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Autoreplicators and hypercycles in typogeneticsq
V. Kvasnicka*, J. Pospichal
Faculty of Chemical Technology, Department of Mathematics, Slovak Technical University, 812 37 Bratislava, Slovak Republic
Received 13 November 2000; revised 9 March 2001; accepted 9 March 2001
Abstract
A simplied formal system typogenetics, closely related to concepts of molecular genetics and introduced by Hofstadter in
his seminal bookDialogues with Godel, Escher, Bach: An Eternal Golden Braid[Basic books, New York, 1979 (Chapters XVI
and XVII)] is discussed. Concepts of autoreplicators and hypercycles, dened within typogenetics, belong to basic entities in
current perception of articial life. A metaphor of chemical reactions (chemostat) is applied to study emergence of autorepli-
cators and hypercycles. The initial version of evolutionary approach, designed for construction of autoreplicators, is able to
produce only small hypercycles composed of two or at most three autoreplicators. An emergence of larger hypercycles
represents extremely complicated combinatorial optimization problem. Therefore, we turn our attention to a sequential technique
of their construction, where a smaller hypercycle is enlarged by another autoreplicator.Both components are thus integrated into one
hypercycle. This method of successive construction of hypercycles substantially reduces combinatorial complexity of the original
approach where whole hypercycles are simultaneously optimized. q 2001 Elsevier Science B.V. All rights reserved.
Keywords: Typogenetics; Strand; Autoreplicator; Hypercycle; Evolutionary method
I met Joe Paldus for the rst time in the middle of
the sixties, when I started my PhD study at Heyrovsky
Institute of Physical Chemistry. Joe together with Jiri
Cizek already worked in diagrammatic perturbation
theory. I remember when Joe showed me Hugenholtz
diagrams (he preferred this type of graphology) and I
was fascinated by these drawings looking like a secret
Caballa. This was the main motivation, why I have
started to study many-body perturbation theories.
Many thanks, Joe.
1. Introduction
A typogenetics is a formal system initially devised
by Hofstadter in his famous book Dialogues with
Godel, Escher, Bach: An Eternal Golden Braid [15]
(see Refs. [22,30,31]). In typogenetics a string (called
the strand) codes a sequence of elementary operations
so that their sequential application to the strand trans-
forms this strand (parent) onto another strand
(offspring). Typogenetics was discussed by Hofstadter
in connection with his attempt to explain or classify a
`tagled hierarchy' of DNA considered as replicative
systems. In particular, a DNA strand contains, amongother things, instructions prescribing a production of
enzymes that are capable of different types of opera-
tions acting on the strand itself. A part of information
contained in sequences of bases of DNA strands
prescribes a synthesis of enzymes that are capable to
make a copy of the DNA strand itself.
Typogenetics as presented by Hofstadter [15] was
not formulated in a very precise and exact way, many
concepts and notions were presented only in a `fuzzy'
Journal of Molecular Structure (Theochem) 547 (2001) 119138
0166-1280/01/$ - see front matter q 2001 Elsevier Science B.V. All rights reserved.
PII: S0166-1280(0 1)00464-X
www.elsevier.com/locate/theochem
q In honour of Josef Paldus on the occasion of his 65th birthday.
* Corresponding author. Tel.: 1421-7-59325294; fax: 1421-7-
5249-3198.
E-mail address: [email protected] (V. Kvasnicka).
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verbal form and the reader was left to an improvisa-
tion and an ad-hoc additional specication of many
notions of typogenetics. Morris [22] was rst who
seriously attempted to formulate the typogenetics in
a precise manner and presented many illustrative
examples and explanations that substantially facili-
tated an understanding of typogenetics. Almost 10
years ago Varetto [30] has published an article
where he demonstrated that typogenetics is a proper
formal environment for a systematic constructive
enumeration of strands that are able of an autoreplica-
tion. Recently, Varetto [30,31] published another
paper where typogenetics was applied to a genera-
tion of the so-called tanglecycles that are simpli-
ed version of hypercycles [7,8] of Eigen and
Schuster.The purpose of the present paper is to present a
simplied version of typogenetics that will be still
capable to form a proper environment for articial
life studies of autoreplicators and hypercycles, both
entities that belong to basic concepts of modern efforts
[13,5,9,11,13,16 18,2325,28,29] to simulate life
in-silico. Simplication of our version of typogenetics
consists mainly in trimming of an instruction set,
where all instructions that introduce or delete bases
in strands were omitted. It is demonstrated that a
construction of autoreplicators and hypercyclesbelongs to very complicated combinatorial problems
and therefore an effort of their systematic constructive
enumeration is hopeless. This is the main reason why
we turned our attention to evolutionary methods of
spontaneous emergence of autoreplicators and hyper-
cycles. One of objectives of the present paper is to
demonstrate an effectiveness of a simple version of
evolutionary algorithm to create autoreplicators and
hypercycles in a way closely related to Darwinian
evolution.
The paper is organized as follows: basic principles
of simplied version of typogenetics are described inSection 2. Strands are determined as strings composed
of four symbols A, C, G, and T. Then a DNA is speci-
ed as a double strand composed of a strand and its
complementary strand. An expression of strands by
enzymes is discussed in Section 3. A simple way
how to assign an enzyme to an arbitrary strand is
demonstrated. The enzyme is composed of a sequence
of elementary instructions and the so-called binding
site. In our simplied typogenetics, we retain only
those instructions that do not change the length of
strands, which excludes for example instructions for
insertion or deletion of bases. An action of enzyme
upon the strand is strongly deterministic, it is applied
to the binding site which rst appears when going on
the strand from the left to the right. Section 4 is
devoted to a specication of autoreplicators. These
entities are determined as double strands with such a
property that each of its strands is replicated by appli-
cation of an enzyme. Firstly the double strands are
separated. Then each strand produces an enzyme,
which is in turn applied to the same strand and
produces its complementary DNA copy. The enzyme
is produced from the code by a prescription `start from
the left, translate a couple of entries into an instruction
and move to the right', creating a sequence of `instruc-tions' from neighboring couples of strand entries. This
sequence of instructions, which is a sort of metacode
of an enzyme, is in this formalism equated with
enzyme. Instructions of such an enzyme usually do
not make the copy of its `parental' strand by a
straightforward `start from the left, copy and move
to the right'. They work more like a Turing machine
on a tape (a metaphor from computer science), where
the instructions can move the enzyme to the left or to
the right on the strand. Such a copying process can
create the copy, e.g. with starting from the middle andjumping back and forth to the left and right, adding
entries to the copy of a strand from both sides in turn.
