184
On the Performance of Particle Swarm
Optimization for Parameterizing Kinetic Models
of Cellular Networks
Natalie Berestovsky
Riya Fukui
Luay Nakhleh
Department of Computer Science
Rice University
6100 Main Street, Houston
TX 77005, USA.
Email: ny1@rice.edu
Department of Computer Science
Rice University
6100 Main Street, Houston
TX 77005, USA.
Email: rf7@rice.edu
Department of Computer Science
Rice University
6100 Main Street, Houston
TX 77005, USA.
Email: nakhleh@rice.edu
Abstract—Recent advances in high-throughput technologies
and an increased knowledge of biochemical systems have
enabled the reconstruction of cellular regulatory networks.
While these reconstructions often consist of the connectivity
patterns among the network’s components, an important task
in this area is to derive values for the kinetic parameters of
the system so that its dynamics can be elucidated and testable
hypotheses can be generated. The problem here is to reverse
engineer the model (in our case, values of kinetic parameters)
given the data that consists of a network connectivity and
time-series data. Particle swarm optimization (PSO) has been
recently used for this reverse engineering task. In this paper,
we introduce three scoring metrics for assessing the optimality
of the solution found by PSO. Using biological data sets, we
study the performance of the PSO framework under the three
scoring metrics. Our results show that PSO achieves very good
parameterization of the cellular networks we consider, regardless of the complexity of the dynamics patterns generated by the
underlying kinetics. Given the flexibility of the PSO framework,
as well as its natural amenability to parallelization and general
high-performance implementations and execution, our results
indicate PSO may be a good candidate for parameterizing
cellular networks.
Index Terms—Particle swarm optimization, cellular networks, kinetic models, time series data.
I. I NTRODUCTION
Understanding biology at the system level requires examining not only the properties of individual molecular species,
but also their organization into a tightly-connected and finetuned network of structural and dynamic interactions [1].
Advanced biotechnologies are amassing data on molecular
interactions from a variety of organisms at an exponential
rate [2]. These data typically take the form of graphs, where
nodes represent the molecules, such as genes or proteins,
and the edges represent interactions. However, the dynamic
behavior may not depend only on the wiring, but also
on the values of the kinetic parameters associated with
the strength of interactions (though views that the wiring
patterns can solely determine the dynamics exists; e.g., [3]).
Typically, inferring parameter values is done by obtaining the
978-1-4673-1191-5/12/$31.00 ©2012 IEEE
known values from the literature, and using experiments to
evalute the remaining ones [4] (using “educated guessing”
of parameter values is also common; the process is very
much ad hoc, indeed). However, the literature has sparse
information regarding the parameters and those available
depend highly on the the experimental conditions, making
them insufficient or inappropriate for analytical purposes in
general.
This triumvirate of issues—the systems biology view, the
availability of network connectivity, and the need to parameterize networks for dynamics—underlies the significance of
the problem of parameter estimation of nonlinear dynamic
biochemical pathways and motivates the emerging efforts on
solving it in a systematic way [5].
To address this problem, several global optimization techniques have been developed and applied to learning the
values of kinetic parameters in cellular networks. Moles et
al. provide a review and comparison of various global optimization techniques applied to biochemical systems [5]. The
authors showed that not all global optimization techniques
work well for parameter estimation of regulatory networks.
Nonetheless, they showed that the general evolutionarybased strategy was able to solve the problem successfully.
In this work, we examine the applicability and performance of Particle Swarm Optimization (PSO), a biologically
inspired algorithm, for parameter estimation of biochemical
pathways using time-series data. Combining aspects of genetic and evolutionary algorithms, PSO uses the concept of
fitness and utilizes random values to search the (solution)
space. Adjustment towards local and global optima by
the particles in the swarm is conceptually similar to the
crossover operation utilized by genetic algorithms [6]. We
employ PSO in conjunction with scoring metrics that we
devise for assessment of the optimality in each search iteration. We show that for some systems these metrics perform
equally well, but for others, some metrics outperform the
others. As such, we analyze the time-series data to elucidate
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the patterns the govern and affect the performance of each
metric, so as the produce guidelines for which metric, or
metrics, to use given the observable trends in the data.
