lengths or shape. In contrast, when statistically blocky DHPs in the vicinity ofthis surface are simulated, P

ndoes not develop any structure. This is because this ensemble of

DHPs cannot self-organize (the free energy/"tness landscape does not have the structure shown inFig. 14). Experimental studies of the physically realizable situation that we have described mayprovide insights (by analogy) to issues pertinent to models of evolution.

The "ndings reviewed in this section also lead us to speculate whether in the early origins of liferecognition was a!ected by statistical pattern matching, with speci"c sequences for binding andrecognition resulting from evolutionary re"nement of ensembles of statistically pattern matched


sequences. This speculation can be explored by studying whether an ensemble of statisticallypattern matched sequences can evolve by mutations to a few speci"c sequences. This type ofresearch may also prove useful for applications. Given a particular surface pattern, can we startfrom the ensemble of statistically pattern matched sequences and systematically design speci"ctypes of sequences that would bind most e$ciently? These issues can be addressed by usingsequence space Monte-Carlo annealing of DHP sequences. Some work along these lines for theadsorption of DHPs on chemically homogeneous surfaces has recently been reported [62].

3. Branched DHPs in the molten state } model system for studying microphase orderingin systems with quenched disorder

The behavior of in"nitely long linear DHPs in the compact state [the volume approximation(e.g., [1}3,63}65])] and that of chains of "nite length in the molten state has been studiedextensively [66,67]. Two issues have been of interest. The "rst concerns the connection between thephysics of DHPs and protein folding. Su$ciently sti!DHPs can undergo a phase transition where,below a certain temperature, the thermodynamics is determined by a few dominant conformations.This transition (often called a freezing transition) from a phase where multitudes of conformationsare sampled to a low entropy state with only a few important conformations occurs because offrustration due to competing intersegment interactions and the quenched disordered sequence.This transition is akin to the folding of proteins to form the native state. Understanding howproteins fold is an important issue in biophysics. The thermodynamics and kinetics of the freezingtransition in DHPs has been studied extensively via theory and computation with a view towardunderstanding the physics of protein folding. These e!orts have been reviewed in several journalsrecently (e.g., [3}5]), and it is interesting to consider the parallels between these studies and thesubject reviewed here in Section 2.

The second issue pertaining to the behavior of globular DHPs and molten DHPs that has beenstudied concerns microphase ordering. Mixtures of incompatible homopolymers undergo macro-scopic phase segregation when cooled below a certain temperature. Molten copolymers withincompatible segments, however, cannot segregate on macroscopic scales because of chain con-nectivity. They undergo an order-disorder transition and form microdomains when cooled belowthe microphase segregation temperature (MST). Mircophase ordering for copolymers with orderedsequence distributions has been studied extensively via experiment, theory, and computation.Interesting physics and exotic microstructures have been revealed by these studies, especially in thecontext of studying diblock copolymers (e.g., [68}72]). These are polymers with two types ofsegments; a polymer of A-type segments joined at one end with a polymer of B-type segments.DHPs are polymers with a disordered sequence distribution. Thus, in addition to connectivity,another source of frustration hinders the propensity for the two types of segments to separate. Thee!ects of frustrating quenched randomness on microphase ordering has been considered bytheorists by studying linear DHPs. A recent review by Shakhnovich and Gutin in this journal [6]and references therein describe these e!orts in detail. The most signi"cant "nding is that theoptimal wave vector (qH) corresponding to the length scale of ordered domains depends strongly ontemperature below the MST. This is in contrast to the behavior of diblock copolymers, whereinqH is essentially temperature independent below the MST [68]. At temperatures su$ciently below


the MST, qH for DCPs does decrease with temperature because of chain stretching and strongsegregation e!ects (e.g., [73,74]).

Until recently, however, there had been no experimental studies of microphase ordering inmolten polymers with quenched sequence disorder. This has largely been because of the di$cultiesencountered by polymer physicists in synthesizing linear DHPs with controlled sequence statistics.Recently, one group [75] has succeeded in synthesizing and characterizing a class of polymerswhich embodies competing interactions and quenched disorder. Speci"cally, there isa homopolymer backbone of length N onto which are grafted p branches of another type ofhomopolymer of length M. The branch points are randomly located on the backbone, and theirlocations are quenched after synthesis is complete. Each chain in a melt of these copolymers hasa di!erent sequence. Following Qi et al. [76], we refer to these materials as randomly branchedheteropolymers (RBHs).

The behavior of this material when it is cooled below the MST has been considered by opticalbirefringence experiments, small angle neutron scattering (SANS), and a "eld-theoretic model.Let us begin by describing a simple theoretical model to consider how this material behaves astemperature is scanned.

Consider Np

chains with ri(n) being the spatial location of the nth segment on the backbone of

the ith chain, and rij(m) be the mth segment of the jth branch on the ith chain. Let the set Mn

ijN

represent the quenched locations of the branch points. The microscopic Edwards Hamiltonian forthe problem can then be written as:

!bH[Mri(n)N, Mr

ij(m)N]

"!

32l

NP

+i/1

PdnAdr

i(n)

dn B2!

32l

No+i/1

P+j/1PdmA

drij(m)

dm B2

!E[Mri(n)N, Mr

ij(m)N] (30)

where E is the energy corresponding to intersegment interactions, and depends explicitly on thechemical identities of the interacting segments.

