P system model optimisation by means of evolutionary based search

P System Model Optimisation by Means of
Evolutionary Based Search Algorithms
Carlos García-Martínez
Dept. Computing and
Numerical Analysis
14071, University of Córdoba
Córdoba, Spain
cgarcia@uco.es
Claudio Lima, Jamie Twycross Natalio Krasnogor
Centre for Plant Integrative
Biology
University of Nottingham
Sutton Bonington Campus,
Nottingham, LE12 5RD
ASAP Research Group
School of Computer Science
Jubilee Campus, Nottingham,
NG8 1BB
nxk@cs.nott.ac.uk
claudio.lima@nottingham.ac.uk
jamie.twycross@cpib.ac.uk
ABSTRACT
1. INTRODUCTION
Executable Biology, also called Algorithmic Systems Biology, uses rigorous concepts from computer science and mathematics to build computational models of biological entities.
P systems are emerging as one of the key modelling frameworks within Executable Biology. In this paper, we address
the continuous backward problem: given a P system model
structure and a target phenotype (i.e. an intended biological behaviour), one is tasked with finding the (near) optimal parameters for the model that would make the P system model produce the target behaviour as closely as possible. We test several real-valued parameter optimisation
algorithms on this problem. More specifically, using four
different test cases of increasing complexity, we perform experiments with four evolutionary algorithms, and one variable neighbourhood search method combining three other
evolutionary algorithms. The results show that, when there
are few parameters to optimise, a genetic and two differential
evolution based algorithms are robust optimisers attaining
the best results. However, when the number of parameters
increases, the variable neighbourhood search approach performs better.
Living cells are the product of an exquisite interplay between a multitude of processes that span many spatial and
temporal scales. These processes involved a vast number
of small molecules, macromolecules (e.g. RNA, DNA, proteins), as well as cell membranes and organelles [1]. The
availability of advanced modelling methodologies is essential
for the successful practice of systems and synthetic [2, 3] biology. Computational modelling serves several purposes: it
enables the essential features of biological entities and phenomena to be captured; understanding behind those features
and their interactions to be disambiguated; and, in some
cases, it allows systems/synthetic biologists to move away
from qualitative descriptions towards quantitative predictions [19].
One of the most applied modelling techniques for cellular
systems has been that of ordinary differential equations [4].
However, since this technique assumes that the dynamics of
the cellular processes are continuous and deterministic, its
application is inappropriate in many cases. Currently, discrete and stochastic approaches, such as those employed in
executable biology [11] and algorithmic systems biology [21],
are receiving a great deal of attention because they overcome the aforementioned assumptions. In particular, P systems [22] is a formalism able to integrate stochastic and discrete modelling into a computational framework [7, 8, 25]. P
systems abstract from the structure and operation of living
cells, representing cell-like membrane structures with compartments, molecules with multisets of objects, and molecular interactions with sets of rules. The temporal evolution
of a stochastic P system is determined by the application of
algorithms based on Gillespie’s Stochastic Simulation Algorithm (SSA) [13] to the multi-compartmental structure of P
systems models [20, 24]. However, as with any modelling approach, P system model building can be a complex endeavour, especially for large models, involving the specification
of the model structure and also of the model parameters.
For such models, the space of all possible model topologies
and parameter values is vast, and sophisticated optimisation methods are required to generate models that fit the
behaviour of the target system.
Evolutionary Algorithms (EAs) [10] are optimisation procedures inspired by the concepts of biological evolution and
Darwinian natural selection. Nowadays, they are widely applied on many different real-world problems. In particular,
Categories and Subject Descriptors
I.2 [Artificial Intelligence]: Problem Solving, Control Methods, and Search; J.3 [Life and Medical Sciences]: Biology
and genetics
General Terms
Experimentation
Keywords
Executable biology, P systems, Evolutionary algorithms, Realvalued parameter optimisation
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187
latter case, multisets o1 outside membrane l, and multisets o2 inside membrane l, are replaced by multisets
o1 and o2 , respectively. Parameter c is a constant associated with each rule, which determines the propensity
according to Gillespie’s theory of stochastic kinetics
[13]. Rewriting rules are applied according to a multicompartment extension [24] of Gillespie’s Stochastic
Simulation Algorithm (SSA) [13].
certain EA approaches are regarded as the current state-ofthe-art methods to deal with real-valued parameter optimisation problems. Evolution Strategies, Differential Evolution (DE), and Real-coded Genetic Algorithms (GAs) are,
amongst others, three of the most important EA families
being applied to such problems.
In this paper, we analyse the application of EAs on the
backward problem: the optimisation of the real-valued parameters of a P system model so that the model behaviour
matches a predefined target behaviour. This target behaviour is specified by time series that record some biological
measurement such as cell optical density, gene expression
levels, molecular concentrations, and so on. We have selected four different EA approaches and a hybrid algorithm
combining several inner EAs within the variable neighbourhood search algorithm (VNS) [18]. We perform experiments
on four test cases of increasing complexity and compare the
results of the different algorithms for a limited number of
fitness evaluations.
