DISORDERED HETEROPOLYMERS: MODELS FOR BIOMIMETIC POLYMERS AND POLYMERS WITH FRUSTRATING QUENCHED DISORDER

Transcription

1 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 1 DISORDERED HETEROPOLYMERS: MODELS FOR BIOMIMETIC POLYMERS AND 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

2 Physics Reports 342 (2001) 1}61 Disordered heteropolymers: models for biomimetic polymers and polymers with frustrating quenched disorder Contents 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 1. Introduction 4 2. Biomimetic recognition between DHPs and multifunctional surfaces Theory of thermodynamic properties Monte-Carlo simulations of thermodynamic properties Kinetics of recognition due to statistical pattern matching Connection to experiments and issues pertinent to evolution Branched DHPs in the molten state } model system for studying microphase ordering in systems with quenched disorder 44 Acknowledgements 52 Appendix 52 References 59 Abstract The ability to design and synthesize polymers that can perform functions with great speci"city would impact advanced technologies in important ways. Biological macromolecules can self-assemble into motifs that allow them to perform very speci"c functions. Thus, in recent years, attention has been directed toward elucidating 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 sequence distributions (disordered heteropolymers) can mimic the ability of biological macromolecules to recognize patterns. Results of this body of work suggests that frustration due to competing interactions and quenched disorder may be the essential physics that can enable such biomimetic behavior. These results also show that recognition between disordered heteropolymers and multifunctional surfaces due to statistical pattern matching may be a good model to study kinetics in frustrated systems with quenched disorder. We also review work which demonstrates that disordered heteropolymers with branched architectures are good model systems to study the e!ects of quenched sequence disorder on microphase ordering of molten address: arup@lolita.cchem.berkeley.edu (A.K. Chakraborty) /01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S (00)

3 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 3 copolymers. The results we describe show that frustrating quenched disorder a!ects the way in which these 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 connections with existing experimental work and suggest future experiments that are expected to yield further insights Elsevier Science B.V. All rights reserved. PACS: Aa; #t

4 4 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 1. Introduction Synthetic polymers have enormously impacted societal and economic conditions because they are commonly used to manufacture a plethora of commodity products. This is one of the driving forces that continues to spur fundamental research aimed toward understanding the physics of macromolecules and learning how to chemically synthesize them. Another motivation for such research is, of course, intrinsic interest in the fascinating behavior of macromolecules. Research conducted by several physical and chemical scientists has led to substantial advances in our ability to synthesize macromolecules and understand their physical behavior. In recent years, technological advances have begun to demand materials which exhibit very speci"c properties. If polymers are to continue to impact society in important ways, they must meet this need. Polymers are good candidates for materials which can perform functions with a high degree of speci"city. We can make this claim because it is well-established that biological macromolecules are able to carry out very speci"c functions. One feature that allows biological macromolecules to perform speci"c functions is their ability to self-assemble into particular motifs. Polymeric materials could impact advanced technologies in important ways if we could learn how to design and synthesize macromolecular systems that can selfassemble into functionally interesting structures and phases. One way to confront this challenge is to take lessons from nature since millenia of evolution have allowed biological systems to learn how to create functionally useful self-assembled structures from polymeric building blocks. By suggesting that we take lessons from nature, we do not imply copying the detailed chemistry which allows a biological system to carry out a speci"c function that we seek. This would be impractical in many contexts. Rather, we suggest asking the following questions: are there underlying universalities in the design strategies that nature employs in order to mediate a certain class of functions? If so, can we exploit similar strategies to design 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 implement in synthetic systems than the detailed chemistries of natural systems, and may illuminate the essential physics. However, it is also important to realize that universal strategies will also lead to 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!ect assembly into functionally interesting materials is to exploit multifunctionality and disordered sequence distributions (e.g., [1}3]). Disordered heteropolymers (DHPs) constitute a class of synthetic polymers that embodies these features. These are copolymers containing more than one type of monomer unit, with the monomers connected together in a disordered sequence. The monomers may also be connected with di!erent architectures; e.g., branched versus linear connectivity. An important point is that once synthesis is complete, the sequence and the architecture cannot change in response to the environment. Since DHPs embody multifunctionality and quenched disorder, they serve as excellent vehicles to explore the suggestion that these features may be essential elements for mediating certain types of biomimetic function in synthetic systems relevant to applications. Competing interactions (due to the presence of di!erent types of monomer units), connectivity, and the quenched character of the disordered sequence also make DHPs quintessential examples of frustrated systems.

5 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 5 In the latter half of this century, physical scientists interested in the condensed phase have directed considerable attention to two broad classes of problems: those involving biological phenomena (e.g., protein folding) and frustrated systems (e.g., the e!ects of frustrating quenched randomness in spin glass physics). In addition to being a way of exploring how biomimetic synthetic soft materials can be designed, studying DHPs also allows us to study those aspects of biopolymer 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 for careful experimental studies of the manifestations of frustrating quenched disorder. In this article, we try to illustrate (via examples) how DHPs can serve as biomimetic polymers and/or as model systems to study the physics of frustrated systems with quenched randomness. We begin (Section 2) by discussing the adsorption of DHPs from solution, and these considerations show 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 systems for 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 experiments and some provocative ideas being considered in evolutionary biology. In Section 3, we discuss theoretical and experimental work which demonstrates that DHPs with branched architectures are good 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 protein folding (e.g., [1}5]), and a recent review in this journal on theoretical considerations of microphase ordering in molten DHPs with linear architectures [6] provide a glimpse of much of what is known about 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 in a certain part of the surface of a cell membrane. By recognition we imply that the protein adsorbs strongly on the pattern-matched region, and not on other parts of the surface; furthermore, it evolves to the pattern-matched region and binds strongly to it in relatively fast time scales without getting trapped in long-lived metastable adsorbed states in the wrong parts of the cell surface. If synthetic polymers were able to mimic such recognition, it would indeed be useful for many advanced applications. Examples of such applications include sequence selective separation processes [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] for recent reviews). Most studies have been concerned with the adsorption of polymers with ordered sequence distributions (e.g., homopolymers and diblock copolymers). These studies have taught us many important lessons. One lesson pertinent to our concerns can be illustrated by considering the example of a homopolymer interacting with a chemically homogeneous surface. In this case, once we have chosen the chemical identity of the polymer segments, di!erent surfaces are characterized by 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

6 6 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 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 "rmly established the nature of this plot. For small values of E/k¹, there is no adsorption because the energetic advantage associated with segmental binding is not su$cient to beat the entropic penalty associated with chain adsorption. For su$ciently large values of E/k¹, adsorption does occur. The transition 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 favored because this increases the entropy. These loop #uctuations cause the adsorption transition to be continuous. The practical consequence is that thermodynamic discrimination between surfaces is not sharp } a requisite feature for recognition. The adsorption characteristics of diblock copolymers on striped surfaces have also been understood (e.g., [15,16]). At equilibrium, they adsorb at the interface of the stripes with each block adsorbed on the stripe that is energetically favored. This phenomenon is di!erent from what is meant by recognition in important ways. Firstly, the chain is not localized in a region commensurate with chain dimensions. This is so because it is entropically favorable for the chain to sample the entire interface. Secondly, it seems highly likely that the diblock copolymer chains would be kinetically trapped in regions away from the interface. This is so because adsorbing one block on an energetically favorable stripe while allowing the non-adsorbed block to sample many conformations appears to be a deep free energy minimum. Thus, this system does not seem to exhibit the hallmarks of recognition either. (Much work has also been done on the behavior of molten diblock copolymer layers on patterned surfaces. We do not discuss these studies here because the focus of this section is on adsorption from solution. Readers interested in this topic are directed to a recent review [17] and references therein.) In short, recognition implies a sharp discrimination between di!erent regions of a surface and localization of the chain to a relatively small pattern matched region without getting kinetically trapped in the `wronga parts of the surface. In biological systems it also usually entails adsorption in a particular conformation or shape. Synthetic polymers with ordered sequence distributions do not seem to exhibit these characteristics. Similar conclusions can be reached by perusing interesting studies of DHP adsorption on homogeneous surfaces [18}21] and homopolymer adsorption on chemically disordered surfaces [22]. One way to make synthetic systems mimic recognition is to copy the detailed chemistries which allow natural systems to a!ect recognition. This is not a practical solution in most cases. Recently,