We demonstrate the validity of PSO as a method for
parameter estimation from time-series data. Further, its
flexibility (in terms of customizing and fine-tuning the
search parameter values), as well as its amenability to highperformance techniques, make it a good candidate for reverse
engineering biological systems from time-series data.
II. PARTICLE SWARM OPTIMIZATION (PSO) FOR
PARAMETER ESTIMATION
PSO is a probabilistic search algorithm that is based on a
social model of, e.g., a swarm of insects searching for a food
source. It was first introduced by Kennedy and Eberhart in
1995 [6], and since then PSO has been studied by several
authors in terms of performance and applications [7], [8].
The basic swarm optimization algorithm works as follows:
k particles are uniformly positioned within given bounds and
initialized with a velocity. At each search step, each particle
determines its new position by, first, updating its velocity
vector using the memory of its own previous experience and
the experience of the best particle in the swarm, and, then,
calculating the new position vector. The particle’s update
steps are stochastic (not deterministic). A more detailed
description of the general PSO framework is shown in
Algorithm 1.
Algorithm 1 Particle Swarm Optimization (PSO)
1) Input: A swarm of k particles: {ρ1 , . . . , ρk }, each represented by an N dimensional initial position vector pρ and initial velocity vector vρ .
2) Repeat:
a) For each particle ρ, update the velocity vector vρ for time t + 1
according to
t+1
vρ
t
t
t
= ωvρ + c1 r1 (lρ − pρ ) + c2 r2 (g − pρ )
(1)
where for particle ρ
ω
is the inertia of the particle.
t
vρ
is the velocity at time t of this particle.
t
pρ
is the position at time t of this particle.
lρ
is the best solution found by this particle thus far (local
memory).
g
is the best solution found by the swarm thus far (global
memory).
c1 , c2
are parameters representing a particle’s “trust” into itself
and the swarm, respectively.
r1 , r2
are random numbers in the range (0, 1).
b) For each particle ρ, update its position at time t + 1 according to
t+1
pρ
t
t+1
= pρ + vρ
(2)
c) For each particle ρ, score the position vector pρ in the context of the
problem being addressed and update:
lρ
if the current position produces the best local score thus far.
g
if the current position of ρ produces the best global score thus
far.
3) Output: The global best position g.
Aside from the stochasticity contributing to the individual
behavior of the particles, there are two other aspects that
influence the outcomes of the optimization. First, there are
three parameters that could be fine-tuned specifically to the
problem of interest. The inertia parameter ω, controls the
exploration properties of the algorithm with higher values
favoring global behavior and lower values focusing on the
local search. The best results can be achieved by dynamically
changing ω: increasing it if no score progress has been made
for several iterations, invoking global search, or decreasing
it for better local refinement. Shi and Eberhart suggest using
0.8 < ω < 1.4 as an optimal inertia range [9]. Other two
PSO parameters are c1 and c2 , which indicate how much
“trust” the particle has in itself and in the swarm, respectively. These values weight the importance of “local best vs.
global best”. The parameter selection for PSO can highly
depend on the type of the problem being addressed, and
many modified approaches have been proposed to address
individual problems [10], [11], [9], [12]. In this work we
use the settings 0.1 < ω < 1.1 and c1 = c2 = 1.49. This
choice of parameters has shown to be appropriate for PSO
parameter estimation for signaling networks in recent work
by Iadevaia et al. [13].
The second factor that influences the success of the swarm
is its initialization. To achieve best results the particles in the
swarm should be uniformly distributed across the solution
space. It is a common practice to use random initialization,
even though it does not always guarantee perfect coverage.