The copy can even be created in nonadjacent parts
with the conjunctive entries copied at the end. This
specication of autoreplicators represents, in fact,
hard constraints, so that their construction is nontrivial
combinatorial problem. Fortunately, it can be effec-
tively solved by making use of evolutionary methods.
Typogenetic articial chemistry is discussed in
Section 5. Recent concepts of articial life [6] often
deal with a metaphor of chemical reactions to study
processes running on the border of biotic and abioticsystems. Loosely speaking, methods of articial
chemistry are closely related to evolutionary methods,
probabilities that specify chemical reactions are often
expressed by parameters that are very similar to
tness of reaction constituents. Hypercycles
composed of double strands are studied in Section 6.
The notion of hypercycles [7,8] is a generalization of
autoreplicators such that a hypercycle is a cyclic
kinetic structure, where a replication of its ith
V. Kvasnicka, J. Pospichal / Journal of Molecular Structure (Theochem) 547 (2001) 119138120
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constituent is catalyzed by an enzyme produced by the
previous (i1)th constituent. Hypercycles are consid-
ered in recent efforts of articial life [27] as a proper
formal tool suitable for specic explanation of a
phenomena called the increase of complexity. We
show that evolutionary algorithms are capable of
inducing an emergence of hypercycles from a popula-
tion initialized by random strands. More complicatedhypercycles (composed of three or four replicators)
represent for evolutionary algorithms very hard
combinatorial problems. This is the main reason
why we turned our attention to a sequential step-by-
step method of their construction, a given hypercycle
is evolutionary enlarged to a larger hypercycle by
adding one additional replicator.
Finally, we would like to emphasize that a theore-
tical (computational) study of autoreplicators and
hypercycles was recently supported on a serious
(bio)chemical basis by `wet chemistry' experimentalworks of Biebricher [4,26] and McCaskill [20].
2. Basic principles of typogenetics
Let us consider a set B {A; C; G; T}; where
elements are bases called adenine, cytosine, guanine,
and thymine, respectively. These four elements are
further classied as purines (A and G) and pyrimi-
dines (C and T), i.e. B Bpur
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For illustration, let us consider two strands
S CGTT###AAT;
R TGT###AAAG:
According to Eqs. (4a) and (4b), the distance
between them is
dS;R 121
1001 11 11 01 0
1 01 01 11 11 0 124
10
3
5:
A zero distance between two strands Sand R means
that they are identical and do not contain hash
symbols.
3. An expression of strands by enzymes
The purpose of this section is to specify one of the
most important concepts of typogenetics, an expres-
sion of a strand by a sequence of instructions, that is
called euphemistically the enzyme. Let us consider a
set
B2 {AA; AC; AG; ; TT} 5
composed of 16 base pairs (doublets). Each strand S
X1X2Xn [ Bp can be expressed by making use of
doublets of (5) as follows:
S D1D2Dp for n 2p
;
S D1D2DpX2p11 for n 2p1 1; 6
where the rst (second) possibility is applicable if the
length of S is even (odd). Let us consider two
mappings
instruction : B2 ! {mvr; mvl; cop; off; rpy; }; 7a
inclination : B2 ! {s; l; r}; 7b
where the rst mapping instruction assigns to each
strand a sequence of instructions that will be sequen-
tially performed over the strand when an enzyme
(specied by the strand and the second mapping incli-
nation) is applied. Details of these mappings will be
specied later.
If doublets of a strand are mapped by Eqs. (7a) and
(7b) (see Table 1), we arrive at the so-called primary
structure of the enzyme that is specied by a sequence
of instructions
instructionS instrD12 instrD22
2 instrDp: 8a
A tertiary structure (2D) of the enzyme is deter-
mined by the mapping inclination, it offers the follow-
ing sequence of inclinations assigned to doublets (see
Table 1)
inclinationS inclinD12 inclinD22
2inclinDp: 8b
Both sequences (8a) and (8b) that are assigned to a
strand specify a transformation of the original (parent)
strand onto a derived (offspring) strand. Loosely
speaking, this transformation is considered as an
application of the corresponding enzyme specied
by sequences (8a) and (8b), where the enzyme is
visualized as a robot arm operating on the given
strand, carrying out the commands that are coded by
sequence (8a), which is unambiguously determined by
V. Kvasnicka, J. Pospichal / Journal of Molecular Structure (Theochem) 547 (2001) 119138122
Table 1
Specication of mappings instruction and inclination
No. Doublet Instruction Inclination No. Doublet Instruction Inclination
1 AA mvr l 9 GA rpy s
2 AC mvl s 10 GC rpu r
3 AG mvr s 11 GG lpy r
4 AT mvl r 12 GT lpu l
5 CA mvr s 13 TA rpy r
6 CC mvl s 14 TC rpu l
7 CG cop r 15 TG lpy l
8 CT off l 16 TT lpu l
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mapping (7a) based on the strand doublets (see also
Table 1). Single instructions are specied by Table 2
and Figs. 24.
What remains to be determined is a starting position
on the strand, where a sequence of enzyme actions is
initialized. Such a position is called the binding site
and it is represented by a base. An application of
enzyme is then started on the rst base (going fromthe left to the right) on the strand. If the strand does
not contain such a base, then we say that the given
enzyme is inapplicable to the strand. The binding site
X is specied by the sequence of inclinations (Eq.