7 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 7 some work has been done to explore whether there are any universal strategies that may allow synthetic polymers and surfaces to mimic recognition [23}31]. (Such universal strategies may be simpler to implement in practical situations, and may also shed light on the minimal ingredients, or principles, required for synthetic systems to mimic recognition.) This body of work is the primary focus of this section. The purpose of this work has not been to explain the physical and chemical mechanisms that allow recognition in biological systems. However, in order to deduce possible universal strategies, some coarse-grained observations about biological systems have provided the inspiration. 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 models for amino acid sequences in proteins. All models exhibit a common feature. In order to illustrate this feature, consider the H/P model [32] wherein amino acids are considered to belong to two classes } hydrophobic and polar. This (and other) models have been used to characterize protein sequences, and it has been found [33}35] that the pattern of H- and P-type moieties is usually not periodically repeating. Similarly, examination of cell and virus surfaces reveals that the chemically di!erent binding sites that constitute receptors (which are recognized by proteins) are also not arranged in a periodically repeating pattern. These observations suggest that disorder and competing interactions (due to preferential interactions between polymer segments and surface sites) may be key ingredients for recognition between synthetic polymers and surfaces. Heteropolymers with disordered sequences carry a pattern encoded in their sequence distribution. The information content is statistical, however, since the sequences are characterized statistically. For example, for a 2-letter DHP (say, A- and B-type segments), the simplest way to describe the disordered sequence distribution is by specifying the average fraction of segments of one type ( f ), and a quantity λ that measures the strength of two-point correlations in the chemical identity of segments along the chain [36]. λ is directly related to the synthetic conditions and the matrix of reaction probabilities, P. Elements of this matrix, P are the conditional probabilities that a segment of type j directly follows a segment of type i. Clearly, λ depends upon the choice of the chemical identity of the segments and synthesis conditions. Consequently, synthesis conditions and the choice of chemistry determines the statistical pattern carried by DHPs. If λ'0, within a correlation length measured along the chain, there is a high probability of "nding segments of the same type. We shall refer to such an ensemble of sequences as statistically blocky. If λ(0, within a certain correlation length measured along the chain, there is a high probability of "nding an alternating pattern of segments. The absolute magnitude of λ measures the correlation length. For example, λ"0 corresponds to perfectly random sequences, and λ"1 implies the correlation length is the entire chain length. Characterization of sequence statistics using f and λ implies that we are only looking at two-point correlations to describe the statistical patterns. More elaborate statistical patterns can be described by considering higher order correlations and/or more than two types of segments. Consider the interaction of DHPs with surfaces bearing more than one type of site, with the sites being distributed in a disordered manner. Examples of such surfaces with two kinds of sites distributed in a disordered fashion are shown schematically in Fig. 2. The distribution of these sites on the neutral surface can be characterized statistically. For example, a simple way would be to specify 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

8 8 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 Fig. 2. Examples of statistically patterned surfaces. White represents a neutral background. The two types of `activea surface sites are depicted using light and dark grey dots. In the panel on the left, within some length scale, there is a high probability of "nding sites of opposite types adjacent to each other. Such surfaces are referred to in the text as statistically alternating. In the panel on the right, within some length scale, there is a high probability of "nding sites of the same type adjacent 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 shows speci"c realizations of two surfaces bearing simple statistical patterns. In nature, recognition (with all its hallmarks noted earlier) occurs when the speci"c pattern encoded in its sequence distribution and that carried by the binding sites is matched (i.e., related in a special way). DHP sequences and the surfaces we have described in the preceding paragraph carry statistical patterns. The question we now ask is: will statistically patterned surfaces be able to recognize the statistical information contained in an ensemble of DHP sequences when the statistics characterizing the DHP sequence and surface site distributions are related in a special way? In other words, is statistical pattern matching su$cient for recognition to occur? This question is interesting for three reasons: (1) the answer may tell us what the minimal ingredients are for the occurrence of a phenomenon akin to recognition; (2) if recognition can occur via statistical pattern matching, the phenomenon might be pro"tably exploited in applications; (3) DHPs interacting with functionalized surfaces bearing statistical patterns may be good model systems to study the physics of frustrated systems with quenched disorder. In order to answer this question, we have to address several issues. We must determine whether competing interactions and disorder are su$cient for sharp discrimination between di!erent statistical patterns, and whether the inherent frustration allows localization (in reasonable time scales) on a relatively small part of the surface which is statistically pattern matched. Addressing these issues requires that we study both thermodynamic and kinetic behavior. Let us begin by describing the thermodynamics Theory of thermodynamic properties Srebnik et al. [23] analyzed a bare bones version of the problem in order to examine whether frustration due to competing interactions and quenched disorder are su$cient to obtain sharp discrimination between surfaces with di!erent statistical patterns. In this model, the 2-letter DHPs

9 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 9 are considered to be Gaussian chains in solution. The surface is comprised of two di!erent types of sites on a neutral background, and each type of site interacts di!erently with the two types of DHP segments. In the in"nitely dilute limit, the physical situation described above corresponds to the following Hamiltonian:!βH"! 3 2l dn dn dr! dn drk(r)δ(r(n)!r)θ(n)δ(z) (1) where r(n) represents chain conformation, k(r) is the interaction strength with a surface site located at r, the factor of δ(z) ensures that these sites live on a 2-D plane, l is the usual statistical segment length, and θ(n) represents the chemical identity of the nth segment. For a two-letter DHP, this quantity is $1 depending upon whether we have an A- or a B-type segment. Eq. (1) implies that a surface site which exhibits attractive interactions with one type of DHP segment has an equally repulsive interaction with segments of the other type. This is assumed for simplicity. What is important is that interactions between a surface site and the two di!erent types of segments are di!erent. Since the DHP sequence and the surface site distribution are disordered, k(r) and θ(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 the essential physics, let us consider the #uctuations in k(r) and θ(n) to be uncorrelated. Further, in order to simplify the analysis, these #uctuations are described by Gaussian processes. This latter approximation should not a!ect the qualitative physics, and generalizes the results to a physical situation 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 value of zero), and the variance of the #uctuations in k is σ. This implies that σ is the only variable which measures the statistical pattern carried by the surface. Di!erent values of this quantity represent di!erent surfaces. Physically, σ is proportional to the total number density of both types of sites on the neutral surface. The uncorrelated sequence distribution is described by θ(n) having a mean value of (2f!1), where f is the average composition of one type of segment. The variance is σ, and is also related to the average composition (it equals 4f (1!f )). Thus, in this simple case the statistical pattern carried by 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 the sites on the surface can anneal in response to the adsorbing chain molecule, then the partition function is self-averaging with respect to the #uctuations in k. This could be the physical situation if the 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 partition function 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 over disorders external to the #uid of interest are equivalent when the medium is su$ciently large, and the 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 the moment, we carry out an analysis that holds for annealed disorders under all circumstances, and

10 10 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 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[!βh ]" Dk(r) exp! 1 2σ dr dr k(r)δ(r!r)k(r)δ(z)δ(z) exp!3 2l dn dn dr! dn drk(r)δ(r(n)!r)θ(n)δ(z). (2) The partition function is not self-averaging with respect to the quenched sequence #uctuations under any circumstances. Replica methods [41,42] provide one way to carry out the quenched average. We will consider f"0.5, thereby "xing the statistics of the DHP sequences. We then study adsorption as a function of the surface statistics (i.e., the variance of the distribution that characterizes the #uctuations in k(r)). Replicating the e!ective Hamiltonian in Eq. (2), and carrying out the functional integral corresponding to the average over the distribution of θ obtains the following m-replica partition function: G" Dr (n) Dk (r) exp 3! 2l dn dr dn exp! 1 2σ dr dr k (r)δ δ(r!r)k (r)δ(z)δ(z) exp σ 2 dn dr dr k (r)k (r)δ(r (n)!r)δ(r (n)!r)δ(z)δ(z). (3) This replicated partition function can be written in a form that is more convenient both for thinking and computing. De"ne the following order parameter, Q (r!r), which measures the conformational overlap on the surface between the replicas: Q (r, r)" dn δ(r (n)!r)δ(r (n)!r)δ(z)δ(z). (4) This de"nition allows us to rewrite Eq. (3) as a functional integral over this overlap order parameter in the following way: G" DQ exp(!e[q ]#S[Q ]),

