the_ant_problem - Laboratory of Evolutionary Modeling

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CO-EVOLUTION OF GENOMES WITH THEIR GENOMIC
PARASITES IN GENETIC ALGORITHMS:
APPLICATION TO THE ANT PROBLEM
Alexander V. Spirov1, 2, *, Alexander B. Kazansky1, Alexey S. Kadyrov1, Juan J. Merelo3 and Vladimir F.
Levchenko1
1The
Sechenov Institute of Evolutionary Physiology and Biochemistry, 44 Thorez Ave., St. Petersburg, 194223,
Russia
2Dept.
of Applied Mathematics and Statistics, The State University of New York at Stony Brook, Stony Brook NY
11794-3600, USA
3Departamento
de Arquitectura y Tecnología de Computadores, University of Granada, Granada, Spain
*Corresponding
author: Dept. of Applied Math and Statistics, The University at Stony Brook, Stony Brook NY
11794-3600
Email: [email protected]; v:631-632-8370; f:631-632-8490
ABSTRACT
Genetic Algorithms (GA) and Genetic Programming were inspired by classic Darwinian ideas of evolution.
However modern evolutionary biology is far beyond classic Darwinism. The application of other algorithms and
modern biological ideas may substantially improve the performance of Evolutionary Computations (EC).
Macroevolutionary mechanisms based on the mobile selfish genetic elements - transposons are good candidates
for this breakthrough. The processes in the world of transposons, living on a substratum of genomes of whole
biological communities, are thought to be the main source of evolution creativity.
In this communication we propose a strategy of construction of a new scheme for EC exploiting the
most essential aspects of co-evolution of the genomes with their genomic parasites. We named this strategy as
construction of the Two-layered Co-evolving Worlds. Our approach exploits implicitly hybrid character of
evolutionary mechanisms discovered by modern evolutionary biology, where global search combined with local
one, as well as random mutagenesis combined with targeted one.
We apply our approach to one of known benchmark tests - the ant problem. We found that our
enhancement of GA technique by the artificial transposons obviously increase the efficacy of searching of the
ant's navigation algorithm. We investigate in details the transposons activity as example of sophisticated local
search.
INTRODUCTION
Many areas of evolutionary computation, especially genetic algorithms (GA), and genetic programming (GP),
were inspired by ideas from evolutionary biology. However modern evolutionary biology has since advanced
considerably, revealing that Darwinian evolution (microevolution) apparently is a particular case of essentially
more complicated mechanisms of macroevolution. Many branches of modern evolutionary computation research
realize evolution of mechanisms, such as functional programs, neural networks, decision trees, cellular automata,
L-systems, finite state automata. For these domains, recent achievements in genomic seems more appropriate as
an inspirational model than classic set of Darwinian algorithms [Cf. Altenberg, 1994; Luke et al., 1999; Lee and
Antonsson, 2001; Lones and Tyrrell, 2001].
This stimulates us to select, formalize and apply to EC some new evolutionary mechanisms that could
help to simulate the creative, heuristic and self-organizing character of (biological) evolution [Spirov, 1996a;
1996b; Spirov and Samsonova, 1997; Spirov and Kadyrov, 1998; Spirov et al., 1998; Spirov and Kazansky,
1999; 2002a; 2002b; 2002c].
We concentrate on mechanisms of Natural Genetic Engineering, whose key actors are mobile selfish
genetic elements. The basic idea can be outlined as follows:

The genome of every organism has got very special mechanisms of genomic rearrangements;

These mechanisms are activated in the period of evolutionary crisis;
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
The mechanisms cause multiple systemic rearrangement of genomes in a few generations.
The mobile selfish genetic elements (synonymous or related terms are jumping genes, selfish DNA,
transposons and retroelements [Makalowski, 1995; 2200; Lozovskaya, 1995; Hurst & Werren, 2001]) are good
candidates for this breakthrough. Many biologists speculate that processes in the world of transposons, living on
a substratum of genomes of the whole biological communities, are the main source of macroevolution creativity
[Brosius, 1991; King, 199?; Shapiro, 2000; 2002].
There are several groups of genomic parasites. Transposons form the most sophisticated one. “…it is
now recognized that a significant portion of the genome of any eukaryote is composed of 'selfish' or 'parasitic'
genetic elements, which gain a transmission advantage relative to other components of an individual's genome,
but are either neutral or detrimental to the organism's fitness” [Hurst & Werren, 2001]. Arguments have been
made that the structure of eukaryotic genomes, including the abundance of transposons, repetitive DNA and
introns, provide high evolvablity [Shapiro, 2000; 2002].
It has been estimated that typically 98% of the DNA in higher organisms is neither translated into
proteins nor involved in gene regulation, that it is simply ``junk'' DNA. The bulk of the junk DNA is
intermediate-repeats comprised of DNA elements that are able to move (or transpose) throughout the genome.
These mobile DNA elements are sometimes termed "selfish DNA," since they do not appear to directly benefit
the host. Their behavior is purely parasitic: they jump between different parts of a genome in order to propagate
themselves, and, from the first sight, this is usually to the detriment of their host. But they do appear to benefit
the host organism by providing genomic rearrangements that permit the evolution of new genetic networks.
Transposons are ubiquitous and may comprise up to 45% of an organism's genome [Lozovskaya, 1995;
Makalowski, 1995; 2200; Hurst & Werren, 2001]. DNA of transposon origin can be recognized by their
palindrome endings flanked by short non-reversed repeated sequences resulting from insertion after staggered
cuts. Many transposons have a unique DNA site, which acts as a forwarding address, directing the transposon to
a complementary DNA site in its host genome. There are usually multiple copies of any given DNA site in the
host genome and transposon will attach the proper site in random manner. Transposons may grow be acquiring
more sequences; one such mechanism involves the placement of two transposons into close proximity so that
they act as a single large transposon incorporating the intervening code.
Some data was obtained recently which evidence that transposable elements are a major source of
genetic change, including the creation of novel genes, the alteration of gene functions, and the genesis of major
genomic rearrangements [Fadool et al., 1998; Hurst & Werren, 2001;Lozovskaya, 1995; Makalowski, 1995;
2200]. The long coexistence of transposable elements in the genome is expected to be accompanied by host transposon co-evolution. In this connection, special interest is attracted by known examples of both competitive
and cooperative strategies in populations of transposons. That is why transposons are treated as higher-level and
intelligent mutators. Besides, transposons can be considered as evolutionary “SOS-crew”, cooperatively acting
“in emergency”, at host’s genomic stress [McClintock, 1984; Lozovskaya, 1995].