In order to construct a tractable theory we now transform to a representation in terms of oA(r)

and oB(r), which are macroscopic "elds corresponding to the local volume fractions of A and

B-type segments.The energy E is easily written in such a representation, and is independent of sequence

distribution and architecture. It simply re#ects that there is an energetic driving force for the twotypes of segments to segregate, and that surface tension penalizes segregation on small scales. Thestandard way to write down the energy corresponding to given macroscopic "elds is

E"

12P drM!2sm2(r)#c2[+m(r)]2N (31)

where s is the Flory parameter [73] which measures the energetic driving force for segregation, c2 isthe surface tension, and m(r)"f

BoA(r)!f

AoB(r) where f

Aand f

Bare the average volume fractions

of A- and B-type segments.The partition function for a given realization of Mn

ijN can be expressed as

z"exp[!E#S] (32)


where the entropy S is de"ned by

S"ln(R)"lnGNP

<l/1

P<j/1

PDri(n)Dr

ij(s)]P

0[r

i(n)]P

0[r

ij(s)]d(r

ij(0)!r(n

ij))H (33)

where P0

are the Gaussian functions in Eq. (30) which enforce connectivity.The energy does not depend on the quenched distribution of branch point locations. The entropy

does. We need to perform a quenched average over the branch point locations. This implies that wemust average Eq. (33); i.e., we must average a logarithm.

We can use the celebrated replica trick to carry out this average [41]. It is important to note thatother approaches such as cumulant expansions could be used to carry out such an average (e.g.,[6,66,67]). Since we are concerned here with microphase ordering, the pertinent quantities bearonly one replica index. Therefore, the issue of broken replica symmetry is not relevant (as discussedin the context of microphase ordering in linear DHPs [6]).

Details of the derivation are provided in the appendix. Here, we merely sketch the steps. Let usintroduce "elds c(r) and /(r) conjugate to the single chain "elds, oA and oB in order to resolve thedelta functions in Eq. (32), and then replicate the same equation to obtain

SRnT"n

<a/1PPD/a (r)Dca (r) expGi Pdr[ca (r)oAa (r)#/a (r)oBa (r)]H

Tn<a/1PPDra (n)<

jPPDr

ja(m)P[rja (m)]P[ra (n)]<

j

d(rja(0)!ra (nj

))U (34)

with the averaged entropy SST"limn?0

(SRnT!1)/n. Eq. (34) has been written with therealization that, although each chain has a di!erent sequence, since the distribution functiondescribing the #uctuations in branch point locations is the same we need only use the symbolnj

to denote branch point locations. We have also taken Np(N#pM)/<"1, and the angular

bracket denotes the quenched average; P[ra(n)]"P0[ra (n)] exp[!i:dn ca (ra (n))], P[r

ja (m)]"P0[r

ja (m)] exp[!i+j:dmUa (rja (m))]; and i"J!1.

We now expand the functionals P in powers of the conjugate "elds up to fourth order, with thebare propagators being Gaussian (P

0). Carrying out the resulting integrals, performing the average

over the distribution of nj, and evaluating the functional integrals over the conjugate "elds using

saddle points we "nd the replica symmetric solution for the entropy to be:

SST"!

12Pdq o=

Mo!

18CPdqo=

MoD

2

! +i,j,k/A[B

Pdq1Pdq

2oi(q

1)o

j(q

2)o

k(!q

1!q

2)C(3)

ijk(q

1, q

2, q

3)

!

18Pdq

1Pdq2

oT(q1)=M

(q1)[SM

0(q

1)=M

(q1)oT(q

1)o(q

2)=M

(q2)M

0(q

2)T


Fig. 24. Optimal wave vector qH as a function of the product sM(&1/¹).

!oT(q1)o(q

2)]=M

(q2)o(q

2)! +

i,j,k,l/A[BPdq

1Pdq2Pdq

3oi(q

1)o

j(q

2)o

k(q

3)

]ol(!q

1!q

2!q

3)C(4)

ijkl(q

1, q

2, q

3) , (35)

where SM0T"=~1, oT"(o

A, o

B), and underlined symbols are matrices.

Combining Eq. (34) with Eq. (30) obtains the free energy functional that we seek. Mathematicalformulas for M

0, C(3), and C(4) are provided in the appendix. In order to see the essential issues

clearly, here we remark on their mathematical forms before presenting detailed results. Thequadratic term (F

2) in our free energy functional contains terms proportional to q2 and 1/q2. The

physical origin of the coulombic term in F2

is the stoichiometric constraint that forces the branchesto be connected to the backbone at the branch points. This is so because the condition that allM segments of a branch must lie within a certain distance (equal to the radius of gyration) from thecorresponding branch point is similar to the neutrality condition in a system of interacting charges.Since such a stoichiometric constraint is absent in linear DHPs, a coulombic term does not arise inthe theory of microphase ordering of linear DHPs [68}72,77]. F

2(q) depends upon temperature,

but exhibits a minimum with a temperature independent value of q at the minimum. Shinozakiet al. [78] and Fredrickson and co-workers [79] obtained a free energy functional for molten RBHsup to quadratic order, and studied the structure factor in the disordered phase as a function ofvarious parameters. Up to quadratic order, the free energy functional obtained by Qi et al. [76] isidentical to that obtained in [78,79]. However, as we shall see the quartic terms derived by Qi et al.[76] play a crucial role in determining the physics of microphase ordering in molten RBHs. So, letus discuss the form of the quartic terms before considering the cubic term. The fourth term inEq. (34) (F

