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Current Opinion in Solid State & Materials Science xxx (2003) xxx–xxx
Review
Biomimetic sequence design in functional copolymers
Alexei R. Khokhlov a,b,*, Pavel G. Khalatur b,1
a Department of Physics, Moscow State University, Moscow 119992, Russiab Department of Polymer Science, University of Ulm, Ulm D-89069, Germany
Received 13 August 2003; accepted 29 August 2003
Abstract
We review recent studies on the methods of design of sequences in synthetic copolymers aimed to achieve given functional
properties. Our approach is biomimetic in the sense that we have as an example some functional features of biopolymers (such as
solubility of globules or selectivity in complex formation with a substrate). In particular, the methods to obtain so-called proteinlike
copolymers (which give soluble globules with core-shell structure) and molecular dispensers (which are able to absorb selectively
nanoparticles of a given size) are discussed. The problems of evolution of sequences in copolymers are considered from the view-
point of emerging of information complexity in the sequences in the course of this evolution.
� 2003 Elsevier Ltd. All rights reserved.
Keywords: Heteropolymers; Proteinlike globules; Sequence design; Biomimetic approach; Molecular evolution
PACS: 61.41.+e; 87.15.Aa; 87.15.Cc; 82.35.Jk; 82.35.Lr
*Co
fax: +7
E-m1 Te
1359-0
doi:10.
Contents
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1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00
2. Proteinlike copolymers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00
3. Long-range correlations in proteinlike copolymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00
4. Experimental realization of proteinlike copolymers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00
respon
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4.1. Surface �coloring’ of a globule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00
4.2. Copolymerization with simultaneous globule formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00
5. Some generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00
6. Molecular dispenser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00
7. Evolution of sequences and their information content. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00
7.1. Ascending and descending branches of evolution of sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . 00
7.2. Information complexity of copolymer sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00
8. Other problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00
9. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00
ding author. Address: Department of Physics, Moscow State University, Moscow 119992, Russia. Tel.: +7-095-939-1013/135-7910;
9-2988/135-5085.
resses: [email protected] (A.R. Khokhlov), [email protected] (P.G. Khalatur).
731-50-23103; fax: +49-731-50-31399.
see front matter � 2003 Elsevier Ltd. All rights reserved.
ossms.2003.08.001
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Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00
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1. Introduction
For a long time chemical industry was interested in
polymers mainly from the viewpoint of obtaining unique
construction materials (plastics, rubbers, fibers, etc.).
Couple of decades ago the main focus of interest shifted
to functional polymers (superabsorbents, membranes,adhesives, etc.). In the nineties scientific and industrial
polymer community started to discuss �smart’ or �intel-lectual’ polymer systems (e.g. soft manipulators, poly-
mer systems for controlled drug release, field-responsive
polymers etc.); the meaning behind this term is that the
functions performed by polymers become more sophis-
ticated and diverse [1–9]. This line of research concen-
trating on polymer systems with more and morecomplex functions will be certainly in the mainstream of
polymer science in the 21st century.
One of the ways to obtain new polymers for sophis-
ticated functions is connected with the synthesis of novel
monomer units where the required function is linked to
the chemical structure of these units. However, the po-
tential of this approach is rather limited, because com-
plicated and diverse functions of polymer materialwould then require a very complex structure of mono-
mer units, which normally means that the organic syn-
thesis is more expensive and less robust.
The alternative approach is to use known monomer
units and to try to design a copolymer chain with given
sequence of these units. There are practically infinite
possibilities to vary sequences in copolymers: from the
variation of some simple characteristics like compositionof monomer units, average length of blocks (for the
chains with blocky structure), availability of branching,
etc. to more sophisticated features like long-range cor-
relations or gradient structure. Therefore, in this ap-
proach a wide variety of new functional copolymers can
be tailored.
It is important to emphasize that the nature has
chosen this way in the evolution of main biologicalmacromolecules: DNA, RNA and proteins. These
polymers in living systems are responsible for functions,
which are incomparably more complex and diverse than
the functions, which we are normally discussing for
synthetic copolymers. The molecular basis for this
ability to perform sophisticated functions is associated
with unique primary sequences of units in biopolymers,
which emerged in the course of biological evolution.
Thus, one of the promising approaches in the se-
quence design of functional copolymers is biomimetic in
its nature: it is tempting to look at the main features of
sequences of monomer units in biopolymers, understand
how these sequences define functional properties, and
then try to implement similar ideas for synthetic co-
polymers. The present paper is dedicated to the reviewof the results obtained recently in this direction, i.e.,
dealing with biomimetic sequence design of copolymers.