Modern evolutionary biology and evolutionary genetics (and genomics) compile a lot of knowledge
about very complicated mechanisms of genome rearrangements governed by transposons [Agrawal et al., 1998;
Brosius, 1991; Hurst & Werren, 2001; King, 199?; Makalowski, 1995; 2200; Lozovskaya, 1995; Shapiro, 2000;
2002]. From EC point of view the most sophisticated of these mechanisms can be treated as local evolutionary
search. First of all, we mean mutational / recombinational hot spots caused by transposons. It is very promising
to include the world of selfish elements in the basic frames of Evolutionary Computations (EC) and to use the
objects/procedures for maintaining and manipulation of the parasites as a new prospective branch of EC. In this
article we study in details co-evolution of artificial transposons with their hosts for GA version of the ant
problem.
State-of-the-Art Biologically Inspired EC
There is a feeling that the field of EC is getting more inspired with the latest achievements in biology, trying to
make the evolutionary algorithms more effective. These attempts have been termed Genomic algorithms,
bacterial algorithms or even “genetic” algorithms (quotes intended). Many other techniques taken from biology,
such as transposition, host-parasite interaction and gene-regulatory networks have also been applied to
evolutionary computation. There is no uniform nomenclature, however, and sometimes the same terms are
applied to different methods, or the other way round.
These techniques can be divided broadly into three groups:
•Host-parasite methods. These methods are based on the co-evolution of two different populations, one
of them acting as “parasite”, and the other acting as “host”; the parasites usually encode a version the problem
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domain, and the hosts encode the solution to the problem [Hillis, 1990; Potter and De Jong, 1994; 1995; De Jong
and Potter, 1995; Olsson, 1996; 2001]. They have been used mainly to evolve sorting networks; but similar
approaches have been used, for instance, for evolving players of the Othello game. In this case, the approach is
not significantly different from other co-evolutionary approaches, in which solutions and instances of a problem
are co-evolved.
•Transposition operators. They are sometimes known as “bacterial” algorithms [Harvey, 1996, Nawa et
al., 1996; Simoes and Costa, 2001]. The basic idea of both approaches is to make intra-chromosome crossovers,
that is, crossover of a chromosome with another part of itself, or else asymmetric crossover, in which a donor
chromosome transfers part of its genetic material to an acceptor chromosome. In some cases, these operators
seem to be better than classical genetic algorithms for combinatorial optimization problems, and also in the case
of a fuzzy rule base. Similar transposition operators have also been used in other contexts, mainly in
combinatorial optimization problems.
•Other biological approaches. Luke et al. [Luke et al., 1999] use a method similar to genetic regulatory
networks to evolve finite state automata that represent a language grammar; this kind of objects cannot be easily
represented using serial, bitstring, genetic algorithms. Burke et al. [Burke et al., 1998] try to make the
evolutionary process closer to its real molecular genetics base, by having a 4-letter genetic alphabet, transduction
processes, and variable-length genetic algorithms. In both cases, it is not a general-purpose techniques; they only
draw some elements from biology to solve problems that would be difficult to solve in other cases.
So far, it has not been found in the literature a technique that is general enough to be applied to a wide
range of problems, and that, in some cases, is able to yield as good or better results than evolutionary algorithms.
In this paper we try to outline such an approach, and apply it to one of known benchmark problems – John Muir
ant’s trail test [Jefferson et al., 1991; Koza, 1992].
In this article we propose general strategy of construction of a new scheme of EC exploiting the most
essential sides of genomes - genomic parasites co-evolution. We named this strategy as construction of the Twolevel Evolving World. In the next parts we give the definition of the strategy. One of the key features of the
approach is very special local search performed by populations of artificial transposons. After description of the
local search we demonstrate the efficacy of a new approach on the example of John Muir ant’s trail test.
Key Characters of Natural Transposons as Inspirational Model
Mobile (transposable) elements encode the ability to move to new positions in the genome and can therefore be
accumulated in genomes. These genomic parasites live on a substratum of gene pool of the whole biological
community. They are the autonomous selfish or parasitic codes, which co-evolve with their hosts. They are
transmissible horizontally (from one host to another) and vertically (from the host-ancestor to the descendants).
Transposons act as intelligent and sophisticated mutators for the host codes (genes). They cause
different types of mutations when they transpose themselves. These mechanisms are based both on enzymes
encoded both by transposon own genes and host genes.
Class I transposable elements transpose through an RNA intermediate. After the element is transcribed,
the RNA copy is converted into DNA, frequently as a result of reverse transcriptase activity encoded in the
element itself. This DNA copy now reinserts in the genome at another location. Transposition of class II
elements involves DNA excision and homologous repair. Each class II element encodes a transposase, which
excises the element from the chromosome. The point of excision is repaired using the sequence of homologue or
sister chromatid as a template, which creates a duplicate of the transposable element. The excised copy is free to
insert at another site within the genome.
Different mechanisms of transposon activity cause different types of mutations:
1)
Non-random insertions. The insertion of transposons (by above sketched mechanisms) is
generally non-random but targeted. Certain regions of the host DNA are favored for insertion; these are referred
to as hot spots and tend to be specific to a transposon or its family. The insertional specificity based on factors
encoded by transposon genes.
2)
Recombinational hot-spots. Transposable elements serve as recombinational hot-spots
allowing the exchange of genetic material between unrelated chromosome sequences [Makalowski, 1995].
Because they are so abundant, they can mediate large scale chromosome restructurings by means of homologous
recombination between similar transposable elements at distant locations.
3)
Mutational hot-spots. For instance, two enzymes key to generating the immune system's
inexhaustible variety of antibodies are relics of an ancient transposon. A great deal of this variability is due to the
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way antibody genes are assembled: by joining three separate sequences (denoted V, J, and D), each of which
comes in many variants, into a single antibody gene. The enzymes that do this assembly, Rag1 and Rag2, work
just like enzymes transposases that mobilize transposons [Agrawal et al., 1998]. The transposases recognize the
ends of the transposon and cut it out of a chromosome. That freed-up piece of DNA, with the transposase still
attached, can loop around itself and move to a new place in the genome. That's just what Rag1 and -2, which act
together in a complex, do.
4)
Rearrangements of host genetic networks. In addition to moving themselves, all types of
transposable elements occasionally move or rearrange neighboring DNA sequences of the host genome.
Transposons “strew” the genome different regulatory sequences. By so doing, transposons can rearrange hosts
genetic networks [Makalowski, 1995; 2000]. Transposons may capture genes and move them wholesale to new
parts of the genome. They make possible DNA shuffling that can place genes in new regulatory contexts and,
possibly, new roles. But what is more, when two transposable elements recognized by the same transposase
integrate into neighboring chromosomal sites, the DNA between them can become subject to transposition.
Because this provides a particularly effective pathway for the duplication and movement of exons, these
elements can help to create new genetic ensembles [Alberts, 1994].
It has been estimated that 80% of spontaneous mutations are caused by transposons. Repeated sequences,
resulting from the activity of mobile elements, range from dozens to millions in numbers of copies per genome.
Transposons co-evolved with their host genome as result of selection, favoring transposons, which
introduce useful variation through gene rearrangement. In this manner, "smart" genetic operators evolved.