11 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 11 where E"!ln Dk (r) exp!1 2 dr dr k (r)p (r, r)k (r), S"ln Dr (n) exp! 3 2l dn dr dn δ[q (r, r)! dn δ(r (n)!r)δ(r (n)!r)] (5) and P (r, r) is P (r, r)" δ(r!r)δ σ δ(z)δ(z)!σ Q (r, r). (6) The quantity S is clearly the entropy associated with a given overlap order parameter "eld since it is the logarithm of the number of ways in which the DHP can organize itself in 3D space with the constraint that the overlap between replicas on the 2D surface is Q. Then, E is the associated energy. As has been demonstrated in the context of protein folding (e.g., [1}5]) and the behavior of DHPs in 3D disordered media [43], these polymers with quenched sequence distributions can exhibit behavior akin to the REM model, the Potts glass with many states, or p-spin models. One consequence of this is that, under certain circumstances, the thermodynamics is determined by a few dominant conformations. This is because frustration due to competing interactions and quenched disorder makes these few conformations energetically much more favorable compared to all others. Since the physical situation that we are considering also embodies the frustrating e!ects of competing interactions and quenched disorder, we must allow for the possibility of such a phenomenon (we will refer to it as freezing for convenience). In fact, since the competing interactions in our case occur on a 2D plane, this e!ect might be enhanced. It is very important to understand that the preceding sentence does not imply that the problem we are considering is one wherein the dimensionality of space is 2. The polymer conformations can (and do) #uctuate in 3D space by forming loops and tails; only the competing interactions in this simple scenario are manifested 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 we must allow for broken replica symmetry. Parisi [44] pioneered the way in which to compute and think about broken replica symmetry in the context of spin glasses. For SK spin glasses, replica symmetry 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 noted earlier, 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 by evaluating the logarithm of the determinant of the matrix P. Mezard and Parisi [44] have

12 12 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 provided formulas for this quantity when there is broken replica symmetry. Using their formula and a 1-step RSB scheme the energy is computed to be: E" 1 2!ln σ # 1 x ln(1!c px ), p "p/n, C "σ σ N/A, (7) where x is the number of replicas in a group, p is the number of contacts with the surface, and A is the surface area of the solid. In writing Eq. (7), the density of adsorbed segments on the surface has been approximated to be uniform. In order to compute the entropy a physically transparent method can be employed. The "rst quantity that we need to compute is the number of ways in which x replicas can be arranged such that their surface conformations overlap perfectly. When polymers adsorb, the conformations are characterized by loops, trains, and tails (see Fig. 1). In the long chain limit, we may ignore tails. Let f (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 trains are incorporated in the computation of the entropy. With this de"nition, the restricted partition function for x replicas in a group can be written as: Z(r, )" dr, 2 dr, f (r,!r,)2f (r,!r, )δ(n #n #2#n!N). (8) 2 The delta function conserves total chain length while allowing loop length #uctuations. We do not integrate over the position of the "rst adsorbed segment, r, for later mathematical convenience., The entropy for x replicas in a group is obtained by integrating Z over r and then taking the, negative logarithm. In order to compute this partition function, let us "rst introduce a Laplace transform conjugate to N; i.e., z(μ)" dnz(n)e (9) where μ is the Laplace variable conjugate to N. The product of the functions that describe the loop probabilities exhibits a convolution structure, and so it is convenient to introduce a 2D Fourier transform conjugate to the 2D spatial coordinate that de"nes position on the surface. The Fourier}Laplace transform of the restricted partition function is Z(k, λ)" f (k)e. (10) The loop probability factors must be of two types. Following Hoeve et al. [45], the factor f for a loop of unit length is taken to be f (r)"ωδ(r!l) (11)

13 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 13 where ω is the partition function for one adsorbed segment, and depends upon the chemical details of chain constitution. In the simple model that we are considering now, the DHP segments do not interact with each other. So, the loop probability factor for longer loops is Gaussian: f (r)" C n exp! 2nl r (12) where C is a normalization constant that depends upon chain sti!ness, n is the loop length, and r is the distance between loop ends. Substituting the Fourier transforms of the loop probability factors into Eq. (10), integrating over r,, and inverting the Fourier and Laplace transforms yields the partition function that we seek. Now, noting that there are m/x groups of replicas, the entropy in Eq. (5) is calculated as the product of m/x and the negative logarithm of this partition function. In carrying out these manipulations, the sum over loop lengths in Eq. (10) can be taken to be an integral since we are concerned with the long chain limit. Combining the result of the entropy calculation with Eq. (7) for the energy obtains the following free-energy functional: F" 1 2!ln σ # 1 x ln(1!c p x )! 1 x N ln p N q 2πl 3x Γ((4!3x )/2) Cω M Γ(q(4!3x )/2) [N!(p N!q)]. (13) Srebnik et al. [23] also added a term that represents contributions from non-speci"c three-body repulsions to the energy. It is not essential to add this term, but at high values of the adsorbed fraction it may be necessary for stability. A mean-"eld solution for the order parameters, p and x, is obtained by extremizing the free energy functional with respect to them. It is worth noting that the free energy functional must be minimized with respect to p and maximized with respect to x. The reason that the free energy functional has to be maximized with respect to x is that when the mp0 limit is taken in the replica calculation the lowest-order correction to the free energy evaluated at the saddle point value is negative; this is because the dimensionality of the integral is m(m!1), which is negative when the replica limit is taken [42]. A simple computer code allowed Srebnik et al. [23] to obtain the saddle point values of the two order parameters pn and x. By following p and x we can learn about the adsorption characteristics of DHPs onto disordered multifunctional surfaces. The order parameter p is simply the fraction of adsorbed segments. It acquires values greater than zero when adsorption occurs. The order parameter x has been interpreted to be 1! P, where P is the probability with which conformation i occurs. When a multitude of conformations are sampled, each of these probabilities is very small and x acquires the asymptotic value of unity. This is the usual situation in polymer physics because entropic considerations lead to large conformational #uctuations. Natural polymers seem to be

14 14 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 designed such that, under appropriate circumstances, conformational #uctuations are suppressed and a few dominant conformations determine the thermodynamics. DHPs can also exhibit similar physics (e.g., [1}5,43]). In fact, our quest for biomimetic recognition can only be successful if adsorption occurs in a few dominant conformations. Let us "x all the parameters that determine the nature of the chains. Then, let us study the variation of the order parameters p and x with C.Di!erent values of this parameter correspond to di!erent surfaces. Speci"cally, each value of C corresponds to a di!erent total number density of sites on the surface. The number of sites per unit area on the surface can be adjusted experimentally by a number of means, the simplest being the adsorption of functional groups onto a surface from solutions of varying concentrations [7]. Fig. 3 shows how the two relevant order parameters vary with C. A uniform neutral surface corresponds to C "0. Therefore, when this parameter is small no adsorption occurs. The order parameter x is unity since in the absence of intersegment interactions and adsorption all conformations are energetically equally likely. Fig. 2 shows that above some value of C weak adsorption occurs with a multitude of conformations being sampled. The transition from no adsorption to weak adsorption is continuous. The theory also predicts that at a higher threshold value of C a sharp transition from weak to strong adsorption occurs. This adsorption transition is accompanied by x becoming less than unity. This signals that the polymer adsorbs in only a few dominant conformations (at least as far as the adsorbed segments, and hence, the loop structure is concerned). As noted earlier, in the simple model that we have been considering, each point on the abcissa corresponds to a di!erent surface. Thus, the sharp transition from weak to strong Fig. 3. The order parameters p and x plotted as a function of C. For the calculation described in the text, each point on the abcissa represents a di!erent statistically patterned surface.

15 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 15 adsorption implies sharp thermodynamic discrimination between surfaces bearing di!erent statistical patterns } one of the hallmarks of recognition. The physical reason for the sharp transition from weak to strong adsorption (and the freezing into few dominant conformations that accompanies it) can be understood by "rst discussing what happens when the continuous transition to weak adsorption occurs. When there are only a few sites on the surface (small C ), the energetic advantage associated with chain segments binding to preferred sites is not su$cient to overcome the entropic penalty for chain adsorption. For higher values of C, adsorption occurs because now the number of sites is su$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 avoid unfavorable interactions. Furthermore, as shown schematically in Fig. 4, because it is easy to avoid unfavorable interactions, the chain can obtain the same energetic advantage in many di!erent conformations. Thus, the system minimizes free energy by sampling a multitude of conformations which have roughly the same energy. As the loading of surface sites increases, however, it is intuitively obvious that it becomes increasingly di$cult to avoid unfavorable interactions. In fact, it is clear that above some threshold loading of surface sites, most arbitrary adsorbed conformations will be subjected to many unfavorable interactions. Thus, for a su$ciently high loading (and hence, adsorbed fraction), most adsorbed conformations constitute a continuous spectrum of high energy states. However, there will be a small ensemble of conformations that are signi"cantly lower in energy. These conformations are the few that carefully avoid unfavorable interactions (as best as possible given the disorder in the sequence and the surface site distributions). These few Fig. 4. Schematic representation of DHPs interacting with a surface bearing just a few sites. The two panels depict di!erent conformations with the same energy.