(8b)) such that going successively from left to right,
we construct recurrently a sequence of arrows
oriented to right, left, up, or down. This process is
initialized by the rst position such that it is automa-tically set to arrow ) , see Fig. 5, so that the rst
inclination is not enacted. When the sequence of incli-
nations is constructed or analyzed, we get the direc-
tion of the last arrow. The binding site is
unambiguously determined by the rst inclination
symbol and by the last arrow (see Table 3)
X ffirst inclination symbol; last arrow: 9
This formula simply determines the binding site on
the strand, e.g. according to Table 3, a sequence of
arrows presented by diagram E in Fig. 5 determines
the binding site X A; it means that a correspondingenzyme is initially applied to a base A (going rst
from the left on strand). Many different enzymes
can have the same binding site. Formally, the whole
procedure of construction of an enzyme assigned to a
strand S is expressed by
enzymeS instructionS;X; 10
where its rst component corresponds to an instruc-
tion sequence (8a) and the second component
V. Kvasnicka, J. Pospichal / Journal of Molecular Structure (Theochem) 547 (2001) 119138 123
Table 2
Description of single instructions from Table 1
No. Instruction Description
1 cop Enzyme turns on copy mode,
until turned off, enzyme
produces complementary bases
2 off Enzyme turns off copy mode
3 mvr Enzyme moves one base to the
right
4 mvl Enzyme moves one base to the
left
5 rpy Enzyme nds nearest
pyrimidine to the right
6 rpu Enzyme nds nearest purine to
the right
7 lpy Enzyme nds nearest
pyrimidine to the left8 lpu Enzyme nds nearest purine to
the left
Fig. 2. Diagrammatic interpretation of instructions cop (A) and off
(B), an enzyme is represented by an oval rectangle attached both to
the lower strand which is copied, and the upper, which is the new
unnished copy. An active copy mode (on) is represented by the
dark rectangle, whereas its inactive copy mode (off) is represented
by the light rectangle. If an enzyme turns on copy mode (applying
the instruction cop), then enzyme produces on the upper strand
complementary bases.
Fig. 3. Diagrammatic interpretation of the instruction mvr, where
both a case of enzyme inactive copy mod (A) and a case of active
copy mode (B) are separately distinguished. The enzyme moves to
one base to the right, its action depends whether the enzyme is in an
inactive mode (A) or in an active mode (B). If the enzyme is in an
inactive mode, then it does not affect the second upper strand. On
the other hand, if the enzyme is in the active mode, then its movecreates a complementary base on the second upper strand. We have
to note, that if an enzyme is in the active state and the corresponding
position in the second upper strand is already created, then an
application of this operation is ignored. The same diagrammatic
scheme can be used also for the instruction mvl.
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species a binding site. This above relatively compli-
cated way of determination of the binding site was
introduced by Morris [22]. Original Hofstadter's
approach [15] is much simpler, the binding site is
specied only by the last arrow in the 2D enzyme
structure, i.e. the type of last arrow directly species
a binding site.
For a given strand S and its enzyme(S) we may
introduce the so-called replication process consisting
in an application of the enzyme(S) to the strand S. Thisreplication process is formally composed of the
following two steps:
Step 1. Construction of an enzyme composed of a
sequence of instructions (amino acids)
instructionS instrD1instrD2instrDp
11a
and a binding site X, i.e.
enzymeS instructionS;X 11b
Step 2. Enzyme enzyme(S) is applied to the strand S
so that its application is initialized at the base X
incoming rst from the left and then instructions are
step-by-step performed over the strand.
This simple process of transformation of the
(parent) strand Sonto another quasistrand (in general,
it may contain also hash symbols) R is called the
replication
replicationS R: 11cA strand R (offspring) is created in the course of
replication as a result of the replication process if at
some replication stage enzyme was switched to on
mode. In general, this strand R may be composed of
a number of empty hash symbols that appear in
the resulting strand when its length is smaller than
a length of the parent strand S. If the result of
replication is not a continuous strand, a strand R
is dened as the rst continuous part of the result
of replication. Diagrammatic representation of the
V. Kvasnicka, J. Pospichal / Journal of Molecular Structure (Theochem) 547 (2001) 119138124
Fig. 4. Diagrammatic interpretation of the instruction rpy, where both case of enzyme inactive copy mod (A) and active copy mode (B) are
separately distinguished. The enzyme moves by steps to the right of current position, until it nds nearest pyrimidine. Action of enzyme
depends on its mode, which can be inactive (A) or active (B). If the enzyme is in the inactive mode, then it does not affect the second upper
strand. On the other hand, if the enzyme is in the active mode, then each its move creates a complementary base on the second (upper) strand.
We have to note, that if an enzyme is in the active state and the corresponding position in the second upper strand is already created, then an
application of this operation is ignored. If the strand does not contain a pyrimidine to the right of the current position, then the enzyme is
stopped at the rightmost position. The same scheme is also applicable for instruction `rpu'. Slightly different scheme is also applicable to
instructions `lpy' and `lpu', i.e. enzyme nds nearest pyrimidine and purine, respectively, to the left of current position.
Fig. 5. An outline of four different cases of local properties of inclinations (AD), where an initial arrow is specied by a black bold arrow. For
instance, the rst diagram A represents three possible folds (directions of double arrow) created from the bold arrow (oriented from the left to
the right) if inclinations s, l, and r are applied. Diagram E corresponds to a 2D structure produced by an inclination sequence srssl.
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above two-step transformation (replication) is
outlined in Fig. 6.
Finally, we will discuss how to apply an enzyme
enzyme(S) toa strandS. Let us postulate that the enzymeis specied by enzymeS instructionS;X; where
Xspecies a binding site on the strand S. Two different
situations should be distinguished:
1. If X S, then the enzyme is inapplicable to the
strand.
2. If X[ S, then the enzyme is applied to the rst
appearance (from the left) of the base X. Enzyme
instructions (amino acids) are sequentially step-by-
step applied to the strand S.