41(q)) is a temperature-dependent function that decays sharply with q at relatively small

values of q. This term originates from the quenched #uctuations in branch point locations. It issimilar in form to a term that arises in the theory of microphase ordering for linear DHPs [6,65}68]because, if we think of the branch points as another type of monomer on the backbone, the#uctuations in branch point locations are similar to the quenched sequence #uctuations along thebackbone of linear DHPs. The last term in Eq. (34) (F

42(q)) is an Ising-like term. C

4contains


contributions from the mean values of the quenched disorder (i.e., the average number of branchpoints). In this sense, this term resembles the Ising-like term that arises in the theory of linearDHPs. For RBHs, however, this term also contains coulombic terms which arise due to thestoichiometric constraints discussed earlier. F

42(q) resembles F

2(q), with the minimum shifted to

higher values of q.In order to understand the basic physics of microphase ordering in incompressible (o

A#o

B"1)

RBH melts, Qi et al. [76] studied the simplest morphology; viz., the lamellar morphology. Thus, thefunctional form for the density wave is taken to be o

A(r)"D cos(q ) r)#m

A. For this morphology,

the cubic term vanishes. A mean-"eld solution is obtained by minimizing the resulting free energyfunctional with respect to D and DqD. Fig. 24 shows the theoretical prediction for the optimal wavevector as a function of temperature for a speci"c choice of chain architecture. At high temperature,we have a disordered phase, and qH is small and invariant with temperature. Immediately below themicrophase separation temperature (MST), qH increases strongly with temperature, a featurereminiscent of earlier predictions for linear DHPs. At lower temperatures, qH becomes essentiallyindependent of temperature. The range of temperatures over which qH varies strongly withtemperature depends upon the values of the parameters that characterize the architecture and thestatistical distribution of branch point locations. The variation of the length scale over whichordering occurs on temperature depicted in Fig. 24 is very di!erent from that observed for diblockcopolymers and that predicted for linear DHPs. In the former case, there is some decrease inqH with temperature largely due to chain stretching. In the latter situation, theory predicts thatqH continues to increase with decreasing temperature until we approach microscopic scales.

The unusual variation of qH with temperature for RBHs can be understood in simple physicalterms. In the disordered state, D"0, and qH&(N#pM)~1@2. This is to say that #uctuations occuron the scale of the entire chain. After the MST, D acquires "nite values, and the system begins toorder on scales smaller than chain dimensions. However, the length scale corresponding to orderedregions is still much larger than the branch length. On scales much larger than the branch length,the system resembles a linear DHP; the branches now look like segments of another type ofmonomer. In this limit, since the #uctuations in the locations of branch points are uncorrelated,there is no natural length scale set by the sequence and architecture of the polymer (in contrast, forexample, to diblock copolymers). Thus, in this limit, as predicted for linear DHPs, qH increases withtemperature. This is because the entropic penalty for ordering on smaller scales decreases continu-ously, and the system orders on progressively smaller scales due to the energetic driving force. Inlinear DHPs this behavior continues until ordering occurs on microscopic scales. In the systemunder consideration, however, when qH acquires values corresponding to the length scale of thebranches (M) or length scales shorter than the mean value of the backbone sections between branchpoints (N/p), a natural scale emerges. Ordering on smaller scales is prevented because of the largeentropy penalty associated with further squeezing the regions occupied by the branches and thebackbone. Thus, we observe that qH becomes essentially independent of temperature. The relativeimportance of M and N/p in determining the "nal temperature independent value of the optimalwave vector depends upon the details of the architecture [80].

The variation of qH with temperature can also be understood by considering the role of thevarious terms in the free energy functional. For small q, F

41<F

42. Thus, immediately following

the MST, the value of qH is determined by the interplay between F41

and F2. F

41depends strongly

on temperature, and hence the value of qH depends strongly upon temperature in this regime.


Fig. 25. Intensity of the optical birefringence signal (in millivolts) as a function of temperature.

Recall that the quartic term F41

originates from the quenched #uctuations in the locations of thebranch points. Thus, in the region immediately below the MST, the physics is determined by thequenched randomness inherent to this system. As the system orders on smaller scales, the value ofthe terms in the free energy functional at larger values of q are relevant. At larger values of q,F42

<F41

. Thus, the optimal wave vector is largely determined by F2

and F42

. F42

hasa temperature-independent minimum, and hence, the value of qH becomes essentially independentof temperature. The quartic term F

42originates from the mean values of the quenched randomness

rather than the #uctuations. Thus, in this regime, the strength of the quenched #uctuations is notrelevant, and the system behaves as an ordinary comb polymer with the average spacing betweenbranch points. By varying the architecture of the polymer (i.e., the values of p, N, and M) the rangeof temperature over which the quenched #uctuations are relevant can be changed. Thus, byappropriate synthesis, the e!ects of quenched disorder can be studied in a controlled manner byexperimental studies of branched DHPs.

Such experiments have now begun to be done. Xenidou and Hadjichristidis [75] have reportedhow branched DHPs can be synthesized with careful control of sequence statistics and architec-ture. The speci"c chemical system that they have synthesized has a polybutadiene (PB) backboneand polystyrene (PS) branches. The PB backbone is "rst synthesized using standard anionicpolymerization methods and the resulting polymer contains both 1}4 and 1}2 units. Usinga platinum catalyst this polymer is reacted with a silane coupling agent which selectively silates the1}2 units. The number of monomers that are silated can be controlled by the concentration of thesilane coupling agent. This e!ectively controls the value of p since the silated PB is then reactedwith living polystyrllithium; this leads to the PS branches being grafted onto the silated 1}2 units.M and N can be controlled by the anionic polymerization procedures used to synthesize the PBchains and the polystyrllithium.

