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the capabilities of modern computers. A number of ... which consider a huge number of degrees of freedom ... The computer simulation of the Trp cage folding.
ISSN 0965545X, Polymer Science, Ser. A, 2011, Vol. 53, No. 9, pp. 846–866. © Pleiades Publishing, Ltd., 2011. Original Russian Text © V.V. Vasilevskaya, V.A. Ermilov, 2011, published in Vysokomolekulyarnye Soedineniya, Ser. A, 2011, Vol. 53, No. 9, pp. 1603–1626.

REVIEWS

Computer Simulation of Macromolecular Systems with Amphiphilic Monomer Units: Biomimetic Models1 V. V. Vasilevskaya and V. A. Ermilov Nesmeyanov Institute of Organoelement Compounds, Russian Academy of Sciences, ul. Vavilova 28, Moscow, 119991 Russia email: [email protected]; [email protected]

Abstract—The review presents the basic models used to analyze the selfassembly of protein macromolecules and the main results of studying the selforganization of macromolecules in terms of the concepts of amphiphilicity of an individual monomer unit. The features of the coil–globule transition of these macro molecules in solutions with different concentrations are described in terms of the statistics of the distribution of monomer units and chain rigidity. It is shown that this model is efficient for interpreting and analyzing experimental data for the study of synthetic and biological macromolecules. DOI: 10.1134/S0965545X11090148

One of the major problems of biology is the predic tion of the tertiary structure of a protein according to its primary sequence [1–4]. This problem arouses keen interest because the biological functions of pro teins are determined by the structure of their globular conformations, i.e., tertiary structure. At present, despite a large body of research in this domain, the problem remains unsolved [5–19] owing to the extreme complexity of protein systems. A protein molecule is a unique sequence of amino acid residues linked via peptide bonds. Twenty kinds of amino acid residues can participate in the formation of a protein sequence. An individual protein may contain tens to thousands of amino acid residues. The fact that, at room temperature, these complex molecules can rapidly and accurately restore their threedimensional structure after denaturation shows that the free energy of the system has a pronounced kinetically attainable global minimum that corre sponds to the energy of the globular state. This idea is called the thermodynamic hypothesis of folding. All information about protein folding is contained in the Hamiltonian of the system, which describes the intramolecular interactions of protein and its interac tion with water molecules, and in the primary sequence of protein units. Revealing the principles of construction of primary sequences and the character istics of the Hamiltonian of the system would mean that the problem of protein folding is solved. In principle, proteinfolding processes are based on the same principles of quantum mechanics that determine interactions between individual atoms. 1

This work was supported by the Russian Foundation for Basic Research (project no. 110300320a) and the Ministry of Sci ence and Education of the Russian Federation (project P334).

However, the calculation of the free energy of proteins via the methods of quantum chemistry is far beyond the capabilities of modern computers. A number of models with a lower detail level were proposed as a practical alternative to quantummechanical calcula tions of the electronic structure of proteins for descrip tion of the folding process. ATOMISTIC PROTEIN MODELS At present, the socalled atomistic protein models, which consider a huge number of degrees of freedom not including electronic degrees of freedom, have the highest detail level. This approach is based on the Born–Oppenheimer approximation, which involves averaging over the motion of electrons and analysis of only the motion of atomic nuclei and makes it possible to introduce averaged potentials of interaction between all types of atoms. However, despite this simplification, the computa tional complexity of the approach is so high that, at present, with the use of the atomistic simulation, no more than a few microseconds of the life of a protein molecule can be studied, even with the use of modern supercomputers. Moleculardynamics methods with atomistic models made it possible to successfully study only the collapse of the shortest proteins that occurs on a scale of microseconds (Trpcage miniprotein, the WW domain, small proteins from the family of major histocompatibility complex class II (MHC class II), the villinheadpiece subdomain, the Nterminal domain of ribosomal protein L9 (NTL9), and the albumin binding domain [20–30]). For example, the folding of the Trpcage minipro tein, which consists of 20 amino acid residues, occurs

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Fig. 1. Freeenergy surfaces of the Trpcage protein at different temperatures plotted in coordinates of gyration radius Rg and fraction of native contacts ρ [23]. See text for explanations. Published with permission of the US National Academy of Sciences.

on a time scale of ~4 μs [31] and is the fastest among all known proteins. There have been many works that are devoted to the description of computer simulations of the Trpcage folding via the moleculardynamics method that use models with different detail levels [20–25]. In the case of the Trpcage protein, com puter simulation made it possible to reveal the mecha nism of folding and to derive tertiary structures that are in good agreement with those experimentally deter mined via NMR. The computer simulation of the Trpcage folding described in [23] is the most accurate to date; in that study, the atomistic simulation of the Trpcage folding was performed via the molecular dynamics method with allowance for the solvent molecules. Atomic inter actions were assigned with the use of the OPLSAA force field [32, 33]. The simple pointcharge model (SPCM) was employed to simulate the solvent [34]. The simulation was performed via the molecular dynamics method based on the new efficient P3ME/RESPA algorithm [35] with intensive parallel computing. In addition, the replicaexchange method (REM) was used [36]; it is intended to increase the POLYMER SCIENCE

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studied phasespace region in the presence of local minima of the free energy of the system. Figure 1 depicts at different temperatures the free energy surfaces of the Trpcage protein folding that are plotted in coordinates of gyration radius Rg and frac tion of native contacts ρ. It is evident that the free energy surface becomes smoother as temperature increases. At low temperatures (~300 K), along with the global minimum corresponding to globular state F of the protein, a local minimum corresponding to metastable transition state I occurs. On the basis of these data, a twostage mechanism of the Trpcage protein folding was proposed. At the first stage of collapse, the metastable conformation I is formed; it contains two clusters separated by a salt bridge that links amino acid residues Asp9 and Arg6. At the second stage, the clusters merge into a single equilibrium globule; a salt bridge is formed on the sur face of the globule and further stabilizes it. The occurrence of metastable state I can explain this rapid collapse of the Trpcage protein at room temperature. In fact, the system can more easily form a regular packing of the backbone in two hydrophobic

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cores of the clusters. In addition, the potential barrier associated with the rupture of the salt bridge and the transition of the twocluster conformation into a sin gle equilibrium globule is fairly low (~4.2 kJ). Note that the temperature dependences of the free energy surface of the Trpcage are in only qualitative agreement with the experimental data. In particular, the melting point for the Trpcage globule was about 440 K, which is much higher than the experimental value (~315 K). This discrepancy can be eliminated via retuning the parameters of the force field that defines interatomic interactions. At present, the problem of forcefield parameter ization and complete investigation of the phase space of proteins to find a minimum energy conformation significantly restricts the possibilities of atomistic sim ulation, even for small fastfolding proteins. The results for other small proteins are not as successful as those in the case of the Trpcage protein. The authors of [37] performed atomistic simula tions of a longer polypeptide fragment, i.e., the so called albumin binding domain containing 47 amino acid residues. The protein consists of three helical seg ments. During folding, the helical segments are formed independently and then assemble. Owing to this structure, this protein is among the proteins with a fairly high rate of assembly; under favorable physico chemical conditions, this process occurs within a few milliseconds. In [37], the simulation of folding of the albumin binding domain was performed through the use of the AMBER FF03 force field with an implicit assignment of the solvent. It was found that protein folding in the computer experiment can proceed along different trajectories; as a consequence, different ter minal structures are formed. In the best of the 40 derived terminal structures, the rms deviation from the equilibrium conformation was about 2 Å. The 20μs atomistic simulation of the folding of the villin headpiece (HP35), i.e., the villin protein subdo main consisting of 35 amino acid residues, likewise revealed a few terminal conformations [38]. However, the sequence of conformational changes observed in atomistic simulations of the folding of this protein was found to depend on either the conformations used for the calculation of force fields or the starting confor mation [39]. In [29], the folding of the WW domain of the human Pin1 protein was studied; it is likewise charac terized by a high folding rate: Depending on the con ditions, it ranges from 8 to 13 μs. Computations were performed through the use of the CHARMM22 force field with the CMAP correction explicitly taking into account the solvent, i.e., water, which was simulated in terms of the TIP3P approach. The computations were performed for 10 μs; they were a failure in a sense: The experimentally found equilibrium (native) state was never achieved.