It is likely, that transposons to represent the highly coevolved “genetic operators”. The possession of
smart genetic operators must have contributed to the explosive diversification of higher organisms providing
them with the capacity for natural genetic engineering.
In designing artificial evolution, it would be worthwhile to introduce such genomic parasites, in order to
facilitate the sharing of the code that they bring about.
The artificial transposons act as intelligent and sophisticated mutators. They can generate arbitrary
procedures of manipulations with hosts’ chromosomes. In general, these operators can be the unitary, binary or
plural ones. Each host has got the mutators of its own. In the simplest case the transposons are the only source of
the host’s mutations.
If artificial transposon founds hopeful mutation strategy, then both host and parasite will get chance for
reproduction. Virtually we have co-evolution of hosts and their intelligent mutators-parasites.
The Two-level Evolving World
We propose a new approach exploiting some aspects of co-evolution of the hosts and their genomic parasites.
The key feature of the approach is the usage of artificial transposons. Two-level Evolving World consists of
basic level of hosts and upper level of transposons multiplying on a substratum of the host genomes. The
artificial transposons are codes served by their own operators of reproduction, mutation and transmission.
Parasites and parasite ensembles always accompany biological evolution. Tom Ray simulated this
process in his Tierra [Ray, 1991]. Parasites tend with time to reduce their organization for the safe of host’s
resources. In a result, they acquire the possibility of fast (essentially faster then hosts) multiplication and
horizontal and vertical transmission. As a general result, they can evolve drastically faster then hosts (influenza
virus is a sound example).
There are examples of evolvable virtual worlds such as Swarm, Creatures, Network Tierra [Daniels,
1999; Cliff and Grand, 1999; Ray, 2001]. In the course of evolution the worlds of that type can split over
parasite and host co-evolving worlds, i.e. they can become the two-leveled. It is the question of time and such
worlds’ complexity. In less complex virtual worlds similar splitting could be realized “by hand”, as in the case of
developing world of computer viruses.
Parasites are so common that hosts soon co-evolve immunity to them. Then eventually the parasites coevolve strategies to circumvent that immunity. Then eventually the hosts co-evolve defenses to repel them again,
and so on. This is evolutionary Arms Race.
A special kind of parasite is genomic parasite living in the host genome. This special kind of coevolution is the arms race between genomic parasites and their hosts.
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Genomic parasites are the main source of host mutagenesis. If the parasite founds a good strategy for
desirable host mutations, both the host and the parasite will get a chance for reproduction. The parasite rides a
new turn of evolution on the transformed host.
We assume that the simplest realization of the two-layer evolving worlds would be as follows:
the hosts-world is GA-like system (standard GA in the simplest case). The manifold of hosts’
chromosomes-strings is the environment for parasitic codes. In the simplest case these GAs don’t have any
mutation operators of their own;
the parasites are the LISP-like programs, manipulating with the hosts’ strings. (For our applications
these programs must include the search function performing the search of sequences in the host strings).
the parasites live in hosts, they are transmitted vertically (when host reproduces) and horizontally (from
one host to another, as infection or computer virus, by the transposition operator).
Artificial Transposons for GA
Analogously with their biological prototype, we can define artificial transposon as marked block of code in
host's chromosome. The code block marked as artificial transposon still functionally belongs to host. It is still the
part of host code. However this code block is transmittable now from host to host and able to mutate and growth
according different rules than remaining parts of host’s code.
The way to distinguish transposon code from other code depends from problem. In case of ant problem
we use strict definition. Special routine scan each chromosome in population to find appropriate code blocks. In
the case of Royal Road functions we don’t use any definitions or search routines, but we introduce in initial
population initial transposons [Spirov and Kazansky, 2002a; 2002b].
To mark the beginning and the end of the transposon's code block, we could use special ''signaling``
sequences or signaling symbols in the host's chromosome. But we found more convenient to use double-string
chromosomes. The main string (binary and non-binary) is used for codes, while additional one (binary) - for
marks. The possibility to use such chromosomes there is, for instance, in C++ library Evolving Objects (EO)
[Keijzer et al., 2001].
Let mark will be 1 and the absence of it - 0. The block (cluster) of marks (...1111111...) distinguishes
transposon elements from the host's ones. In other word, instead to mark the beginning and the end of a
transposon, we mark all its elements in a sequence.
The main string containing the host's and transposon's code blocks can be binary, symbolic, floatingpoint or other. In the case of binary main string transposon looks as follows:
The additional string 1111111100000000...
The main string
11111111********...
-------transposon
where * is either 0 or 1.
We used such scheme for evaluations the performance of our artificial transposons approach relative to
standard GA on a class of Royal Road fitness functions [Spirov and Kazansky, 2002a; 2002b].
In the case of ant problem treated in this article we can represent our artificial transposon as follows:
The additional string
1 1
1 . . . 1
1
1
The main string
s1 s2 s3 . . .sn-2 sn-1 sn
---------------------transposon
5
0
*
0
*
0
*
0
*
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Here the main string is symbolic one; Si is one of symbols (R, L, F, N, #); R, L, F, N are the symbols of
navigational commands for animate - ant and # is turning point referring at one of previous positions of the
string. R means turn left, L - turn left, F means one step forward, N -is NOP.
Mutagenesis Caused by Artificial Transposons
As in the case of biological prototype, artificial transposons act as sophisticated high-level mutators acting
dependently on the transposon own code and on the transposon “context” (surrounding host's code). Owing from
different examples of known mechanisms of transposon-caused mutagenesis we can test several schemes of
artificial transposons activity. Some of these schemes have been testing by other authors also inspired by modern
evolutionary ideas [Burke et al., 1998; Luke et al., 1999; Simoes and Costa, 2001; Pereira and Costa, 200?; …].
Our approach is different from others because we exploit co-evolution of hosts-transposons based on the
genomic parasites populational dynamics.
Mutational Hot Spots
One of the most general and simple schemes of transposon action as mutators is random point mutagenesis of the
sequence covered by transposon, coupled with transmission of the transposon. Apparently it would be an
example of local search distributed through host population via transmission. We tested this general scheme on
two benchmark problem – Royal Road functions and the ant problem and found substantial improvement of
evolutionary search comparing with standard GA. These results will be published elsewhere. Apart from this
scheme of mutagenesis we investigated some more, which we describe here.
Let we define as array S the transposon own code with it's nearest surrounding. For instance, the S for
the Royal Roads test is:
S:
111...1110
111...111*
where * is either 0 or 1.
Hence, in the case of our Royal Roads tests, the context includes only one host’s element, following the
most right element of transposon [Spirov and Kazansky, 2002a; 2002b].
General scheme of transposon action is: S' = F (S). Concrete view of the F function depends on
the concrete problem.
In case of Royal Road tests described in [Spirov and Kazansky, 2002a; 2002b] we used such simple
definition of F function:
if
S:
111...1110
111...1110
then S':
111...1110
111...1111
if
S:
111...1110
111...1111
then S':
111...1110
111...1110
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Hence this F function gives flip-flop mutation of the first host's element following the most right
element of transposon. In other words, transposon conditions mutational hot spot.