16 16 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 conformations are energetically more favorable than all others. Thus, in a manner reminiscent of the random energy model of spin glasses, the energy spectrum develops a gap between the small ensemble of conformations that adsorb in a pattern-matched way and the multitude of others. By pattern matching we mean registry between adsorbed segments and the preferred sites. As the loading of surface sites increases, the system becomes more frustrated, and the energy gap between the pattern-matched adsorbed conformations and all others increases. Beyond a threshold value of the loading, the energy gap becomes much greater than the thermal energy, k¹. This causes the polymer chain to sacri"ce the entropic advantage of sampling a multitude of conformations, and it adsorbs in the few pattern-matched conformations. Of course, these pattern-matched conformations are strongly adsorbed. Thus, we get a phase transition with a sharp increase in the adsorbed fraction and passage to a thermodynamic state where only a small ensemble of conformations are sampled. Mean-"eld theory predicts that this transition is "rst order. There are two free-energy minima, one corresponding to a replica symmetric solution of the equations and the other to a solution with broken replica symmetry [46]. Before the transition from weak to strong adsorption, the minimum corresponding to the replica symmetric solution is the global minimum. When the 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 not large [5]. Signi"cantly larger gaps are obtained for designed sequences. In the situation we have been considering, interaction with the disordered distribution of surface sites adds another source of frustration which makes the energy gap in a REM-like picture quite large when the surface loading exceeds a threshold value even for random sequences and surface site distributions. At least this appears to be true for the thermodynamic behavior. We shall see later that kinetic considerations require delicate design of the statistics of heteropolymer sequence and surface site distribution statistics. The preceding discussion provides a compelling physical argument for the existence of a transition from weak to strong adsorption accompanied by the adoption of a few dominant conformations when the statistics of the DHP sequence and surface site distributions are related in a special way. However, we have provided no physical reason for the transition to be sharp (or "rst order as predicted by the mean-"eld calculation). The order of the transition can be established rigorously only by carrying out a renormalization group calculation. A proper calculation of this sort has not yet been performed. From a fundamental standpoint, it is important that the order of the transition be established in a rigorous way. From a practical standpoint what is important is that the transition is sharp and hence can display one of the hallmarks of recognition } a sharp discrimination between surfaces to which the chains bind weakly and others to which it binds very strongly. The physical reason for the sharpness of the transition is suggested by a simple model of the phenomenon under consideration; Monte-Carlo simulations for "nite size systems is also indicative of a "rst-order transition. First, let us discuss a model [26] which complements the replica "eld theory that we have discussed and is motivated by simple physical considerations. Again, consider DHP chains comprised of two types of segments (A and B) interacting with a surface functionalized by two di!erent types of sites. The segments of type A prefer to interact with one type of surface site, and those of type B exhibit the opposite preference. Thus, from an energetic standpoint, there are two types of segment-surface contacts: good and bad contacts. Good contacts are those that

17 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 17 involve preferred segment-surface site interactions. Let us try to develop a free-energy functional with the order parameters being the total number of adsorbed segments (p) and the number of good contacts (q). The energy corresponding to given values of p and q is: E k¹ "[qδe#pe ] (14) where δe is the energy di!erence between good and bad contacts, and E is the energy of a bad contact. We now need to compute the entropy for a chain of length N and p adsorbed segments of which q are good contacts. This entropy can be partitioned into three separate contributions. Firstly, there is a `mixinga entropy (S ) associated with the number of ways to choose p adsorbed segments out of N. There is also the entropy loss (S ) associated with segmental binding, and the entropy (S ) 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 yields S "!p ln p!(1!p ) ln(1!p ) (15) N where p "p/n is the fraction of adsorbed segments. As is usual, the loss in entropy upon segmental binding is given by S /N"!wp (16) where ω is a constant related to chain #exibility and solution conditions. Now consider the computation of the loop entropy. When a homopolymer adsorbs on a chemically homogeneous surface, the energetic advantage associated with every segment-surface contact is the same. Thus, the adsorbed chain exhibits large #uctuations in the loop structure to maximize entropy. It is important to note that the loops live in three-dimensional space, and any description of the physics must account for the loop #uctuations properly. The problem that we are considering is distinctly di!erent from homopolymer adsorption because the segment}surface contacts are of two types. The existence of good and bad contacts implies that there are two types of loops. There are loops associated with forming good contacts at both ends, and those that are associated with forming the other contacts. These two types of loops are fundamentally di!erent in character. Each loop is characterized by the loop length and the distance between loop ends on the surface. For the loops associated with forming good contacts at both ends, only certain values of these quantities are allowed. This is because the two segments that correspond to the loop ends must be bound to surface sites with which they prefer to interact. The allowed loop lengths and distance between the loop ends on the surface are intimately related to the probabilities of "nding certain types of sites and segments at di!erent locations along the chain and on the surface. Thus, the allowed #uctuations of loops associated with good contacts are determined by the statistics that characterize the chain sequence and the surface site distribution. Loops associated with contacts that are not good are not restricted in this manner, and the usual

18 18 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 #uctuations in loop length and distance between loop ends are allowed. The above argument suggests that competing interactions and disorder cause loop #uctuations to be suppressed by the formation of good contacts. It is reasonable to suspect that if the statistics of the chain sequence and the surface site distribution are such that there is a high probability for the formation of good contacts, loop #uctuations are strongly suppressed. Suppression of loop #uctuations makes the chain e!ectively sti!er. Su$ciently sti! chains are known to undergo sharp ("rst-order) adsorption transitions [14]. These arguments suggest that strong suppression of loop #uctuations resulting from frustration due to competing interactions and quenched disorder, and statistical pattern matching are the origins of the sharp adsorption transition. The meaning of statistical pattern matching also is made more clear; we mean that the statistics that describe the DHP sequence and the surface site distribution are such that the probability of making good contacts in certain adsorbed conformations becomes su$ciently high. In order to explore the veracity of these arguments, we need to write down a mathematical formula for the entropy corresponding to loops associated with good and bad contacts when they coexist. (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 with each type of loop when it is the only type of loop that exists. Once these formulas are available, it is relatively 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 hence Gaussian) in our model, the loop factor for a loop of length n returning to the plane with the two ends separated by a distance d is again (see Eq. (12)) given by P(n, d)" C n exp! 2nb d. (17) The quantities n and d depend upon the statistics that describe the chain sequence and surface site distributions. For example, in the case of uncorrelated #uctuations in the surface site distribution, the most probable value of d&1/σ, where σ is the width of the distribution and is proportional to the surface loading of both types of sites. Since we are considering a situation where only quenched loops exist, if there are q adsorbed segments, there are q quenched loops with the average loop length being N/q. Shortly, we shall see how the average loop length (and hence n) is closely related to the statistics which describe the sequence and surface site distributions. In view of the considerations noted above and Eq. (17), it is easy to write down an expression for the entropy corresponding to q quenched loops. Speci"cally, taking n and d to equal their most probable values, obtains S N "q ln(cq)! aq (18) 2bσ where q "q/n. Now consider the entropy associated with forming annealed loops only. Toward this end, consider the well-known problem of a homopolymer adsorbing on a chemically homogeneous surface since herein the loops are annealed. Let the energy bonus for segmental adsorption be represented by a potential (z) which is zero everywhere except at the surface; z is the coordinate normal to the surface. The e!ective energy bonus for segmental adsorption, ln β is then given by the

19 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 19 following formula: ln β"ln dz[e(!1]. (19) As noted earlier, in this case, the loop lengths and the distance between loop ends can #uctuate with the only constraint being the "xed chain length. This problem is similar to the adsorption of a 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!erence between the problem considered in [14] and our concern is that the potential is imposed by an impenetrable two-dimensional manifold rather than a point. This imposes certain additional symmetries. Exploiting these symmetries, it is easy to show [26] that the following Schrodinger-like equation describes the problem under consideration: [1#βδ(z)]g( ψ(z)"νψ(z) (20) where g is the standard connectivity operator, and the eigenfunction ψ and the eigenvalue ν have their usual meaning. Very simple manipulations (described in [14]) lead to the following relationship between β and ν: 1 β " 2π 1 dk g(k) [ν!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 β and P. In order to "nd this relationship, it is convenient to de"ne a generating function z(n, β) as follows: z(n, β)" βz(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., ν"z(n, β). 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 approximation. In other words, P"β R ln z Rβ. (23) With this approximation, making use of the relationships between β and ν (Eq. (21)) and that between ν and the generating function allows us to obtain the equilibrium value for P/N. Speci"cally, we "nd that P N "1 dkg(k)/[ν!g(k)] ν dkg(k)/[ν!g(k)]. (24)