3.1. Illustrative example of a strand replication
Let us have a strand S AA CG GG GA AG
TA TT composed of seven doublets. According to
Table 1, it is possible to construct a sequence of
instructions and inclinations that are assigned to the
given strand
instructionS mvrcoplpyrpymvrrpylpu
inclinationS lrrssrl:
Step 1. A sequence of folds (directed arrows)
constructed from inclination(S) looks as follows:
The rst inclination (s) and the last arrow ( ( )specify the binding site Xs; ( G:
Step 2. An application of sequence of instructions
instruction(S) starting from the binding site G gives the
following sequence of `intermediate' double strands:
V. Kvasnicka, J. Pospichal / Journal of Molecular Structure (Theochem) 547 (2001) 119138 125
Table 3
Different possibilities for binding site determination
No. First inclinationa Last arrow Binding site No. First inclinationa Last arrow Binding site
1 s ) A 7 l ( G
2 s * C 8 l * T
3 s + G 9 r * A
4 s ( T 10 r ( C
5 l + A 11 r ) G
6 l ) C 12 r + T
as (straight), l (left), and r (right).
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where the underlined bold letters in the lower strands
correspond to the current position of the enzyme. As a
result of its application to the strand S we get a
quasistrand (upper strand in the last double strandA7) composed of three hash symbols and 11 bases;
after removing the hash symbols we get
replicationS GCCCCTTCATA:
4. Autoreplicators
One of the central notions of articial (or algorith-
mic) chemistry [13,5,79,11,13,1618,2325,28
31] are autoreplicators, initially introduced in thebeginning of seventies by Eigen and Schuster [7,8]
as hypothetical biomacromolecules that are endowed
with standard `mass-law' kinetics and that are capable
of autoreplication catalyzed by themselves. These
authors demonstrated that in this `abiotic' level it is
already possible to observe phenomena closely resem-
bling Darwinianevolution basedon thesurviving of best
tted individuals (i.e. bestadapted biomacromolecules).
A double strand
A
R
S2 3
12
is called the autoreplicator, if the replication process
applied to both its parts results in
replicationS R and replicationR S
13a
in a composed form
replicationreplicationS S; 13b
i.e. the strand S is replicated to R, and the strand R is
replicated to S. In typogenetic environment, we
manipulate always with single strands, the above
presented denition should be considered as a two-
step process: in the rst step the strand S is repli-
cated to an offspring R, and then R is replicated to
the next offspring identical with the parent strand
S, see Fig. 7.
4.1. An example of autoreplicator
It is quite apparent that the above specication of
autoreplicators is very restrictive, pairs of comple-
mentary strands are subjected to two severe
constraints, in particular that each of them is repli-
cated exactly onto the other one. The main purpose
of this subsection is to present an illustrative example
of a simple strand
S GCCGTCTTTTCTCA
and to demonstrate that this simple nontrivial strand is
an autoreplicator. First, we construct enzymes of the
strands S and its complementary form R
CGGCAGAAAAGAGT; we get
enzymeS rpucoprpulpulpuoffmvr; G;
enzymeR coprpumvrmvrmvrrpylpu ; C:
Second, applying both of them to strands S and R
we get the following two sequences of replications
V. Kvasnicka, J. Pospichal / Journal of Molecular Structure (Theochem) 547 (2001) 119138126
Fig. 7. Schematic outline of an autoreplication process of a strand S, it may be considered as a double application of a scheme presented in Fig.
6. If the strand S is an autoreplicator, then an output from two replications is again the same strand S.
Fig. 6. Schematic outline of a replication process of a strand S. At
the rst stage an enzyme enzyme(S) is constructed, then, at the
second stage this enzyme is applied to the strand S by a set of
instructions ins(S) at a binding site X. Loosely speaking, we may
say, that each strand contains a necessary information for its repli-
cation process.
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that demonstrate an ability of the strand S to be an
autoreplicator:
We see that in both cases we have achieved in the
last replication an upper strand equal to the original
strand or its complementary form, which was to be
demonstrated.
4.2. An evolutionary construction of autoreplicators
For an application of evolutionary methods
[10,14,21] to construction of autoreplicators we
need a quantitative measure of a fact whether a strand
is autoreplicator or not. We introduce the so-calledtness of strands that achieves the maximal value if
the strand is an autoreplicator. Let us have a strand S,
its tness will reect its ability to be an autoreplicator.
In particular, let R S be a complementary strand to
the original strand S, applying to these two strands
independent replication processes we get
replicationS R 0 and replicationR S0:
14
Then a tness of S is determined as follows:
fitnessS 1
222 dS; R 02 dR; S0 15a
with values ranged by
0 # fitnessS # 1: 15b
Its maximal value fitnessmax 1 is achieved for
S R 0 and R S0 (i.e. strands S, R 0 and R, S0 are
complementary). This means that the maximal tness
value is achieved for strands that are autoreplicators.
A mutation represents very important innovation
method in evolutionary algorithms. In particular,
going from one evolutionary epoch to next epoch,
individuals of a population are reproduced with
small random errors. If this reproduction process
was always without spontaneously appearing errors,
than the evolution does not contain `variations' that
are a necessary presumption of the Darwinian evolu-
tion.
Letus consider a strandS X1;X2; ;Xn; this strand
V. Kvasnicka, J. Pospichal / Journal of Molecular Structure (Theochem) 547 (2001) 119138 127
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is transformed onto another strand T Y1; Y2; ; Yn;
(where Y's are bases or empty symbols) applying a
stochastic mutation operator OmutT Omut S : 16
This operator is realized in such a way that on going
successively from the left to the right, each element
(base) with a small probability Pmut is either changed
(mutated) to another base, or deleted from the strand,
or enlarged from the right by a new randomly selected
base
OmutAACGTTA
TACGTTA mutation;
AAOGTTA deletion;
AACGATTA insertion;
AACGTTA exact copy;
VbbbbbbbbbbX
17
where the rst three particular cases (mutation,
deletion, and insertion) are realized with the same
probability.
Evolution of strands towards an emergence of
autoreplicators in population (composed of strands
that are considered as objects of Darwinian evolution)
is simulated by the following simple evolutionary
algorithm (see Fig. 8):
1. a population is represented by a population of
single strands, and
2. in reproduction process a mutation operator is
applied to a randomly selected parent strand,
creating one offspring.