2. Proteinlike copolymers
The first ideas connected with design of sequences in
functional copolymers were formulated by us in 1998
[10–13]. They were based on the simple and well-knownfact that the functioning of all globular proteins depends
on two main factors: (i) they are globular; (ii) they are
soluble in aqueous medium. The combination of these
two factors is non-trivial, e.g., for homopolymers and
random copolymers the transition to globular confor-
mation is usually accompanied by the precipitation of
globules from the solution [14,15]. Protein globules are
soluble in water because of the special primary sequence:in the native conformation most of hydrophobic
monomer units are in the core of the globule while hy-
drophilic and charged monomer units form the envelope
of this core. Having in mind the biomimetic approach
described above, we can formulate the following prob-
lem: whether it is possible to design such sequence of
synthetic HP-copolymer (copolymer consisting of
monomer units of two types, H and P) that in the mostdense globular conformation all the hydrophobic
H-units are in the core of this globule while hydrophilic
(polar) P-units form the envelope of this core? This
question was first addressed in Ref. [10] (see also Refs.
[11–13]) and the corresponding polymers were called
�proteinlike’ copolymers.
The proteinlike HP-sequences were first obtained in
computer experiments [10–13], which can be describedas follows. We start with arbitrary homopolymer glob-
ule conformation formed due to the strong attraction of
monomer units (Fig. 1a) and perform for it a �coloring’procedure (Fig. 1b): monomer units in the center of the
globule are called H-type (hydrophobic) units, while
monomer units belonging to globular surface are as-
signed to be P-type (polar) units. Then this primary
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Fig. 1. Main steps of the sequence design scheme for proteinlike co-
polymers: (a) homopolymer globule; (b) the same globule after �col-oring’ procedure; (c) proteinlike copolymer in the coil state.
A.R. Khokhlov, P.G. Khalatur / Current Opinion in Solid State and Materials Science xxx (2003) xxx–xxx 3
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structure is fixed, attraction of monomer units is re-moved and proteinlike copolymer is ready for the fur-
ther investigation (Fig. 1c).
It was shown [10–13,16] that the coil-globule transi-
tion for such copolymers, induced by the attraction of
H-units, occurs at higher temperatures, leads to the
formation of denser globule and has faster kinetics than
for random and random-block counterparts. The reason
for this is illustrated in Fig. 2a and b where the typicalsnapshots of globules formed by proteinlike and random
HP-copolymers with the same HP composition are
shown.
One can see that the core of proteinlike globule is
much more compact and better formed; it is surrounded
by the loops of hydrophilic units, which stabilize the
core. Apparently, this is due to some memory effect: the
core which existed in the �parent’ conformation (this isthe term introduced in Ref. [10] to describe the confor-
mation of Fig. 1b where the coloring is performed) was
simply reproduced upon refolding caused by the at-
traction of H-units. One may say that the features of
parent conformation are �inherited’ by the proteinlike
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Fig. 2. Typical snapshots of globular conformation for (a) proteinlike
and (b) random copolymers. Hydrophobic H units are shown in light-
gray color and hydrophilic P units in dark-gray color.
HP-copolymer. Looking at the conformations of Fig.
2a and b, it is natural to argue that proteinlike copoly-
mer globule should be soluble in water and thus open to
further modification in the course of biological evolu-tion, while random copolymer globules will most prob-
ably precipitate, and thus drop out of the evolution.
EDPROO
F3. Long-range correlations in proteinlike copolymers
Sequences of monomer units in proteinlike copoly-
mers generated as described above look pretty random,
and at the first sight it is impossible to notice any long-
range correlations. However, one may guess that such
correlations should exist, because assigning of the typeof monomer (H or P) at the parent conditions (Fig. 1b)
depends on the conformation of globule as a whole, not
on the properties of some small part of the chain.
In Refs. [17–19] it was shown, both by exact analyt-
ical theory and by computer simulations, that it is in-
deed the case and that the long-range correlations in the
proteinlike sequences can be described by the so-called
Levy-flight statistics [20,21].The possibility for exact analytical description of se-
quences resulting from surface coloring (Fig. 1b) comes
from the fact that the statistics of polymer chains inside
dense globule is Gaussian, i.e., it is described by the
ordinary diffusion equation. One has only to worry
about correct boundary conditions, and this problem
was resolved in Ref. [18].