Note that the ultrafast folding of the above proteins (Trpcage, the albumin binding domain, the villin headpiece subdomain, and the WW domain) is the exception rather than the rule. The folding of most of the important proteins occurs on the scale of millisec onds, and large protein molecules that contain thou sands of atoms undergo folding on a time scale of a few seconds pass through a series of metastable states [3]. To date, the maximum time scale of the simulation of protein collapse was achieved in [28]. A few folding trajectories for the NTL9 protein were derived on a time scale of ~1 ms. Thus, despite the steady increase in computing power, the atomistic simulation of protein folding remains an extremely complicated task [40], the solu tion of which requires not only a significant increase in computing power but also substantial refinement of the used forcefield potentials that govern the interac tion between amino acid residues and solvent mole cules. BEAD MODELS An alternative approach to describe the behavior of proteins is the construction of simplified models. The approach is based on the assumption that the degree of freedom of individual atoms that constitute the pro tein is not very important for describing the folding processes. Protein folding occurs on much longer time scales than those that characterize atomic motions. This circumstance makes it possible to decrease the “resolution” of the model and, hence, to simulate some groups of atoms as beads and to introduce between them effective interaction potentials that are the result of the averaging of atomic interactions. However, the exact averaging of atomic interactions, similar to the averaging of electronic interactions in the Born–Oppenheimer approximation, is not the objective in constructing these coarsegrained models. Effective interaction potentials between the beads are introduced on the basis of information derived from experiments or theoretical assumptions. In particular, in a higherlevel analysis, the confor mational properties of a protein are determined by a combination of interactions, such as hydrophobic– hydrophilic, dipole–dipole, and electrostatic interac tions between its constituent amino acid residues, as well as by the processes of hydrogen bonding between solvent molecules and atoms of polar groups and by the processes of formation of intramolecular hydrogen bonds in the protein molecule and hydrogen bonds between the protein and solvent molecules. The number of types of beads used in the coarse grained model to define the primary protein structure determines the socalled simulation alphabet. To study the processes of protein folding, models with different alphabets are used; they range from 20letter models, in which each amino acid residue corresponds POLYMER SCIENCE

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to its own bead [40–43], to simplified 2letter models. (They will be described in detail below.) A proteinfolding simulation using the coarse grained model was recently reported in [26], the authors of which succeeded at quantitatively describ ing the folding mechanism for the simulated protein. Previously, this was possible only with the use of atom istic models. In [26], the computer simulation of the Trpcage protein folding was performed with the use of a sixlet ter coarsegrained model. The model Hamiltonian contained six members that corresponded to the fol lowing interaction potentials: hydrophobic interac tions, formation of the salt bridge, interactions between aromatic amino acid residues, interactions between proline and aromatic amino acid residues, and formation of hydrogen bonds between backbone beads and between backbone beads and side groups. The folding simulation via molecular dynamics resulted in a tertiary structure of the Trpcage protein that is in good agreement with the structure found in NMR experiments. The values of the rms deviation (RMSD) of the position of the macromolecular back bone from the experimental values were less than 1 Å. The simulation made it possible to assume that this fast folding of Trpcage is caused by the specific struc ture of its primary sequence. For example, the pres ence of a large amount of proline in the primary sequence restricts the region of accessible conforma tional space and increases the number of proline–aro matic contacts, which make the main contribution to the stabilization of the globular conformation in the case of the Trpcage. This specific feature of the Trp cage may be one of the causes of successful atomistic simulations of the folding of this protein. However, this circumstance means that the folding mechanisms revealed in the case of Trpcage cannot be generalized to a wider class of proteins. The methods used in sim ulation of the Trpcage will probably not give satisfac tory results in the case of other proteins. In summary of the above, it should be emphasized that coarsegrained models have an advantage over atomistic models not only with respect to reducing the computational cost. It is more important that the con struction of successful coarsegrained models makes it possible to single out the main factors that determine the mechanism of protein folding and some key inter actions from the general Hamiltonian of a real system. However, the complexity of atomistic simulation and the absence of a clear technique for the construc tion of coarsegrained models mean that the problem of predicting the tertiary structure of a protein accord ing to its primary sequence cannot be currently solved only via computer simulation. To solve this problem, a number of effective complementary experimental and theoretical methods are necessary. POLYMER SCIENCE

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ANALYSIS OF PRIMARY SEQUENCES One of these methods is the analysis of primary sequences and the comparison of the found features of the sequence with the experimentally derived tertiary structure corresponding to this sequence. The protein represented as a sequence of 20 different amino acids is studied as a text written in a 20letter alphabet and is analyzed as a “text,” for example, with respect to the frequency of occurrence of “words” [44]. In this case, either the “word” is understood as a welldefined (often tandem) sequence or it is assumed that all words in the text are of equal length. In the lastmentioned case, the sequence text is arbitrarily divided into defi nitelength sections (usually no more than five letters) and then analyzed in terms of the frequency of occur rence (incidence) of other words [44–46]. In [47–49], it was assumed that proteins can con tain words with different lengths, and proteins of dif ferent spatial groups were analyzed in terms of this assumption. It was found that the maximum length of words and the frequency of their occurrence are differ ent for different classes of proteins. The longest words are found in fibrillar proteins; the shortest, in mem brane proteins. In the analysis of protein sequences, amino acids are divided into groups, for example, into hydropho bic and hydrophilic groups [48] or into groups con taining nonpolar, polar uncharged, charged, and aro matic groups [46]. Other classifications can be used. In any case, each method converts the 20letter protein sequence into a sequence consisting of fewer letters. The analysis of these sequences reveals the presence of periodicity or other distributions of groups. The simplest model is the socalled twoletter clas sification in which all amino acid residues are divided into two groups, i.e., hydrophobic and hydrophilic. The analysis showed that proteins with different spa tial structures have features of the distribution of hydrophobic and hydrophilic units in the primary sequence [46–50]. Fibrillar proteins usually have a regular structure; the distribution of hydrophilic and hydrophobic groups in them is subject to periodicity. For example, the main motif of the primary structure in silk fibroin protein is the repetition of eight blocks, with each block exhibiting an alternation of small glutamine res idues and larger residues. Membrane proteins are composed of a fragment located in an anhydrous membrane and a portion sub merged in water. Hydrophobic and hydrophilic units in membrane proteins are arranged in blocks. The intramembrane hydrophobic portion of the protein has a regular structure (in common with fibrillar pro teins, it is supported by hydrogen bonds); however, the size of these regular units is limited to the membrane thickness. Globular (watersoluble) proteins have a less regu lar structure (especially small proteins). This structure

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is due to interactions of the protein chain with itself. In this case, interactions of hydrocarbon (hydrophobic) groups that are distant along the chain but close in space, as well as interactions of the protein chain with cofactors (small molecules, ions, sugars, nucleotides, etc.), are significant. The linguistic analysis of primary sequences of real proteins reveals a relationship between the properties of the primary sequences of proteins of different classes and the structures of their globular conformations without resorting to the con struction of a mathematical model of the protein and performing computations. The derived information can be useful for constructing and analyzing mathe matical models. TWOLETTER HYDROPHOBIC–POLAR MODEL OF AN AMPHIPHILIC COPOLYMER In 1985, Dill became the first to use a simplified hydrophobic–polar (HP) model for studying the pro teinfolding processes [51, 52]. He proposed to ana lyze a protein model in the form of a simple model of a copolymer comprising a linear chain of two types of monomer units, i.e., hydrophobic H and hydrophilic P units. H