To address this, Venter et al. carried out a series of numerical
experiments showing that space filling design approach is not
significantly superior to the random initialization [8]. This is
due to the swarm’s great dynamics during the execution. In
this work we use random initialization for particle swarms.
Next, we discuss the application of PSO to parameter
estimation of cellular networks from time-series data.
A. PSO for cellular networks
The general optimization problem, to which PSO can be
easily applied, consist of some objective functions, where
the independent variables comprise the problem dimension,
and the dependent variables serve as a scoring metric. The
convergence of the algorithm is assumed if for a specified
number of consecutive iterations, the change in the objective function score is less than some pre-defined expected
change.
In the context of cellular networks, each particle in the
PSO tries to find the parameter set that best fits the timeseries data. The particle’s position vector represents a single
parameter set. At each iteration, the particles update their
position vector according to Equation 2 in Algorithm 1,
trying to find the set that results in the fitness score higher
than the current global best. The best position vector is
determined by the highest scores from a metric associated
with the PSO execution. The scoring of a particular position
vector follows a three step procedure: set, simulate, and
compare. First, the values in the position vector are set as
the model parameters; second, the model is simulated to
produce the dynamic profiles associated with this parameter
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set; finally, the simulated time-series data are compared
with experimental time-series data using one of the metrics
proposed below. The processes is repeated until convergence
is achieved. In the next section, we propose three distinct
scoring metrics for scoring the solutions found by the
particles in each iteration.
B. Comparison metrics
For the comparison metrics, we are give two sets of
time-series data, one produced by the PSO-parameterized
model, labeled sP SO below, and the other is the experimental
(“true”) data, labeled strue below. We assume each data
set consists of n molecules, the concentration of which is
measured at m different time points.
1) Euclidean distance: One scoring metric is the average
Euclidean distance between sP SO and strue . While this
metric has the highest lower bound, it does capture the
correlation between parameter inconstancy and difference in
the curves. This score is calculated according to
n
qP
(3)
score = P
n
m−1 P SO
true )2
(s
−
s
j
j
i=1
j=0
In order to focus on the comparison of the systems’ dynamics, both times-series data sets were trimmed to exclude
steady-state tails and then were adjusted to have the same
size by using the Piecewise Aggregate Approximation (PAA)
algorithm [14].
2) Tangent slope distance: Another metric measures the
distance between the linear features of the two time-series
data sets. Assuming the concentrations were measured at
time points t0 , t1 , . . . , tm−1 , we define the vector of slopes
for each of the time-series data sets, α, using the formula
sj+1 − sj
,
(4)
αj =
tj+1 − tj
where 0 ≤ j ≤ m − 2. Then, we define the score as
n
qP
score = P
n
m−2
P SO − αtrue )2 )
j
i=1
j=0 (αj
(5)
3) Feature extraction by Discrete Fourier Transform
(DFT): A third scoring metric is based on feature extraction.
The Discrete Fourier Transform is the projection of a signal
from the time domain into the frequency domain by
n
−2πif t
1 X
cf = √
f (t)e n
(6)
n t=0
√
where f = 1, . . . , m and i = −1. Each of these complex
numbers cf , represent the amplitude and the phase of a
sinusoidal function term in a decomposed time series-curve.
DFT assumes the curve to be periodic, and the complex
numbers describe the global frequencies of the curve. The
low frequencies describe the general feature or the trend
of the curve, whereas the high frequencies describe the
details of the curve, which are often obstructed by noise.
By using only the first k complex numbers for comparison,
the metric can compare the general characteristic of the two
curves, since the first complex numbers correspond to low
frequencies of the curve [15]. The score is then calculated
according to
n
qP
(7)
score = P
n
k
P SO k − kctrue k)2
(kc
j
j
i=1
j=0
where k = m
4 . DFT was calculated using a Fast Fourier
Transform (FFT) algorithm.