In the case of ant problem, one of the tested F functions is defined as follows:
If S:
1 1 1 . . . 1
1
1
s1 s2 s3 . . . sn-2 sn-1 N
then
S':
1
1
1 . . .
1
1
s1 s2 s3 . . . sn-1 sn
If S:
1 1 1 . . . 1
s1 s2 s3 . . . sn-1
1
#
1
1
then
S':
1
1 . . .
1
s1 s2 s3 . . . sn-1 sn-5
Here si means ith symbol in the transposon's code; N is NOP and # is internal cycle reference (turning point).
The F function substitutes the NOP element of the transposon code by, for instance, the fifth element of the
transposon, counted in order, while if the transposon ends on the internal cycle reference, this element is
substituted by the fifth element of the transposon, counted backward from the end.
The Growth of Artificial Transposon
It is known that transposons tend to duplicate themselves and form clusters in host's chromosomes. We simulate
this feature as outgrowth of our transposon by one element in a time. Of course, it is not the only way to simulate
multiplication and spreading transposons in host's chromosomes, but this concrete scheme works well in our
applications. More precisely, it mimics known feature of real transposons to grow by including more genes.
Namely, time by time, an artificial transposon adds one more element to it's edge:
time T1:
1111...1111000
****...*******
time T2:
1111...1111100
****...*******
where * is appropriate element of the main string.
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Depending on application, the new-recruited transposon element may be arbitrary one (say 0 or 1 in our Royal
Road tests [Spirov and Kazansky, 2002a; 2002b]) or it must have predetermined values, as in the case of our
version of ant problem.
The Ant Problem
The artificial ant problem is the simulation of an ant navigation aimed at passing through the labeled trail
(nicknamed “The John Muir Trail” in the UCLA experiment [Jefferson et al., 1991]). The problem has been
repeatedly used as a benchmark problem [Jefferson et al. 1992; Koza, 1992; Lee and Wong, 1995; Chellapilla,
1997; Harries and Smith, 1997; Luke and Spector, 1997; Ito et al., 1998, Kuscu, 1998]. In [Jefferson et al.,
1991], the artificial ant has to path the John Muir trail placed in a grid world. The ant’s task is to pass through the
labeled cells one by one (the more the better) for the limited time period. The ants are simple finite-state automata
or an artificial neural network which can move along the grid world and test their immediate surroundings. The trail
starts off quite easy to follow, and gradually gets more difficult, as the turns become more unpredictable and gaps
appear (Fig.1). Each black (labeled) cell is numbered sequentially, from the 1st which is settled directly next to
the starting cell, through to the 89th, the last cell. The ant’s task is to follow this trail and move across each black
cell in sequence. Therefore, the successful ant’s program must be quite sophisticated.
<FIGURE 1>
Artificial ant problem for two variants of trail (the “Santa Fe” and the “Los Altos”) is well tested and
discussed in publications. In both cases the trail has been carefully designed in such a way that it would be easy
to navigate at start but as ant proceeds through the trail, gaps and unexpected turns emerges progressively. By
the end of trail there are more gaps then labeled cells. But at the beginning, the successful tactic is very simple only to pick up trail and to move forward. A little further, the ant will need to learn the trick of turning to the
right with the trail. Some next moment they will need to learn the secret of turning occasionally left, and so on.
So, ant is progressively self-learning and perfecting algorithm of search in a course of passing it through.
The ant stands on a single cell and directed to the north, south, east, or west. It is capable of sensing the
state of the cell directly in front of it. In each time step, the ant must take one of four actions. It may turn left,
turn right, move forward one step, or stand still. (When an ant steps on to a black cell, the cell turns white.) After
the ant performs its action, it shifts to a new state. Only the internal state of the ant and the state of the cell in
front of the ant are used as inputs to the decision table which determines (a) the ant’s action and (b) the state it
will assume in the next time step. The decision table itself is coded as the ant’s genotype (Fig. 1B). The ant’s
score is the number of cells passed by the ant for the fixed time period.
METHODS AND APPROACH
While the ant test was implemented at least in two different C++ libraries [Zongker and Punch, 1995],
we gave preference to the Peter Brennan’s version [Brennan, 1994]. This “ANT program” was designed in such a
way that to isolate, as far as possible, the components of the genetic algorithm from the trail-following
experiment and the ant representation. This is a deliberate decision intended to foster an easy transition for the
GA code itself, out of the ANT application, and into whatever application may find a use for the GA. The
development of this program as example of the MGE-approach was the aim of magisterial thesis of one of us
(A.S.K).
ANT Program
The program consists of three major subroutines: Expose, Select, and Reproduce [Brennan, 1994].
In the Expose subroutine, the each ant in the population is run against the John Muir Trail. Each ant’s
score is recorded.
In the Select subroutine, statistics are generated for the previous Expose run. Each ant’s score is
compared to the maximum score attained in the population. One of two selection strategies is employed to
choose a given ant for reproduction. If the user has selected a truncation strategy, the fraction of ants only with
the highest scores are marked for reproduction. If the user has selected a roulette-wheel strategy, an ant with a
higher score has a greater chance of being marked, although there is a chance that even the highest-scoring ants
may not be marked.
In the Reproduce subroutine, the genes of ants which are not marked for reproduction are overwritten
by copies of those which are marked, and then crossover and mutation are applied.
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Technique of Mobile Genetic Elements (Transposons)
Mobile genetic elements (MGEs) are akin to computer viruses. They are the autonomous programs, which are
transmissible horizontally (viz., from one site to another one on the same or another chromosome) or vertically
(from the ancestor to the descendants in the reproduction process). These autonomous parasitic programs
cooperate with the host genetic programs, thus realizing process of self-replication - the only aim, which can be
associated with that activity. We developed some new operators which are the computer program procedures,
performing processes of replication, mutation and invasion of transposons into specific sites on chromosomes, as
well as interactions of transposons with the chromosome (interrelations of parasite - host type).
It is appropriate here to make some notes, concerning the terminology. MGE-technique comprises the
procedures for initialization of mobile genetic elements and procedures for operating with these elements.
Hereinafter in remaining sections mobile elements will be referred to as “transposons”, whereas the procedures,
operating with them will be termed as “MGE-operators”. There are only two types of operators. The one-place
operator is an analogue of point mutation and the two-place (binary) operator realizing the procedure of
transmission of transposon from one chromosome (host) to another chromosome (another host).
Artificial Transposons
Let us recall (see the previous sections) that the ant binary string - chromosome is coding a state transition table
of finite state automation. Altogether there are 32 finite states of automation, ranging from state#0 up to state#31
(See Fig. 1B). All operators start reading and interpreting the table beginning from the state#0. For example,
state#0 determines one of the four actions or instructions (FWD - “forward”, RGT - “to the right”, LFT - “to the
left” or NOP - “do-nothing”) and the number of the next state, depending on binary input value (0 or 1). This
finite state automation can be represented as a state transition diagram and interpreted as a decision tree but, as
far as references to already passed by states are permissible, that tree can have loops.