P

H

P

H

P

The results derived with this generalized model can be attributed not only to the folding of proteins but also to the selforganization of other amphiphilic mol ecules. Most biopolymer molecules are amphiphilic mac romolecules: DNA single strands, RNA, proteins, and lipids. Examples of synthetic amphiphilic polymers are poly(1vinylimidazole) and poly(Nisopropy lacrylamide). Amphiphilic macromolecules exhibit the ability to selfassemble; that is, during a change in external con ditions that is related to deterioration in the quality of the selective solvent, they undergo conformational transitions to form dense nanostructures whose struc ture and properties depend on the primary sequence of the amphiphilic polymer. These structures can be micellelike globules, vesicles, lamellae, and fibrils [53–55]. The selforganization due to amphiphilicity is one of the most important factors that govern the proteinfolding process. All amphiphilic polymers exhibit one common property, the intramolecular seg regation of hydrophobic and hydrophilic units, which leads to formation of the characteristic globule struc ture of the core–shell type, in which hydrophobic units form aggregates protected from contacts with the solvent by the shell of hydrophilic units. In particular, an important feature of globular pro teins that makes them distinct from globules of syn thetic polymers is solubility and stability (absence of any aggregates) in solutions, even at relatively high concentrations. This feature is due to the presence of

the core–shell structure in the protein globule. The density of the shell of hydrophilic amino acid residues is sufficient to protect the hydrophobic cores of pro teins from interacting with each other [2]. In [52], using a twoletter HP model, Lau and Dill studied the full conformational spaces and full sequence spaces of short (with a length of N = 10 units) HP chains on a twodimensional lattice. The folding process was simulated as the random walk of a chain with a given primary sequence over a 2D lattice. After that, the total conformational energy resulting from the random walk was calculated. The hydropho bic effect was simulated via assignment of a negative weight to the interactions of nonbonded H units. It was assumed that the globular state that corresponds to the primary sequence of H and P units is the mini mum energy conformation. The full conformational spaces and full sequence spaces of chains with a length of N = 10 were studied. This chain can assume 2034 different conformations on the 2D lattice. The num ber of possible twoletter primary sequences of the chain with a length of N = 10 is 210 = 1024. The study of all the above states of the system and the subsequent analysis of the data on the phase space of these chains made it possible to theoretically confirm the concep tual issues of protein physics that existed in the form of hypotheses. The ability of a chain to fold into a dense globule with a hydrophobic core depends not only on the per centages of H and P units but also on their relative positions along the chain (on the primary sequence of the chain). Primary sequences can be “bad” or “good” in terms of the value of energy consumption for forma tion of a dense globule with a hydrophobic core. Most primary sequences exhibit a large number of globular states that greatly differ from each other. However, there are primary sequences that exhibit only a few possible globular states or even only a single globular state. It is these sequences that occur in pro teins in the sense that their tertiary structure is com pletely determined by the primary sequence. The study of HP copolymers can provide much more information about the folding of real proteins than might be expected from this minimalist model. This fact led to formation of a direction of computer science that is actively being developed today: the sim ulation of “lattice proteins.” Some studies in terms of this approach were published in [56, 57]. Algorithms were developed to find the minimum energy conformations of long HP chains of a given sequence (the socalled native state, by analogy with proteins) with a length up to 100 monomer units. In addition, HP copolymers are used to study the secondary structure of proteins. Although HP copoly mers do not have the secondary structure in the con ventional sense, with the use of the socalled theorem of corresponding states, the spatial structure of an HP POLYMER SCIENCE

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chain with the primary sequence corresponding to a certain real protein can be unambiguously extrapo lated to the structure of the real protein. Thus, to understand the proteinfolding process, it is necessary to answer the following questions: How can we derive a “good” sequence that corresponds to a single nondegenerate globular state with minimum energy? How could these sequences be formed during the evolutionary process? Progress in the understand ing of these issues can be achieved through the use of the simplest HP models combined with the approach first described in [58–59]. CONFORMATIONDEPENDENT SYNTHESIS OF PROTEINLIKE MACROMOLECULES The conformationdependent synthesis of macro molecules is a method for the preparation of a het eropolymer in which a required conformation is given

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first and then units are modified, depending on their location in the structure (a review of data derived via this approach is given in [60]). In pioneering studies [58, 59], the conformation dependent synthesis was used to find a primary sequence that maximally contributes to the formation of a watersoluble globule. Unlike the above studies of HP copolymers, this approach was not aimed at inves tigation of the folding of macromolecules with definite primary sequences of H and P units in order to find sequences that, similar to proteins, can form water soluble globules; it was focused on the formation of these primary sequences. Amphiphilic HP copoly mers with primary sequences derived through this method are called proteinlike macromolecules. The procedure for the computeraided preparation of proteinlike copolymer HP macromolecules consists of a few steps.

(iii)

(i) A dense globule is formed from a homopolymer coil. (ii) The “coloring” procedure is performed. All units in the globule are divided into two types: hydro phobic H units, which lie in the core of the globule and have the least contacts with the solvent, and hydro philic P units, which are located on the surface of the globule and have the most contacts with the solvent. (iii) The resulting primary sequence is fixed, and the formed HP copolymers are studied. In terms of distribution statistics for H and P units, the sequences obtained via the conformationdepen dent design are not random. The analysis of the pri mary structure conducted in [58] showed that average block length L differs from the average block length of a random sequence and depends on the length of the macromolecule. A detailed analysis of the statistics of distribution of units in a proteinlike sequence was performed in [61]. Via construction of an analytical theory and computer simulations, it was shown that proteinlike sequences are characterized by the presence of largescale corre lations in the distribution of H and P units. It is this property of a sequence that determines the possibility of forming a soluble globule with the core–shell struc ture. The statistics of the distribution of units in protein like, block–random, and random sequences was ana POLYMER SCIENCE

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lyzed. To find correlations in the arrangement of units, a method developed by Stanley et al. was used [62, 63]. The method is as follows. The sequence of an HP copolymer is transformed into sequence ui of values +1 and –1, so that ui = 1 for a P unit or –1 for an H unit. Sums of the elements of sequence γk(L) located in a “window” with length L, which moves along the entire sequence with a step of one monomer unit, are calculated: k +L

γ k (L) =

∑u , i

i =k

where k is the number of sequence unit that defines the beginning of the calculation window. The data on the presence of correlations in the sequence can be derived through calculation of the dispersion of γk(L): k+L 2

DL =

∑ ( 〈u u 〉 – 〈u 〉 〈u 〉) i j

i

j

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i, j = k

The resulting sequence ui consisting of +1 and –1 can be regarded as a onedimensional random walk. By analogy with Brownian motion, if the sequence has a random statistics, the dispersion is proportional to the average displacement during a random walk, DL2(L) ~ L. If the order of units in the primary

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log10(DL) 1.6

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0

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sequence is not random and their arrangement exhib its largescale correlations, then estimate DL2(L) ~ LS , S > 1 is valid for the dispersion. Figure 2 depicts characteristic dependences DL2(L) for three different types of distributions of units: ran dom, block–random, and proteinlike copolymers (the content of dissimilar units is 1 : 1, the total length of the macromolecule is N = 1024, and the average block lengths of the block–random and proteinlike copoly mers are the same) [61]. It is evident that DL2(L) for random and block–random sequences has asymptot ics L, which corresponds to the random statistics of distribution of units, whereas DL2(L) for proteinlike sequences does not follow this dependence, even at a substantial value of length L. This fact confirms the presence of largescale correlations in proteinlike sequences. In addition, in [61], the theory of proteinlike mac romolecules based on the description of a macromol ecule as a random walk in a volume bounded by two spheres was proposed. The units falling within the inner sphere were considered hydrophobic; those fall ing within the outer sphere, hydrophilic. This model made it possible to calculate analytical dependences of the distribution of H and P units over blocks. It was shown that proteinlike macromolecules have a specific statistics of distribution of H and P units: The proba bility density functions of the length of blocks of H or P units are described by power functions (individual for each block) and depend on the total length of the macromolecule. This probabilistic process of a ran dom walk in which the length of each step x (in this case, block length K) is defined by a probability density −µ function that varies as power function f (x) ∝ x , (1 < µ < 3) was described by the French mathematician Paul Lévy and is called the Lévy flights.