4.3. Results of computer simulations of evolutionary
emergence of autoreplicators
The above formal denition of the autoreplicator is
relatively complicated, it requires two-step process to
verify whether a strand is an autoreplicator. Varetto
[30] studied a systematic constructive way for the
construction of autoreplicators, which is applicable
for shorter strands or for strands with the same
repeated `motif'. In order to demonstrate full capacity
of typogenetics for AL studies, a simple evolutionary
algorithm is applied to achieve an evolutionary spon-
taneous emergence of autoreplicators (see Fig. 8). The
basic parameters of the algorithms were set as
follows: size of population N 1000, minimal and
maximal lengths of strands lmin 15 and lmax 30:
Probability Pmut was set variable during the course of
evolution, at the beginning of evolution its value is
maximal Pmaxmut and then it decreases to a minimal
value Pminmut A current value of probability for an
evolutionary epoch t is determined by
Pmut Pmaxmut 2 Pmaxmut 2 Pminmut t
tmax; 18
where tmax is the length of evolution (maximal number
of epochs). In our calculations, we set Pmaxmut 0:01
and Pminmut 0:001:
In order to get a better insight into numerical results
we introduce the following set of parameters that is
successfully used in simulated annealing [19]. The
interval 0; 1 of tness values is decomposed onto
N subintervals Ik xk21;xk; where xk k=N;
V. Kvasnicka, J. Pospichal / Journal of Molecular Structure (Theochem) 547 (2001) 119138128
Fig. 8. A diagrammatic visualization of a simple model of Darwinian evolution, where a population is composed of strands evaluated by tness.
A strand is quasirandomly selected to a reproduction process, a probability of this selection is proportional to the strand tness, strands with a
greater tness have a greater chance to be selected to the reproduction process. The reproduction process consists in simple copy process, where
a strand is simply reproduced with possibility of appearance of stochastic mutations (specied by the probability Pmut). If new population Q
composed of offspring of reproduction process has the same number of individuals as the original population P, then the population P is updated
by the population Q, P Q.
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for k 1; 2; ;N: A population P is specied by aprobability distribution 0# wk# 1, it determines a
fraction of strands from the population P with tness
values from the interval Ik,
N21k0
wk 1: 19
Let us dene the following four entities:
1. Mean value of tness
fh i Nk1
xkwk: 20a
2. Mean value of the second power of tness
f2
h i
Nk1
x2kwk: 20b
3. Dispersion
s fh i22 f
2h i
: 20c
4. Entropy
S 2
N
k1
wk lnwk: 20d
The last two statistical parameters (dispersion and
entropy) tend to zero when a population is evolved
towards a state composed substantially of identical
strands.
Another proper method to visualize evolution is a
plot of distance between the temporarily best strand
Stbest (specied for the evolutionary epoch t) and the
best strand resulting from the whole evolution Sall
best :
V. Kvasnicka, J. Pospichal / Journal of Molecular Structure (Theochem) 547 (2001) 119138 129
Fig. 9. Four different plots that characterize evolutionary emergence of autoreplicators. Diagram A shows plots of maximal tness, mean
tness, and a frequency of appearance of temporarily best strand. Diagram B shows plots of mean tness (already presented in diagram A) and
the mean of second power of tness. A difference of these two tness determines the so-called dispersion, displayed in diagram C. Its initial low
values are caused by low initial values of mean tness. Its big positive values indicate very intensive `structural transitions', i.e. new strands,
which are identical with the temporarily best strand, permanently emerged in population. At the end of evolution, when the population is
already composed almost entirely of copies of one of the best strands, dispersion tends to small positive numbers. Similar formal interpretation
has also entropy displayed in diagram D. Its values `monotonously' decrease to small values.
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Since the strands Stbest and Sallbest may be, in general, of
different length, the distance specied in Section 2
(see Eqs. (4a) and (4b)) is not applicable for this
consideration. It means that we have to determine a
notion of distance in a more general way than the one
mentioned in Section 2. Let us consider two strands
S X1X2Xn and R Y1Y2Ym; their lengths are
uSu n and uRu m; respectively. Let p min{m; n}be a minimal distance of strands S and R, then analternative distance between them is determined by
DS;R uSu1 uRu2 2pi1
dXi; Yi
; 21
where d is an analog of Kronecker's delta already
dened by Eq. (4b). A positive value of this new
distance reects a measure of difference between
strands S and R, its vanishing value corresponds to a
fact that both strands are identical. A plot of
DStbest; Sall
best visualizes a way of approaching oftemporarily best strands through the evolution to the
nal and resulting best strand that may be considered
as a result of the evolutionary emergence of
autoreplicators.
Different plots are shown in Fig. 9. The rst
diagram A corresponds to plots of maximal and
mean tness and a frequency of appearance of
temporarily best strand. At the beginning of evolution
there appeared a mixture of different strands. As the
population was more evolved (say starting from 500
epochs), where a nal solution (an autoreplicator) was
already created, its fraction of appearance almost
monotonously increased to unit value. Diagrams B
and C are closely related, diagram B shows a plot of
the mean tness kfl and the mean of second power oftness kf2l, whereas diagram C shows a plot of disper-sion (derived as a difference of the previous two
tness, see Eq. (20c)). The last diagram D shows a
plot of entropy, its big positive values (similar proper-
ties has also the dispersion) indicate that the popula-
tion is very far from an equilibrium state composed
entirely of identical autoreplicators. Fig. 10 shows a
plot of a distance D between temporarily best strand
and the best nal strand (autoreplicator) produced by
the evolution of population. We see that the distancedecreases with small uctuations so that starting from
the half of evolution this distance is vanishing, i.e. the
correct strand (or strands) has emerged from the
evolution. The following set of observations from
our numerical results can be formulated (see Ref.
[12]):
1. There do not exist dramatic changes in the compo-
sition of best strands throughout the whole popula-
tion period. Rather, we see that evolution of
autoreplicators is very opportunistic, it containsonly small changes in compositions of strands
such that whole evolution is inherently directed
to an emergence of autoreplicators.