To our knowledge, this is the first study in evolving parameters for P systems. The work in [8] represents the state
of the art for evolving structures for P systems, which includes one of the EAs analysed herein (see Section 3.4). For
the state of the art of optimisation of systems biology models
based on other than P systems please refer to [9].
The structure of the paper is as follows. The next section
introduces P systems and their modular assembly. In Section 3, we describe the analysed algorithms. In Section 4,
we describe the empirical framework used to investigate the
application of EAs to the problem of optimising P system
parameters. In Section 5, we present and discuss the results.
Finally, conclusions are drawn in Section 6.
2.
Cellular behaviour is usually the result of the networked
interactions of many molecular species that follow specific
connectivity motifs [2]. Thus, biological motifs and modularity are one of the cornerstones of a principled approach
to modelling in systems and synthetic biology [3].
A biological module is a separable discrete entity that defines a particular biological function [16]. A P system module
is a set of rewriting rules (1) in which some of the objects, kinetic constants, or labels, might be variables. This increases
reusability because large models can be constructed by combining common modules instantiated with specific values.
Formally, a module M is specified as M (V, C, L) where V
are the object variables, which can be instantiated with specific objects, C are variables for the kinetic constants, and L
are variables for the labels of the compartments appearing
in the rules.
There are two major advantages when using modules to
automatically design biological cellular models: 1) they help
produce biologically valid and plausible models because modules are fundamental building blocks obtained from biological
knowledge; 2) they promote model diversity because, while
the number of modules is limited, they can be instantiated
with different values for their variables, and therefore combinatorially combined to provide a huge variety of candidate
models. As an example, we introduce one of the P system
modules used in our test cases. Details on the remainder of
the modules can be found in [3, 5, 6, 8].
Constitutive or Unregulated expression: This module describes how a gene, gX, is first transcribed into its mRNA,
rX (in this case without any transcriptional regulatory factor), and then, how the mRNA is translated into the corresponding protein pX. Cell machinery can also degrade the
mRNA and protein. These processes take place in compartment l according to the kinetic constants c1 , . . . , c4 .
P SYSTEMS
We use a variant of P systems (in their original form introduced by Paun [22]) called stochastic P systems, which
are designed for specifying and simulating cellular systems
[20]. A stochastic P system is a tuple
Π = (O, L, μ, Ml1 , . . . , Mln , Rl1 , . . . , Rln )
where:
• O is a finite alphabet of molecules.
• L = {l1 , . . . , ln } is a finite set of labels representing
compartment types.
• μ is a membrane structure that contains n ≥ 1 membranes defining hierarchically arranged compartments.
There is a one-to-one correspondence between the membranes and the labels in L, which determine their types.
• Mli , 1 ≤ i ≤ n, is the initial configuration of membrane i, which consists of a multiset of objects over
O placed inside the compartment of membrane with
label li .
• Rli = {r1li , . . . , rklil }, 1 ≤ i ≤ n, is a finite set of
i
rewriting rules associated with the compartment with
label li . Their form is as follows:
o1 [o2 ]l −
→ o1 [o2 ]l
c
=
U nReg({X},
{c1 , c2 , c3 , c4 }, {l})9
8
c1
>
>
r
:
[gX]
−
→
[gX
+
rX]
l
l >
> 1
>
>
c2
=
<
r2 : [rX]l −→
[rX + pX]l
=
c3
>
>
>
> r3 : [rX]l −→ [ ]l
>
>
c4
;
:
r4 : [pX]l −→
[ ]l
As shown in the module definition, X is a variable that
can be instantiated with a specific identifier, and the kinetic
constants c1 , . . . , c4 can be assigned specific values representing different transcription, translation and degradation
rates.
3. EVOLUTIONARY ALGORITHMS
FOR REAL-VALUED PARAMETER OPTIMISATION
(1)
o1 , o2 , o1 , o2
where
are multisets of objects over O and
l ∈ L is a label. These rewriting rules transform objects inside their associated membrane, and transport
objects either inside or outside the membrane. In the
EAs [10] are algorithms that mimic aspects of biological
evolution and constitute a powerful tool for solving complex
high-dimensional problems. They have been successfully applied to many real-world problems. EAs rely on a population
188
of individuals that represent potential solutions to a given
problem. These individuals undergo probabilistic operations
such as mutation, recombination, and selection, to evolve
into better solutions. The aptitude of the individuals is usually measured according to a given fitness function, which is
defined according to the problem in hand.
Solution encoding is an important aspect of EAs. Solutions (or individuals) generally have to be represented according to the nature of the problem being solved. In particular, when the problem involves the optimisation of several real-valued parameters, the individuals of the population are usually represented as a vector of real numbers
(c1 , c2 , . . . , c ) where ci is the ith parameter of the problem
and is the total number of parameters.