III. C ASE STUDIES
In this section we examine the performance of PSO
parameter estimation for signaling pathways from timeseries data. We consider three systems of different levels of
complexity: MAPK cascade, MAPK with negative feedback,
and Jak/Stat. The input for all test cases consist of a COPASI
model [16] and a text file containing the experimental time
series data. In the first two cases, the experimental data
is synthetic and generated from the corresponding COPASI
model given the parameter sets described in their sources.
The last system, on the other hand, contains real experimental data. For it, we devise a simple mass-action model
that incorporates all experimental species with known upstream/downstream interactions and attempt to parametrize
it using PSO method. Both MAPK systems have the initial
concentrations corresponding to the ones used in generation
of synthetic “true” data, Jak/Stat has initial concentrations
set to zero for all species except the source.
For all systems, the inertia parameter is in range 0.1 <
ω < 1.1 and changes dynamically by the factor of 2 according the the improvement counter δ. Each particle keeps
track of its δ by incrementing it with each improvement
and decrementing when no improvement happens. Inertia is
incremented when δ falls below 3 to favor global search
and decremented when δ is above 5 to favor local search.
The “trust” parameter c1 and c2 both are set to 1.49 [13].
All simulation use 200 particles, random initialization, and
executed until no improvement was observed for 2000 PSO
iterations.
We show that in some cases all comparison metrics
perform equivalently well, and in others, some metrics
are more appropriate than others. In addition, we confirm
that multiple runs of the PSO converges to a plethora of
parameter estimates of varying quality of fit with respect to
the profiles [17].
A. A MAPK pathway
The mitogen-activated protein kinase (MAPK) cascade
has been heavily studied in the past two decades. This
pathway is very significant and has been shown to participate
in regulating gene expression in a larger variety of biological
processes such as cell differentiation, cell survival/apoptosis,
adaptation to drugs, etc. Ultra-sensitivity of MAPK cascade
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Fig. 1. Time-series data of the MAPK pathway. Left: synthetic time-series data generated by the curated COPASI model [18]. Right: replication of
MAPK sensitivity experiment demonstrated in [18] using the COAPSI model.
(a) Euclidean
(b) Tangent slope
(c) DFT
Fig. 2. Convergence of PSO on the MAPK pathway using the three scoring metrics. Each colored line corresponds to a score convergence for a
single run of PSO parameter estimation. Each reaches a different optimal solution, with some processes converging within few hundred iterations, while
others taking thousands of iterations.
was first studied in great detail both experimentally and
computationally by Huang and Ferrell in 1996 [18]. Since
then, over a dozen of MAPK models arose from Huang and
Ferrell work [19].
Huang and Farrell developed a system of ordinary differential equations (ODEs) to model the MAPK pathway
using experimental data to parametrize it. Using this model,
they demonstrated how the cascade organization of the
mitogen-activated protein kinase (MAPK) greatly enhances
the sensitivity of cellular targets to external stimuli. The
curated version of this model is available in the BioModels
database [20]. Figure 1 shows the raw time-series data
generated by the curated model (left) and the replication
of the MAPK cascade sensitivity validation demonstrated in
their work (right).
We utilize this computational model to assess the performance of PSO parameter estimation. Time-series data in
Figure 1(left) is used as the synthetic “true” data. The range
for possible parameter values is restricted to [0.001, 10000].
Each of the three comparison metrics is examined and the
experiments are repeated five times.
Figure 2 shows the convergence of five experiments for
each of the three metrics. It is clear that each PSO parameter
estimation run reaches different optimal solutions, with some
converging within a few hundred iterations and others taking
thousands of iterations. Next, we examine the quality of
those solutions by visually validating the highest and lowest
scoring solutions from each metric.
Figure 3 shows the the COPASI simulation using the
parameters corresponding to the highest and the lowest
score associated with each comparison metric. All metrics
consistently generate the parameters resulting in very good
fit to the original data set. It is worth noticing that all PSOgenerated parameter sets achieve the desired dynamics in a
very short amount of time. The reason for this could be that
the PSO was given the large parameter search range, and
considered values significantly different from those in the
curated model. We also repeated the experiments considering
the range closer to that of curated model and PSO was much
less successful in finding optimal parameter set.