20 20 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 The entropy is now easy to calculate as the free energy equals!n ln ν and the energy bonus associated with segmental adsorption is ln β. We"nd that S N "! F N¹ # E N¹ "ln ν!p N ln 2π 1 dk [ν!g(k)]. (25) Eqs. (21) and (24) can be solved simultaneously to obtain P as a function of β, and Eqs. (21) and (25) yield the relationship between the entropy and β. 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 physical grounds, it is clear that the total number of segment}surface contacts is greater than or equal to the number of good contacts. This implies that the annealed loops live within the quenched ones. This is illustrated schematically in Fig. 5. The annealed loops can redistribute among the quenched ones in an unconstrained manner. It is most convenient to consider this physics in the grand canonical ensemble, whence we can say that the chemical potential of the annealed loops is the same in all the quenched loops. The chemical potential can be easily calculated from the equations we have derived 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 equal in the quenched loops, leads to the conclusion that the concentration of annealed loops is the same in each quenched loop. By concentration, we mean the quantity p "number of annealed loops/length of quenched loop. These remarks allow us to properly combine the formulas for the entropies of quenched and annealed loops (Eqs. (18), (21), (24) and (25)). Noting that there are q quenched loops (q good contacts), that p "p!q, and that p is the same in each quenched loop lead us to the following expression for the loop entropy corresponding to p adsorbed segments of which q are good contacts: S N "q ln P(1/q, d)#1 s(pn q ) (26) where P is the probability written down in Eq. (17), and s(p ) is the entropy of annealed loops with concentration p divided by n. The latter quantity is obtained from Eqs. (21), (24), and (25) with one Fig. 5. Schematic depicting quenched and annealed loops. The darkly shaded loops correspond to loops with both ends being good contacts. The lightly shaded loops do not have good contacts at the end and exhibit signi"cant #uctuations.

21 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 21 modi"cation. The calculation leading to these equations considered a uniform surface. We are concerned with a situation where adsorption can only occur on particular surface sites which do not uniformly cover the surface. The concomitant entropy loss is proportional to ln σ, 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"qδe#(ω#e )p #p ln p #(1!p ) ln(1!p )! q ln(cq)# aq 2σb!s(p!q )!(p!q )ln σ b a. (27) These two order parameters can be further related by noting that q "p P (28) where P is the probability of making a good contact. At in"nite temperature, when entropy is irrelevant, 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 " P (m)p (m). Here P (m) is the probability of "nding a block of length m of like segments on the chain sequence, and P (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 site distributions, P (m) and P (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: e P "P P e#(1!p )e (29) where F 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-"eld solution. The purpose of this exercise is to obtain some insight into the origin of the sharp transition. Thus, let us consider results for the simplest possible scenario } uncorrelated sequence and 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 p and q with σ, the width of the distribution that characterizes the surface site #uctuations. We have taken the statistics of the DHP sequence to be "xed in constructing Fig. 6. Thus, points on the abcissa in this "gure correspond to di!erent surfaces. Fig. 6 shows that when σ is small, both pn and qn are zero. This is simply a re#ection of the fact that the energetic advantage associated with segmental binding is not su$cient to overcome the concomitant entropic penalty. This is because the number of binding sites available on these surfaces is not su$cient. When σ becomes su$ciently large, adsorption does occur. However, it is

22 22 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 Fig. 6. The order parameters p (solid line) and q (dotted line) plotted as a function of σ. 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 σ, our theory predicts a sharp transition from weak to strong adsorption. This transition coincides with a jump in q, signifying that now the preponderance of contacts 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 q grows faster than p, and ultimately approaches p. 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 evident from 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 have a situation resembling the adsorption of a sti! chain, a case for which the adsorption transition is known to be "rst order [14]. In some ways, the phenomenon we are considering resembles protein folding (or heteropolymeric models of folding). In the latter situation, a "rst-order transition called the coil}globule transition occurs wherein the preponderance of contacts become native ones. This is followed by a continuous transition to the "nal low entropy folded state. The sharp transition we see may be considered the analog of the coil}globule transition. The fact that in the strongly adsorbed state the quenched loops dominate is very signi"cant. The dominance of quenched loops implies that the chain adopts a small number of conformations characterized by a certain distribution of loops speci"c to the sequence and surface site distributions. Only small #uctuations around these conformations (shapes) of the adsorbed chain occur after the transition. This adoption of a small class of shapes makes the phenomenon we are considering richer than other successful e!orts to elucidate strategies that can localize chains to certain regions of a surface [29}31]. This feature was also signaled in the replica "eld theory by broken replica symmetry coinciding with the transition from weak to strong adsorption. In the simple model that we have just discussed, the structure of the quenched loops is measured only by the average length, q. Notice that even this quantity is determined by the probability of good contacts; i.e., the statistics of the sequence and surface site distributions. This suggests that the class

23 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 23 of shapes that are adopted upon strong adsorption is determined by the statistical patterns on the chain 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 pattern recognition, and the suggestions of the thermodynamic model we have been considering will become vivid Monte-Carlo simulations of thermodynamic properties The predictions of the models we have been considering are consistent with a series of Monte-Carlo simulations designed to compute thermodynamic properties [24,25,27]. These studies were carried out using an adaptation of the non-dynamic ensemble growth method pioneered by Higgs and Orland [48,49]. The simulations were carried out on a cubic lattice. The introduction of the lattice does a!ect quantitative predictions. However, the phenomenology is expected to be the same because of reasons that have been explained in detail in [48]. A particular sequence is "rst drawn from the statistical distribution under consideration. A particular realization of the surface site distribution is also drawn from the statistical distribution under consideration. M monomers are then placed randomly with Boltzmann probabilities dictating the positional probabilities. These positions are allowed to vary between 0 and 2N where N is the length of the polymer we wish to simulate. This is equivalent 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 particular realization 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 by intersegment interactions and interactions with the surface sites; excluded volume interactions are enforced. This process is continued until chains with the desired length have been grown. For M<N, a canonical distribution results [48}50], and properties are computed as non-weighted averages over the ensemble. If necessary for statistical accuracy, many populations of the same sequence are simulated. The same procedure is repeated for several realizations of the quenched sequence distribution. Averages over di!erent realizations of the surface site distribution are computed in two di!erent ways. In principle, if the DHP samples a su$ciently large surface over asu$ciently long observation time, then treating the external disorder that constitutes the surface as quenched or annealed yields equivalent results (e.g., [37}39]). The annealed case is easier to simulate because the e!ects of the #uctuating surfaces sites can be integrated out to obtain an e!ective Hamiltonian which can then be simulated. However, the conditions for this equivalence to hold are di$cult to establish in a quantitative manner. Thus, explicit quenched averages over the disordered surface sites have been computed and compared to results obtained by simulating a pre-averaged Hamiltonian [25]. It was found that, for N"32, once the surface sites live on a lattice larger than , both procedures yield equivalent results. The results we discuss have typically been obtained by taking M to be for N"32, 64 and 128. In order to compare the simulation results to the basic phenomenology predicted by the analytical models, the "rst simulations [25] were carried out for situations where both the DHP sequence and surface site distributions exhibit uncorrelated #uctuations. Thus, two-letter DHPs with the sequence statistics characterized by the average composition f were examined. Consider surfaces with two types of sites with the average composition always being equal to half. As in the

24 24 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 Fig. 7. Results of Ensemble Growth Monte-Carlo simulations. Adsorbed fraction p and the parameter x as a function of σ. 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 total number of sites per unit area (i.e., loading). Fig. 7 depicts the results of the ensemble growth MC simulations for these situations by plotting the adsorbed fraction p as a function of σ; σ is the width of the distribution characterizing the surface site distribution, and is proportional to the loading. The absorbed fraction p is vanishingly small for small loadings of the surface. For su$ciently large values of σ, we"nd that a gradual transition to weakly adsorbed states occurs. We also observe that beyond a higher threshold value of σ a sharp transition occurs from weak to strong adsorption. In Fig. 7, we also plot the variation of a parameter x with σ. This quantity is de"ned in a manner analogous to the way in which x was de"ned in the replica "eld theory. The quantity x"1! P, where P is the probability of "nding a conformation with energy E. Due to degeneracy, x and x are not identical. However, simulation results [25] show that x faithfully reproduces the qualitative trends obtained by computing x.itis much easier to compute x from the simulation data as E is easily obtained from the simulation results. Fig. 7 shows that x equals unity for values of σ for which we have no adsorption or weak adsorption. However, the transition from weak to strong adsorption is accompanied by x becoming less than unity. (The simulation results suggest that x become less than unity for a value of σ that is slightly larger than that corresponding to the weak to strong adsorption transition, and some reasons have been discussed to explain this [26]. We shall not focus on this minor detail here.) The simulation results shown in Fig. 7, therefore, reveal the same phenomenology as the theoretical models. For uncorrelated sequence and surface site #uctuations, and for "xed DHP sequence statistics, when the surface loading exceeds a threshold value a sharp change from weak to strong adsorption occurs with the thermodynamics being determined by a few dominant conformations when strong adsorption occurs. The transition is rounded in the Monte-Carlo simulations because of "nite size e!ects. Unlike the theoretical predictions, however, it is di$cult to ascertain the order of the transition from the simulation results. This question can be explored by examining "nite size e!ects. Thus, the quantity (δp) has been computed. This quantity exhibits a peak when the sharp adsorption transition occurs. Furthermore, the width of the peak narrows