It was shown [61] that, in the case of proteinlike macromolecules, the distribution functions f(K) for both hydrophobic H and hydrophilic P decrease with an increase in the block length according to the power function f(K) ~ K–3/2 for relatively low values K < Kcr. For high values of block lengths K > Kcr, this depen dence is replaced with an exponential dependence; in addition, the value of Kcr is determined by the length of the macromolecule and structural details of its globule [61]. Correlations in the arrangement of H or P units rapidly decrease with an increase in the distance between base pairs, and the information function becomes equal to that of the random sequence. There fore, it is assumed that the pattern of the distribution of units in these macromolecules is quasirandom [64]. The statistics of the primary sequence has a signifi cant effect on the properties of the coil–globule tran sition and the structure of the globular conformation. In [59], the hydrophobically induced (caused by the attraction of H groups) collapse of HP copolymers with different primary sequences was studied via the molecular dynamics method. It was shown that the coil–globule transition in proteinlike copolymers occurs at higher temperatures and leads to the forma tion of denser globules than those in random and block–random copolymers with an average block length equal to that of a proteinlike copolymer. Below, we show snapshots of copolymer globules (a degree of polymerization of N = 512) with the protein like,

,

Random,

and block–random

statistics of the distribution of units [59]. It is evident that the globule of a proteinlike copolymer differs from the globules of random and block–random mac POLYMER SCIENCE

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romolecules: We can see one dense hydrophobic core and a shell of hydrophilic loops. A theoretical study of the stability of globules of HP copolymers in the presence of a small number of hydrophilic units on the surfaces of the globules was presented in [65], where the selfconsistent field approximation was used to study the globular state of proteinlike, regular, and random copolymers as well as of HP–H copolymers comprising one long and many short hydrophobic blocks. It was shown that the free energy for HP–H copolymers is the lowest among all types of distributions of units along the chain because the long hydrophobic block has an almost unperturbed set of conformations in the bulk of the globule, and an HP–H macromolecule proves to be the most stable. The structure of the globules of HP–H copolymers and their aggregates was theoretically studied in [66]. It was found that, for a regular distribution of a small number of P units along the chain, their effect on the conformation of the macromolecule was negligible. In the case of a definite sequence of a small number of P units (a long H block and a few short H blocks alter nating with P units), aggregates of these copolymers can have the shape of a disk, cylinder, torus, sphere, or star. For multiblock HP copolymers, structural transi tions are accompanied by changes in the surface area of the globule and the appearance of variform branches from the main globule [67]. Proteinlike macromolecules can be experimentally prepared via modification of the units of the backbone or via crosslinking of hydrophilic chains to the hydro phobic backbone. In the above theoretical studies, this “coloring” procedure was performed instantaneously [58, 59]. In [68–70], the synthesis of proteinlike mac romolecules in which the coloring procedure is simu lated as a chemical reaction proceeding in time was studied with the use of Monte Carlo computer simula tion. The authors of [68] showed that the statistics of units of the resulting sequence heavily depends on the conformational state of the molecule during the color ing reaction. According to expectations, during color ing of a molecule in the loosenedcoil state, the statis tics of the distribution of units is random. If the color ing reaction is conducted for a molecule in the globular state, then the sequence has the properties of selfsimilarity, a circumstance that is indicative of the presence of largescale correlations in it. In this case, proteinlike sequences similar to those discussed in [58, 59] occur. In [69], the conformationdependent synthesis of an HP copolymer was studied via discrete molecular dynamics. It was shown that the properties of the pri mary sequence of an HP copolymer derived via the coloring reaction heavily depends on system parame ters, i.e., temperature, solubility of H and P units, and the quantitative relationship of H and P units. With variation in the ratio between the number of H and P units and their affinity for the solvent, it is possible to POLYMER SCIENCE

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control the blockiness of the resulting sequence and to form a wide range of sequences from random to block–random. With improvement of the solubility of hydrophobic H units or an increase in the number of hydrophilic P monomers in the solvent, the average length of the block in the sequence of the resulting HP copolymer decreased and the statistics of the distribu tion of units approached a random pattern. Con versely, worsening of the solubility of hydrophilic P units led to formation of a sequence with the random– block structure. Experimental data on the preparation of protein like molecules are presented in [71–77]. In [71–73], the statistics of the distribution of hydrophilic groups was varied via a change in the conformational state of the precursor of the backbone. Thus, the authors of [71, 72] synthesized macromolecules of poly(Niso propylacrylamide) with poly(ethylene glycol) pen dants. The pendants were grafted onto the macromol ecule of poly(Nisopropylacrylamide) at different temperatures (15 and 29°С), i.e., at a distance and in the immediate vicinity of the critical separation point of poly(Nisopropylacrylamide). The properties of the resulting molecules greatly differ: In the first case, the backbone is in the coiled state and the grafting points of pendants are distributed randomly; in the second case, a conformation appears from the hydrophobic globule surrounded by a shell of hydrophilic pendants. The result is that the polymers modified at 29°С exhibit a higher stability in terms of solubility. In recent paper [74], the precursor for obtaining a pro teinlike copolymer was polystyrene whose units were modified via bromination of the phenyl ring in differ ent organic solvents and at different temperatures. Another method for the experimental derivation of proteinlike copolymers is the conformationdepen dent copolymerization of hydrophobic H and hydro philic P units in solution; it was formulated and first used by Lozinsky et al. in [75, 76]. The synthesis was successfully performed owing to the use of vinylcapro lactam (VC) as a hydrophobic monomer. During homopolymerization, VC yields polyvinylcaprolac tam, a thermosensitive polymer whose solubility in water decreases during heating. Vinylimidazole (VI) was selected as a hydrophilic monomer. The lower critical solution temperature (LCST) of polyvinylcaprolactam is 34°C. Above this tempera ture, hydrophobic intramolecular interactions begin to dominate over polymer–water interactions. The result is that polyvinylcaprolactam precipitates from concentrated solutions and forms dense globules in dilute solutions. During synthesis above the LCST, i.e., at 65°C, VC exhibited a behavior similar to that of a strongly hydrophobic monomer. As a consequence, shortly after the beginning of copolymerization, mac roradicals formed proteinlike globules that presum ably comprised a hydrophobic core of VC units and a polar shell of VI fragments.

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Subsequently, Lozinsky studied the conformation dependent copolymerization for another pair, i.e., iso propylacrylamide–VI [77]. The authors of studies [75, 76] and computeraided studies [78, 79] proposed the following process mech anism that explains the formation of proteinlike copolymers. Short macroradicals formed at the initial stage of synthesis grow according to the laws of statis tical solution copolymerization. However, after achieving the critical chain length, they collapse into monomolecular micelles owing to the hydrophobic effect on the monomer units of VC. Further copoly merization continues in these micelles, the hydropho bic cores of which sorb VC monomers from solution. The active site of the growing chain can be located in the globule core, where the site, with a high probabil ity, reacts with hydrophobic monomers, or it can emerge on the globule surface, thereby providing for the addition of two monomers from solution. Thus, the hydrophobic core and the polar shell of the micelle can grow simultaneously. The computer simulation of this process was pre sented in [78, 79], in which the Monte Carlo and moleculardynamics techniques were used to analyze the irreversible radical copolymerization of hydropho bic H and hydrophilic P units that occur in a poor sol vent for H units. The polymerization was simulated as the reaction of successive addition of H and P units to a growing macroradical. The probabilities of addition of H and P units to the chain depended on their aver age concentrations for reaction time τR in reaction volume Vτ near the active chain end. The calculations showed that conformation dependent copolymerization results in the formation of spherical globules with a pronounced core–shell structure and that the peaks of radial distributions of H units and P units are located at the center and on the outer edge of the globule, respectively. It was shown that the composition of the resulting HP copolymer depends on the ratio of the initial con centrations of H and P units in solution and on the degree of hydrophobicity of H units. For long reaction times τR (in the socalled kinetic control mode), the probability of addition of one type is determined by the conformation of a macroradical on the whole and its primary structure. This cooperat ivity results in the formation of a sequence with large scale correlations in the distribution of H and P units that is described by the Lévy statistics and is similar to sequences derived via the coloring methods. The authors of [80, 81], using the HPcopolymer model and the continuummoleculardynamics tech nique, proposed and implemented another method to obtain proteinlike primary sequences, i.e., the confor mationdependent evolution of the primary sequence of copolymers. This approach can be regarded not only as an attempt to simulate some features of the evolution of real protein sequences but also as a tech