Furthermore, we examined whether the PSO-determined
parameters preserve the ultra-sensitivity in the MAPK cascade that has been shown by Huang and Farrell [18].
Using the parameters produced by the highest-scoring PSO
execution, we conducted the sensitivity experiment. Figure 4 shows the replication of sensitivity experiment with
PSO generated parameters. All metrics preserved the ultrasensitivity property. However, the sensitivity of the cascade
is triggered by different concentrations of MAPKKK activator compared to the original system. This is due to the
difference in the magnitude of the parameter values in the
system.
B. A MAPK pathway with negative feedback
Next, we examined the model of Kholdenko [21] on the
dynamic properties of MAPK cascade. Kholdenko demon-
188
(a) Euclidean highest
(b) Tangent slope highest
(c) DFT highest
(d) Euclidean lowest
(e) Tangent slope lowest
(f) DFT lowest
Fig. 3. Visual validation of the highest and lowest scoring solutions for the MAPK pathway. Top row shows the results for highest scoring solutions
under all three metrics (a-c), and the bottom row shows the results for the lowest scoring solutions (d-f). The color coding of the species correspond to
those in Figure 1 (left).
(a) Euclidean
(b) Tangent slope
(c) DFT
Fig. 4. MAPK ultra-sensitivity to the MAPKKK activator. The experiment is replicated using the PSO generated parameters using each of the three
metrics.
strated that a negative feedback loop combined with intrinsic
ultra-sensitivity of the MAPK cascade can bring about sustained oscillations in MAPK phosphorylation. We obtained
the curated COPASI model of the MAPK pathway of [21]
from the BioModels database [20]. The oscillatory patterns
produced by this model are shown in Figure 5. Using it as the
synthetic “true” data, we apply particle swarm optimization
to determine the parameters that best fit this system. Figure
6 shows the highest and the lowest scored solutions out of
the five runs of the algorithm.
It is clear from the results, the Euclidean distance is not
appropriate metric for this highly oscillatory system. While
tangent slope approach performed reasonably well, it is not
surprising that DFT comparison produced the best results.
At its core, DFT uses a series of sinusoidal curves to fit
the data and it is much superior to other metrics in this
case. For all metrics, PSO success is less consistent in the
case of such complex system. Additionally, the parameter
range for allowed values was restricted to [0.001, 100],
since with the larger range, the particles generated many
parameter sets that failed to converge in the simulation step
of optimality scoring. One reason for this may be the use of a
simple underlying dynamic model, with too few parameters
to capture the true dynamics.
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Fig. 5.
[20].
A MAPK pathway with negative feedback. The time-series data correspond to the model of [21] as obtained from the BioModels database
(a) Euclidean highest
(b) Tangent slope highest
(c) DFT highest
(d) Euclidean lowest
(e) Tangent slope lowest
(f) DFT lowest
Fig. 6. Visual validation of the highest and lowest scoring solutions for the MAPK pathway with negative feedback. Top row shows the results
for highest scoring solutions under all three metrics (a-c), and the bottom row shows the results for the lowest scoring solutions (d-f). The color coding
of the species correspond to those in Figure 5.
C. A Jak/Stat pathway
For the last case study, we applied the PSO algorithm
to experimental data (as opposed to the two cases above,
where the data was generated synthetically from curated
dynamic models). We used a Jak/Stat signaling network with
negative regulatory feedback loops. The dynamics of this
pathway were explored by Bachmann et al. to determine the
roles of the two transcriptional negative feedback regulators
Socs and CIS [22]. We obtained the time-series data for this
system from [22] (shown in Figure 7). The experimental
data shows a spike in the activity of the phosphorylated
signaling components (pJak, pEpoR, and pStat) at the initial
time points, but later suppression by the inhibitory feedback
from the expressed genes Socs and CIS.