Optical birefringence and small angle neutron scattering (SANS) experiments [76] have beencarried out on such a molten branched DHP with the following parameters de"ning the sequenceand architecture: M"2138 statistical segments, N"285 statistical segments, and p"16. Theseare the parameters used to obtain the results displayed in Fig. 24. The temperature dependence of


Fig. 26. Small angle neutron scattering pro"les at selected temperatures.

Fig. 27. Intensity at the peak position of the SANS data as a function of sM. The circles correspond to experimental dataand the line is theoretical result.

the measured optical birefringence signal is shown in Fig. 25. It is well established that theorder-disorder transition in molten copolymers is announced by a discontinuous drop in thebirefringence signal to less than 1 mV [81]. The birefringence signal from the molten branchedDHP is large over the entire range of temperatures that has been studied, indicative of an orderedphase. This shows that in the experiments that have been performed thus far, the order-disordertransition for branched DHPs has not been accessed. However, extrapolation of the birefringencedata suggests that the MST occurs at sM&O(10). This is consistent with the theoretical predictionshown in Fig. 24.

The SANS experiments [76] allow much closer comparison with the theory that we have justoutlined. Fig. 26 shows the scattering pro"les (which were found to be independent of thermal


history) at a few selected temperatures. A single scattering peak is observed at all temperatures, andthe coherent scattering intensity goes to zero in both high and low q limits. These features,coupled with the "nite birefringence signals, are classic signatures of microphase ordering. Thewidth of the scattering peaks are substantially wider than that observed for diblock copolymers (asystem that has been extensively studied). This is obviously because of the disordered sequence ofbranch points, and the resulting distribution of the lengths of PB segments between the branchpoints.

The theory that we have described predicts the peak position, qH, and the intensity at the peak,IHJDDD2 as a function of s. In order to predict IH as a function of temperature we need to know thedependence of s on temperature. Taking this dependence to be of the familiar form, A#B/¹, thevalues of the constants A and B can be adjusted to best "t the experimental data. Fig. 27 shows thatthe theory "ts the data remarkably well with the values of A and B being 0.003 (vanishingly small)and 20, respectively. The constant B re#ects the energy of interactions between PS and PBsegments, and should be roughly the same for copolymers of various architectures and sequences.Thus, the value of B obtained by "tting the experimental data for branched DHPs should besimilar to that which is well established for PS}PB diblock copolymers. The literature value ofB for the PS}PB diblock copolymer is 19. This essentially exact comparison lends further credenceto the theory for branched DHPs developed by Qi et al. [76].

The value of B thus obtained makes the temperature range over which experimental data hasbeen collected correspond to sM values between 13 and 15. Fig. 24 shows that this places theexperimental data in the region corresponding to the crossover from strongly temperature depen-dent to essentially temperature independent values of qH. Thus, a very small (but "nite) variation ofqH with temperature is expected. The experimental data does exhibit a small (but systematic)increase of qH with temperature. However, because it is small, it cannot be unequivocally claimedthat the variation is outside the limits of experimental error. The data in Fig. 24 shows that theaverage value of qH is 0.032#0.001As ~1 in this range of temperatures. Using the statistical segmentlength of PS ("5.02As ) theory predicts [76] that the value of qH should vary between 0.027 and0.028As ~1 in the range of temperatures over which experimental data exists. The asymptoticvalue of qH deep in the ordered region is predicted to be 0.033As ~1. The agreement betweentheory and experiment is reasonable considering the uncertainties in the experimental values ofl, M, N, and p.

As noted earlier, the study of DHPs in the molten state can provide important insights into thephysics of frustrated systems with quenched disorder. The discussion in this section was aimed toillustrate this potential. While past theoretical studies of linear DHPs were extremely useful inmany contexts, the lack of experimental information on these systems has allowed only limitedprogress. The ability to routinely synthesize branched DHPs with controlled sequence statisticsand architecture opens up many new possibilities for careful experimental studies and analyses thatshould enhance our fundamental understanding of the e!ects of frustration due to quenchedramdomness on microphase ordering. The advantage of using polymers to study the physics (ascompared to say spin glasses) is that the statistics of the disorder can be carefully controlled and theself-assembled ordered structures that may form have length scales that are large and can beexamined with relative ease (e.g., via scattering experiments and microscopies). Further, because ofthe slow dynamics of long chain systems, careful studies of the dynamics should also be possible.The work described in this section suggests that branched DHPs may be particularly good model


systems to study the physics of frustrated systems with quenched disorder. By varying the branchlength and the statistics of branch point locations, the behavior can be varied from the linear DHPlimit to that of simple comb polymers. So far, only the lamellar morphology has been studied forthe ordered phase. Future work should focus on studying the entire phase diagram using the "eldtheory and the type of experiments described in this section. It may also prove fruitful, both froma fundamental standpoint and from the viewpoint of applications, to study the behavior ofbranched and linear DHPs when they are components of a mixture (e.g., with homopolymers,solvent mixtures, and block copolymers). Following the completion of these e!orts, the dynamicsof the order}disorder and order}order transitions in these systems with quenched disorder shouldbe studied. These studies may be most exciting and may allow us to develop a deep understandingof dynamics in frustrated systems with quenched disorder.