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4. Experimental realization of proteinlike copolymers
After the idea of sequence design of proteinlike co-
polymers was presented and realized in computer sim-
ulations, several teams started experimental research
aimed to obtain such copolymers in synthetic chemical
laboratory.
4.1. Surface ‘coloring’ of a globule
This is a most obvious method, which exactly follows
Fig. 1. At first we have a homopolymer chain and we
transfer it to a poor solvent where dense globular con-
formation is formed. Then the surface of the globule is
experiencing a chemical reaction (polymer–analogous
transformation); as a result, monomer units on the
surface, instead of being lyophobic, are acquiring lyo-philic properties. Thus, emerging copolymer adopts a
core-shell proteinlike structure in the globular state.
These experiments were first performed by Virtanen
and Tenhu (University of Helsinki) who studied grafting
of short poly(ethylene oxide) (PEO) chains to the co-
polymer of thermosensitive N -isopropylacrylamide
(NIPA) and glicydil methacrylate [22–24]. At room
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Fig. 3. Normalized radial distribution functions of H and P mono-
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drophobic and hydrophilic monomers in a selective (polar) solvent,
using a Monte Carlo simulation technique [28].
4 A.R. Khokhlov, P.G. Khalatur / Current Opinion in Solid State and Materials Science xxx (2003) xxx–xxx
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temperatures, such copolymer is in the coil state and
grafting takes place in a random manner. At elevated
temperatures, the transition to globule occurs, and
grafting proceeds mainly in the globular surface, thusleading to its hydrophilization and to the creation of
proteinlike copolymer in the sense described above. In-
deed, it was shown that proteinlike copolymer prepared
in this way exhibits solution turbidity at higher tem-
peratures than the random one, and gives smaller ag-
gregates in the turbid solution.
Several other attempts along the same lines were
performed, but in general this method was shown to berather unreliable, because of the impossibility to stabi-
lize dense globules in the solution for the time sufficient
to implement a polymer–analogous transformation.
4.2. Copolymerization with simultaneous globule forma-
tion
This method turn out to be much more robust and it
was first proposed in Ref. [25], where the redox-initiated
free-radical copolymerization of thermosensitive N -vi-
nylcaprolactam (NVCa) with hydrophilic N -vinylimi-
dazole (NVIz) was studied at different temperatures. At
room temperature, such polymerization gives a randomcopolymer. On the other hand, when polymerization
takes place at elevated temperatures (�65 �C) growingchains form globules, and the concentration of mono-
mers around the active radical is influenced by this fact.
The conditions were found when proteinlike copolymers
are emerging as a result of such synthesis. These co-
polymers were not precipitating at all when the solution
is heated up to 80 �C; on the other hand, dense globuleswere formed already around 30 �C. Recent results ob-
tained by our group [25], as well as by the groups of
Prof. Mattiasson (University of Lund, Sweden) [26] and
of Prof. Chi Wu (Chinese University of Hong Kong)
[27] show the universality of this approach of obtaining
proteinlike copolymers. It was called copolymerization
with simultaneous globule formation, and the corre-
sponding theoretical background was given by our re-cent simulations [28].
The main idea behind this approach of the synthesis
of proteinlike copolymers can be explained as follows.
Let us assume that we are performing a copolymeriza-
tion of moderately hydrophobic (thermosensitive) and
hydrophilic monomers in aqueous medium. The condi-
tions for copolymerization (e.g., temperature) should be
chosen in such a way that when the emerging chain islong enough it forms a globule. Globule formation will
lead to a redistribution of monomers: hydrophobic ones
should be predominantly concentrated within the glob-
ules, while hydrophilic monomers will be mainly located
in the outside solution. Therefore, when a growing chain
end is inside the globular core, mainly hydrophobic
units will be added, while when the end is located in the
PROO
F
outer shell of a globule, the addition of hydrophilic units
becomes more preferable. In this way, the proteinlike
structure with predominantly hydrophobic core and
hydrophilic outer envelope should emerge automati-cally.
Using a Monte Carlo simulation technique, we have
modeled the process of copolymerization of hydropho-
bic (H) and hydrophilic (P) monomers, which lead to
the globule formation [28]. The preferential sorption of
hydrophobic monomers in the core of globule was ex-
plicitly taken into account. Typical snapshots of the
globular conformation of a resulting copolymer chainare similar to that presented in Fig. 2a. Already from
this fact one can see that in this way we do indeed end
up with a proteinlike copolymer having a dense hydro-
phobic core surrounded by a hydrophilic shell.