There exist a large variety of EA approaches for realvalued parameter optimisation problems. We briefly describe four representative examples, and a VNS [18] combining three EAs, that we use to address the continuous
backward problem of optimising the kinetic constants of P
system models to match a prespecified target behaviour.
these five schemes to generate each solution. Like other
EAs, DE also applies recombination and selection to evolve
the population of solutions. For more details about these
operators see [26].
3.3 Opposition-based Differential Evolution
Opposition-based differential evolution (ODE) [23] is a
modification of DE inspired by opposition-based learning
which aims to accelerate convergence speed. More concretely, ODE explores opposite solutions to those present in
the current population and keeps the best alternative. Given
a point x = (c1 , c2 , . . . , c ), where ci ∈ [ai , bi ], the opposite
˘ = (˘
point x
c1 , c˘2 , . . . , c˘ ) is calculated as c˘i = ai + bi − ci .
While the opposite population is always inspected in population initialization, during the evolutionary search the process
of opposition-based generation jumping is only performed
with a given probability Jr . Further details can be found in
[23].
3.4 A Real-coded Genetic Algorithm
This method is a steady-state genetic algorithm (GA) for
real-value optimization similar to that employed in [8]. The
GA uses multi-parent crossover as the sole genetic operator as follows: First, randomly select M > 2 individuals
x1 , x2 , . . . , xM from the parameter population with xi =
(ci1 , ci2 , . . . , ci ) for i = 1, 2, . . . , M . Then, randomly generate M coefficients αi satisfying: (1) αi ∈ [a, b] where a and
b are control parameters
of the algorithm such that a < 0
P
and b > 1; (2) M
α
=
1. Finally, generate a new vector
i
i=1
of constants xnew
PMas a non-convex linear combination of xi
using xnew = i=1 αi · xi . If the fitness of xnew is better
than that of the worst individual in the current population
then this individual is replaced by xnew .
3.1 Covariance Matrix Adaptation Evolution
Strategy
CMA-ES [15] is a derandomised evolution strategy that
adapts the complete covariance matrix C of the normal mutation search distribution. The algorithm adapts the covariance matrix based on the evolution path p of previous
successful directions. In this way, C is changed to make
steps in the direction p more likely. The version of CMAES used in this paper combines weighted recombination and
rank-μ-update of the covariance matrix [15]. Therefore, the
λ new candidate solutions are generated according to
“
”
2
(t+1)
(t)
xk
∼ N xW , σ (t) C(t) , k = 1, . . . , λ,
(2)
3.5 Variable Neighbourhood Search with Evolutionary Components
where N (m, C) stands for a normally distributed random
vector with mean m and covariance matrix C. The recombination point is given by the weighted mean of the selected
Pμ
(t)
(t)
individuals: x
i=1 wi xi:λ , where wi > 0 for all
PWμ =
i = 1 . . . μ and i=1 wi = 1.
VNS with evolutionary components (VNS-ECs) [12] is an
example of a novel methodology to design hybrid algorithms
which applies specialised EAs as inner operators [17]. VNSECs combines the following three EAs:
1. CMA-ES (Section 3.1): used to generate an initial
VNS solution. The applied parameter values are the
ones suggested in the C code available from the webpage of the author1 (see Table 3).
2. Micro Cross-generational elitist selection, Heterogeneous
recombination, and Cataclysmic mutation [12]: used
to perturb the current best solution in order to escape local optima. Following the idea in the original
VNS, the intensity of the perturbation performed (Nk )
is increased when the previous improvement does not
outperform the current best solution, and decreased
otherwise.
3. Continuous Local EA [12]: used to locally optimise the
current VNS solution so that a local optimum is accurately reached. Importantly, Continuous Local EA
employs a specific replacement strategy and mating
scheme to gather and exploit information from the
search space, allowing it to enhance its efficiency and
efficacy.
3.2 Differential Evolution
Differential evolution (DE) [26] is a population-based method that uses differences between current solutions to bias
the search. Essentially, a new parameter vector is generated
by adding the weighted difference between two (or more)
solutions to a third vector. Two typical schemes used are
DE/rand/1 and DE/rand/2:
xi = xr1 + F · (xr2 − xr3 )
(3)
xi = xr1 + F · (xr2 + xr3 − xr4 − xr5 )
(4)
where xi ∈ R is the ith generated solution, r1, r2, r3, r4, r5 ∈
[0, N ] are randomly generated and mutually exclusive solution indexes, and F > 0. Two other well-know schemes,
DE/best/1 and DE/best/2, can be obtained by simply replacing xr1 with the current best solution xbest . Combining
these different schemes, DE/rand-to-best/1 was also proposed:
xi = xr1 + λ · (xbest − xi ) + F · (xr1 − xr2 )
(5)
Steps (2) and (3) are repeated until a halting condition
(in our case a fixed number of iterations) is fulfilled.
