Download Comparing Two Evolutionary Algorithms for Inducing Chaos in a

COMPARING TWO
EVOLUTIONARY
ALGORITHMS FOR
INDUCING CHAOS IN A
GENE REGULATORY
NETWORK
1
2
Héctor Guardado Muro , Eunice Esther Ponce de León Sentí ,
Francisco Diego Acosta Escalante1 , Aurora Torres Soto2
1 División Académica de Informática y Sistemas, Universidad Juárez Autónoma de Tabasco, Av.
Universidad s/n, Zona de la Cultura, Tel. +52 (993) 358 15 00. Villahermosa, Tabasco, México.
2 Departamento de Ciencias de la Computación, Universidad Autónoma de Aguascalientes, Av.
Universidad # 940, Ciudad Universitaria, C. P. 20131, Tel.: +52 (449) 910 74 00, Aguascalientes, Ags.
INTRODUCTION
The homeostatic process in a cell adapts the internal state to a
changing environment by adjusting the concentrations of proteins
related to expression genes. The changes in the concentrations of
proteins in the cell can be represented like a dynamical system,
particularly by means of a system of ordinary dif ferential
equations (ODE’s). Consequently, there is an evident relation
between breaking homeostasis and inducing chaos to the
dynamical system representing the proteins in the Gene Regulatory
Network (GRN). Since chaos induction can be very harmful, and
lead important damages, its study represent a way to model a
destroying of an organism and design strategies of intervention.
The aims of his paper is to discussing, and compare two dif ferent
optimization algorithms from evolutionary computation to inducing
chaos, using an specific gene regulatory network , the V-System.
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GENE REGULATORY NETWORK (GRN)
At molecular level, the functioning of organisms depends of which
genes are expressed. The regulation of gene expression is obtained
through genetic regulatory systems that are composed by many
interactions between DNA , RNA , proteins and small molecules.
Specially, the models of measurable properties of Gene Regulatory
Systems together with the regulatory relationships among them is
called Gene Regulator y Networks(GRN). The basic models of GRN's
are the Boolean Networks, the Bayesian Networks, the Dif ferential
and dif ference equations models and the Association Networks.
When the concentration of RNA's, proteins and other metabolites
changes over the time is very useful their representation by means
of Ordinary Dif ferential Equations (ODE's), considering that their
evolution can be modeled in a continuous way.
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THE V-SYSTEM
Fig. 1. The grafic interaction in the
V-System
Poignard in [1] induces chaos
in a GRN that is represented by
a
four
equations
system,
named
V-System.
The
interaction between the four
proteins is represented by
graph in the figure 1 . The Vsystem is the matching of two
sub-systems, composed of the
two first equations and the last
two ones. The first sub-system
(in which 𝐴 3
is this time
considered as a parameter)
admits a Hopf bifurcation,
which creates an oscillating
behavior near the associated
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bifurcation point.
The second one (in which 𝐴 1 is seen as parameter) admits a
hysteresis-type dynamics which makes it jump from a stable steadystate to the other one. In the graph, the nodes represent the
concentration of the proteins, named 𝐴 1 , 𝐴 2 , 𝐴 3 and 𝐴 4 . The four
equation system that represents the evolution of the concentrations
is the next:
𝐴1 2
𝐴3 2
Where
𝐴𝑖 , 𝑖 =
𝑘1 + 𝑘11 𝑗
+ 𝑘13 𝑗
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13
𝐴1 =
− 𝛾1 𝐴1 1 … 4are proteins
2
2
𝐴1
𝐴2
𝐴3 2
1+ 𝑗
+ 𝑗
+ 𝑗
and the 𝑗 𝑖𝑘 terms
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12
13
are
constants,
𝑘 𝑖𝑗 are kinetic
𝐴1 2
𝑘21 𝑗
21
constant, 𝛾 𝑖 are
𝐴2 =
2 − 𝛾2 𝐴2
𝐴
the degradation
1+ 𝑗 1
21
terms,associated
at every 𝐴 𝑖 . All
𝑘3
𝐴3 =
− 𝛾3 𝐴 3
parameters have
𝐴4 2
1+ 𝑗
been
taken
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in ℝ + .
𝐴1 2
𝑘4 + 𝑘4 𝑗
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𝐴4 =
2
𝐴
𝐴
1+ 𝑗 1 + 𝑗 3
41
43
2
− 𝛾4 𝐴4
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OBJECTIVE
The optimization problem optimum represents the set of
parameters values for the ODE’s in the V-System and the initial
concentrations for the proteins 𝐴 𝑖 , 𝑖 = 1 … 4 where the system is
more chaotic and is maximized the fitness function. These values of
parameters, and proteins concentrations, represent the internal
state when the chaos is present. The individuals with a bigger
fitness value are nearer to the values obtained by Poignard. The
principal objective is to compare two methods of optimization that
promotes the chaos behavior. It’s our interest to know which have
more accuracy and converges faster on the search of the values
calculated by Poignard.