2. Moreover, there exist long evolutionary periods in
which the maximal tness is kept xed and small
changes appear in composition of strands. Such
evolutionary periods are called the neutral periods,
in which evolution `gathers' an information for
changes that lead to a substantial increase of
quality of strands towards their ability to be
autoreplicators.
5. Typogenetic articial chemistry
Recently, in AL has become very popular the so-
called articial (or algorithmic) chemistry [6], where
a metaphor of `chemical reaction' (an elementary
interaction between molecules) is applied to simulate
an emergence of autoreplicators or more complicated
V. Kvasnicka, J. Pospichal / Journal of Molecular Structure (Theochem) 547 (2001) 119138130
Fig. 10. Plot of distance DStbest; Sallbest ; where S
tbest is a temporarily
best strand (for an epoch t) and Sallbest is a best strand (an
autoreplicator) produced by the evolution of population. The
distance D is determined by Eq. (21). The displayed plot indicates
that distance D monotonously decreases (with some small uctua-
tions due to random genetic drift in population) to zero value, whichindicates a spontaneous emergence of an autoreplicator (Sallbest)
at the end of evolution.
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`molecular' structures (e.g. hypercycles), see Fig. 11.
We apply this interesting idea in an attempt to formu-
late an abstract chemostat that will simulate an emer-
gence of autoreplicators.
A chemostat is formally considered as a multisetcomposed of n strands
P {S1; S2; Sn}: 22
Each strand S[ P is evaluated by a tness that
reects its ability of autoreplication (see Eqs. (14)
(15b))
fitnessS 1
222 dS; S02 dS; S00 23
where S0 replicationS and S00 replication S.
For a randomly selected parent strand S[ P (a prob-
ability of the selection is merely proportional to its`concentration' in the chemostat) a chemical reaction
is applied
S !prob
S1 S0; 24
where offspring strand S0 is created from the parent
strand S by a mutation operation, S0 OmutS: The
above reaction is performed with a probability
probS e2bfitnessmax2fitnessS; 25
where tnessmax is a maximal value of tness achieved
in the previous history of the chemostat until now and
b is a positive parameter specifying a sensitivity of
the probability prob to the size of difference between
currently maximal tness and a tness of the given
strand S. In particular, an increasing value ofbmeans
that the probability is progressively smaller for
strands with tness much smaller than its currently
maximal value. A pseudo-Pascal implementation of
the articial chemistry approach to an emergence ofautoreplicators in the chemostat is outlined in
Algorithm 1.
What resemblance or differences can be found
between evolutionary algorithm presented in Section
4 and a metaphor based on chemostat device? They
may be formulated as follows:
1. In evolutionary algorithms, selection of strands to
reproduction process is proportional to strand
tness, where: (a) all selected strands participate
automatically in a reproduction process and (b) a
return of offspring created by reproduction process
to population is performed so that strands with
lower tness are eliminated from the population.
2. In chemostat approach, selections of strands to
chemical reaction process are performed fully
randomly (independently of their tness). (a)
Unlike the evolutionary approach, a reaction is
applied to a randomly selected strand with a prob-
ability depending on tness, e.g. strands with a
higher tness participate in the chemical reaction
V. Kvasnicka, J. Pospichal / Journal of Molecular Structure (Theochem) 547 (2001) 119138 131
Algorithm 1. A pseudo-Pascal code of typogenetic articial chem-
istry. The algorithm is initialized by a random creation of the
chemostat P and by evaluation of all strands by tness. The evolu-
tion of chemostat is composed of kmax elementary acts (chemical
reactions). An operator Oselect performs a random selection of a
strand (probability of this selection is proportional to its concentra-
tion) from the chemostat.
Fig. 11. An evolution of strands in articial chemistry is simulated
by a chemostat (well stirred chemical ow reactor vessel). The
chemostat contains a homogeneous `solution of strands', a strand
is fully randomly selected (with a probability determined by its
concentration), then this strand undergoes a chemical reaction
(with probability determined by the tness of strand) that consists
of a reproduction (copy) of the strand with small errors (mutations).
The produced copy (offspring) is returned to the chemostat in such a
way that it eliminates a randomly selected strand.
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with a higher probability, and (b) a product of
chemical reaction (offspring) is returned to the
population so that a randomly selected strand is
eliminated.
It is evident that the differences between evolution-
ary approach and the approach based on the metaphor
of chemical reaction are not of substantial character.
These differences depend mainly on the point of view,
which determines features of the given approach that
are accented or suppressed. We believe that the most
important difference between evolutionary and arti-
cial-chemistry algorithms exists in a shift of tness
usage from the selection to the reaction probability
(an analog of rate constant in chemical kinetics). In
chemostat approach a selection of an individual
intended for chemical transformation should be fully
determined by a `concentration' of the individual(there is used the so-called mass-action low kinetics).
The basic parameters of the present articial-chem-
istry algorithm were set as follows: size of chemostat
N 1000; mutation probability Pmut 0:001; mini-
mal and maximal lengths of strands lmin 15 and
lmax 30; and the evolution of chemostat was
watched one million epochs (i.e. tmax 106). The
obtained results displayed in Figs. 12 and 13 are very
similar to those ones obtained by the evolutionary
V. Kvasnicka, J. Pospichal / Journal of Molecular Structure (Theochem) 547 (2001) 119138132
Fig. 12. Four different plots that characterize chemostat emergence of autoreplicators. All comments to single diagrams are the same as in Fig. 9.
Fig. 13. Plot of distance DStbest; Sallbest ; for chemostat emergence of
autoreplicators, see comment in Fig. 10.
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algorithm presented in Section 4. We see that the
chemostat approach offers results that are closely
related to our evolutionary approach to an emergence
of autoreplicators outlined in Section 4. Summarizing,
both approaches, evolutionary as well as chemostat,are able to perform a spontaneous emergence of auto-
replicators. An evolution of population or chemostat
runs in such a way that strands with tness slightly
below one are very quickly created at rst stage of
evolution, and then almost all remaining evolution is
spent to create strands (autoreplicators) with unittness.