nique for the multistage synthesis of proteinlike amphiphilic copolymers. The conformationdepen dentevolution method implies the multiple use of the coloring technique on one macromolecule in succes sive iterations. Every iteration step involves macro molecule refolding due to a strong attraction of H units and the performing a new coloring procedure for the resulting globule. Each “recoloring” leads to cer tain changes in the sequence of H and P units of the copolymer, thereby making it possible to regard this iterative process as an evolution of the primary sequence. The amount of information in these pri mary sequences was measured with the use of the so called Jensen–Shannon divergence [82, 84]. It was shown that, at different parameters of interaction between H and P units, there can be either ascending branches or descending branches of evolution, where the amount of information in the sequences either increases or decreases, respectively. The calculations showed that the type of branch of evolution imple mented depends on the force of attraction between P units. The descending branch is implemented for weak attraction between hydrophilic units (the energy of attraction of hydrophilic units is ε PP ≤ 0.3kT ); in addition, owing to the abovedescribed iterative pro cess, the proteinlike HP sequence degenerates into a trivial diblock or triblock sequence of H and P units. In the case of a fairly strong attraction of P units (ε PP ≥ 0.3kT ), the ascending branch was imple mented. The evolutionary process in this case led not only to the preservation of the proteinlike structure of sequences but also to an increase in largescale corre lations in the arrangement of H and P units. The highest values of the Jensen–Shannon diver gence, which correspond to the greatest amount of information in the primary sequence, were found for the proteinlike copolymers derived via the conforma tiondependent evolution at an energy of attraction of hydrophilic units of ε PP = 0.3kT . The resulting pro teinlike globules had a denser shell of hydrophilic units than the globules prepared via the conventional color ing technique [58, 59]. Experimentally synthesized proteinlike globules exhibit high stability and do not aggregate with each other, even in highly concentrated solutions. However, it was found that proteinlike HP globules prepared via computer experiments are not fully protected from aggregation [58–59], although their aggregating abil ity is significantly lower than that of polymers with random and random–block sequences of units. This circumstance is due to the fact that long hydrophilic loops covering the core are sparse and prone to strong fluctuations; as a consequence, the hydrophilic shield does not provide for thermodynamic stability of the globules (see above). This situation means that, in terms of the simplest HP model in the computer experiment, it is impossi ble to obtain a globule similar to globules of real pro POLYMER SCIENCE

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(а)

(b)

(c)

(d)

(e)

(f)

855

Fig. 3. Snapshots of conformations.

teins, i.e., with a dense hydrophobic core and a dense hydrophilic shell. This problem was solved via intro duction of a new model of the monomer unit.

P

MATHEMATICAL SIMULATION OF MACROMOLECULES THAT ARE AMPHIPHILIC AT THE LEVEL OF THE MONOMER UNIT In [85], Khokhlov et al. noticed that hydrophilic units often contain not only hydrophilic groups but also hydrophobic groups. In this case, we can state that a macromolecule is amphiphilic at the level of an indi vidual unit. These macromolecules, which are amphiphilic at the level of the individual monomer unit, include many watersoluble polymers: for exam ple, synthetic macromolecules, such as poly(1 vinylimidazole), poly(Nisopropylacrylamide), and poly(2ethylacrylic acid), as well as biological macro molecules, i.e., single strands of DNA macromole cules and some amino acids. Because units of amphiphilic macromolecules contain both hydrophobic and hydrophilic groups, it may be expected that, if they are placed in a mixture of incompatible polar and nonpolar solvents, the amphiphilic macromolecules and lowmolecular mass precursors of their units will concentrate at the polar/nonpolar solvent interface. This concept of the structure of monomer units made it possible to introduce a new twodimensional classification of synthetic macromolecules and amino acids [85–87] that is based on two parameters: surface activity and the hydrophobicity parameter. A model of an amphiphilic unit was proposed in [88]. According to this model, amphiphilic monomer unit A, which contains hydrophobic H and hydro philic P groups, is represented as a dumbbell consist ing of hydrophobic H and hydrophilic P beads linked through a fixedlength bond. POLYMER SCIENCE

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Hydrophobic beads form the backbone of the chain, and hydrophilic beads constitute side groups attached to the backbone.

In [88], the coil–globule transition of an amphiphilic homopolymer macromolecule composed of amphiphilic monomer units of the A type was stud ied. Figure 3 shows snapshots of the conformations of these macromolecules. The quality of the solvent in the figure deteriorates in alphabetical order. In a good solvent, this macromolecule occurs in the coiled state (Fig. 3a). As the quality of the solvent deteriorates, the conformation of the macromolecule undergoes several stages. Hydrophobic units begin to form clusters and yield the socalled necklacelike conformation (Figs. 3c, 3d). With further deterioration in the quality of the solvent, the clusters aggregate to form a single cylindrical globule (Figs. 3e, 3f). These morphologies of the chain result from the intramolecular segregation of the chemically different H and P groups; the distri bution of units within the structure apparently tends to reduce the contact area between the hydrophilic and hydrophobic groups. Because it is impossible to com pletely separate these groups, owing to the presence of a tight chemical bond between some of them, the shape of the globule changes; it transforms from

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free energy that is insignificant relative to the contri bution to the aggregation term. This circumstance makes the intrachain micelle formation in polyam phiphilic molecules advantageous at the initial stage of collapse, when the chain is in the loosenedcoil state.

spherical to cylindrical because the surface of the glob ule increases. It was shown that the size of the globule, Rg, as a function of solvent quality can exhibit behavior that is not quite usual behavior; i.e., it can increase with deterioration in the solvent quality for hydropho bic parts. The sizes of cylindrical globules linearly depend on the degree of polymerization of the chain. Note that the physical picture of the formation of a necklacelike conformation at the first stage of collapse is largely similar to the process of micelle formation in a dilute surfactant solution. The structure of intrac hain micelles constituting necklacelike conformations is similar to that of micelles of a conventional low molecularmass surfactant. However, in the case of a lowmolecularmass surfactant, micelle formation occurs only after the solution has achieved a certain concentration at which the translational entropy of surfactant molecules becomes too small to prevent the aggregation. In the case of an amphiphilic macromol ecule, the formation of intramolecular micelles does not require any critical concentration of monomers, because, owing to the binding to the chain, their trans lational entropy is low and makes a contribution to the

H

H H

H

In addition, the preparation of the hydrophobic– amphiphilic proteinlike copolymer was performed in a few steps.

H

P P H H H H H H H H P P H H H H H H H H H H H P H P H H P P P P P

H

H H H H H H H H H H H H

(i)

The proposed algorithm for designing the protein like HA polymer was similar to the above method of surface coloring; however, it was slightly different.

P

P

H

H

H

H

In [89], semidilute solutions of HA macromole cules composed of amphiphilic monomer A units and hydrophobic H units in a poor solvent for H beads were studied. Three types of macromolecules were examined: regular copolymers with a repeating block consisting of amphiphilic A and hydrophobic H units ([HA]n), a regular multiblock copolymer in which the repeating element comprised a few H and A units ([HLAL]n at L = 3), and polymers with a specially pre pared proteinlike sequence of H and A units.

P

H

H P

(ii) In the second step, a certain amount of hydro philic monomer units, Np > Nh/2, was added into the cell. These units were further adsorbed on the surface of the hydrophobic core. (iii) In the third step, the attraction between hydro philic and hydrophobic monomer units was “turned on”; the hydrophilic units approached the hydropho bic core. When the distance between them is less than the bond length, a chemical bond is formed and fixed; thus, an amphiphilic unit is formed. Each hydrophilic monomer unit could form a bond only with one

H

H

H

H P

H

H

H

H

P H

H H H H H H H H H P

(ii)

(i) First, a single homopolymer chain composed of hydrophobic monomer units was placed in a cubic cell with periodic boundary conditions and dimensions greater than the radius of the hydrophobic coil. A decrease in temperature was used to make the hydrophobic macromolecule fold into a dense glob ule; the resulting globule was allowed to relax for some time.