We will take into consideration that half of the decision tree which corresponds zero binary input value,
i.e. when ant see white (empty) square in front of it. Namely this part of ant’s transition state governs ant’s
search of black squares. In so doing, we reduce ant’s genome to symbolic string and substantially simplify
following treatment of the problem. This is an example of such symbolic string:
L,R,R,F,R,L,L,F,#0
where L is LFT, R is RGT, F is FWD and #0 is turning point referring at the first element of the string. The
navigation program encoded by this string is optimal for classic Santa Fe trail.
It is remarkable, that these strings are really variable length ones, but the maximum length is 32
symbolic elements. For instance, this string
R,L,L,F,#0
is shorter but equivalent to the previous one.
For further consideration it is essential that two abovementioned examples of symbolic strings are good
as navigation program for our trail also, but up to 64 th element only. Hence we can treat these programs as
building blocks for search of more sophisticated ant’s navigation algorithms.
If we use two-string implementation, then it seems perspective for following search to mark this
sequence of symbols as a transposon. Namely we could imagine such artificial transposon:
The additional string
1 1 1 1 1
0
0
0
0
The main string
R L L F #0 *
*
*
*
---------transposon
Generally speaking, in accordance with all what we know about transposons activity in rearrangement
of host genomes, we treat it as mutational / recombinational hot spots. Besides, one of the very special features
of transposons is that they are jumping hot spots. Apparently there are many ways to implement these features in
artificial transposons. In this article we pay attention to that transposon features which allow us to treat it as kind
of generator and multiplicator / distributor of building blocks. In such a way artificial transposon activity is keen
example of local search in limits of global GA search. Namely, initial population of artificial transposons
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disseminate and mutate short code blocks till them find the first building block. The transposon caring and
covered this building block (for example as it is the case of abovementioned R,L,L,F,#0 string) have got
selective advantage, spread and become dominant in host population. In the long run another transposon will
find one more, or large, or better building block and will get selective advantage, etc.
Obviously the simplest way to perform such local search would be the general scheme o random point
mutagenesis of transposon sequence coupled with the transposon horizontal transmission, as mentioned in
“Mutagenesis Caused by Artificial Transposons” section. However, to study in details populational dynamics of
artificial transposons as sophisticated mutators we decide to restrict maximally the transposon variability. To
achieve this we use a definition of transposon. For all tests discussed in this article we use this concrete
definition. Namely, transposon is the sequence of symbols, having the following properties:
the sequence should include symbols which number lie in the range between minimum and maximum
values;
the sequence should not contain NOP elements and internal circles;
the sequence should be finished up with command NOP or with a reference to the initial state. The
transitions cycle will be executed until only white squares remain ahead of the ant.
If the fitting sequence is a cycle, it will be referred to as a mature transposon. But, if the fitting
sequence finishes up with NOP, we will name it the immature transposon. The examples of mature and
immature transposons are given on Fig. 2.
The usage of such sort of transposon definition drastically restricts transposon’s search space and
essentially facilitates monitoring of transposon population dynamics. By usage of the definition we exploit very
small part of transposon sequences manifold, but this small part contains solutions we looking for. Apparently
this restriction is one of some (or many) others. We tested it and found appropriate for our purposes. What’s
more, the usage of the transposon definition makes unnecessary to mark the transposon sections, because
transposon have definition.
MGE - operators
MGE - operators scan the predetermined quota of chromosomes in population. Successively decoding
chromosome record, this operator is seeking for sequences, which are identified as transposon (mature or
immature).
Two-place MGE-operator provides the transmission of the transposon from an ant to another one, thus
realizing the reproduction procedure of this transposon in gene pool of the host (ant) population. This procedure
performs the following operations.
First, a pair of ants is chosen at random. Then, the chromosome of any of them is scanned in search of
the transposon. If the transposon is found, it is replicated in the partner chromosome, irrespectively of initial
record character in that chromosome. The chromosome scanning starts from the zero line (state#0) and goes on
as far as the first transposon is met. If no transposon is met, scanning finishes up only when the chromosome
record ends. So, scanning ceases irrespectively of the remaining chromosome un-scanned part content.
One-place MGE-operator is a sort of point mutation, realized under particular conditions (Cf. the
previous section). This is what we call an intelligent mutator. In detail, the operator acts in such a way. If it finds
a sequence in the predetermined length range, and the action NOP completes this sequence, then this instruction
is substituted for the one of the three other actions (FWD, RGT or LFT). Specifically, this NOP is substituted for
the action from the fifth element of the sequence, counted in order. But, if the found sequence is completed by
the reference to the one of the elements inside sequence (internal cycle), then we have the following. The action
of this element is substituted for the action of the fifth element, counted backward from the end, the reference
being substituted for found at random reference to the element outside of the sequence.
The following peculiarity should be marked here. The reference from the last symbol to the first one
(i.e., by definition, a mature transposon proper) is processing with the mutator using the same rules. In other
words, the mutator breaks the cycle, thus destroying transposons!
So, the action of an intelligent mutator results in accumulation in the population of large length
immature transposons. Point mutations are needed for their maturation. In a result, these new mature transposons
get chance of spreading in host population.
We should emphasize, that the two-place MGE-operator operate only with mature transposons!
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RESULTS
The test trail, used in this work is illustrated in Fig. 3. It can be seen that up to the 64 th element our trail coincide
with the Santa Fe one, but the next part of the trail includes chaotically scattered elements of high complexity.
Being trained on much simpler preceding trail part, the ant is not prepared to surmount the subsequent,
complicated sector (biologists would say that the ant is not pre-adapted to new conditions it faced with in this
sector). More specifically, problems arise at attempts to get over gaps between the 64 th and the 65th, or the 67th
and the 68th cells.
<FIGURE 3>
Transposons Really Accelerates the Evolutionary Search
The preliminary computer experiments showed that it is not possible to construct trail, which ants could not
come through for the real feasible computer time. The accelerating effect of transposons is especially noticeable
for small populations, when the probability of the effective navigation algorithm finding by applying standard
crossover and mutation operators is low.
On this basis, the following experiments were carried out on populations of 100 ants. The choice of
such a small population is also explained by our aim to carry out a comprehensive analysis of transposons
dynamics. Such an analysis is not feasible for large populations of ants because of great number of transposons.
For example, thousands of transposons appear in a population of only 100 ants for 1000 generations.
Besides limiting value of ant population abundance, the following parameters were fixed in all test and
control runs: the maximum number of generations (330) and the part of population to reproduce (15%). The
truncation strategy of reproduction was used when the copies of chromosomes with score exceeding the average
value replace all chromosomes having score less then average. The analysis of preliminary runs showed that the
experimental results concerned and the inferences made are not sensitive to the variation of these three
parameters, i.e. the parameters are not critical in this case.