where λ and F control the balance between exploitation and
exploration. In this paper, DE randomly chooses one of
1
189
http://www.lri.fr/∼hansen/cmaes_inmatlab.html
Table 1: Target models
Test case
Test case 1
/
Test case 2
13 kinetic
constants
Test case 3
18 kinetic
constants
Test case 4
38 kinetic
constants
Target model
Π = (m1 , m2 , m3 )+Fixed modules in Weiss library [3, 5, 6, 8]
m1 = P const({X = LuxR}, {c1 = 0.1})
m2 = P lux({X = CI}, {c1 = 1, c2 = 1, c3 = 10000)})
m3 = P luxP R({X = GF P }, {c1 = 1, c2 = 1, c3 = 1, c4 = 1, c5 = 5, c6 = 10−7 , c7 = 5, c8 = 10−7 , c9 = 4})
simulation time: 170 sec.
interval: 10 sec.
Π = (m1 , m2 , m3 , m4 )
m1 = UnReg({X = gene1}, {c1 = 4.5, c2 = 1, c3 = 0.03, c4 = 0.001})
m2 = P osReg({X = gene1, Y = gene2}, {c1 = 1, c2 = 100, c3 = 5, c4 = 1, c5 = 0.15, c6 = 0.6})
m3 = P osReg({X = gene1, Y = gene3}, {c1 = 1, c2 = 10, c3 = 8, c4 = 1, c5 = 0.15, c6 = 0.6})
m4 = N egReg({X = gene2, Y = gene3}, {c1 = 1, c2 = 0.1})
simulation time: 118 min.
interval: 1 min.
Π = (m1 , m2 , m3 , m4 , m5 )
m1 = UnReg({X = LuxR}, {c1 = 0.15, c2 = 0.0004, c3 = 0.03, c4 = 0.001})
m2 = P luxR({X = LacI}, {c1 = 0.1, c2 = 0.175, c3 = 1, c4 = 0.0063, c5 = 1, c6 = 0.0063, c7 = 0.01875,
c8 = 1, c9 = 1, c10 = 0.001, c11 = 0.004, c12 = 0.03, c13 = 0.001})
m3 = P luxR({X = CI}, {c1 = 0.1, c2 = 0.175, c3 = 1, c4 = 0.0063, c5 = 1, c6 = 0.0063, c7 = 0.01875,
c8 = 1, c9 = 1, c10 = 0.1, c11 = 0.004, c12 = 0.03, c13 = 0.001})
m4 = P R({X = LacI}, {c1 = 0.15, c2 = 0.004, c3 = 0.03, c4 = 0.001, c5 = 0.000166, c6 = 0.002,
c7 = 0.1666, c8 = 0.002, c9 = 0.0083, c10 = 0.0002})
m5 = P lac({X = F P }, {c1 = 0.15, c2 = 0.004, c3 = 0.03, c4 = 0.001, c5 = 0.000166, c6 = 0.01,
c7 = 11.6245, c8 = 0.06})
simulation time: 3600 sec.
interval: 36 sec.
Plac::gFP=1
PR::gLacI=1
Plux::gCI=1
Plux::gLacI=1
gLuxR = 1
Ps3OC6ext = 5
of any object does not appear in the table, it is set to the default value 0. Notice that, in some test cases, some rules appear more than once because of the particular instantiations
of their modules. These repeated rules must be removed and
therefore, the number of kinetic constants to be optimised is
reduced.
The first two test cases consist of a gene regulatory network of three genes that simulate a pulse generator. The
target output is the number of molecules of GFP for four
different initial states (source). These first and second test
cases only differ in the parameter ranges defined for the kinetic constants (Table 2). While for the first case, a narrow
search is performed based on biological knowledge about the
magnitude of particular kinetic constants, for the second
case, most parameter ranges span several orders of magnitude. These two test cases allow us to investigate the importance of having specific knowledge about the parameters in
order to find good parameter configurations. The third test
case represents a more general network where three genes
are able to produce a pulse in the expression of a particular
gene. The target output is the number of molecules of the
corresponding three types of proteins. The fourth case is the
most complex one and represents a network with five genes
that behave as a bandwidth detector. The target output is
the number of molecules of pLuxR2, pCI, pLacI, and pFP.
Table 3 gives the parameter settings for each algorithm
(please, refer to the cited publications for descriptions and
discussions of each parameter). In the case of VNS-ECs,
we performed experiments with two configurations. VNSECs-1 applies the parameter values suggested in the original publication [12], whereas VNS-ECs-2 employs the following changes suggested by preliminary experiments (see
Table 3 for details): certain parameters of Continuous Local EA have been modified; execution of CMA-ES at the
beginning of the run is replaced by sampling a random solution because CMA-ES was generally not able to rapidly
converge fast to one solution; the initial population of Continuous Local EA is not evaluated to save some resources,
since this component is not biased by the fitness function to
select the individuals performing crossover, its performance
is not adversely affected.
4.