P

P

P

H H

P

P

P

(iii)

hydrophobic monomer units and vice versa. This con tinued up until the number of adsorbed monomers was equal to one half of the monomer units of the hydro phobic core. After that, the attraction between hydro phobic and hydrophilic units was “turned off”; the free polar units remaining in the cell were removed, and the primary structure of the macromolecule was fixed. As was shown in [89], the average block size and the statistics of the distribution of units of the sequence derived in accordance with this procedure are identi cal to those of macromolecules synthesized via the coloring of units and with the use of the analytical dependences theoretically calculated in [60]. This outcome means that the distribution of monomer units in this macromolecule is in fact subject to the Lévy flight statistics. The conformational properties of these macromol ecules were studied; it was shown that they greatly depend on the statistics of the distribution of H and A POLYMER SCIENCE

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units. Below, there are snapshots of globules of copol ymer HA macromolecules with regular

857

〈M〉 3 1 2

2

1

and proteinlike

0.25

0.5

1

2

4

~ χ

Fig. 4. Average aggregation number M as a function of solvent quality χ for HA macromolecules with (1) regular and (2) proteinlike statistics of the unit distribution [89].

statistics of the distribution of monomer units. The degree of polymerization of the macromolecules is N = 128; the fraction of amphiphilic units in either chain is f = 0.5. It is evident that regular copolymers form cylindrical globules; proteinlike copolymers yield globules with a spherical shape. However, in either case, a dense globular core is surrounded by a dense hydrophilic fringe. It may be expected that these globules, which have a dense hydrophobic core and a dense hydrophilic shell, are stable and do not aggregate. To verify this assumption, semidilute solutions of globules of mac romolecules composed of amphiphilic A and hydro phobic H monomer units were studied in [89]. Figure 4 depicts the dependences of average aggre gation number M on parameter χ that characterizes the quality of the solvent for a semidilute solution of regular [HA]n and proteinlike macromolecules. Aggregation number M was defined as the fraction of hydrophobic groups of the chain that constitute the aggregate. Thus, if M is lower than 1, each chain −1

contains, on average, M hydrophobic clusters. If M ~ 1, then all hydrophobic groups are assembled into a single aggregate; however, the intermolecular aggregation is absent. Finally, at M greater than 1, the system undergoes intermolecular aggregation, and the value of M determines the average number of chains in one intermolecular aggregate. It is evident from the figure that M is low in a good solvent because only a small number of H groups associate with each other; then the value of M increases: It monotonically grows for the polymer with a regular POLYMER SCIENCE

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distribution of units and exhibits a plateau at M = 1 for proteinlike macromolecules. This result means that proteinlike globules do not associate, even in a very poor solvent. However, the analysis showed that regular HA copolymers do not merge with each other either; they form threadlike aggregates of a few cylin drical globules linked through their edges, on which the hydrophilic shell is insufficiently dense. Conformational transitions of amphiphilic comb macromolecules with long (more than one monomer unit) side chains with an increase in attraction between hydrophobic groups of the backbone were studied via computer simulation in [90–92]. Macro molecules with a regular crosslinking of side chains were examined in [91, 92]; macromolecules with both regular and proteinlike statistics of the distribution of crosslinking points of side chains were studied in [90]. The calculation results qualitatively agree with those described above: The coil–globule transition proceeds through the stage of formation of the necklacelike conformation, and, at a fixed length of pendants, the shape of the globule depends on the degree of poly merization of the backbone, the solvent quality, and the statistics of the distribution of crosslinking points of side chains. An increase in the degree of polymer ization of side chains leads to a shift in the coil–glob ule transition to the region of a poorer solvent and to a change in the shape of the globule: The higher the degree of polymerization of side chains, the more elongated the resulting globule. It was shown [90] that coil–globule transition temperature T tr decreases as degree of polymerization m increases and, regardless of the statistics of the distribution of crosslinking points, is approximated as Ttr ~ m −0.16±0.02. This scaling estimate was close to the theoretical estimate calcu lated for regular comb macromolecules in [93], according to which, Ttr ~ m −3/21 ~ m −0.14.

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In [94, 95], globular structures of polymers similar to those derived via computer simulation were experi

n

x

n

n

O

O

mentally observed. The chemical formulas of macro molecules studied in [94, 95] are presented below.

OH

N O

OH

O

O O

O x

x

OH

O

In the first case, the side group of the monomer unit contains hydrophilic and hydrophobic parts sepa rated by a rigid benzene ring that severely restricts their mutual orientation. In the second case, hydrophobic and hydrophilic parts are separated by a flexible frag ment of the alkyl chain. Studies showed that, regard less of the structural details of the amphiphilic mono mer unit, these macromolecules form globules with a micellar core–shell structure in a selective solvent. In the third case, the side group of the polymer has two hydrophilic parts separated by a flexible hydrophobic alkyl chain (CH2)m, whose length m varied: m = 5–15. These macromolecules were formed in a vesicle solu tion which likewise had a core–shell structure and a cavity inside. In addition, macromolecules comprising a neutral backbone and a side group containing hydrophilic and hydrophobic parts separated by a benzene ring were studied in [95]. It was found that this polymer forms micelles in water; in the case of the collapse of the polymer in a nonpolar solvent, it forms inverse micelles in which the core of polar units is protected by the shell of hydrophobic units. In [96], the theory of the structure of globules of flexible amphiphilic macromolecules was developed with allowance for the fact that the hydrophilic groups of amphiphilic units tend, on the one hand, toward maximum exposure to the solvent and, on the other hand, to minimum contact with each other and a par ticular orientation. These circumstances lead to changes in the nature of the surface tension and in the shape of the globule. The state diagrams constructed in [96] included regions of spherical globules, disk shaped and toroidal structures, and necklacelike con formations. Subsequently, the statistical theory describing the formation of aggregates in dilute solutions of amphiphilic polymers with amphiphilic monomer units was developed [97]. Depending on parameter γ, which characterizes surface tension, and parameter k, which defines the curvature of the surface of the poly mer phase, phase diagrams were constructed for solu tions of amphiphilic polymers. As a result, it was shown that, at different values of the system parame ters, a large number of different characteristic confor mations of macromolecules predicted in computer simulations occur in solutions of amphiphilic

OH

N

homopolymers. According to expectations, the main region in the phase diagram is occupied by conforma tions of the “beads on a necklace” type, which are characteristic of amphiphilic polymers. In addition, cylindrical globular conformations and gyroid aggre gates can be formed in the system; in the case of nega tive values of γ, lamellar aggregates may be observed. However, models taking into account only the amphiphilicity of monomer units cannot describe the diversity of conformations of real amphiphilic poly mers. Conformations of real macromolecules are sig nificantly affected by chain rigidity and fixed valence angles. The dependence of properties of amphiphilic mac romolecules on chain rigidity was studied in [98–101]. It was shown that the coil–globule transition of amphiphilic rigidchain macromolecules is a first order phase transition; it occurs in an extremely nar row temperature range and usually ends with the for mation of cylindrical globules. In cylindrical globules, the chain is structured in blobs; that is, the chain sequentially fills a cylindrical globule, moving from one section of the cylindrical globule to another. Blobs of flexible circuits are spher ical. As the chain rigidity increases, the blobs elongate, their number decreases, and at a high rigidity, different parts of the chain inside the blob undergo a collagen like ordering; that is, different fragments of the chain become entangled. Collagenlike globules (globules comprising a single blob with tangled chains) coexist with toroidal structures in which different fragments of the macromolecule likewise entangle. The coil–globule transition of amphiphilic macro molecules can pass through the stage of formation of necklacelike conformations with beads, the shape of which changes as the rigidity of the macromolecule increases. In the case of flexible chains, beads are spherical (Fig. 5a); more rigid macromolecules form collagenlike beads (Fig. 5b). Figure 6 shows the state diagram for a chain of N = 256 units in the Kuhn segment Lk versus solvent qual ity variables, which are characterized by the energy of attraction of hydrophobic groups, εHH. The higher |εHH|, the worse the solvent. For a small chain length (N < 64) in the poor sol vent region, both flexible and rigid amphiphilic mac romolecules form spherical globules. In this case, the POLYMER SCIENCE

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(b)

Fig. 5. Snapshots of necklacelike conformations of amphiphilic macromolecules for Kuhn segment lengths of Lk = (a) 2.9 and (b) 12.9. The unit length of the Kuhn segment is the diameter of a monomer bead [98].

