How Symbiosis
Can Guide Evolution?
Vladimír Kvasnička
Department of Mathematics
Slovak Technical University
September 1999
What is the symbiosis?
(1) A symbiosis is simply the living together of
different organisms.
(2) A special case of the symbiosis is the
mutualism, where symbionts (organisms in
symbiotic relationship) are mutually beneficial.
(3) Symbiosis can lead to the symbiogenesis, i.e.
the creation of new species via the genetic
integration of symbionts (eukaryote cells have
a symbiogenic origin).
(4) In modern evolutionary biology, the symbiosis
is considered as an integral part of evolution
and a fundamental source of innovation.
V. Kvasnička: How Can Symbiosis Guide Evolution?
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Model
 The present model of symbiosis is a modification
of Hinton and Nowlan model of Baldwin effect,
that was published in 1987 in their seminal paper
"How Leraning Can Guide Evolution".
 The present lecture is based on the recent paper
"How Symbiosis Can Guide Evolution" by R.
A. Watson and J. B. Pollack (see home page of J.
Polack).
 The concept of "lifetime plasticity" of cognitive
device of organisms is replaced by "lifetime
interaction" between organisms. The present
model has two periods:
 First period, symbiotic groups find the solution
of a problem more quickly than the solution can
be found by a single organism.
 Second period, after a group has found the
solution, the evolution of individual organisms is
continued towards their ability be independent of
other members of group.
V. Kvasnička: How Can Symbiosis Guide Evolution?
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Chromosomes and population
A chromosome x is an ordered n-tuple composed
of three symbols 0, 1, and #
a
0 ...
R
|1 ...
x S
|T# ...
fk p
x  x1 , x2 ,..., xn  0,1,#
i
n
incorrect symbol
correct symbol
neutral symbol
A population P is a multiset composed of
chromosomes
k
p
P  x  , x  , x  ,...
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For a given chromosome x a group of chromosomes is determined as an ordered g-tuple of
chromosomes randomly selected from the population P
af
G x  x0 , x1 ,..., x g 1
where x0=x. All chromosomes from the group G(x)
are unified at a form of n-dimensional vector chromosome
a
f
 Gaf
x  x  x ... x
~
x ~
x1 , ~
x2 ,..., ~
xn 
0
2
g 1
 k  if xi j   #, for j  0,1, 2,...,k  1,
 xi  


k 


and xi  #
xi  



k 
#
if
x
i , k  0 ,1,...,g  1



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Illustrative example
af
~
x  Gaf
x x
G x0  x0 , x1 , x2 , x3
x0
x1
x2
x3
~
x
1
#
1
0
1
#
#
0
1
0
0
1
#
1
0
#
1
#
#
1
0
 x1  x2  x3
0
1
#
0
0
#
#
#
0
0
#
#
#
#
#
1
1
0
1
1
1
#
0
1
1
#
#
0
1
0
Rule: For each column we look for "non-hash"
symbol going successively by top-down way.
V. Kvasnička: How Can Symbiosis Guide Evolution?
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Reducible and irreducible group
af
x  G x is
A group G(x)=[x0, x1,…, xg-1] with ~
called reducible, iff a removing of its last vector
xg-1 outside the group does not change its
unification.
In particular, let us considered two groups G(x)
and G'(x), where the group G'(x) was formed from
the group G(x) by removing its last vector xg-1
af af kp
G x  G x  x
The group G(x) is reducible iff
af
af
G x  G x
If the group G(x) is not reducible, then it is called
irreducible.
x0
x1
x2
x3
~
x
1
#
1
0
1
#
#
0
1
0
0
1
#
1
0
#
1
#
#
1
0
1
#
0
0
#
#
0
#
0
#
#
#
#
#
1
1
0
1
1
1
#
0
1
1
#
#
0
1
0
this vector may be removed, i.e. group G(x) is reducible
V. Kvasnička: How Can Symbiosis Guide Evolution?
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Fitness of chromosomes
For each chromosome xP we generate randomly
a prescribed number of times (M) a group G(x) , if
x  G x is correct (all its entries are 1),
its union ~
then a fitness f(x) is increased by one.
af
PseudoPascal implementation of
the fitness evaluation
for each xP do
begin f(x):=1;
for i:=1 to M do
begin randomly create
group G(x);
~
x:  G x ;
if ~
x is correct then
f(x):=f(x)+1;
end;
end;
af
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An evolution of population
of chromosomes
Two stochastic operators are introduced:
(1) Selection operator
x=Oselect(P)
It means that from the population a
chromosome x is quasirandomly selected,
where a probability of the selection is
proportional to the fitness f(x) (see Goldberg's
roulette wheel).
(2) Mutation operator
x'=Omut(x)
Each entry of x is changed with probability
Pmut to a feasible value.
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PseudoPascal implementation of
the evolutionary algorithm
time:=0;
P:=randomly generated population;
while time<timemax do
begin time:=time+1;
evaluate all chromosomes
of P by fitness;
Q:=;
while |Q|<|P| do
begin x:=Oselect(P);
x':=Omut(x);
Q:=Q {x'};
end;
P:=Q;
end;
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Size of randomly generated group
A group G(x) of size g (it contains g chromosomes) is generated with probability
af
Pr ob g 
1
z g 1
where z>1.
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Let x(0,1) be a random number with uniform
distribution, then the size g is determined by
solving the equation x=Prob(g), we get
ln x O
L
gM
1
ln z P
where  is an integer part of .
Conclusion: The parameter z models a "density"
of population
z  1  population is crowded
z 
population is scattered
V. Kvasnička: How Can Symbiosis Guide Evolution?
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Illustrative results
V. Kvasnička: How Can Symbiosis Guide Evolution?
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An application to a minimization of
binary functions
kp
f
min F af
x
kp
F : 0,1  0,
n
xopt  arg
x 0 ,1
n
An unification of group chromosomes
x0
x1
x2
x3
~
x
~
xmod
1
#
1
0
1
1
#
#
#
0
0
0
0
1
#
1
0
0
#
#
#
#
#
1
0
1
#
0
0
0
#
#
0
#
0
0
#
#
#
#
#
0
1
1
0
1
1
1
1
#
0
1
1
1
#
#
0
1
0
0
hash symbols are randomly
substituted by 0/1
V. Kvasnička: How Can Symbiosis Guide Evolution?
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x  G  x   xmod  Omod  x 
A chromosome x{0,1,#}n is evaluated by a
functional value f(x) so that a minimal value is
look for within a neighborhood U(x) composed of
modified unifications of randomly created groups