6. Hypercycles
According to Eigen and Schuster [7,8], hypercycle
is a kinetic composition of replicators, where a repli-
cation ofAi is catalyzed by enzymes produced by the
previous replicator Ai21
Ai 1 Ei21 ! 2Ai 1 Ei21 for i 1; 2; ; n;
where Ei21 is an enzyme produced by the previous
replicator Ai21, and A0 An; E0 En: Hypercycles
may be considered as multilevel hierarchical catalytic
kinetic systems. They represent an important concept
of the current mental image of an abiotic period of
molecular evolution. Autoreplicators, which emerged
in the rst stage of this evolution, may be integratedinto higher level kinetic systems that represent units
relatively independent from other autoreplicators or
hypercycles. Moreover, hypercycles represent an
uncomplicated example of an increase of complexity
[27], with well described mathematical model and
simple computer implementation [8].
Let us consider a sequence of replicators S1, S2,,Sn,
that are mutually related in such a way that a replication
ofSi is catalyzed by an enzyme enzyme(Si21) produced
by the previous strand Si21 (a previous strand with
respect to S1 is a strand Sn), see Fig. 14. Applying ametaphor of chemical reactions, a hypercycle can be
represented as a sequence of the following reactions:
Si !enzyme Si21
Si 1 Si and Si !enzyme Si21 Si 1 Si
26
for i 1,2,,n. We see that their precise determination
V. Kvasnicka, J. Pospichal / Journal of Molecular Structure (Theochem) 547 (2001) 119138 133
Fig. 14. Diagrammatic visualization of a sequence of n replicators S1, S2,,Sn integrated in a hypercycle structure. The same scheme is
applicable also for complementary strands S1; S2; ; Sn: Symbol e(Si) means enzyme created from the replicator Si, which is used for
construction of complement of the next strand Si11 from itself (see (26)). Bold loops represent this autoreplication, while interrupted arrows
show, that this autoreplication is caused by an enzyme created by a previous replicator in hypercycle.
Fig. 15. An illustrative example of 2-hypercycle. The upper diagram corresponds to a general scheme of hypercycle (see Fig. 14). The lower
diagram represents a scheme of single replication reactions that are catalyzed by enzymes produced by a `previous' replicator. We leave to
readers a verication whether S1 is a replicator catalyzed by an enzyme produced by S2, and reciprocally for S2.
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is highly restrictive and may give rise to very seriousdoubts whether hypercycles can exist and be
constructed (e.g. within Typogenetics).
Recently, Varetto [31] has introduced the so-called
tanglecycles as an alternative to our hypercycles that
were specied in a way closely related to their origi-
nal meaning proposed by Eigen and Schuster [7,8]. In
particular, in a specication of tanglecycles there is
suppressed an autoreplication character of strands,
Varetto only required that there exists a replication
of a strand Si onto another strand Si11 and this process
is catalyzed by an enzyme of Si21 strand (he does not
specify properties of complementary strands takingpart in the tanglecycle)
Si !enzymeSi2 1
Si 1 Si11 i 1; 2; ; n; 27
where S0 Sn and Sn11 S1: The main differencebetween hypercycles and tanglecycles is the fact
that the strands in hypercycles, unlike the tangle-
cycles, are coupled only through enzymatic catalysis,
while in tanglecycles inside a replication ofSi the forth-
coming strand Si11 is created. The present version of our
typogenetics machinery is not applicable to a study of
tanglecycles, since a replication product Si11 of a strand
Si (see Eq. (27)) should be a complementary strand toSi,
i.e. we could not expect that by applying a sequence of
reactions (27) for n$ 3 we get at its end a product
identical with the initial strand S1.
6.1. An illustrative example of 2-hypercycle
Let us consider two doublestrands (DNA
molecules) A1 and A2:
V. Kvasnicka, J. Pospichal / Journal of Molecular Structure (Theochem) 547 (2001) 119138134
Fig. 16. Four different plots that characterize evolutionary emergence of hypercycles composed of two replicators (2-hypercycle). All
comments to single diagrams are the same as in Fig. 9.
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Their strands create a hypercycle (see Fig. 15)
composed of two elements (2-hypercycle) specied
by the following sequence of reactions:
S1 !enzymeS2
S1 1 S1; S1 !enzyme S2 S1 1 S1;
S2 !enzymeS1
S2 1 S2; S2 !enzyme S1 S2 1 S2:
6.2. An evolutionary construction of hypercycles
A population P is composed of hypercycles (or
hopefully future hypercycles) that are composed of
the same number n of strands. Each hypercycle of
the population is evaluated by a tness that reects
its ability of all its components to autoreplicate itself.
Hypercycles are selected quasirandomly (with a prob-
ability proportional to their tness) to simple repro-
duction process with a possibility of stochastic
mutations (controlled by a probability Pmut). The
design of the evolutionary algorithm is the same as
for the evolution of single autoreplicators.
The tness of a hypercycle is determined as
follows: let us consider a hypercycle and its
complementary form
x S1; S2; ; Sn and x S1; S2; ; Sn
28a
Si !eSi2 1
Si 1Ri and Si !e Si21 Si 1R
0i: 28b
Each ith component (Si and Si) is evaluated by a
`local' tness
fitnessi 1
222 dSi; Ri2 d Si; R
0i: 29
A tness of the hypercycle x is determined as a
minimum of local tness of its constituents
fitnessx mini
fitnessi: 30
Loosely speaking, a tness of a hypercycle is deter-
minedby a local tness of its weakest replicator (a chain
is as strong as its weakest link). A Darwinian evolu-
tion of strands towards an emergence of hypercycles
in a population is simulated by a simple evolutionary
algorithm used for an evolution of autoreplicators (see
Fig. 8).