Acknowledgements

Several people have in#uenced my thinking on the physics of DHPs. I would especially like tothank those who have in#uenced my ideas through collaborations: Prof. Eugene Shakhnovich(Harvard University), Prof. V. Pande (Standford University), Dr. L. Gutman, Dr. S.Y. Qi,Dr. S. Srebnik, and Mr. A. Golumbfskie. Dr. Qi and Mr. Golumbfskie were also kind enough tocomment on this manuscript. I am deeply grateful to the National Science Foundation and the USDepartment of Energy for generous "nancial support of my work on disordered heteropolymers.

Appendix

In order to compute the entropy as a function of the macroscopic order parameters witha proper quenched average over the branch point #uctuations, we note the identity

1"k

<a/1PDoa

A(r)Doa

B(rl )dCoa

A(rl )!Pdn d[rl !rl a

A(n)]DdCoa

B(rl )!

P+j/1Pds d[rl !rl a

j(s)]D

"

k<a/1PDoa

A(rl )Doa

B(rl )Dca

A(rl )Dca

B(rl ) expGiPdrl [ca

A(rl )oa

A(rl )#ca

B(rl )oa

B(rl )]

! iCPdn caA(rl a(n))!

P+j/1Pds ca

B(rl aj(s))DH . (A.1)

where c and / are "elds conjugate to oA(r) and o

B(r). Substituting this identity into the replicated

form of R (Eq. (32)) obtains

SRkT"P<a/1PDoa

A(rl )Doa

B(rl )Dca

A(rl )Dca

B(rl ) expGiPdrl [ca

A(rl )oa

A(rl )#ca

B(rl )oa

B(rl )]H

Tk

<j/1PDrl a(n)Drl a

j(s)P[rl a(n)]P[rl a

j(s)]d[rl a

j(0)!rl a(q

j)]U , (A.2)


where the angular brackets denoted the quenched average, and

P[rl a(n)]"P0[rl a(n)] expG!iPdn ca

A(rl a(n))H , (A.3)

P[rl aj(s)]"P

0[rl a

j(s)] expG!iPds ca

B(rl aj(s))H .

Let us denote the expression that needs to be averaged by Q. So,

SQkT"Tk<a/1

P<j/1PDrl a(n)Drl a

j(s)P[rl a(n)]P[rl a

j(s)]d[rl a

j(0)!rl a(q

j)]U

"TGP<j/1PDrl (n)Drl

j(s)P[rl (n)]P[rl

j(s)]d[rl

j(0)!rl (q

j)]H

k

U. (A.4)

We now expand SQT in powers of the conjugate "elds c and /, up to quartic order; i.e.,

Q"Q0#Q

1#Q

2#Q

3#Q

4#2, (A.5)

Q0"<, and Q

1contributes an irrelevant constant to the free energy formula. The "rst non-trivial

term is Q2, which after introducing Fourier transforms is

Q2"!

12Pdql cl (q)M(ql )cl (q) (A.6)

where c"(cA, c

B)T and M is a 2]2 matrix:

M11

(q)"2x2

[Nx#exp(!Nx)!1]"N2g2(Nx) ,

M22

(q)"PM2g2(Mx)#P(P!1)M2g

1(Mx)2g

2(Nx) , (A.7)

M12

(q)"M21

(q)"NPMg1(Mx)g

2(Nx)

with s"q2b2/6, g1(y)"[1!exp(!y)]/y and g

2(y)"2/y2[y#exp(!y)!1], and g

2(y) is the

so-called Debye function.Similarly, expressions for Q

3and Q

4as functionals of conjugated "elds can be derived. Qi et al.

[79] provide detailed derivation of these terms using a graphical method. Their result is

SQT"<!12Pdql cl (ql )M(ql )cl (ql )#

i6Pdql

1dql

2

[C30

(ql1, ql

2)c

A(ql

1)c

A(ql

2)c

A(!ql

1!ql

2)#C

31(ql

1, ql

2)c

A(ql

1)c

A(ql

2)c

B(!ql

1!ql

2)

#C32

(ql1, ql

2)c

A(ql

1)c

B(ql

2)c

B(!ql

1!ql

2)#C

33(ql

1, ql

2)c

B(ql

1)c

B(ql

2)c

B(!ql

1!ql

2)]

#

124Pdql

1dql

2dql

3[C

40(ql

1, ql

2, ql

3)c

A(ql

1)c

A(ql

2)c

A(ql

3)c

A(!ql

1!ql

2!ql

3)


#C41

(ql1, ql

2, ql

3)c

A(ql

1)c

A(ql

2)c

A(ql

3)c

B(!ql

1!ql

2!ql

3)

#C42

(ql1, ql

2, ql

3)c

A(ql

1)c

A(ql

2)c

B(ql

3)c

B(!ql

1!ql

2!ql

3)

#C43

(ql1, ql

2, ql

3)c

A(ql

1)c

B(ql

2)c

B(ql

3)c

B(!ql

1!ql

2!ql

3)

#C44

(ql1, ql

2, ql

3)c

B(ql

1)c

B(ql

2)c

B(ql

3)c

B(!ql

1!ql

2!ql

3)] (A.8)

where the functions C3i

(q1, q

2) are

(2p)3@2C30

"6N3h2(Nx

1, Nx

2) ,

(2p)3@2C31

"6(PM)N2g1(Mx

2)[2h

2(Nx

1,Nx

2)#h

2(Nx

1, Nx

3)] ,

(2p)3@2C32

"3GPM2Ng2(Nx

1)H

2(Mx

1!Mx

2, Mx

2)#4(PM)2Ng

1(Mx

2)g

1(Mx

3)