Initial values
GFP=1
LuxR=1
LacI=1
TetR=1
CI=1
source=0/1/2/16
gene1 = 1
gene2 = 1
gene3 = 1
EXPERIMENTS
4.1 Test Cases
In this section, we describe the empirical framework used
to study the application of the EA-based optimisation methods described in the previous section to the backward problem of optimising P system kinetic constants.
Four test cases of increasing complexity have been selected
to perform experiments. While the first two test cases were
designed for this paper, the more complex problems have
been taken from [8], where a detailed analysis of several
objective functions for P system structure and parameter
optimisation is presented. These test cases model bacterial systems and employ a membrane structure consisting
of a single membrane. Therefore, it is sufficient to indicate the vector of modules that compose a rule set Rl , i.e.,
Π = (O, {l}, [], Ml , Rl ) is represented by (m1 , . . . , mn ). Table 1 gives details on the target models for the four test
cases. The target models were simulated in silico to generate the target time series for each test case. Then, the
obtained time series were used to attempt to reverse engineer the models that produced them. When the initial value
4.2 Running Conditions
Candidate solutions for all the algorithms are encoded as
follows: each candidate solution consists of a real-valued
vector (c1 , . . . , c ) where ci is the kinetic constant associated
with rule ri . Table 2 shows the ranges and scale types of the
kinetic constants optimised for each test case. If the scale
is linear, the constant can take any value within the given
range. If the scale is logarithmic, the constant can take any
power of ten whose exponent is within the given range. In
the latter case, less knowledge is available and we are more
interested in the order of magnitude of the corresponding
constant, e.g., the second test case concerns the same P
system model as the first test case with, however, the scale
190
Table 3: Parameter setting for the algorithms
Table 2: Range and precision of the kinetic constants of the rules for each test case
Test case
Test
case 1
/
Test
case 2
Test
case 3
Module
PConst
Plux
PluxPR
UnReg
PosReg
NegReg
Test
case 4
Plac
PR
PluxR
UnReg
Constant
c1
c1 ,c2
c3
c1 to c8
c9
c1
c2 to c4
c1
c2
c3
c4 to c6
c1
c2
c1
c2
c3
c4
c5
c6
c7
c8
c1 ,c7
c2 ,c9
c3
c4
c5
c6 ,c8
c10
c1 ,c2
c3 ,c5 ,c8 ,c9
c4 ,c6
c7
c10 ,c13
c11
c12
c1
c2
c3
c4
Scale
linear
linear / log
linear
linear / log
linear
linear
linear
linear
log
linear
linear
linear
log
linear
linear
linear
log
linear
linear
linear
linear
linear
linear
linear
log
linear
linear
linear
linear
linear
linear
linear
log
linear
linear
linear
linear
linear
log
Algorithm
CMA-ES
Range
(0,10)
(0,10) / (-10,2)
(0,10)
(0,10) / (-10,2)
(0,10)
(0,10)
(0,2)
(0,10)
(-3,3)
(0,10)
(0,2)
(0,10)
(-3,3)
(0.1,0.3)
(0.001,0.01)
(0.01,0.05)
(-4,-1)
(0.0001,0.0003)
(0,0.02)
(9,12)
(0.01,0.08)
(0.1,0.3)
(0.001,0.01)
(0.01,0.05)
(-4,-1)
(0.0001,0.0003)
(0,0.005)
(0,0.0005)
(0,0.2)
(0,2)
(0,0.01)
(0,0.02)
(-4,1)
(0.001,0.006)
(0.01,0.05)
(0.1,0.3)
(0.001,0.01)
(0.01,0.05)
(-4,-1)
DE
ODE
GA
VNS
Continuous Local EA
1 / 2 versions
muCHC
Value
10
20
rangei /3
20
0.9
0.5
20
0.9
0.5
0.5
0.3
20
4
-0.5
1.5
20
1e-2 / 1
100 / 50
100
0.5 / 2
0.5
20
0.5
during the optimisation and 1000 for the final solution), and
an average model output is obtained for each time series
T
ˆ1 , X
ˆ2 , . . . , X
ˆ N ), where X
ˆ j = (ˆ
(X
x1j , x
ˆ2j , . . . , x
ˆM
j ) . In other
words, each time series consists of M data points, ∀j =
1, . . . , N . The output time series are then compared to the
corresponding target series (X1 , X2 , . . . , XN ), where Xj =
T
(x1j , x2j , . . . , xM
j ) , by means of the randomly weighted sum
method used in [8]. This method was shown, in contrast to
the root mean square error, to guide search algorithms to
reliable final solutions even when different time series range
over significantly different scales.
All algorithms are run on each test case for 1000 fitness
evaluations. The reason for this short evaluation budget is
twofold. In the first place, the evaluation of a solution for
the tackled problems requires a simulation process that is
very time-consuming, i.e. solution evaluation is the bottleneck of the search process. Therefore, we are particularly
interested in the capability of the algorithms to reach highquality solutions in a small number of fitness evaluations.