coil–globule transition of rigid chains undergoes three successive stages: coil–diskshaped globule–spherical globule. In the case of relatively low Kuhn segment values, the macromolecule passes through the stage of formation of a necklace of spherical micelles in addi tion to the above stages in the coil–globule transition. The studies conducted in [101] showed that the properties of concentrated solutions of amphiphilic macromolecules depend on chain rigidity and the method of solution preparation. During the concen tration of a dilute solution of the preformed globules of amphiphilic macromolecules, globules of both flexible and rigid macromolecules do not aggregate, even at concentrations comparable to globuleoverlap con centrations. Longterm calculations showed that there is no merging of individual macromolecules and that there is no aggregation of globules through edge units into long intermolecular aggregates that is observed in the case of the alternating HA copolymer (Fig. 4). The cause of this behavior is the dense shell of hydrophilic units over the entire globule surface. Even during direct contact of globules, this shell hinders the inter action of hydrophobic groups. The snapshots of the system taken during deterio ration in the quality of the solvent in the concentrated solution of polymer coils are shown in Fig. 7. It is evi dent that, in this case, relatively flexible amphiphilic macromolecules likewise form a solution of individual nonaggregating globules (Fig. 7a). In addition, the structure of the globules is almost identical to that of the globules formed in dilute solutions. During deteri oration in the quality of the solvent of the concen trated solution of coils, more rigid amphiphilic mac romolecules associate into bundles of a few entangled macromolecules that buckle on a scale on the order of the cell size (Fig. 7b). The number of chains in a bun dle depends on chain rigidity and solvent quality and can vary along the bundle. In [102], which was about the analysis of the pro cesses of structure formation in rigidchain polyelec trolytes, elongated cylindrical intermolecular aggre gates similar to the abovedescribed macromolecular POLYMER SCIENCE

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bundles were experimentally revealed. It was shown that, in aqueous solutions of dodecylsubstituted poly(pphenylene) sulfonate (PPS), cylindrical micel lar aggregates with a constant diameter and a length that depends on the degree of polymerization of PPS chains are formed. As the PPS concentration increases, these micelles form higher order structures, i.e., ellipsoidal clusters with internal lyotropic order, a nematic phase, and a hexagonal packing of cylindrical micelles. In the resulting cylindrical micelles, rigid back bones of PPS chains are oriented along the axis of the cylinder, the hydrophilic ionic groups are exposed to the solvent region, and the hydrophobic aliphatic groups fill the inner space of the cylinder. A review of other studies on this subject is given in [103]. It was Lk 35 4 25 1 15

5 1.5

2.5

3.5

2

3''

2'

3' 4.5

5.5

6.5 −εHH

Fig. 6. Phase diagram of an amphiphilic macromolecule with a degree of polymerization of N = 256: (1) coiled state, (2) necklace conformation with (2') spherical and (2'') collagenlike micelle beads, (3) cylindrical globules with (3') unordered and (3'') collagenlike blobs, and (4) region of coexistence of collagenlike and toroidal structures [99]. Published with permission of IAPC Nauka/Interperiodica.

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VASILEVSKAYA, ERMILOV (а)

(b)

Fig. 7. Selforganization of rigid amphiphilic macromolecules in concentrated solutions in cases of (a) low rigidity (Kuhn seg ment Lk = 2.9) and (b) high rigidity (Lk = 29.2). The unit length of the Kuhn segment is the diameter of a monomer bead [101].

shown that other modified poly(pphenylenes) also form threadlike clusters. The aggregation number of these clusters in diameter varies from two to ten, and the total length of the aggregate is a few times higher than the length of one chain. Properties of the more flexible mpoly(phenylene ethynylenes) (PPEs) with and without alkyl side groups were studied in [104, 105]. It was shown that PPE with these groups in a poor solvent for the phe nylene backbone, 90% H2O–DMSO, forms individ ual globules that remain soluble in even a fairly con centrated solution (5000 µg/ml) for a few months, a circumstance that is due to the presence of NH3 groups grafted onto phenylene rings. PPE globules with alkyl side groups aggregate at much lower poly mer concentrations (50 µg/ml), form extended linear aggregates, and precipitate after a few days. Note that molecules with a phenylene backbone are among the most rigid molecules. Thus, the Kuhn segment is ~115 Å for sulfonated PPS, 96 Å for PPE with grafted alkyl groups, and ~88 Å for PPE without alkyl groups, values that significantly exceed the value of the Kuhn segment (~20 Å) for typical flexible chains [103]. The selforganization of a few chains in fibrillar clusters is often observed in nature, for example, the DNA double helix, various fibrillar proteins, and some polysaccharides [1–3]. The aggregates formed from macromolecules of modified sulfonated poly(pphenylene) were studied via computer simulation in a series of papers [106– 109]; there, electrostatic interactions between charged groups and counterions were explicitly taken into account via calculation of the lifetime of specially pro duced micelles in which hydrophobic units formed the

middle, while charged units, which provided solubil ity, were brought to the surface. The dependences of the thermodynamic stability of these micelles on the aggregation number, hydrophobic interactions, and the valence of counterions were studied as well. In [109], macromolecules of rigidchain poly(γ benzylLglutamate) with grafted side chains of PEG in mixtures of tetramethylfuran and water were exam ined. At a certain composition of the solvent that con tains an amount of water sufficient for the swelling of PEG, these macromolecules likewise form fibrillar aggregates in which helix bundles of a few chains asso ciate into long threadlike entities. Studies based on computer simulations of rigid amphiphilic macromolecules were continued in [110], where, in addition to the potential that defines bend ing angle θ0 of a macromolecule, internal rotation angle ϕ0 was fixed (Fig. 8a). Angles θ0 and ϕ0 corre sponding to the minimum potential were selected so that the chain was locally ordered into a secondary helical structure; the rigidity of this fixation was defined through rigidity parameter εst. The molecular dynamics method was used to study the coil–globule conformational transition of macromolecules with a secondary local helical structure as a function of bend ing angle θ0, internal rotation angle ϕ0, and fixation rigidity εst. As above, the coil–globule transition is due to the sequential deterioration in the quality of the sol vent for hydrophobic groups. Open helices (θ0 = 29°, ϕ0 = 154°), with large spaces between the helical turns, and packed helices (θ0 = 57°, ϕ0 = 166°), in which neighboring turns directly adjoin each other, were studied (Figs. 8c, 8e). In the case of open helices, the coil–globule tran sition occurs abruptly in a narrow range of changes in POLYMER SCIENCE

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(a) (c)

(d)

(b)

Fig. 8. Model of a macromolecule with fixed bending and rotation angles: (a) θ is the bending angle, and ϕ is the internal rotation angle; (b) amphiphilic macromolecule; (c, d) backbone of the amphiphilic macromolecule at (c) θ0 = 29° and ϕ0 = 154° and (d) θ0 = 57° and ϕ0 = 166° [110].

〈Rg2 〉 180

〈Rg2 〉 (a) 1 2 3 4 5

300

200

120

60

100

0

(b)

0

0.5

1.0

1.5

2.0

2.5 −εHH

0

0.5

1.0

1.5

2.0 −εHH

2

Fig. 9. Dependences of gyration radius Rg of a macromolecule on solvent quality –εHH at different values of the potential rigid ity of a chain of 128 monomer units at (a) θ0 = 29° and ϕ0 = 154° and (b) θ0 = 57° and ϕ0 = 166° [110]; εst = (1) 0, (2) 8, (3) 16, (4) 24, and (5) 32.

solvent quality. In addition, the higher the chain rigid ity, the higher the amplitude of changes in sizes at the transition point (Fig. 9a). The collapse of these mac romolecules passes through the stage of formation of necklacelike conformations. Flexible chains form a necklace of spherical unstructured beads; rigid chains form a necklace of helical beads. It is shown that glob ules of rigid chains are a complex of two or three entangled branches. Moreover, in the case of a ternary helix, the branches are entangled at each turn; in the POLYMER SCIENCE

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case of a double helix, entanglements are scarce or absent. The two structures have the same energy, and the formation of one state or the other depends on only the kinetic factors. The sizes of packed helices smoothly change as the quality of the solvent deteriorates; the amplitudes of these changes are low. In addition, as chain rigidity increases, the more insignificant the changes in its size become during a deterioration in the solvent quality (Fig. 9b). In the case of rigid chains with an almost

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VASILEVSKAYA, ERMILOV (a)

(b)

(c)

(d)

Fig. 10. Snapshots of AP macromolecules. The quality of the solvent deteriorates in alphabetical order [112].

defectfree secondary structure along the entire length of the macromolecule, deterioration in the solvent quality leads to a decrease in the distance between the turns and to an increase in the number of units in a turn. In addition, the chain becomes more rigid and forms a highly structured rodlike globule. Globules of the flexible chains are compact and composed of heli cal chain fragments linked together through chain fragments with a broken helical structure, e.g., a “hairpin,” in which two helical fragments are arranged parallel to each other. Analysis showed that, in all studied cases, at all val ues of the rigidity of the chain and the parameters that fix its local structure, the compaction of a macromol ecule during deterioration of the solvent quality leads to stabilization of the secondary helical structure. In [111, 112], copolymer macromolecules com posed of amphiphilic A and hydrophilic P units were analyzed. It was shown that the coil–globule transi tion of these macromolecules likewise passes through the stage of the necklacelike conformation (Fig. 10); its formation is accompanied by a sharp change in sizes and occurs at approximately the same quality of the solvent as that for macromolecules with strongly different degrees of polymerization. The fact that the collapse of the system into the necklacelike conformation of the copolymer of amphiphilic and hydrophilic units for all degrees of polymerization of the chain is observed for approxi mately the same solvent quality can be explained by the noncooperativity of the coil–necklace transition; that is, an individual bead is formed only via interac tion of neighboring units along the chain. In fact, if the formation of a bead has no effect on neighbors, the energy of hydrophobic attraction at which the bead is formed must not depend on the degree of polymeriza tion of the chain.