25 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 25 and shifts to the left as the chain size increases. Unfortunately, no "rm conclusions can be drawn about 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 for the lack of simulation data is that the important result is that a sharp transition occurs from weak to strong adsorption when the statistics of the sequence and surface sites are related in a special way. The order of the transition is fundamentally very important, but not crucial for examining the physics of discriminatory pattern recognition exploiting the notion of self-assembly driven by statistical pattern matching. We now turn to these exciting questions, for which the preceding discussions serve as a prelude. We begin our considerations of how the basic physics described above can be exploited to cause DHPs bearing certain statistical patterns to recognize surfaces that bear complementary statistical patterns by discussing some simulation results that pertain to thermodynamics. This will be followed by detailed considerations of the kinetics of the self-assembly process that leads to recognition driven by statistical pattern matching. Consider DHPs with symmetric average compositions that carry two di!erent types of statistical patterns encoded in their sequence. The two types of statistical patterns are characterized with two-point correlations only. As we have noted earlier, these two-point correlations are speci"ed by a parameter λ. Positive values of λ imply that within a certain length along the chain, there is a high probability of "nding segments of the same type. We shall call the ensemble of DHPs with such sequences statistically blocky. Negative values of λ imply that within a certain length along the chain there is a high probability for "nding the two types of segments arranged in an alternating fashion. 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 to situations where the composition specifying the relative amounts of the two types of sites is symmetric. Surfaces that we call statistically patchy have a high probability for sites of the same type to be adjacent to each other within some correlation length, κ, measured on the twodimensional surface. Those that we call statistically striated have a high probability for sites of the opposite type to be adjacent to each other within a correlation length κ. Speci"cally, the density}density correlation function specifying the distribution of sites on the surface for the two situations are: σ exp(!κr) and σ(!1) exp(!κr) for statistically blocky and striated surfaces, respectively. r is distance measured on the two-dimensional surface, and x and y are displacements in the two orthogonal cartesian coordinates used to specify position on the surface. σ 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 can be varied conveniently in experiments. Ensemble growth MC simulations have been performed to study whether statistically blocky DHPs can recognize statistically patchy surfaces more easily than statistically striated surfaces, and vice versa [24]. The two types of statistically patterned DHPs were characterized by λ"0.4 and λ"!0.4. The statistically patterned surfaces are characterized by a value of κ"0.7/l, where l is the statistical segment length. The way the statistics of surfaces within a class (statistically patchy or striated) is varied is by changing the total loading, i.e., σ. Figs. 8a and b show the results of the ensemble growth MC simulations for statistically alternating and patchy surfaces. The adsorbed fraction pn is plotted as a function of σ for statistically blocky and alternating DHPs in each panel. In all four cases, we "nd that a sharp adsorption

26 26 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 Fig. 8. Results of Ensemble Growth Monte-Carlo simulations for the interactions of statistically alternating (open circles) 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 threshold value. For the statistically patchy surfaces, we see that the sharp adsorption transition occurs at a smaller value of σ for the statistically blocky chains. The opposite is true for the statistically striated surface. Thus, a statistically patchy (striated) surface with a loading that will strongly adsorb the statistically blocky (alternating) ensemble of DHPs will not strongly bind the statistically alternating (blocky) chains. These results show that, at least from a thermodynamic standpoint, 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 the following. For pseudo patchy surfaces, arbitrary adsorbed conformations of blocky DHPs will experience more unfavorable interactions at smaller surface loadings when compared to sequences which are statistically alternating. Further, pattern matched adsorbed conformations with low energies are statistically far more probable for blocky DHPs interacting with pseudo patchy surfaces than for statistically alternating DHPs. These reasons cause the energetic driving force to

27 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 27 adsorb in a few pattern-matched conformations (Δ) to be larger for statistically blocky chains interacting with pseudo patchy surfaces (at the same value of loading). The larger value of Δ when combined with the fact that it is entropically less costly for the statistically blocky chain to organize itself in a pattern matched conformation when interacting with a pseudo patchy surface leads such statistical sequences to adsorb strongly to pseudo patchy surfaces at smaller loadings compared to statistically alternating sequences. Similar arguments explain why statistically alternating chains recognize pseudo striated surfaces at smaller loadings compared to statistically blocky chains. The simulation results we have just described demonstrate that, from the standpoint of thermodynamics, statistically patterned chains can recognize statistically patterned surfaces when the statistics characterizing the sequence and surface sites are related in a special way. The simulation results have been found to be qualitatively captured by the simple formula described earlier with P " P (m)p (m). Static ensemble growth MC simulations have also been carried out to investigate the competition 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 intersegment interactions. The "rst is a non-speci"c excluded volume interaction. The strength of these nonspeci"c interactions is (< #< #2< )/2, where < are the strengths of the short-range intersegment interactions between segments of type i and j. The sequence speci"c intersegment interactions which encourage segregation of like-type segments and freezing into a few dominant conformations are also taken to be short-ranged. The strength of these interactions is b"(< #<!2< )/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 describing the sequence and surface site #uctuations. When b is small, the variation of p and x with σ is not qualitatively di!erent from that we have discussed earlier for b"0. This is shown in Fig. 9 for DHPs with symmetric composition and for surfaces wherein the average composition of the two types of surface sites is also symmetric. Fig. 10 shows results for a relatively large value of b"4. Even before adsorption occurs x(1. This is because of the well-known phenomenon Fig. 9. Results of Ensemble Growth Monte-Carlo simulations. Adsorbed fraction p (full line) and x (dotted line) plotted against the surface disorder strength σ. The results are for a small value of b.

28 28 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 Fig. 10. Results of Ensemble Growth Monte-Carlo simulations. Adsorbed fraction p (full line) and x (dotted line) plotted against the surface disorder strength σ. The results are for a relatively value of b"4k¹. (e.g., [1}5]) wherein, for large enough values of b, frustration due to connectivity and the quenched sequence disorder causes a few energetically favorable conformations to determine the thermodynamics. The situation we are studying is thus analogous to the adsorption of a folded protein from solution. The results displayed in Fig. 10 show that the variation of p with σ is of the same form as that when b is small. Comparison of Figs. 9 and 10 shows that the sharp transition from weak to strong adsorption occurs at higher values of σ when b is larger. Physically, this is so because stronger speci"c intersegment interactions imply that the DHP chains are in lower energy states in solution compared to situations where b is small. Thus, compared to situations where the strength of the intersegment interactions are small, stronger (more favorable) segment-surface interactions are required for strong adsorption to become favorable. A signi"cantly more interesting e!ect of speci"c intersegment interactions is revealed by the variation of x with σ displayed in Fig. 10. The few conformations that are adopted in solution are determined by the nature of the intersegment interactions and the sequence statistics. As the loading 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 same frozen conformations as those in solution, adsorption to the surface eliminates a few more conformations from being sampled. When the loading is increased beyond the point where a sharp transition from weak to strong adsorption occurs, the MC simulation results show a rather unusual variation of x with σ. Fig. 10 shows that x "rst increases and then decreases again. This is interpreted as follows. As the interactions of the chain with the surface increase because of the higher density of surface sites, these interactions compete favorably with the intersegment interactions. Thus, the few dominant conformations favored by the intersegment interactions alone are no longer energetically far more favorable than all other conformations. Thus, the system minimizes free energy by gaining the entropy associated with sampling conformations other than the low-energy conformations determined by the intersegment interactions alone. In other words, the few dominant conformations adopted by the chains due to intersegment interactions unravel upon strong adsorption to the surface. As the loading, and hence interactions with the surface, are