Secondly, our future goal is to design an approach to optimise both the structure as well as the constants of P system
models, which will involve running an algorithm many times
to optimise many different model candidates. In this case,
each run of an algorithm will need to consume the minimum resources possible for the global approach to work in
an acceptable amount of time.
For each algorithm, we calculate the mean, standard deviation, and median, of the best fitness values found over
50 independent runs (i.e. of the fittest candidate solution
found by each run). We also record the fitness value of the
best solution found over these 50 runs. The same statistics
are also obtained for the root mean square error (RMSE)
between the fittest solution and the target solution in each
of the 50 independent runs. The RMSE is a better metric to use in evaluating overall algorithm performance than
the fitness values. Because fitness is calculated using the
randomly weighted sum method mentioned above, different
fitness values can be assigned to identical solutions, which is
not the case with the RMSE. Therefore, conclusions based
solely on the fitness values of the final solutions could be
misleading.
of most of the kinetic constants set to logarithmic instead of
linear. This makes the problem more difficult because the
interval of final possible values is larger (e.g. c2 of Plux is
optimised in (0, 10) for the first test case, but in (10−10 , 102 )
for the second test case).
Sometimes, the algorithms will produce candidate solutions containing out-of-bounds parameters. These are dealt
with as follows: GA, DE, and ODE repair an out-of-bounds
solution by assigning a correctly-bounded random value to
any out-of-bounds parameters. VNS-ECs versions apply the
mechanism described in [12]. CMA-ES resamples each solution until all parameters are correctly bounded. To avoid
significant computational effort in this task, a maximum
number of resamples is set to 103 . If an invalid solution
is still obtained after this number of resamples, the fitness
of the candidate solution is set to a very high fitness value
(in case of fitness minimization), where solutions closer to
the feasible space are less penalized than solutions further
away. Specifically, the penalized fitness is given by:
f (x) = f (xrepaired ) + 109 · x − xrepaired ,
Parameter
μ
λ
(0)
σi
|P op|
CR
F
|P op|
CR
F
λ
Jr
|P op|
M
a
b
kmax
mating threshold
maxIters
|P op|
Initial α
alpha
bits/variable
factorEvals
(6)
where xrepaired is the closest feasible solution to x.
The fitness value of a given candidate solution is computed
as follows: firstly, the corresponding candidate model Π is
constructed by setting the kinetic constants to the values
proposed by the solution. The Multicompartment Gillespie
SSA [24] is applied to run Π a given number of times (20
191
Table 4: Results of the algorithms on the four test cases
Test
case
1
2
3
4
5.
Method
1. DE
2. VNS-ECs-2
3. GA
4. ODE
5. CMA-ES
6. VNS-ECs-1
1. GA
2. DE
3. ODE
4. CMA-ES
5. VNS-ECs-2
6. VNS-ECs-1
1. DE
2. ODE
3. GA
4. VNS-ECs-2
5. VNS-ECs-1
6. CMA-ES
1. VNS-ECs-1
2. VNS-ECs-2
3. CMA-ES
4. ODE
5. DE
6. GA
Fitness
Mean
316.05
326.58
342.36
340.89
365.89
398.5
373.36
453.73
498.05
595.49
656.27
792.98
258.38
284.23
274.11
304.58
298.98
411.76
633.8
743.38
754.38
870.98
889.56
1139.87
St. Dev.
70.69
91.54
82.4
82.51
81.27
57.96
97.99
120.08
165.13
170.75
380.04
405.36
35.47
53.34
97.79
65.12
104.73
88.16
210.73
189.73
125.71
230.29
260.27
237.36
Median.
319.64
332.96
355.48
341.35
376.77
408.4
379.55
427.17
474.82
571.82
528.94
671.4
250.62
277.17
261.21
296.07
282.42
411.68
582.07
714.81
749.43
899.73
881.1
1128.6
Best
193.25
117.99
105.07
184.31
204.5
223.65
154.36
240.52
199.86
371.23
224.33
322.44
195.27
195.58
146.58
197.3
163.09
265.78
245.7
387.46
430.25
307.86
367.1
576.68
RESULTS AND DISCUSSION
RMSE
Mean
23.36
24.08
24.96
25.17
27.79
29.97
27.41
34.13
36.49
44.55
47.71
58.52
3.51
3.64
3.83
3.95
4.09
4.78
8.87
10.15
10.97
12.37
12.97
18.57
Ranking
St. Dev.
5.76
6.89
6.34
6.32
7.43
4.93
7.29
8.53
12.46
12.78
26.56
28.29
0.33
0.41
1.41
0.75
1.21
0.87
3.42
3.03
2.21
3.88
4.66
5.61
Median.