More quantitative results are presented in Fig. 3
where the typical radial distribution of hydrophobic and
hydrophilic units (with respect to the center of the
globule) is presented for copolymer chain of 512 unitswith 1:1 HP composition. These findings are explicitly
proving the core-shell (proteinlike) structure of the
globule obtained via copolymerization in a poor solvent.
Our simulations show that this result is quite universal
and robust [28].
TE5. Some generalizations
The approach to sequence design of functional co-
polymers presented above can be generalized in many
different ways.
Fig. 1 represents only one of possible ways of real-ization of coloring procedure. If our intention is to mi-
mic not globular proteins––enzymes, but rather
membrane proteins, we can apply another coloring
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A.R. Khokhlov, P.G. Khalatur / Current Opinion in Solid State and Materials Science xxx (2003) xxx–xxx 5
procedure for parent homopolymer globule, namely, we
can assign the type H to monomer units lying inside the
central cylindrical slice and the typeP to monomer units
lying in two hemispheres to both sides of H-layer. Itwas shown [17,19] that such coloring procedure leads to
formation of HP-copolymer chain, which exhibits a
number of unusual properties. For example, such a
chain shows the effect of stability of parent micro-seg-
regated structure: after refolding the segregation of both
hemispheres of P-units is reestablished.
Furthermore, to design sequences with special prop-
erties it is not necessary to perform coloring of a denseglobule. In fact, any special macromolecular confor-
mation can play the role of a parent one. For example,
in Ref. [29] the conformation of a polymer chain ad-
sorbed on a plane surface was considered. The monomer
units closest to the surface in some instant snapshot
conformation were assigned to be A-units, others be-
came B-units. The AB-chain thus obtained was called
an adsorption-tuned copolymer. It was shown that thiscopolymer adsorbs on another plane surface (to which
only A-units are attracting) more efficiently than ran-
dom and random-block copolymers with the same ABcomposition and the same degree of blockiness.
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6. Molecular dispenser
The idea of molecular dispenser is a further devel-opment in the direction of conformation-dependent se-
quence design. Namely, we consider the conformation
of a homopolymer chain adsorbed on a spherical col-
loidal particle (Fig. 4) and perform design of sequence
for this state of macromolecule. The motivation behind
this design procedure is that if we eliminate the �parent’colloidal particle after the design is completed (e.g., by
etching), the resulting copolymer will be hopefully tunedto selectively adsorb another colloidal particle of a pa-
rental size rp. For example, if such copolymer is exposed
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Fig. 4. Stages of preparation of copolymer envelope: (a) adsorption of hydro
of the polymer chain and introduction of crosslinks to stabilize hollow-spheri
are shown in light-gray color, hydrophilic P units are depicted in dark-gray
TEDPROO
F
to a polydisperse colloidal solution of particles of dif-
ferent size, it will selectively chose to form a complex
with the particle having the same radius as that in pa-
rental conditions. That is why such copolymer can becalled a molecular dispenser. This idea was realized in
computer simulations [30].
First we considered a homopolymer chain attracting
to a colloidal nanoparticle. Such chain is forming a
complex with the particle (its typical conformation is
shown in Fig. 4a). Only part of chain segments are in
direct contact with colloidal particle, while other seg-
ments form flower-like loops. Let us now �color’ thesegments in the loops in �blue’, while the segments near
colloidal particle remain �red’, i.e., they are attracting to
this particle (Fig. 4b). If the sequence design is stopped
at this stage, the pronounced selectivity of the complex
formation with another particle of parental size rp is not
reached. However, if additional crosslinks are intro-
duced between �red’ units, thus fixing the cage structure
of the central cavity (Fig. 4b), the macromoleculeemerging after elimination of colloidal particle (Fig. 4c)
does indeed show the features of a molecular dispenser
[30].
To characterize the complex formed between dis-
penser and particle, we calculated the probability
P ðr; T Þ of finding a complex made from the copolymer
envelope and a particle of a given size, r, at the tem-
perature T . Typical results are shown in Fig. 5a. One cansee that the selectivity of the complex formation with the
particle of a certain size is indeed reached, i.e., the idea
of molecular dispenser works.