29 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 29 Fig. 11. Schematic depiction of the adsorption of DHPs with di!erent values of b: (a) moderate values of b. Here for large enough surface disorder strength, the conformations preferred by the intersegment interactions can unravel, (b) large values of b. 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 interactions. Thus, the physics should be similar to that observed when the intersegment interactions are weak. That is, the strong segment}surface interactions coupled with the frustration due to disorder in DHP sequence and surface site #uctuations should cause the DHPs to freeze into a few dominant conformations determined by the statistics of the chain sequence and surface site distributions. Consistent with this picture (shown schematically in Fig. 11), we see that x decreases again in Fig. 10. The analytical and simulation results that we have described so far suggest the occurrence of a phenomenon akin to recognition in biological systems due to statistical pattern matching between the sequences of DHPs and the distribution of sites on multifunctional surfaces. Speci"- cally, these thermodynamic studies show that frustration (due to competing interactions and disorder) and statistical pattern matching lead to one of the hallmarks of recognition: a sharp discrimination between surfaces to which a given ensemble of DHP sequences binds strongly and those to which they do not. These studies also suggest that when strong adsorption occurs, the chains adsorb in a small class of conformations or shapes. These suggestions notwithstanding, it is unclear as to whether statistical pattern matching is su$cient to realize the other hallmarks of recognition and hence be pragmatically useful and of further scienti"c interest. The basic question that the aforementioned studies do not address is: can DHPs bearing a statistical pattern discriminate between various parts of a single surface which bears di!erent statistical patterns in di!erent regions? Furthermore, can this happen in time scales of practical interest? If the answer to

30 30 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 these questions is to be a$rmative, then DHPs must not get kinetically trapped in the `wronga regions of the surface. By wrong we mean those regions of the surface which bear statistical patterns of sites that are not matched to the DHP sequence statistics. To address these issues kinetic Monte-Carlo simulations have been performed [28]. The results of these simulations show that the answers to the questions posed above are `yesa provided the statistical patterns are properly designed. Furthermore, they reveal that the dynamical behavior of this system is typical for frustrated systems characterized by rugged free energy landscapes. As we shall discuss, the results of the simulations demonstrate that the system we are considering may be a good model system for experimental studies of kinetics in frustrated systems. Intriguing connections can also be made to Kau!man's ideas [51] regarding the interactions between selection and the propensity for self-organization in evolution Kinetics of recognition due to statistical pattern matching In order to discuss the kinetics of the phenomenon under consideration, let us begin again by considering 2-letter DHPs which carry statistical patterns. As before, let us concern ourselves with statistically alternating and blocky patterns with the values of λ being #0.4 and!0.4. The compositions of the two ensembles of sequences are symmetric. As discussed earlier, surfaces with two types of sites on a neutral background can also bear simple statistical patterns. The simplest statistical measures of the patterns carried by statistically alternating and patchy surfaces are the correlation length, the total density of A and B type sites on the surface (loading), and the ratio f of 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 this type 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 interactions, and MC moves which are exchanges of the identity of two sites. Typically 100 million MC steps 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 those between sites of the opposite type are repulsive (attractive). Neutral sites are non-interacting. The correlation 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 mixed molecular adsorbents [52}56]. Consider a surface comprised of four quarters, each of size lattice units. Let two quarters have essentially random distributions of A (red) and B (yellow) type sites. Let the other two quarters be realizations of statistically patchy and alternating surfaces generated by the methods described earlier. Such an arrangement is shown in Fig. 12, where all quarters of the surface are characterized by an average total loading of 20%, and the correlation length is &1.4 for the statistically patterned regions. Golumbfskie et al. [28] have carried out MC simulations with the Verdier}Stockmayer algorithm 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 situations where the segment}surface site interactions are much stronger than the intersegment

31 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 31 Fig. 12. Surface bearing four types of statistical patterns. White denotes the neutral background, and light and dark grey dots correspond to the two types of active surface sites. The right top corner is statistically alternating and the left bottom corner 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. The preferred segment}surface interaction strengths are taken to be!1 and the ones that are not preferred equal #1 in units of k¹, where ¹ 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, and 128. The results show that for "xed f and surface loading the important design variables are λ, ¹ and ¹/¹ (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 alternating DHPs 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 containing a mixture of statistically blocky and statistically alternating DHPs, will the chain molecules selectively 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 be strongly favored for binding thermodynamically, but also that the `wronga regions of the surface not serve as kinetic traps.

32 32 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 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. Fig. 13 depicts typical trajectories at ¹/¹ "0.6. All points on these trajectories do not correspond to adsorbed states. Both the statistically alternating and blocky DHPs begin on randomly patterned parts of the surface and ultimately "nd their way to the region of the surface that is statistically pattern matched with its sequence statistics. The length of the simulations roughly corresponds to 1 s, over which separation or recognition is achieved with over 90% e$ciency (out of 1000 trials). These results show that biomimetic recognition between polymers and surfaces is possible due to statistical pattern matching. It seems possible to exploit this notion to design inexpensive devices that can separate a large library of macromolecules into groups of statistically similar sequences. Local motion on the scale of monomers occurs in 10}10 seconds depending upon solvent conditions and monomer size (see [47]). The MC moves employed by Golumbfskie et al. [28] are for motions of a statistical segment length. 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 associated with the MC moves in [28] ranges between 10}10 seconds. This is in agreement with previous estimates. The numbers reported by Golumbfskie et al. [28] are based on using 10 seconds as the time scale for primitive MC moves.

33 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 33 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 artifacts of 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 the strongly adsorbed state. Once a chain is strongly adsorbed onto a region that is statistically pattern matched with its sequence the chain center of mass essentially does not move on simulation time scales (vide infra). The reason for this is made clear by the results shown in Fig. 14. The `wronga regions of the surface correspond to local free energy minima which are separated by relatively small barriers from each other. In contrast, in the statistically pattern matched region there exist a few deep global minima; each of these minima, in turn, is very rugged. It is important to note that ¹/¹ 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 appreciable extent because the entropic penalty associated with adsorption dominates the free energy. At su$ciently low values of this ratio, the `wronga regions of the surface serve as kinetic traps, and long-lived metastable adsorbed states exist in these regions. Thus, it is clear that due to both kinetic and thermodynamic considerations there should be an optimal value of ¹/¹. Simulations have been carried out for ¹/¹ "0.45, 0.6, and Of these conditions, Golumbfskie et al. [28] "nd that 0.6 leads to the best discrimination and recognition. More work is needed in order to provide quantitative criteria for the optimal value of ¹/¹. The importance of both kinetics and thermodynamics for statistical pattern matching is further emphasized by the following point. We have provided a detailed discussion of thermodynamic arguments which suggest that DHPs with random sequences (λ"0) sharply discriminate between surfaces 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 high when uncorrelated sequence and surface site distributions are statistically pattern matched by adjusting the loading. In particular, let us compare the following two situations: (a) a surface with random site distribution bearing regions with loadings of 20% and 40% being exposed to DHPs

34 34 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 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 the equilibrium adsorbed fraction in the region with a loading of 40% in case (a) and that for the statistically 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 for successful recognition ('80% e$ciency) is 50% wider for the latter case (b). The reason for this is that in the former situation the free energy landscape that the chains negotiate is virtually uncorrelated, and thus rife with local optima that lead to kinetic trapping [51]. Having illustrated the basic phenomenology of biomimetic recognition due to statistical pattern matching, Golumbfskie et al. [28] have also examined a situation that is a step closer to applications. If the notion of statistical pattern matching is to be used to a!ect molecularly selective separations, we need to separate more than two types of statistical patterns. A more diverse set of statistical patterns can be obtained by increasing the number of letters that code the statistical patterns. Using a 3-letter code, Golumbfskie et al. [28] generate four di!erent statistical patterns to illustrate a more complex separation than that depicted in Fig. 13. The surfaces are generated using a lattice where the identity of each site can be A, B, C or neutral. For all cases, in the MC simulations that generate statistically patterned surfaces the interaction energies (<, where i and j denote type of site) are symmetric and equal in magnitude, with the neutral sites being Fig. 15. Statistically patterned surface with three types (depicted in light grey, dark grey and black) of sites. Four di!erent statistical patterns (described in the text) exist along with regions having a random site distribution.

35 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 35 non-interacting. For statistical pattern number 1, < "< "!1, < "< "#1. For statistical pattern number 2, < "< "!1, < "< "#1. For statistical pattern number 3, < "< "#1, < "< "!1. For statistical pattern number 4, < "< "#1, < "< "!1. The realizations of the statistical patterns shown in Fig. 15 correspond to a total loading of 30%, a 1 : 1 : 1 ratio of di!erent types of sites, and a correlation length of &1.4. The correlation 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 are statistically pattern matched with the surface patches. The DHP sequences employed by us belong to the "rst three excited states in terms of the energy spectrum. Just the energy is clearly not a complete speci"cation of the sequence statistics, and points to the need to develop better design criteria 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 pattern matched 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., biomimetic recognition due to statistical pattern matching is possible in this case also. However, unlike the 100% e$ciency reported for the two letter cases, over a simulation time of 5 s, Golumbfskie et al. Fig. 16. Starting (1,2,3,4) and ending (1,2,3,4) points of typical trajectories of four di!erent types of statistically patterned DHPs on the surface shown in Fig. 15. For these trajectories statistical pattern matching leads to successful recognition of complementary surface patches.