With further deterioration in the quality of the sol vent, the aggregation number increases and micelle beads associate into a single cluster and form a globule. The size of a bead of the necklacelike conformation is determined by the structure of the macromolecule and the features of interaction between hydrophobic groups. Therefore, the higher the degree of polymer ization of the macromolecule, the greater the number of micelle beads in the initially formed necklace and the poorer the solvent quality at which the micelle beads merge into a single globule. The process of formation of a single globule in this case can slightly differ from the processes that occur in systems with spherical micelle beads (compare Figs. 3 and 10). In an AP copolymer, a decrease in the total number of micelle beads during deterioration in the solvent quality is not accompanied by the redistribu tion of units between them; this decrease results from the sequential merging of the beads themselves, so that spherical and cylindrical beads with strongly different aggregation numbers coexist in the macromolecule. Globules formed in a poor solvent likewise have a cylindrical shape, which is typical of the longest homopolymer macromolecules with amphiphilic A units. The formation of cylindrical globules in all the abovedescribed cases is due to the fact that this shape of globule provides a more complete segregation of hydrophobic H and hydrophilic P groups owing to an increase in the total surface area. An AP macromole cule contains hydrophobicgroupincompatible hydrophilic groups not only in the monomer A unit but also in the backbone; therefore, this effect becomes even more pronounced. In fact, to provide the maximum contact of hydrophilic groups with the solvent and between hydrophobic groups, the chain folds to form turns. POLYMER SCIENCE

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C

thread 0 nm

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Fig. 11. Chromatin packaging (a) in the case of nucleosomes with histone tails and (b) in the case of removed histone tails [113]. Published with permission of Elsevier.

The result is that the packing of the backbone of this chain in the globule is clearly helical [111]. The anal ysis showed that these structures are quasihelical in the sense that both lefthanded and righthanded turns coexist. This is not surprising, because the above model is completely ambidextrous. If we draw rough minimalist parallels with biologi cal macromolecules, then we find a similar structure in, e.g., polypeptides, in which amino acids are linked via polarized peptide bonds, and in chromatin, which is a complex of DNA macromolecules and histone proteins. Chromatin comprises special supramolecular structures, i.e., nucleosomes, in which the positively charged core of histone protein molecules is wrapped with a DNA molecule. Nucleosomes are linked via free DNA fragments into a nucleosome string. It is known that a nucleosome string in cells of living organisms forms an irregular helical fibril [3]. The mechanism of chromatin packaging into helical fibrils is still not sufficiently understood; however, it is known that nucleosomes exhibit, in addition to electrostatic repulsion, a shortrange attraction potential. In terms of our model that uses shortrange potentials, nucleo somes can be extremely roughly simulated as amphiphilic A groups, while DNA macromolecule fragments that link them can be regarded as hydro philic P chains. Recent studies have shown that the pattern of chro matin packaging can be changed through variation in the properties of histone tails or their complete removal [113]. During the compaction of chromatin with histone tails (their analog in our model is hydro POLYMER SCIENCE

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philic side groups), helical fibrils form; during the compaction of chromatin without histone tails, unor dered aggregates form (Fig. 11). The presence of helical fragments in proteins is well known. We can draw a parallel between the model PA copolymer and the structure of real proteins, if we assume that amphiphilic A units in the model corre spond to hydrophilic amino acid residues and that hydrophilic P units correspond to atomic groups of polarized peptide bonds [1–3]. Note that, in these systems, the processes that lead to the formation of helical conformations are fairly complex and cannot be reduced only to hydrophobic– hydrophilic interactions. In particular, an important role in the formation of α helices in polypeptides is played by intramolecular hydrogen bonds. In the case of chromatin, we deal with complex interactions of histone proteins that are currently the subject of intense research. Nevertheless, we can infer that the combination of amphiphilic and hydrophilic units in the same chain with shortrange interaction potentials contributes to the intramolecular structuring with a quasihelical local order and to the formation of extended fibrillar structures. This phenomenon is apparently used sometimes in nature as well. CONCLUSIONS In this review, we have described the presentday approaches to the analysis of the processes of self assembly of protein molecules. The main focus has been on models with a high degree of simplification, which today are among the basic tools of computer studies of protein folding and selfassembly of syn thetic polymers. In particular, we have described the model of an amphiphilic monomer unit and the results derived in terms of this model. The data of computer experi ments presented in the review clearly show that the coil–globule transition in macromolecules with amphiphilic monomer units greatly differs from the coil–globule transition of both homopolymer and

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copolymer macromolecules, in which hydrophobic H and hydrophilic P units are linked in a linear chain. Macromolecules with amphiphilic units can form a much greater number of different morphological con formations, such as necklacelike, spherical, cylindri cal, diskshaped, and toroidal conformations. In addi tion, macromolecules with amphiphilic monomer units are particularly structured in semidilute solu tions: During deterioration in the solvent quality, flex ible amphiphilic macromolecules form soluble glob ules; rigidchain macromolecules form fibrillar aggre gates. In recent studies [114–116], melts of amphiphilic macromolecules were investigated in computer simu lations in terms of the degree of incompatibility between the backbone and pendants. It was found that, during a significant incompatibility of groups, these macromolecules form bicontinuous microphases that are stable over a wide temperature range. This circumstance means that, to all appear ances, it is useful to take into account the amphiphilic ity of the structure of a monomer unit not only for a more complete and consistent description of confor mational transitions in synthetic and biological mac romolecules but also for constructing the simplest models of other phenomena of selforganization in polymer systems. REFERENCES 1. M. V. Vol’kenshtein, Biophysics (Nauka, Moscow, 1988) [in Russian]. 2. A. V. Finkel’shtein and O. B. Ptitsyn, Physics of Protein (Knizhnyi Dom “Universitet,” Moscow, 2002) [in Russian]. 3. D. L. Nelson and M. M. Cox, Lehninger Principles of Biochemistry (Worth, New York, 2000). 4. C. B. Anfinsen, Science (Washington, D. C.) 181, 223 (1973). 5. A. R. Fersht and E. I. Shakhnovich, Curr. Opin. Struct. Biol. 8, 478 (1998). 6. M. Levitt, M. Gerstein, E. Huang, et al., Annu. Rev. Biochem. 66, 549 (1997). 7. J. N. Onuchic, Z. LutheySchulten, and P. G. Wolynes, Annu. Rev. Phys. Chem. 48, 545 (1997). 8. V. S. Pande, A. Y. Grosberg, and T. Tanaka, Rev. Mod. Phys. 72, 259 (2000). 9. K. W. Plaxco, D. S. Riddle, V. Grantcharova, and D. Baker, Curr. Opin. Struct. Biol. 8, 80 (1998). 10. E. I. Shakhnovich, Curr. Opin. Struct. Biol. 7 (1P), 29 (1997). 11. V. I. Abkevich, A. M. Gutin, and E. I. Shakhnovich, Biochemistry 33, 10026 (1994). 12. J. D. Bryngelson and P. G. Wolynes, Proc. Natl. Acad. Sci. U. S. A. 84, 7524 (1987). 13. J. D. Bryngelson and P. G. Wolynes, J. Phys. Chem. 93, 6902 (1989). 14. K. A. Dill, Biochemistry 29, 7133 (1990). 15. N. Go and H. Abe, Biopolymers 20, 991 (1981).

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