The reason for the selective adsorption of a colloidal
particle of �parent’ size is explained by the typical
snapshots in Fig. 5b and c. One can see that the particle
of �parent’ or smaller size (r6 rp) is fully absorbed by
the central cavity (Fig. 5b), because the correspondingfitting was ensured by the sequence design procedure
(Fig. 4b). On the other hand, particle of a larger size
(r > rp) turns out to be too big for a central cavity (Fig.
phobic homopolymer chain on a �parent’ colloidal particle; (b) coloringcal structure; (c) elimination of the core particle. Hydrophobic H units
color, and H–H crosslinks are presented as gray sticks.
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Fig. 5. (a) Probability of finding a complex made from a 512-unit
copolymer envelope and a particle of a given size, r, at the temperature
T for the case when the copolymer envelope has 48 crosslinks. (a,b)
Snapshots of the complexes made from a 512-unit copolymer envelope
for (b) r=rp ¼ 0:8 and (c) r=rp ¼ 1:8, where rp is the size of the
�parent’ particle.
6 A.R. Khokhlov, P.G. Khalatur / Current Opinion in Solid State and Materials Science xxx (2003) xxx–xxx
REC5c), and thus the complex formed does not saturate all
the possibilities for the attraction of �red’ units to the
surface of the particle. As to small particles, they easily
penetrate inside the molecular dispenser, but the com-
plex formed is not stable (especially at high temperature)
because of small surface of such particles.
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7. Evolution of sequences and their information content
The concept of evolution is one of the cornerstones of
modern natural sciences: in cosmology the evolution of
the Universe is discussed, in geology the evolution of the
Earth, in life sciences biological evolution (driven by
selection) [31–34]. This concept can be also applied in
polymer science. The corresponding statement of theproblem is very clear. The present day biopolymers
(proteins, DNA, RNA) possess complicated sequences
of monomer units, which encode their functions and
structure (e.g., unique tertiary structure of globular
proteins). Therefore, these sequences (in 20-letter al-
phabet for the case of proteins and in four letter al-
phabet for the cases of DNA and RNA) should be
TEDPROO
F
statistically very different from random ones and often
exhibit significant correlation on different scales. In
other words, it is natural to expect that the contents of
information in these sequences is relatively high incomparison with random sequences (e.g., DNA se-
quences contain all genetic information). On the other
hand, the formation of first copolymers at the very be-
ginning of molecular prebiological evolution could lead
only to random sequences or sequences with trivial
short-range correlations. That is, the information con-
tent of these sequences was practically zero. One can
argue that in the course of molecular evolution the co-polymer sequences became more and more complicated
until they reached the stage of information complexity
of present day biopolymers. The study of various pos-
sibilities of this evolution of copolymer sequences is just
the area where the evolution concept can be used in the
context of polymer science.
It is worthwhile to note that since the information
content of a sequence is a mathematically definedquantity, the whole process of evolution of biopolymer
sequences can be specified in exact mathematical terms,
which is not always the case for other examples of
evolution.
On the other hand, the formulated fundamental
problem is extremely difficult because of the absence of
direct information on the early prebiological evolution.
Therefore, of particular interest are �toy models’ ofevolution of sequences, which show different possibilities
for appearance of statistical complexity and long-range
correlation in the sequences [35–40]. Since by random
mutations it is impossible to increase the information
contents of a sequence, such �toy models’ should take
into account the coupling between polymer chain con-
formation (defined by the interactions between mono-
mer units of different type) and evolution of sequence. Inother words, we have to explore the possibilities of
conformation-dependent evolution of copolymer se-
quences.
7.1. Ascending and descending branches of evolution of
sequences
The aim of our studies [41,42] was to introduce ex-
plicitly the concept of evolution of sequences into the
scheme of generation of proteinlike copolymers.
Using a molecular dynamics-based algorithm, we
have simulated the conformation-dependent evolution
of model two-letter (HP) copolymer sequences [41].The sequence evolution mechanism involved generation
of initial proteinlike sequence by inspecting of a ho-
mopolymer globule and by attributing H type to the
monomer units in the core of this globule and P type to
the units on the surface of the globule. The resulting
copolymer was then transferred to a coil conformation
and then refolded due to the strong attraction of H
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Fig. 6. Snapshots of two typical conformations of designed copoly-
mers obtained after long conformation-dependent evolution of a 128-
unit chain: (a) core-tail (tadpole-like) structure at ePP=e�PP ¼ 0 and (b)
core-shell structure at ePP=e�PP J 1. Hydrophobic H units are shown
in dark-gray color and hydrophilic P units in light-gray color. (c) The
Jensen–Shannon divergence measure as a function of ePP=e�PP. The
characteristic energy of H–H interactions is fixed at eHH ¼ 2kBT ,thus stabilizing a dense globular core, while the attraction energy ePP
between hydrophilic P units is considered as a parameter. We adopt
the simplest choice for the cross-parameter: eHP ¼ ðeHH � ePPÞ1=2.