With the aim of demonstrating of the MGE-technique efficiency we performed 100 independent runs of
the program, 5000 generations each. The results of test and control runs (population with transposons and
without transposons correspondingly) were compared in several series with the different values of standard
mutation parameters. Everywhere in this section we will accept that the effective navigation algorithm should
overcome the level of maximum score in 64 for 330 time steps. Different populations reached different
maximum scores (65, 67, 69, 72, 74, 76, 78, 79, 80, 81) for 5000 generations. But the best result was the score
81, reached for 330 generations.
The results of program runs with the MGE-operator and without it are illustrated in Fig. 4. It can be
seen, that MGE-technique obviously increases the probability of finding of effective navigation algorithm for
small populations and for a little number of generations.
<FIGURE 4>
As it is evident from the graphs on Fig. 4, the mean and the best-of-generation score in experiment and
in control are growing, to a first approximation, linear in time. But the increment of growth in experiment with
transposons is substantially higher, than in control.
It may be suggested that MGE-operators raise ant variability mainly in nonspecific manner thus
supplementing mutation effect of standard operators. But, this suggestion is not substantiated by the detailed
analysis of mutation process (see below). We carried out control runs with the frequency values of standard
mutations, which are two and ten times higher than in control series given in Fig. 4. Hence, the high level of
standard mutation does not raise the efficiency of the navigation algorithm search; moreover, it decreases this
efficiency (the results are not shown graphically).
No values of mutations and crossover parameters taken from the wide spectrum of combinations had a
pronounced effect on evolutionary search efficiency, if MGE-operators were disabled. (Specifically, the
following combinations of crossover and mutation rates were tested: 0.0001 and 0.01, 0.0 and 0.04, 0.0001 and
0.08, 0.001 and 0.04, correspondingly; the results are not shown graphically).
Analysis of Transposon Populations Dynamics
The transposons population dynamics analysis was studied for host population of 100 ants, the number of
generations to the end of experiment being limited by 500. These limitations were non-critical for the
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12
experiment. At the same time, they made possible the graphical representation of the results, because the number
of new abundant transposons forms in every run did not exceed the value of 200.
Every generation of computer population in 100 ants produces new transposons offspring, consisting of
tens of new mutant forms. We accounted for only forms with abundance, exceeding minimal threshold in 10
individuals. So, for the first 500 steps of artificial evolution we had the order of 100 - 150 new forms.
We performed a comparative analysis of dynamics of the 25 ant populations which were succeeded to
found for the determined number of generations an effective navigation algorithm with the 25 unlucky ant
populations, which could not elaborate such an algorithm. Some regularities in the dynamics of ant and
transposons populations were revealed. These regularities are illustrated in Fig. 5 and summarized as follows.
1) As a rule, only one form of transposons is dominating in the population at a moment; tens of new
forms appear in every host’s generation and disappear as maximum, in a several generations.
2) From time to time, the current dominating form is displaced by the other one. In a large number of
generations the former dominating form can rehabilitate itself (at least, some cases were observed).
3) As a rule, 2-3 (less often, 5-6, sometimes, up to 10) dominant forms have an opportunity to change
each other in population for the chosen period of time (the first 500 generations).
4) From time to time, an explosion of abundance of subdominant forms does occur, which can last for
5-10 generations. Eventually, these subdominant forms can give 2-3 population explosions. One of such
explosions can finish up with the conversion of subdominant form into a dominating one.
<FIGURE 5>
These regularities are inherent equally to the dynamics of stagnant ant populations and to the highevolvable populations, which found an effective navigation algorithm for the period being studied. It turned out,
that dynamics of stagnant transposons populations demonstrate the same four properties as the populations,
succeeded in finding an effective navigation algorithm. The moments of finding of that algorithm correlates with
dominant forms changes.
Comparison of the dynamics of learning of ant population with the dynamics of transposons
populations revealed apparent “coincidence” of the moment of change of transposon dominant form with the
moment of finding of the first effective navigation algorithm (Fig. 5). Just after this moment, mean population
score steeply rises up to the next plateau. The process lasts for several generations. This local S-shape function
of the mean score growth correlate with curves of population size dynamics of dominant transposon forms (first,
the old form, then the new one) (Fig.5). It should be emphasized that this is only a correlation, not a functional
relationship. However, the existence of direct functional relationship between the code of a new transposon
dominant form and the code of an effective navigation algorithm is still should be proved.
Larger Transposons Have More Pronounced Effect on the Rate of Ant
Population Training
It was mentioned in the section “Methods and Approach”, that by default, MGE-operator is working only with
sequences of length, which lie in the range from 5 to 11. In all experiments, illustrated in the Figs. 3-5, the
admissible sizes of transposons lie in the range from 5 to 11. Analyzing transposons population dynamics, we
revealed, that very soon, over some tens generations, all transposons reach the upper length threshold (= 11).
Moreover, sequences of length 12 and more appear in the population. MGE-operator recognizes these large
sequences but leave them intact, i.e. it does not apply to them the mutation procedure. This is explained by the
fact, that the one-place MGE-operator causes mutation not only in the sequences, close to transposons by the
definition (See “MGE-operators” section), but in the mature transposons as well. Selection pressure cause
surprisingly fast elimination of all transposons of length less than 11 (irrespectively, mature or immature it is)
and an accumulation of sequences of length 12 and more (MGE-operator does not act on the latter).
We investigated also the behavior of an ant population, having transposons of lager lengths. It became
clear, that revealed regularity viz., domination of transposons of maximum length is valid for all its sizes up to
32 (this is a physical limit).
Moreover, it turned out, that transposons with higher upper size threshold have more pronounced effect
on the rate of ant training. The averaged score dynamics, being taken from the previous experiments (see Fig. 3)
was compared with the score dynamics, obtained in the experiment with the same parameters but one - the upper
transposons length threshold which was made equal to 32 (Fig. 6).
<FIGURE 6>
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The analysis of transposons population dynamics has shown that observed general features of dynamics
sequences are the same for both cases - transposons with high upper size threshold and transposons with low
one. In a several generations, transposons of maximum possible length are coming to domination. As may be
inferred from the diagram (Fig. 6), transposons of length 32 are more effective mutators, than transposons of
length 11. So, the larger transposons, the “wiser” is it as a mutator. We cannot give any simple explanation of
this effect so far.
Horizontal Transmission of Transposons Is Necessary For Their
Effective Mutation Effect
As far transposons are transmitted vertically (from ancestors to descendants), transposons of the host, that have a
superiority in reproduction success is rapidly spreading in the population and gives new forms. But this process
per se is insufficient for the effective acceleration of ant learning. Two-place MGE-operator, performing
horizontal distribution of transposons from one ant to another is a necessary for rising of ant training ability. In
Fig. 7 we illustrate the results of comparing of the test, presented in Fig.3, with the similar test, in which
frequency of applying of two-place MGE-operator was reduced by the factor of 10 and accounted 5%. This
parameter determines the proportion of population, which is subjected to the action the two-place MGE-operator
in a generation. In all previous experiments, presented in this article, this quota accounted 50%.