]h2(Nx

1, Nx

2)#(PM)2g

1(Mx

2)g

1(Mx

3)g2(Nx

2)!g

2(Nx

3)

x3!x

2H ,

(2p)3@2C33

"6PM3h2(Mx

1, Mx

2)#6MP2M3g

2(Nx

2)g

1(Mx

2)H

2(Mx

2!Mx

1,Mx

1)

# (PM)3g1(Mx

1)g

1(Mx

2)g

1(Mx

3)h

2(Nx

1, Nx

3)N (A.9)

and si"q2

ib2/6 for i"1, 2 and s

3"(q

1#q

2)2b2/6. The functions C

4i(q

1, q

2, q

3) are

(2p)3C40

"24N4h3(Nx

1, Nx

7,Nx

4) ,

(2p)3C41

"96PMN3g1(Mx

4)h

3(Nx

1,Nx

7, Nx

4) ,

(2p)3C42

"12M2PM2N2H2(!Mx

4, Mx

4#Mx

7)[2h

2(Nx

1, Nx

7)#h

2(Nx

1, Nx

2)]

# 4(PMN)2g1(Mx

3)g

1(Mx

4)h

3(Nx

1, Nx

7,Nx

3)

#

(PM)2x4!x

6

g1(Mx

3)g

1(Mx

4)C

g2(Nx

3)!g

2(Nx

6)

x6!x

3

!

g2(Nx

3)!g

2(Nx

4)

x4!x

3D

#

4(PMN)2N(x

6!x

4)g1(Mx

3)g

1(Mx

4)[h

2(Nx

1, Nx

4)!h

2(Nx

1,Nx

6)]

# 2(PMN)2g1(Mx

3)g

1(Mx

4)h

3(Nx

1, Nx

6,Nx

2)N , (A.10)

(2p)3C43

"8G3NPM3g2(Nx

1)H

3(Mx

1!Mx

7, Mx

7!Mx

4, Mx

4)

#

12P2M3

x1

h2(Nx

1, Nx

2)H

2(!Mx

4, Mx

4#Mx

7)g

1(Mx

2)

#3P2M3g1(Mx

4)g2(Nx

5)!g

2(Nx

4)

x4!x

5

H2(!Mx

2,Mx

2#Mx

5)


#6(PM)3

x1

g1(Mx

2)g

1(Mx

3)g

1(Mx

4)[h

2(Nx

7, Nx

3)

!H3(Nx

1!Nx

7, Nx

7!Nx

3, Nx

3)]

#6(PM)3g1(Mx

2)g

1(Mx

3)g

1(Mx

4)h2(Nx

2,Nx

4)!h

2(Nx

7, Nx

4)

x7!x

2H ,

(2p)3C44

"24PM4h3(Mx

1,Mx

7, Mx

4)

#24P2M4g1(Mx

1)H

3(Mx

1!Mx

7, Mx

7!Mx

4, Mx

4)g

2(Nx

1)

#12P2M4g2(Nx

7)H

2(!Mx

1, Mx

1#Mx

7)H

2(!Mx

4,Mx

4#Mx

7)

#72P3M4g1(Mx

1)g

1(Mx

2)h

2(Nx

1,Nx

7)H

2(!Mx

4,Mx

4#Mx

7)

#24(PM)4g1(Mx

1)g

1(Mx

2)g

1(Mx

3)g

1(Mx

4)h

3(Nx

1, Nx

7,Nx

4) ,

where si"q2

ib2/6 for i"1, 2, 3, and s

4"(q

1#q

2#q

3)2b2/6, s

5"(q

2#q

3)2b2/6, s

6"

(q1#q

3)2b2/6 and s

7"(q

1#q

2)2b2/6. The following functions have been de"ned for simplicity:

h1(y

1)"P

1

0

dn2P

n2

0

dn1

exp[!(n2!n

1)y

1]

"

1!g1(y

1)

y1

"

g2(y

1)

2, (A.11)

h2(y

1, y

2)"P

1

0

dn3P

n3

0

dn2P

n2

0

dn1

exp[!(n2!n

1)y

1!(n

3!n

2)y

2]

"

1y1Ch1 (y

2)!

g1(y

2)!g

1(y

1)

y1!y

2D ,

h3(y

1, y

2, y

3)"P

1

0

dn4P

n4

0

dn3P

n3

0

dn2P

n2

0

dn1

exp[!(n2!n

1)y

1!(n

3!n

2)y

2

!(n4!n

3)y

3]

"

1y1Gh2(y

2, y

3)!

1y1!y

2Cg1(y

2)!g

1(y

3)

y3!y

2

!

g1(y

1)!g

1(y

3)

y3!y

1DH ,

H2(y

1, y

2)"P

1

0

dn2P

n2

0

dn1

exp[!n1y1!n

2y2]

"

1y1

[g1(y

2)!g

1(y

1#y

2)] ,


H3(y

1, y

2, y

3)"P

1

0

dn3P

n3

0

dn2P

n2

0

dn1

exp[!n1y1!n

2y2!n

3y3]

"

1y1

[H2(y

2, y

3)!H

2(y

1#y

2, y

3)] .