U  x   xmod  x  G  x 

f  x   min f  xmod 
xmod U x 
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PseudoPascal implementation
for each xP do
begin Fmin:=∞;
for i:=1 to M do
begin randomly create group G(x);
~
x:  G x ;
af
x mod :  O mod  x 
if F( ~
x mod)<Fmin then
Fmin:=F( ~
x mod);
end;
F(x):=Fmin;
end;
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Fitness of chromosomes
Step 1. Functional values f(x) are lowered by the
 
 U x 
so-called multiplicity of the optimal xmod
opt
Uopt  x   U  x 

 
x  Uopt  x  : f  x   f xmod
opt

multiplicity  x   Uopt  x 
af af
af
f  x  f x    multiplicity x
This means that optimal solutions with the greater
multiplicity are preferred.
Step 2. Modified functional values f'(x) are used
for the evaluation of chromosome by fitness.
af a f
af
F af
x  Fa
x f... F a
x f
f  x1  f  x2 ... f  x N 
1
2
N
V. Kvasnička: How Can Symbiosis Guide Evolution?
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PseudoPascal implementation of
the genetic algorithm
time:=0;
P:=randomly generated population;
while time<timemax do
begin time:=time+1;
evaluate all chromosomes
of P by fitness;
Q:=;
while |Q|<|P| do
begin x1:=Oselect(P);
x2:=Oselect(P);
(x'1,x'2):=Orepro(x1,x2);
Q:=Q {x'1,x'2};
end;
P:=Q;
end;
V. Kvasnička: How Can Symbiosis Guide Evolution?
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Kauffman's ragged functions KN
Kauffman's NK function maps binary vectors
N
   1 2 ... N   0,1 of the length N onto
positive real number from the interval [0,1], this
mapping is determined with respect to the table T
and the vector  as follows
1 N
F  x     int Ti1 Ti 2 ...TiK TiK 1
N i 1
 

Plot of Kauffman's function for K=3, N=10
fopt=0.31045 , xopt=882=int(1101111100)
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Illustrative calculation
(Kauffman's function for N=50, K=3)
0,5
mean number of # symbols
0,4
0,3
0,2
0,1
0,0
0
50
100
150
200
Epoch
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A new paradigm of symbiosis
Let us assume that we have p independent
optimization subproblems
fi : 0,1  0, 
 i  1,2,..., p 
n
 