The basic parameters of the present evolutionary
algorithm that was used for an emergence of hyper-cycles are set as follows: size of population N
2000; mutation probability Pmut 0:001; minimal
and maximal lengths of strands lmin 15 and lmax
30; and the evolution of population is watched two
thousands epochs (i.e. tmax 2000). The obtained
results are displayed in Fig. 16. We see that the evolu-
tionary approach offers 2-hypercycles; if the same
approach was used for higher hypercycles, then we
never succeeded in their emergence. Main conclu-
sions from computer simulations of evolutionary
emergence of hypercycles:
1. An evolutionary emergence of hypercycles
composed of more than two autoreplicators is a
very improbable evolutionary event. In other
words, it represents for evolutionary algorithms a
very difcult combinatorial task.
2. More complex hypercycles may be evolutionary
constructed from simpler hypercycles such that
they are enlarged by another autoreplicator with
evolutionarily optimized composition.
V. Kvasnicka, J. Pospichal / Journal of Molecular Structure (Theochem) 547 (2001) 119138 135
Fig. 17. Schematic visualization of an enlargement of a 2-hypercycle (diagram A) onto a 3-hypercycle (diagram C). In the rst period a parasitic
replicator S3 is attached to the hypercycle (diagram B), its replication is catalyzed by an enzyme produced by the second replicator S2. In the
second period, the parasitic replicator S3 is `evolutionary incorporated' into hypercycle, i.e. its enzyme catalyzes a replication of S1. In other
words, it means a structure of the parasitic replicator S3 is evolutionary reoptimized such that it will be incorporated into the greater hypercycle.
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6.3. Creation of larger hypercycles from simpler
hypercycles
The main conclusion of our simulations, outlined in
the previous subsection, is that an evolutionary
construction of hypercycles composed of more than
two replicators belongs to very hard combinatorial
tasks. This is the main reason why we turn our
attention to another evolutionary possibility of their
construction. Fig. 17 outlines a simple way of enlar-
gement of a smaller hypercycle onto a bigger one such
that a replicator is incorporated. This simple approach
may be simply formulated in a form of an evolution-
ary algorithm. Let us consider a hypercycle x
S1; S2; ; Sn composed of n replicators with their
complementary strands, and let it be enlarged by areplicator denoted by Sn11. We postulate that this
new strand Sn11 is incorporated into the hypercycle
such that: (1) its replication is catalyzed by an enzyme
enzyme(Sn) and (2) its enzyme(Sn11) catalyzes a
replication of the strand S1
Sn11 !eSn
Sn11 1Rn11 and
Sn11 !e Sn Sn11 1R
0n11;
31a
S1 !eSn11
S1 1R1 and S1 !e Sn1 1 S1 1R
01: 31b
A tness of the new strand Sn11 is determined as
follows:
fitnessSn11 1
442 dSn11; Rn11
2 d Sn11; R0n112 dS1; R12 d S1; R
01
32
with values ranged by 0#tness(Sn11)# 1. Its maxi-
mal value corresponds to a situation where the strand
Sn11 is exactly replicated to a complementary strandSn11 (catalyzed by enzyme(Sn)) and the strand S1 is
exactly replicated to S1 (catalyzed by enzyme(Sn11)),
and similarly for the replication of complementary
strands Sn11 and S1.
The above approach to construction of larger
hypercycles from smaller ones can be simply imple-
mented within an evolutionary algorithm. The basic
advantage of the suggested method is its capability to
overcome a combinatorial complexity that has
severely plagued standard evolutionary approach
discussed in Section 6.2. This standard approach is
now modied in such a way that from a previouscalculation we know an n-hypercycle. Its enlargement
by a new strand Sn11 is evolutionary optimized
(according to tness (32)) while the original n-hyper-
cycle is kept xed through the whole enlargement
evolution. Since this evolutionary approach is a very
simple modication of the original algorithm
specied in Section 6.2, its numerical properties are
very similar to those ones presented in Fig. 16 and
therefore we do not present here illustrative plots.
V. Kvasnicka, J. Pospichal / Journal of Molecular Structure (Theochem) 547 (2001) 119138136
Fig. 18. An illustrative example of a 3-hypercycle and a 4-hyper-
cycle that were evolutionary constructed from 2-hypercycle
displayed in Fig. 15.
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We present here only a simple illustrative example of
3-hypercycle that was constructed by an evolutionary
enlargement process of the already known 2-hyper-
cycle presented in Section 6.1. This 3-hypercycle is
composed of three DNAs A1, A2, and A3.
These double strands form a 3-hypercycle with
components specied by the following reactions
(see Fig. 18):
S1 !enzymeS3
S1 1 S1; S1 !enzyme S3 S1 1 S1;
S2 !enzymeS1
S2 1 S2; S2 !enzyme S1 S2 1 S2;
S3 !enzymeS2
S3 1 S3; S3 !enzyme S2 S3 1 S3:
7. Summary
It seems, according to our results, that typogenetics
offers new analogies and formal tools for computer
scientists active in articial life. A central `dogma' of
the typogenetics is that strands have twofold role:
First they are replicators, and second, they code an
information about the way of their replication.
Formally, typogenetics may be considered as a
molecular automaton that on its input reads strands
and on its output it replicates strands. To make such
an automaton more interesting, we may endow strands
with additional properties enabling them to behave in
some specic manner. In the present simple approach,
strands have innite resources for their replications. If
we introduce a limited space of resources, then we get
an additional selective pressure (a struggle for raw
materials) with respect to a selection entirely based
on strand tness that reect their capability of
replication. As was already clear in evolutionary
construction of hypercycles, an introduction of a
`geographical' distributions of strands in a population
might be very important. In that case, the population
could not be considered as a homogeneous well-stirred
chemostat. A replication function of strands usually
requires only a fraction of the enzyme that is coded in
the strand; it is then possible, in general, to code addi-
tional strand or enzyme properties that may give rise to
an emergence of new properties and hierarchically orga-
nized structures. Summarizing, typogenetics represents
a very rich and exible formal tool, closely related to
basic concepts of molecular biology, that opens new
possibilities and horizons for articial life activities
and efforts.
Acknowledgements
This work was supported by the grants # 1/7336/20
and # 1/8107/01 of the Scientic Grant Agency of
Slovak Republic. We are also thankful to a referee
for bringing to our attention the biochemical works
of McCaskill and Biebricher.
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