Combining all these expressions yields

TAQ<B

m

U"1#1<

m+a/1G!

12Pdql a cl (ql a)M(ql a)cl (ql a)#

i6Pdql a

1dql a

2

[C30

(ql a1, ql a

2)c

A(ql a

1)c

A(ql a

2)c

A(!ql a

1!ql a

2)

#C31

(ql a1, ql a

2)c

A(ql a

1)c

A(ql a

2)c

B(!ql a

1!ql a

2)

#C32

(ql a1, ql a

2)c

A(ql a

1)c

B(ql a

2)c

B(!ql a

1!ql a

2)

#C33

(ql a1, ql a

2)c

B(ql a

1)c

B(ql a

2)c

B(!ql a

1!ql a

2)]

#

124Pdql a

1dql a

2dql a

3[C

40(ql a

1, ql a

2, ql a

3)c

A(ql a

1)c

A(ql a

2)c

A(ql a

3)c

A(!ql a

1!ql a

2!ql a

3)

#C41

(ql a1, ql a

2, ql a

3)c

A(ql a

1)c

A(ql a

2)c

A(ql a

3)c

B(!ql a

1!ql a

2!ql a

3)

#C42

(ql a1, ql a

2, ql a

3)c

A(ql a

1)c

A(ql a

2)c

B(ql a

3)c

B(!ql a

1!ql a

2!ql a

3)

#C43

(ql a1, ql a

2, ql a

3)c

A(ql a

1)c

B(ql a

2)c

B(ql a

3)c

B(!ql a

1!ql a

2!ql a

3)

#C44

(ql a1, ql a

2, ql a

3)c

B(ql a

1)c

B(ql a

2)c

B(ql a

3)c

B(!ql a

1!ql a

2!ql a

3)]H

#

14

m+aEb Pdql adql bScl (ql a)M

0(ql a)cl (ql a)cl (ql b)M

0(ql b)cl (ql b)T (A.12)

where the matrix M0

is

(M0)11

(q)"P`1+j/1

2x2

M(qj!q

j~1)x#exp[!(q

j!q

j~1)x]!1N

# 2P`1+i:j

exp(qix)!exp(q

i~1x)

x]

exp(!qj~1

x)!exp(!qjx)

x, (A.13)

(M0)22

(q)"PM2g(Mx)#2C1!exp(!Mx)

x D2 P+i:j

exp[!x(qj!q

i)] ,

(M0)12

(q)"(M0)21

(q)"1!exp(!Mx)

xP`1+i/1

P+j/1Kexp(!xDq

j!q

iD)!exp(!xDq

j!q

i~1D)

x K


Combining Eqs. (A.1) and (A.11) yields SRmT as a function of the order parameters and conjugate"elds. We now evaluate the functional integrals over the conjugate "elds using a saddle pointapproximation; i.e.,

d lnSRmT[oA, o

B, c

A, c

B]

dcaA(!ql )

"0 ,

(A.14)d lnSRmT[o

A, o

B, c

A, c

B]

dcaB(!ql )

"0 .

Solving the above equations obtains SS[oA,o

B]T"lim

k?0[SRT

4.1.[o

A,o

B]!1]/k as a functional

of oA

and oB:

SS[oA, o

B]T"!

<2 Pdql ol =ol #

i6<Pdql

1dql

2[C

30(ql

1, ql

2)c

1A(ql

1)c

1A(ql

2)c

1A(!ql

1!ql

2)

#C31

(ql1, ql

2)c

1A(ql

1)c

1A(ql

2)c

1B(!ql

1!ql

2)

#C32

(ql1, ql

2)c

1A(ql

1)c

1B(ql

2)c

1B(!ql

1!ql

2)

#C33

(ql1, ql

2)c

1B(ql

1)c

1B(ql

2)c

1B(!ql

1!ql

2)]

#

124<Pdql

1dql

2dql

3[C@

40(ql

1, ql

2, ql

3)c

1A(ql

1)c

1A(ql

2)c

1A(ql

3)c

1A(!ql

1!ql

2!ql

3)

#C@41

(ql1, ql

2, ql

3)c

1A(ql

1)c

1A(ql

2)c

1A(ql

3)c

1B(!ql

1!ql

2!ql

3)

#C@42

(ql1, ql

2, ql

3)c

1A(ql

1)c

1A(ql

2)c

1B(ql

3)c

1B(!ql

1!ql

2!ql

3)

#C@43

(ql1, ql

2, ql

3)c

1A(ql

1)c

1B(ql

2)c

1B(ql

3)c

1B(!ql

1!ql

2!ql

3)

#C@44

(ql1, ql

2, ql

3)c

1B(ql

1)c

1B(ql

2)c

1B(ql

3)c

1B(!ql

1!ql

2!ql

3)]

!