0
xopt
 arg minn fi  x  and fi xopt
i
i
x0 ,1
Let us define a unified function
p
f   fi
:
0,1
n p
i 1
p
  0 , 
 
f  x    fi x  
i
i 1
x  x    x    ...  x 
1
2
p
Its functional value is determined as a sum of
functional values of p "subfunctions".
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The minimum of function f is determined by
 
p

f  xopt    fi xopt
0
i
i 1

 
 
xopt  xopt
 xopt
 ...  xopt
1
2
p
It means, that for the function f optimization
problem is reduced to p "local" fully independent
optimization subproblems
A structure of chromosome
x  x  1   x  2   ...  x  r 
i
i
i
The chromosome is of a variable length and
composed of r subchromosomes of subproblems i1,
i2,..., ir. If a subproblem appears more than once,
then its first appearance from the left is considered,
while remaining ones are ignored.
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Four different operators over
chromosomes
1.
Concatenation (two chromosomes are concatenated into one chromosome)
x  x  1   ...  x  r 
j 
j
y  y  1   ...  y q
i
i
z  x  y  x  1   ...  x  r   y  1   ...  y
i
i
j
 jq 
 x , y   Oconcat  x , y 
Oconcat is a deterministic operator.
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2.
Crossover (between two chromosomes some
parts are exchanged)
x  x  1   ...  x  a   x  a1  ...  x  r 
j 
j
j
j
y  y  1   ...  y  a   y  a1  ...  y q
i
i
i
i
j 
x  x  1   ...  x  a   y  a1  ...  y q
j
j
i
i
y  y  1   ...  y  a   x  a1  ...  x  r 
i
i
j
 x , y   Ocross  x , y 
Ocross is a stochastic operator.
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3.
Mutation (a few entries in chromosome are
randomly changed)
x  x  1   ...  x  r 
i
i
x  Omut  x   x  1   ...  x  r 
i
i
 
x  k   Omut x  k 
i
i
Omut is a stochastic operator.
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4.
Deletion (from a chromosomes a subchromosome is deleted)
x  x  1   ...  x  k 1   x  k   x  k 1  ...  x  r 
i
i
i
i
i
x  x  1   ...  x  k 1   x  k 1  ...  x  r 
i
i
i
i
x  Odel  x 
Odel is a stochastic operator.
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Fitness of chromosomes
Let us consider a chromosome
x  x  1   ...  x  r 
i
i
A fitness assigned to the chromosome x is
r
F  x   
k 1
1
 
  fik x
 ik 
 d 
1. A summation symbol with a star means that the
sum runs only over different subproblems
(those ones that already appeared are omitted).
2. The second negative terms of the right hand
side correspond to a penalization term, where 
is is a small positive number and d determines a
number of subproblems that are not included in
x.
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Pprokar:=randomly generated population of prokaryotic
chromosomes;
Peukar:=; epoch:=0;
while epoch<epochmax do
begin epoch:=epoch+1;
all chromosomes are evaluated by fitness;
x1:=Oselect(Pprokar); x2:=Oselect(Pprokar);
if random<probprokar-prokar(x1,x2) then
begin y:=Oconcatenation(x1,x2); y is evaluated by fitness;
Peukar:=Peukar{y};
end;
x1:=Oselect(Peukar); x2:=Oselect(Pproukar);
if random<probeukar-prokar(x1,x2) then
begin y:=Oconcatenation(x1,x2); y is evaluated by fitness;
Peukar:=Peukar{y};
end;
x1:=Oselect(Peukar); x2:=Oselect(Peukar);
if random<probeukar-eukar(x1,x2) then
begin y:=Oconcatenation(x1,x2); y is evaluated by fitness;
Peukar:=Peukar{y};
end;
Qprokar:=;
for i:=1 to |Pprokar| do
begin x:=Oselect(Pprokar); x':=Omut(x); Qprokar:=Qprokar{x'};
end; Pprokar:=Qprokar;
Qeukar:=;
for i:=1 to |Pprokar| do
begin x:=Oselect(Peukar); x':=Omut(x); Qeukar:=Qeukar{x'};
end; Peukar:=Qeukar;
end;
Algorithm 1. A psedoPascale code of simple version of evolutionary algorithm (see Fig. 2) applied for a
simulation of emergence of symbiosis. An operator Oselect performs a quasirandomly selection of a chromosome
from population , where the probability of selection of a chromosome is proportional to its fitness. Two different
types of populations are defined. the first population Pprokar is composed of prokaryotic chromosomes represented
by binary vectors of the length n, whereas the second population Peukar is composed of eukaryotic chromosomes
represented by binary vectors of the variable length pn, for p2 ( it means that an eukaryote is composed at
least of two prokaryotes). Both these populations occasionally interact in such a way that two randomly selected
prokaryotic chromosomes x1 and x2 are "inserted" with a probability Pinsertion into one eukaryotic chromosome,
which is added into population Peukar. An evolutionary epoch of both populations consists in a creation of new
populations with the same cardinalities (number of chromosomes) as the original ones, and then original
populations are updated by these new populations.
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