<FIGURE 7>
The obvious lowering of ant learning abilities with the decreasing of frequency of the two-placed
operator application is evident from the diagram. Disabling of the operator lowers the efficacy further and makes
it almost equal to the control (case without transposons).
Ant Populations Adapts to the Selective Pressure of Transposons
In contrast to external, rigidly predetermined operator of mutation for classical GA, wise mutators co-evolve
with their host. It turned out, that the maximum length of transposons is the critical parameter, controlling fitting
of the host to parasite.
Small sized ant populations are predisposed to noticeable fluctuations, what makes them inappropriate
for our analysis. That is why, in this section we present the results of computer experiments with the more
abundant ant population of 1000 individuals. The large population is more stable, so, the fine peculiarities of
dynamics are obvious (Fig. 8).
<FIGURE 8>
The analysis of all computer experiments, carried out and presented in previous sections gave us
possible to propose the following model of the transposons role in the ant population dynamics.
In accordance to our model, transposons subject the population of hosts (i.e., ants) to a selective
pressure only for short period of evolutionary time (as a maximum - some tens of generations). So, as can be
seen from the diagram 8, the quotas of population, subjected to the action of one-place and two-place MGEoperator accounts 17% and 48% correspondingly, over 40 - 50 generations. For this short period of time ant
population is able to adapt to the transposons selection pressure at the expense of loosing of short transposons
with length of less than 11.
DISCUSSION
The problem of programming an artificial ant to follow the Santa Fe trail has been repeatedly used as a
benchmark problem in GP [Koza, 1992; Lee and Wong, 1995; Chellapilla, 1997; Harries and Smith, 1997; Luke
and Spector, 1997; Ito et al., 1998, Kuscu, 1998]. Recently Langdon and Poli have shown that performance of
several techniques is not much better than the best performance obtainable using uniform random search
[Langdon and Poli, 1998]. According to these authors, the search space is large and forms a Karst landscape
containing many false peaks and many plateaus riven with deep valleys. The problem fitness landscape is
difficult for hill climbers and the problem is also difficult for Genetic Algorithms as it contains multiple levels of
deception.
There are many techniques capable of finding solutions to the ant problem (GA, GP, simulated
annealing, hill climbing) and although these have different performance the best typically only do marginally
better than the best performance that could be obtained with random search [Langdon and Poli, 1998]. That is
why the ant problem may be indicative of real optimization problem spaces and so be worthy of further study.
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Dominant Transposon Forms Are the Components of the Effective
Navigation Algorithms
The results of careful analysis of organization of several tens of dominant transposons (11-13 elements in
length), taken from those ant populations, which coped with the navigation task, can be summarized as follows.
1) By the definition, the transposon-program begins and ends with the zero state, i.e., it is a loop,
executed over and over until the ant will meet the black cell. Typical examples of transposon-programs and
corresponding ant movement trajectories are given in Table 1.
2) Four-fold execution of the transposon-program produces in most cases the closed ant trajectories,
i.e., the ant will return to the starting position. The examples of these closed trajectories are given in Table 1. As
a rule, the closed contour is located in domains the size of 44 or 55 cells. Some cases of transposon-programs
are encountered which provide ant movement back and forth along the polygonal path (See Table 1).
3) As a rule, the transposon-program is beginning to work not from the zero state but from the Nth state,
which is specific to every transposon, not beginning with the initial, zero state. This transition into the Nth state
takes place as soon as the ant (host of the transposon) runs against the white cell.
4) Start the transposon-program from the Nth state provides the execution of the simplest navigation
algorithm, necessary for overcoming the simplest gaps, arranged in the first half of the trail (“looking around”,
then one step ahead, “looking around” again and so forth). This algorithm provides the successful passage of trail
up to the 64th cell inclusive.
5) The majority of program-transposons guarantee overcoming of the element of high complexity
between the 64th and the 65th cells (See Table 1).
6) Some transposons are not suitable for the navigation programs. In that case the chromosome
elements, arranged in transposon-free domain take control over navigation.
<<TABLE 1>>
The detailed analysis of the organization of dominant transposon forms in populations, which are
succeeded in finding of the effective navigation programs, showed, that the transposons themselves become the
components of these programs. Namely, the case in point is about the part of navigation program that is used for
effective “snuffing around” in situation, when ant faces with a wide gap.
In other words, transposons in our experiments really found, cover and transmit host’s building blocks.
Wise Mutators Have a Search Space Confining Effect
The Muir’s Trail search space has rugged geometry due to specific and discrete character of the problem. That is
why, the gradient methods are not effective here. Moreover, this ant navigation problem is classified as a GA
hard problem, especially if trail is not designed specially for ant population training. The efficiency of
transposons in the role of intelligent mutators can be measured by their search space domain confining ability.
Therefore, the selection criteria inserted into MGE-operators had to increase the probability of the effective
navigation algorithm finding on the element of high complexity. We proposed, that quasi-periodical state
successions without NOP (“no operation” instruction) and without internal cycles (in conditions, when an ant
have white elements ahead of it) will be picked up by selection and used for universal navigation algorithms
development.
A comparison of mutation frequencies in experiment and control with the according learning rates
confirm multiple reduction of evaluation numbers, needed for reaching of the same required learning in
experiments with transposons. Mutation frequencies for basic experiments (Fig.3) in control accounts: crossover
rate + mutation rate = 0.0001+0.04 P/bit/generation; MGE1 and MGE2 operators add in average 0.0027 and
0.0075 P/bit/generation accordingly. In other words, transposons in average add to value 0.041 about 0.012
P/bit/generation. This addition brings to multiple acceleration of ant population learning! Hence, according to
fig. 5, up to the end of the experiment (4622 time-step) the control set gives max score 6.47, whereas in the test
set this value is attained already on the 451 time-step, i.e. 10 times sooner.
And what is more, though the ant test is now accepted as a classical one, its implementation in the form
of finite state automata is far from optimum. Firstly, the 32 states are obviously too much for the classical trails.
This redundancy brings to enormous extension of search space dimensionality. Secondly, the possibility to
generate various action sequences in the situation, when the ant faces with the black cell is redundant as well. It
is clear that the “step forward” is the only adequate action in this situation. Thirdly, multiple internal cycles,
essentially increasing the size of search space are hardly needed. At last, the action NOP is absolutely useless.