18<2Pdql

1dql

2MScl

1(ql

1)M

0(ql

1)cl

1(ql

1)cl

1(ql

2)M

0(ql

2)cl

1(ql

2)T

! [cl1(ql

1)M(ql

1)cl

1(ql

1)][cl

1(ql

2)M(ql

2)cl

1(ql

2)]N (A.15)

where the matrix = is the inverse of M and c1A

"i<(=11

oA#=

12oB), c

1B"

i<(=12

oA#=

22oB), and c

1"(c

1A, c

1B)T. The modi"ed quartic coe$cients are

C@40

(ql1, ql

2, ql

3)"C

40(ql

1, ql

2, ql

3)!1

3M[C

30(ql

1#ql

2, ql

3)#C

30(ql

3, ql

1#ql

2)#C

30(ql

3, ql

4)]

]=11

(ql1#ql

2)[C

30(!ql

1!ql

2, ql

1)#C

30(ql

1,!ql

1!ql

2)#C

30(ql

1, ql

2)]

# 2[C30

(ql1#ql

2, ql

3)#C

30(ql

3, ql

1#ql

2)#C

30(ql

3, ql

4)]

]=12

(ql1#ql

2)C

31(ql

1, ql

2)#C

31(ql

1, ql

2)M

22(ql

1#ql

2)C

31(ql

3, ql

4)N ,


C@41

(ql1, ql

2, ql

3)"C

41(ql

1, ql

2, ql

3)!1

3M2[C

30(!ql

1!ql

2, ql

1)#C

30(ql

1,!ql

1!ql

2)

#C30

(ql1, ql

2)]

]=11

(ql1#ql

2)[C

31(ql

1#ql

2, ql

3)#C

31(ql

3, ql

1#ql

2)]

# 2[C30

(!ql1!ql

2, ql

1)#C

30(ql

1,!ql

1!ql

2)#C

30(ql

1, ql

2)]

]=12

(ql1#ql

2)[C

32(ql

3, ql

1#ql

2)#C

32(ql

3, ql

4)]

# 2C31

(ql1, ql

2)=

12(ql

1#ql

2)[C

31(ql

1#ql

2, ql

3)#C

31(ql

3, ql

1#ql

2)]

# 2C31

(ql1, ql

2)=

22(ql

1#ql

2)[C

32(ql

3, ql

1#ql

2)#C

32(ql

3, ql

4)]N , (A.16)

C@42

(ql1, ql

2, ql

3)"C

42(ql

1, ql

2, ql

3)!1

3M2[C

30(!ql

1!ql

2, ql

1)#C

30(ql

1,!ql

1!ql

2)

#C30

(ql1, ql

2)]

]=11

(ql1#ql

2)C

32(ql

1#ql

2, ql

3)# [C

31(ql

1#ql

3, ql

2)#C

31(ql

2, ql

1#ql

3)]

]=11

(ql1#ql

3)[C

31(!ql

1!ql

3, ql

1)#C

31(ql

1,!ql

1#ql

3)]

# 2C31

(ql1, ql

2)=

12(ql

1#ql

2)C

32(ql

1#ql

2, ql

3)

# 2[C30

(!ql1!ql

2, ql

1)#C

30(ql

1,!ql

1!ql

2)#C

30(ql

1, ql

2)]

]=12

(ql1#ql

2)[C

33(ql

1#ql

2, ql

3)#C

33(ql

3, ql

1#ql

2)#C

33(ql

3, ql

4)]

# 2[C31

(ql1#ql

3, ql

2)#C

31(ql

2, ql

1#ql

3)]=

12(ql

1#ql

3)

][C32

(ql1,!ql

1!ql

3)#C

32(ql

1, ql

3)]

# 2C31

(ql1, ql

2)=

22(ql

1#ql

2)[C

33(ql

1#ql

2, ql

3)#C

33(ql

3, ql

1#ql

2)

#C33

(ql3, ql

4)]

# [C32

(ql2, ql

1#ql

3)#C

32(ql

2, ql

4)]=

22(ql

1#ql

3)[C

32(ql

1,!ql

1!ql

3)

#C32

(ql1, ql

3)]N ,

C@43

(ql1, ql

2, ql

3)"C

43(ql

1, ql

2, ql

3)

! 13M2C

32(ql

1#ql

2, ql

3)=

11(ql

1#ql

2)[C

31(!ql

1!ql

2, ql

1)

#C31

(ql1,!ql

1!ql

2)]

# 2C32

(ql1#ql

2, ql

3)=

12(ql

1#ql

2)[C

32(ql

1, ql

2)#C

32(ql

1,!ql

1!ql

2)]

# 2[C31

(!ql1!ql

2, ql

1)#C

31(ql

1,!ql

1!ql

2)]

]=12

(ql1#ql

2)[C

33(ql

1#ql

2, ql

3)#C

33(ql

3, ql

1#ql

2)#C

33(ql

3, ql

4)]

# 2[C32

(ql1,!ql

1!ql

2)#C

32(ql

1, ql

2)]

]=22

(ql1#ql

2)[C

33(ql

1#ql

2, ql

3)#C

33(ql

3, ql

1#ql

2)#C

33(ql

3, ql

4)]N ,


C@44

(ql1, ql

2, ql

3)"C

44(ql

1, ql

2, ql

3)!1

3MC

32(ql

1#ql

2, ql

3)=

11(ql

1#ql

2)C

32(!ql

1!ql

2, ql

1)

# 2C32

(ql1#ql

2, ql

3)=

12(ql

1#ql

2)

][C33

(!ql1!ql

2, ql

1)#C

33(ql

1,!ql

1!ql

2)#C

33(ql

1, ql

2)]

# [C33

(!ql1!ql

2, ql

1)#C

33(ql

1,!ql

1!ql

2)#C

33(ql

1, ql

2)]

]=22

(ql1#ql

2)[C

33(ql

1#ql

2, ql

3)#C

33(ql

3, ql

1#ql

2)#C

33(ql

3, ql

4)]N ,

where

ql4"!ql

1!ql

2!ql

3.

References

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