36 36 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 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, and failures are the result of kinetic trapping in the wrong regions of the surface. Simulations have not been carried out for long enough times to report the time scale over which the success rate would approach 100% for all situations. While Fig. 16 demonstrates the feasibility of carrying out molecular scale separations using statistical pattern matching with many such patterns, the e$ciencies found thus far (without detailed design considerations) also illustrate that we have not yet learned how to design statistical patterns with large number of letters in the code such that long-lived metastable states are largely eliminated. Fig. 14 suggests that, for the system we are considering, chain dynamics on various time scales and di!erent spatial locations should be quite di!erent and interesting. Detailed analyses of dynamic MC simulation results serve to elucidate these issues, and provide evidence for macromolecular shape selection when recognition occurs due to statistical pattern matching. Consider the "rst passage time, which is de"ned to be the time taken for a chain starting in the random part of a statistically patterned surface to adsorb in the pattern-matched region with the energy being below a certain cut-o! value. Fig. 17 shows the distribution of "rst passage times for chains of length 32, with the energy cut-o! being!20k¹. It is clear that, after the usual turn-on time, the distribution is decidedly non-exponential; a stretched exponential with a stretching exponent equal to 0.43 "ts the simulation results. The non-exponential character of the distribution of "rst passage times is indicative of highly cooperative chain dynamics. The event we are considering is tantamount to two events that occur in succession. These are chain motion to the edge of the statistically pattern matched region of the surface followed by strong adsorption in one of the deep free energy minima in the statistically pattern matched region. The "rst passage times for these two events have been computed separately [28]. Signi"cantly, the distribution of "rst passage times for the "rst event is exponential, and that for the second is non-exponential. This implies that as the chain traverses the wrong parts of the surface, the center of mass motion is di!usive. This is because the free energy minima and barriers it encounters in these regions of the surface are relatively small. In contrast, it appears that strong adsorption in the statistically pattern matched region is highly

37 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 37 Fig. 18. Distribution of P : (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 of the deep free energy valleys that exist in this part of the surface. What is the physical origin of this behavior in the statistically pattern matched region? It is interesting to observe [28] that single particle MC dynamics on the free energy landscape shown in Fig. 14 is exponential. This indicates something very important for understanding the nature of the dynamics. Single particle dynamics on the free energy landscape in Fig. 14 would reproduce the "rst passage time distribution obtained from the full chain dynamics if motion of the center 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 of these loops #uctuates in time, and hence loop #uctuations are dynamic modes. Prior to considering the dynamics of these modes, let us examine the distribution of these loops as trajectories evolve starting from wrong parts of the surface. Extensive MC simulations have been carried out with chains of length 100. All trajectories show the qualitative features depicted in Fig. 18 where we show the probability distribution P for an arbitrary segment to be part of a loop of length n for a statistically alternating DHP. Each panel shows P 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 the surface P is 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. 18 demonstrate that as the chain center of mass enters the statistically pattern matched region P begins to show structure. Ultimately, it exhibits a spectrum of peaks which correspond to

38 38 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 preferred loop lengths and hence macromolecular shape in the adsorbed state. This observation of adsorption 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, while these types of loops seem to dominate the behavior in the statistically pattern matched region has been discussed earlier. In the wrong region of the surface arbitrary adsorbed conformations with the same number of contacts have roughly the same energy. This is because there are no particular arrangements that lead to much higher degrees of registry between adsorbed segments and their preferred sites, compared to other conformations. Thus, the distribution of loops #uctuates strongly, thereby gaining entropy while maintaining roughly the same energy. In the statistically pattern-matched region of the surface, however, the situation is di!erent. While most adsorbed conformations are of relatively high energy as in the wrong region of the surface, now there exist a few conformations that can bind segments to their preferred sites with high probability in certain regions because the statistics of the sequence and surface site distributions are matched. These few pattern-matched conformations are much lower in energy than all others, and thus the chain sacri"ces the entropic advantage associated with loop #uctuations and `freezesa into one of these conformations. The better the statistical pattern matching, the greater the suppression of loop #uctuations. Dramatic evidence for this thermodynamic argument (and the associated model) is provided by the simulation results displayed in Fig. 18. We shall also make this vivid shortly by showing snapshots of animations of the dynamic simulation results. Let us return to the puzzle of why single particle dynamics on the free energy landscape calculated for the chain center of mass does not reproduce the correct dynamical behavior. Consider the time correlation functions P (t)p (t#τ) 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. 18 shows 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 the time scale on which the chain center of mass moves. Thus, the modes corresponding to loop Fig. 19. Squared and averaged displacement of the center of mass as a function of time measured in Monte-Carlo steps.

39 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 39 Fig. 20. Relaxation time for loops of length 26. Data shown only after the center of mass enters the statistically pattern matched 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 matched region, the chain's center of mass "nds the location corresponding to one of the deep free energy minima in this region. Now, the center of mass essentially does not move during the simulation. As shown in Fig. 19, for chains of length 32 the center of mass ceases to move, and for chains of length 100 the di!usion coe$cient becomes two orders of magnitude smaller than that in the wrong parts of the surface. On much longer time scales, the center of mass of these "nite chains will be able to escape 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, these events would occur over time scales that are not experimentally relevant. The point is that localization of the center of mass implies that this dynamic mode has equilibrated from a practical standpoint. Simulations show that, after the chain center of mass has equilibrated in the statistically pattern matched region, the time correlation functions describing loop #uctuations decay over very long times. In fact, ultimately, once shape selective adsorption occurs, they do not decay to zero and simply oscillate around a "nite value. Thus, these modes, which were the fastest dynamic modes in the wrong part of the surface become the slowest modes once the center of mass reaches a location corresponding to a deep free energy minimum. This explains why single particle dynamics on the free energy landscape appropriate for the center of mass cannot describe the correct dynamics obtained from the detailed simulations. Fig. 20 shows how the relaxation time for loops 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 of the chain is located in the region corresponding to a deep free energy minimum, the chain has to acquire the shape (conformation) that is most favorable on energetic grounds. In order to do this, it must arrange its loops in a speci"c way. The necessary conformational rearrangements occur in a highly cooperative way, and there are important entropic barriers associated with this reorganization. These entropic barriers which correspond to arranging the chain to acquire a speci"c shape (a hallmark of recognition) lead to the non-exponential character of the "rst passage time distribution. In order to make some of the results discussed above vivid, Golumbfskie et al. [28] have animated trajectories of their simulation runs. Fig. 21 shows snapshots from such a movie for

40 40 A.K. Chakraborty / Physics Reports 342 (2001) 1}61 Fig. 21. Snapshots of chain conformations at di!erent times during a typical trajectory for a 128 segment statistically alternating DHP. Dark and light shadings represent the two kinds of surface sites and two kinds of segments that make up the chain. The top three panels correspond to portions of the trajectory in the wrong parts of the surface, and the bottom 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 the statistically random part of the surface. The DHP does adsorb somewhat on this region of the surface. The absorbed conformations are characterized, as usual, by loops, trains, and tails which exhibit large #uctuations. For example, both the loop length distribution and the distance on the surface 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 chain segments adsorb on preferred sites on the surface. The resulting loops are quenched in time in that there are essentially no #uctuations of these loops on the time scale of our simulations. Within these quenched loops live annealed loops that exhibit small conformational #uctuations. The problem of how macromolecules recognize patterns on surfaces has also been studied by Muthukumar and co-workers [29}31] in the context of polyelectrolytes interacting with a surface pattern of opposite sign. These studies complement the work described earlier in this section, and are very important contributions to the conceptual advances that are being made in understanding pattern recognition by macromolecules. We do not, however, provide a detailed review of this work because Muthukumar has recently provided insightful reviews [17,57]. The surfaces considered by Muthukumar and co-workers [29,30] contain a single pattern of typical size made up of charges imprinted on a neutral background (of size < ). The question they ask is: can a polymer consisting of oppositely charged segments distributed on the backbone in a particular sequence recognize (adsorb on) this region of the surface? Muthukumar [29] used a variational method to derive conditions for occurrence of such selective adsorption. Further insight on this issue has been obtained via MC simulations [30]. Recent experiments (see references quoted in [17]) concerning the binding of polyelectrolytes to proteins seem to be consistent with