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Apparently, the most simple and effective method of this navigation problem solving would be using
the symbol strings, composed of combinations of three possible actions (FWD, RGT, and LFT) in the situation,
when the ant faces with the white cell. Applying the MGE-approach to the ant problem, we just perform such a
search space size reduction. In so doing, we deal only with the “white half” of search space which is formed by
all sequences of actions, executed while the ant has white cells in front of it. Besides we escape internal loops,
and exclude the NOP action. We suppose, that mechanism explains the MGE-approach effect i.e., ten-fold
increase of evolutionary search rate. The additional contributory factor for the MGE-technique efficiency raising
is inclusion of transposition procedure, application frequency of which being made high in compare to the other
mutation operators.
Targeted Mutagenesis Coupled With Horizontal Transmission
It is become common place that hybrid optimization approaches are more effective instrument for heuristic
search then solo techniques. In this regard it is remarkable that complicated mechanisms are thought to be the
main power of macroevolution has evident hybrid character. In this paper we demonstrated that the targeted
mutagenesis of transposon sequence coupled with the transposon horizontal transmission qualitatively improve
the GA search performance. High rate of the transmission is crucial for such improvement. As we mentioned
before, recent years some authors have been using several new algorithms for EC improvement owing from fresh
findings and ideas of modern evolutionary biology. For our approach is essential “populational ecological” as
well as co-evolutionary aspects of interactions jumping selfish mutators with chromosomes-solutions. Coevolutionary formation of the world of genomic parasites with own competition and cooperation and own
communication we suppose have very promising perspectives. In this regard, our long term goal is to find
conditions and rules of genomes – genomic parasites co-evolution. It seems quite reasonable that in the long run
of evolution genomes – parasites interaction forms kind of natural “operation system” facilitating manipulations
with genomes. The usage of this meta-language for genome rearrangements could explain fascinating efficacy of
biological evolution that still beyond complete rational explanation.
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A.
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B.
State
Input=0
Input=1
00
01
02
03
04
05
06
07
08
09
0A
0B
0C
0D
0E
0F
10
11
12
13
14
15
16
17
18
19
1A
1B
1C
1D
1E
1F
FWD/0A
RGT/0E
NOP/08
FWD/13
NOP/17
RGT/17
FWD/04
LFT/0A
LFT/12
RGT/1E
RGT/16
RGT/13
LFT/03
LFT/0E
LFT/0C
RGT/15
NOP/0E
FWD/12
LFT/11
RGT/02
RGT/0C
FWD/18
FWD/01
NOP/0B
LFT/13
NOP/0D
NOP/1E
FWD/03
RGT/0A
RGT/06
RGT/0C
RGT/10
NOP/09
FWD/03
NOP/0E
RGT/18
NOP/0B
RGT/0A
NOP/09
LFT/17
FWD/1E
RGT/16
FWD/06
NOP/16
LFT/0B
NOP/14
NOP/12
FWD/1F
LFT/17
FWD/0F
NOP/0C
NOP/1D
LFT/0E
FWD/09
NOP/08
LFT/1A
NOP/11
RGT/01
LFT/1B
FWD/10
NOP/00
LFT/0A
NOP/18
FWD/04
Figure 1. (A). The John Muir Trail (According to Brennan, 1994). Some “milestone” squares are numbered. The trail itself is
a series of black squares on a 32x32 white toroidal (i.e., wraparound) grid. Each black cell is numbered sequentially, from the
1st to the 89th.
(B). An example of organization of an ant as finite state automata. There are 32 states at all. Each state determines
two alternate actions, depending on input signal. The input signal is what an ant sees before him. If the cell before him is
black then the input is 1, in opposite case the input is 0. Each of alternative actions includes one of four possible movements
(FWD, RGT, LFT or NOP) and transition to the next state. (See text for further details.)
20
21
A.
The additional string
1 1 1 1 1 1
0
0
0
0
The main string
L F F L L #0 *
*
*
*
-----------transposon
B.
State
Input=0
0
17
13
21
9
LFT/#17
FWD/#13
FWD/#21
LFT/#9
LFT/#0
C.
The additional string
1 1 1 1 1
0
0
0
0
The main string
L F F L N
*
*
*
*
--------transposon
D.
State
Input=0
0
17
13
21
9
LFT/#17
FWD/#13
FWD/#21
LFT/#9
NOP/#30
Figure 2. Mature and immature transposons.
A. Here is an example of mature transposon. The transposon is a closed five-element cycle of states transitions (0, 17, 13, 21,
9, and again, 0).
B. The state table corresponding to the transposon (A).
C. Here is an example of immature transposon. It differs from the previous one by the presence of procedure NOP in the last,
fifth line and this sequence is not cycle.
D. The state table corresponding to the transposon (C).
21
22
Figure 3. Ant trail used in our computer experiments. The trail itself is a series of squares on a 32x32 white
toroidal grid. Each cell is numbered sequentially, from the 1 st to the 89th. The first two gaps of the higher
complexity are between 64th and 65th and 67th and 68th.
22
23
Figure 4. Numerical experiments, demonstrating statistically certain increasing of the GA efficiency due to the effect of
MGE-operators. A comparison of the mean and the best-of-generation score dynamics (MGE-operator being activated) with
the control (MGE-operator is disabled). The score values are averaged over 100 runs in both cases.
The size of population = 100; the number of generations = 5000; the sequence size varies from 5 to 11; crossover
rate (P/bit)/generation = 0.0001; mutation rate (P/bit)/generation = 0.04; I are the best-of-generation scores and II are the
mean scores for the test runs; III are the best-of-generation scores and IIII are the mean scores for the control runs.
23
24
Figure 5. An example of mobile genetic elements size population dynamics inside gene pool of the host (ant) population (the
first 500 generations). The curve of maximum score dynamics for ant population is given at the top of the diagram (marked
off with squares; the additional score scale is given on the left of the diagram). The moment of finding of the first effective
navigation algorithm and the moment of changing of dominant form of transposon are marked off with the edges of arrows
(see details in the text).
24
25
Figure 6. Comparison of ant populations learning abilities in dependence on transposon length. The curves of
maximum and mean score dynamics (the parameters of sequence size lies in the range of 5 - 32), averaged over
100 computer runs are compared with the corresponding curves, presented on fig.3, where parameter of
transposon sequence size lies between 5 and 11. The values of all other parameters are the same as in the
caption to Fig. 3. I are the best-of-generation scores and II are the mean scores for the runs for large
transposons; III are the best-of-generation scores and IIII are the mean scores for the runs from the Fig. 3.
25
26
Figure 7. The influence of decreasing of frequency of applying of two-place MGE-operator on the ant learning
abilities. The other parameters are the same as in the previous experiments (see caption to Fig. 3). I are the bestof-generation scores and II are the mean scores for the test runs from the Fig. 3; III are the best-of-generation
scores and IIII are the mean scores for the runs with disable MGE-algorithm.
26
27
Figure 8. The dynamics of quotas (in %) of individuals in ant population, subjected to the action of one-place
and two-place MGE-operators (MGE 1 and MGE 2, correspondingly).
27
28
Table 1. Examples of ant’ paths determined by transposons.
R is turn right, L is turn left, > is step forward.
28
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