Auto Replica Tors and Hyper Cycles in Typo Genetics

8/3/2019 Auto Replica Tors and Hyper Cycles in Typo Genetics

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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


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


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:


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


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



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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;


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 :


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


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


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


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.


15/20

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


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.


16/20

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:


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.


17/20

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.


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.


18/20

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.


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.


19/20

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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Auto Replica Tors and Hyper Cycles in Typo Genetics