22.6
23.52
26.27
24.68
27.9
30.82
28.08
32.11
35.57
42.02
38.1
50.31
3.5
3.64
3.48
3.91
3.71
4.64
8.24
9.36
10.41
12.87
12.91
18.61
Best
12.92
9.05
7.88
14.39
14.29
16.36
11.8
16.21
14.37
26.38
17.16
23.73
2.66
2.87
2.63
2.72
2.51
3.52
3.46
6.35
6.96
4.57
4.73
8.36
{1,2,3,4}
{1,2,3,4}
{1,2,3,4}
{1,2,3,4,5}
{4,5}
{6}
{1}
{2,3}
{2,3,5}
{4,5}
{3,4,5}
{6}
{1,2,3}
{1,2,3,5}
{1,2,3,5}
{4,5}
{2,3,4,5}
{6}
{1}
{2}
{3,4}
{3,4,5}
{4,5}
{6}
Finally, results are significantly different on the fourth test
case, in which there is a larger number of parameters. VNSECs versions are the best algorithms with statistical different performance according to the Mann-Whitney U test.
CMA-ES, ODE, and DE seem to perform similarly and the
GA performs the least well.
In general, we see that GA, ODE, and DE perform better
than the other algorithms for problems with few parameters. Among them, GA performs the best when biological
knowledge is reduced. On the other hand, VNS-ECs variants perform better for the problem with larger number of
parameters. Next, we examine why this is the case.
Figure 1 shows the mean convergence graphs of each algorithm on the test cases over 50 independent runs. As said
previously and due to computational constraints, to produce this graph we calculated the average of the best fitness
found so far using 20 simulation runs, instead of 1000 runs.
Therefore, some differences between this figure and the results presented in Table 4 can be expected. Nevertheless,
Figure 1 shows that the algorithms present similar convergence velocities with two important differences: 1) when
the problem has a larger number of parameters to optimise,
e.g. test case 4, GA, DE, and ODE reduce their convergence
speeds because evolving populations of individuals consume
many resources; 2) when the problem involves fewer parameters, the GA, DE, and ODE algorithms are not affected
by resources consumption as much. However, in this case,
VNS-ECs versions are affected by the implicit noise present
in these problems, because they focus the search process on
just one solution. Interestingly, we also note that in the
fourth test case, CMA-ES initially seems to struggle to generate feasible solutions, and is therefore not able to improve
the best randomly generated solution for some time. This
might be related to the fact that, for test case 4, many parameter ranges are of different orders of magnitude, which
is not particularly helpful when using CMA-ES [14].
Finally, Figure 2 depicts the target time series of each test
case together with the average output of the best solutions
for each algorithm. In the first two test cases, the level of
Table 4 shows the results for each of the algorithms on
the different test cases, with best results highlighted in bold.
The algorithms are ranked based on the RMSE average results, as opposed to the single overall best solution obtained,
since we are interested in the ability of each algorithm to robustly and consistently provide good solutions. We applied
the Mann-Whitney U test with p-value=0.05 to compare
and determine the statistical significance of the RMSE average results between algorithms on each test case. Based
on this test, for each algorithm, the final column of Table 4
indicates the algorithms for which no statistically-significant
difference was found.
In the first test case, we can see that most of the algorithms (DE, VNS-ECs-2, GA, and ODE) perform similarly
because the Mann-Whitney U test did not find significant
differences between their RMSE results. CMA-ES and VNSECs-1 clearly performed worse than the other algorithms.
For the second test case, the Mann-Whitney U test finds
more differences between the performance of the algorithms.
GA is the best algorithm and VNS-ECs-1 is the worse one
according to the statistical analysis. The rest of the algorithms obtain similar results between themselves although
DE seems to perform slightly better. Since the first and
second test cases are the same problem but there is less biological knowledge about the parameter ranges in the latter,
we can compare the results of the algorithms on these two
problems. We can see that all the algorithms obtain worse
results when biological knowledge is reduced. However, the
performance of GA is less severely affected than the others.
In the third test case, many algorithms obtain similar performance, as in the first test case. Among the algorithms,
DE, ODE, and GA perform best. In this test case, the scale
of all the parameters but one is linear and subsequently,
their search intervals are not too large. The fact that the
conditions of test cases 1 and 3 are similar may explain why
the results are also similar on both test cases. However, the
results of VNS-ECs-2 are now considerably worse than the
best performing algorithms.