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METHODS
It’s important to notice that the individuals in the population
represent an particular system to the ODE’s in V-System. Since one
of the classics measures of the chaotic dynamics is the biggest
Lyapunov exponent, it was selected to build the fitness function. To
support the calculus of the Lyapunov exponents and their use in the
objective function to calculate the fitness value to every individual,
first it was calculated a numerical solution using the 4th grade
Runge-Kutta method. The multiobjective fitness function is builded
like a linear combination, involving a numerical approximation to
Lyapunov exponents in the system. Using the approximations by
𝜆𝑘 ≈
1
𝑡𝑁
𝑙𝑛
∆𝑓h𝑡
h
with 𝑘 = 1. . . 4, we can define the next function to
optimize: 𝐹 𝑋 = 4𝑖=1 𝑎𝑏𝑠(𝜆 𝑘 ) + 𝑚á𝑥 𝑎𝑏𝑠 𝜆 𝑘 ,
where 𝑋 = 𝛾 1 , 𝛾 2 , … , 𝛾 21
is a vector of values
parameters and the 4 initial conditions in the system.
for the
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Two dif ferent ways to optimization were used: An Evolutive Strategie
(ES) and Estimation Distribution Algorithm (EDA). Evolution
Strategie, ES-(N+N), is an optimization method developed by
Schwefel[2] with the next algorithm:
BEGIN Generating an initial random population of size N.
CALCULATE the fitness of all individual in the population, with the function F(X).
REPEAT from 1 to Number of generations
RECOMBINATION To produce N children from N parents and recombining
parameters and initial conditions.
MUTATION To every of the N .
EVALUATION To calculate the fitness of every N.
SELECTION in a elitist way to determine the N individuals to survive.
END
The Estimation of Distribution Algorithms (EDA’s)[3] and the ES are
similar but EDA’s try to find correlations among variables in an
explicit way. For this problem we choose the Estimation of
Multivariate Normal Algorithm (EMNA).
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The Pseudocode of EMNA that we used is the next:
𝑷𝟎 ← randomly generate M individuals
for k = 1, 2, . . . until a Number of iterations
pool ← select n=(M/2) ≤ M individuals from 𝑷𝒌−𝟏 in an elitist way,
selecting ones with F(X) biggest values
𝒑𝒍 = 𝒑(x | pool) ← estimate the mean μ , and the variance σ from the
selected ones.
𝑷𝒍 ← sample new population from 𝒑𝒍 (x) with normal distribution N~(μ, σ )
end for
With
the
same
fitness
function,
defined
like
a
𝐹 𝑋 = 4𝑖=1 𝑎𝑏𝑠(𝜆 𝑘 ) + 𝑚á𝑥 𝑎𝑏𝑠 𝜆 𝑘
and
two
methods
of
optimization we can approximate the values calculated by Poignard.
The goodness of the approximation and the convergence of the
algorithm can be easily calculated by means of:
𝐸 𝑋 =
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𝑖 =1
𝑎𝑖 − 𝑎𝑖
2
+
4
𝑖 =1
𝐴𝑖 − 𝐴𝑖
2
with 𝑎𝑖 parameters of the system and 𝐴𝑖 initial conditions.
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RESULTS
We took a random initial population of size 35, a number of
generation of 50 and repeat the process 30 times, taking the
squared error committed by the best individual in the population to
approximate the values calculated by Poignard, in each iteration.
The EDA method converge faster than EA algorithm to Poignard
solution but requires to maintain a high mutation rate and also can
diverge in some random cases.
Since the values already have been calculated in [1], we want to
show the convergence of the individuals in the population to the
values calculated by Poignard. Those are the critical points in the
second subsystem.
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The results can be condensed in the next figure:
Fig. 2. Comparison of the convergence EA vs. EDA
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CONCLUSIONS
a)
b)
c)
d)
e)
The problem of induction of chaos in a dynamic system can be
defined like an optimization problem.
The use of the Runge-Kutta method can provide a fast
approximation to the Lyapunov exponents of one system.
The
fitness
function
𝐹 𝑋 = 4𝑖=1 𝑎𝑏𝑠(𝜆 𝑘 ) + 𝑚á𝑥 𝑎𝑏𝑠 𝜆 𝑘
represent one ef ficient way to describe the chaotic dynamic.
The EDA method can converge faster to solution but needs
higher mutation rates
to avoid
fall in local optimum
meanwhile the EA method provides a good approximation to
solution in a few generations without fall in local optimum.
The EDA method provides a better approximation than ES to
the values obtained by Poignard because EDA are faster and
have more accuracy. To keep the goodness of the method is
important to maintain enough variability to avoid local
optimum.
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REFERENCES:
[1] Poignard C., "Inducing chaos in a gene regulator y network by
coupling an oscillating dynamics with a hysteresis-type one", J.
Math. Biol. 69:335–368, Springer (2014), 2.
[2] Schwefel H.P.,”Evolution and Optimum Seeking", John Wiley
and Sons, (1995).
[3] Mühlenbein, H., Paaß, G.: “From recombination of genes to
the estimation of distributions”. Binary parameters. In Eiben, A.,
Bäck, T., Shoenauer, M., Schwefel, H., eds.: Parallel Problem
Solving from Nature, Berlin, Springer Verlag (1996) 178–187
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THANKS FOR YOUR
ATTENTION!!