A.R. Khokhlov, P.G. Khalatur / Current Opinion in Solid State and Materials Science xxx (2003) xxx–xxx 7
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units. The HP sequence is then further modified de-
pending on the position of a monomer unit in the core
or on the surface of a newly formed globule. Such
modifications leading to changing of primary HP se-quence were repeated many times. With this evolution-
ary process, structures and sequences are formed self-
consistently. A 128-unit flexible-chain heteropolymer
with the HP composition fixed at 1:1 was simulated for
the condition when hydrophobic H monomer units
strongly attract each other, thus stabilizing a dense
globular core, while the attraction energy ePP between
hydrophilic P monomer units is considered as a pa-rameter. For this model system, we have calculated
various conformation-dependent and sequence-depen-
dent properties, including information-theoretic-based
quantities.
We have found that there are two regimes (branches)
of evolution (regimes I and II) depending on the at-
traction energy between polar units ePP [41]. If ePP is
smaller than some crossover energy, e�PP (regime I) theevolution can lead to a second order-like transition in
sequence space from the sequences with a proteinlike
primary structure capable of forming a core-shell glob-
ule to the degenerated (non-proteinlike) sequences hav-
ing long uniform H and P blocks (Fig. 6a).
The degenerated primary structure looks like a di- or
tri-block sequence (�core-tail’ conformation, Fig. 6a).
Therefore, when the attraction between hydrophilicunits is not sufficiently strong, we deal with the de-
scending branch of the evolution, which leads to non-
proteinlike sequences having low information content
and low complexity. On the other hand, in the second
regime (at ePP P e�PP) the complexity of proteinlike
structures was found to increase (see Fig. 6b) and
therefore we have an ascending branch of evolution of
sequences.
7.2. Information complexity of copolymer sequences
A common approach for the analysis of the com-
plexity of a system is to use concepts from informationtheory and information-theoretic-based techniques [43–
45]. In general, our aim was to find a measure capable to
indicate how far copolymer sequences generated during
our evolutionary process differ from each other and
from random or trivial (degenerate) sequences. It turned
out that the usual measures of the degree of complexity
(based, e.g., on the Shannon entropy and related char-
acteristics) are non-adequate. To overcome this prob-lem, we have proposed to use the so-called Jensen–
Shannon (JS) divergence measure [41,46].
For the sequences generated in the evolutionary
process described above, it was shown that at ePP P e�PP
(regime II) the degree of complexity, as measured by JS
divergence, can be considerably higher as compared to
that observed for the regime I, at ePP < e�PP. The com-
plexity slightly increases with ePP decreasing, reaches its
maximum in the vicinity of e�PP, and then sharply drops
(Fig. 6c).
8. Other problems
Among other problems related to the field discussed
in this review one should mention: analysis of confor-
mations and aggregation stability of proteinlike poly-
electrolytes by molecular dynamics simulations [47] and
on the basis of polymer integral equation (PRISM)
theory [48]; studies of microphase separation and self-
organization processes in the melts of designed copoly-
mers [49,50].
9. Conclusion
In all cases described above, some functional features
of the �parent’ conformation were �memorized’ by the
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8 A.R. Khokhlov, P.G. Khalatur / Current Opinion in Solid State and Materials Science xxx (2003) xxx–xxx
copolymers generated according to our sequence design
scheme. These features are then manifested in other
conditions. Such an interrelation can be regarded as one
of the possible mechanisms of molecular evolution:polymer acquires some special primary sequence in the
�parent’ conditions and then (in other conditions) uses
the fact that primary structure is �tuned to perform
certain functions’.
566567568569570571572573574575
Acknowledgements
The financial support from Alexander-von-Hum-
boldt Foundation, Program for Investment in the Fu-
ture (ZIP), INTAS (project # 01-607), and Russian
Foundation for Basic Research is highly appreciated.
576577578579580581582583584585586587588589590591592593594595596597598599600601602603604605606607608609610611612613614615616617618619620621622
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