192
Test case 1
Test case 2
800
700
Fitness Average
Test case 3
2000
CMA-ES
DE
ODE
GA
VNS-ECs-1
VNS-ECs-2
600
1500
500
Test case 4
700
CMA-ES
DE
ODE
GA
VNS-ECs-1
VNS-ECs-2
2500
CMA-ES
DE
ODE
GA
VNS-ECs-1
VNS-ECs-2
600
500
CMA-ES
DE
ODE
GA
VNS-ECs-1
VNS-ECs-2
2000
1500
1000
400
400
1000
500
300
300
200
200
400
600
1000 0
800
200
Evaluations
400
600
200
1000
800
200
400
Evaluations
600
500
1000
800
200
400
Evaluations
600
800
1000
Evaluations
Figure 1: Convergence graphs of the algorithms on each test case
Test case 4
Test case 1
120
Target
CMA-ES
VNS-ECs-2
DE
GA
ODE
VNS-ECs-1
8
6
40
20
4000
3500
3000
2500
2000
1500
0
4000
3500
3000
2500
4000
0
4000
3500
3000
2500
0
2000
2
0
1500
4
180
160
140
120
100
80
60
40
0
6
20
0
0
8
40
1000
20
10
3500
60
12
3000
80
Target
CMA-ES
VNS-ECs-2
DE
GA
ODE
VNS-ECs-1
14
pFP
100
40
20
2000
120
500
60
16
Target
CMA-ES
VNS-ECs-2
DE
GA
ODE
VNS-ECs-1
140
pLacI
80
1500
160
Target
CMA-ES
VNS-ECs-2
DE
GA
ODE
VNS-ECs-1
100
2500
Test case 2
120
500
180
160
140
120
80
100
60
40
0
20
0
0
1000
0
2000
2
60
1500
20
80
1000
4
Target
CMA-ES
VNS-ECs-2
DE
GA
ODE
VNS-ECs-1
100
1000
40
0
No. of proteinGFP
140
10
500
60
160
12
pCI
80
14
500
Target
CMA-ES
VNS-ECs-2
DE
GA
ODE
VNS-ECs-1
100
pLuxR2
No. of proteinGFP
120
Test case 3
2
0
protein3
Target
CMA-ES
VNS-ECs-2
DE
GA
ODE
VNS-ECs-1
15
10
5
120
100
80
60
40
20
0
60
0
40
120
100
80
60
40
20
0
0
Target
CMA-ES
VNS-ECs-2
DE
GA
ODE
VNS-ECs-1
8
6
4
20
10
20
0
20
16
14
12
10
120
Target
CMA-ES
VNS-ECs-2
DE
GA
ODE
VNS-ECs-1
30
25
100
40
protein2
protein1
50
20
18
80
60
Figure 2: Average of the simulated output of the solutions of the algorithms on each test case
ter than the average results (Table 4), it is expected that
a more accurate fit to this time series is obtained in some
runs. Finally, considering test case 4, we can see that almost
all the algorithms produce solutions that seem to match the
levels of pLacI and pCI well, but have difficulties matching
the levels of the other two molecules (except pFP for CMAES). It can also been seen that the good performance of the
VNS-ECs variants when tackling this test case is due to a
better fit of the pLacI level, which has a larger scale than the
other time series where the algorithms present differences
(pLuxR2 and pFP). In particular, though CMA-ES fits the
pFP level time series better than the VNS-ECs variants, it
does not match the pLacI one as well as the VNS-ECs variants and as a consequence obtains worse results in Table 4.
GFP achieved from different initial values are presented in
the same graph. In the other cases, different graphs are used
for the levels of each molecules. Comparing the graphs for
the first two test cases, we see that all the algorithms experience a noticeable degradation in their performance when
less knowledge about the problem is available (test case 2).
Among the algorithms, the VNS-ECs variants are the most
affected. Regarding test case 3, all the algorithms present
similar results, accurately matching the level of protein1.
There is a slight difference in matching the level of protein2,
and the level of protein3 is rarely well matched. This fact
indicates that the algorithms usually find difficulties matching this time series. However, since the best results among
the 50 runs of the algorithms are usually significantly bet-
193
6.
CONCLUSIONS AND FUTURE WORK
In this paper, we have analysed the application of EAs to
the backward problem of optimising the kinetic constants of
P systems in order to match a target model. Four different
EAs and a VNS method combining three inner EAs have
been applied to four different test cases of increasing complexity. Limited computational resources (1000 fitness evaluations) have been imposed because of the increased time
consumption when evaluating candidate solutions. Based on
our test cases, the results show that: 1) when the number
of kinetic constants being optimised is small, GA, DE, and
ODE become robust optimisers attaining the best results;
and 2) when the number of parameters to be optimised increases, the VNS-ECs variants obtain better results.
To our knowledge, this is the first study that analyses the
application of EAs to the continuous backward problem. In
the near future, we intend to analyse the way the algorithms
try to match the different target time series throughout their
execution. Moreover, we are interested in the ability of the
studied algorithms to provide diverse solutions for the backward problem, that is, alternative parameter sets that match
the target time series equally well. We also intend to use the
insights we have gained in this paper as a basis for developing efficient algorithms for optimising model structure.
7.
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
ACKNOWLEDGMENTS
Authors acknowledge F.J. Romero-Campero for his helpful discussions and support. C. Garc´ıa-Mart´ınez acknowledge Research Projects TIN2008-05854 and P08-TIC-4173.
N. Krasnogor, J. Twycross, and C. Lima acknowledge the
UK EPSRC project EP/E017215/1 and the BBSRC projects
BB/F01855X/1 and BB/D0196131.
[16]
[17]
8.
ADDITIONAL AUTHORS
Additional authors: M. Lozano (Dept. of Computer Science and Artificial Intelligence, CITIC-UGR, University of
Granada, Spain, email: lozano@decsai.ugr.es).
[18]
9.
[19]
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