The challenge in analyzing this systems lies in the fact
that we have no curated dynamic model associated with it.
Using the structural knowledge of the Jak2/Stat5 system as
described by Bachmann et al., we constructed a small mass
action model corresponding to the high level interactions
among these five species:
pEpoR + Jak ↔ pEpoRJak
pJak + Stat ↔ pJakStat
pStat + Socs ↔ pStatSocs
pStat + CIS ↔ pStatCIS
SocsRNA + pJak ↔ SocsRNApJak
CISRNA + pStat ↔ CISRNApStat
pEpoRJak → pJak + pEpoR
pJakStat → pJak + pStat
pStatSocs → pStat + SocsRNA
pStatCIS → pStat + CISRNA
SocsRNApJak → SocsRNA + Jak
CISRNApStat → Stat + CISRNA
All initial concentration were set to 0, the source con-
190
centration (pEpoR) was set to 1, and the allowed parameter
value range was [0.001, 10000]. Using this model, we applied the PSO algorithm to determine the parameter values.
The results of the parameter estimation are shown in
Figure 8. All comparison metrics capture the expression of
two RNA components, and the activation with further deactivation of pJak. However, all systems missed the activation and suppression of pStat. Additionally, it is interesting
that DFT showed less consistency in the accuracy of its
solution than the other two metrics. This could be due to
the fact that not enough data points were given to capture
the periodic cycle needed to extract the signal frequencies.
IV. D ISCUSSION
Parameter estimation is a central problem in modeling the
dynamics of cellular networks. In this work, we analyzed
the performance of Particle Swarm Optimization (PSO) for
estimating the parameters from a given time-series data.
We showed in many cases that PSO successfully determines optimal solutions for a given system. When multiple
solutions were found, not all final parameter sets showed
good results. However, as we have shown, to achieve best
results, the characteristics of the estimated data set should
match the choice of scoring metric. Systems with simpler
dynamics might not need complex metrics like DFT, but
others with complex dynamics might only perform well
under DFT. To quantify the complexity of dynamics we
calculated the Shannon entropy for each system [23]. The
values are 2.98, 3.48, and 2.25 for the MAPK pathway, the
MAPK pathway with negative feedback, and the Jak/Stat
pathway, respectively. Based on these values, a correlation
emerges between systems with complex dynamics (higher
entropy of the time-series data) and the performance of more
detailed scoring metrics, such as DFT.
In exploring the performance of PSO, we observed that
some of the runs differed significantly in the execution time.
In Table I, we show the average times (in seconds) and their
standard deviation for a single optimization step with 200
particles. It is clear that DFT is a much more computationTABLE I
T HE AVERAGE TIME µ ( IN SECONDS ) FOR
A SINGLE 200- PARTICLE
OPTIMIZATION STEP AND ITS STANDARD DEVIATION σ.
MAPK cascade
MAPK (feedback)
Jak2/Stat5
Euclidean
µ = 18.45
σ = 0.015
µ = 43.29
σ = 1.348
µ = 1.18
σ = 0.003
Tangent slope
µ = 13.62
σ = 0.0197
µ = 60.89
σ = 0.587
µ = 1.08
σ = 0.003
DFT
µ = 32.47
σ = 0.708
µ = 103.80
σ = 0.484
µ = 9.56
σ = 0.038
ally demanding metric. In many cases the Euclidean distance
and the tangent slope distance can achieve good results in
much less time, and the use of DFT can be reserved for
systems with higher complexity in the data.
We also examined how fast PSO runs achieve convergence. We showed that multiple runs potentially lead to
multiple solutions. Sometimes, having an ability to acquire
multiple sets of parameter estimates could be beneficial since
multiple solutions could be examined by the experimentalist
for biological plausibility [17].
V. ACKNOWLEDGMENTS
This work was supported by a Seed Grant from the Gulf
Coast Center for Computational Cancer Research, funded by
John and Ann Doerr Fund for Computational Biomedicine,
as well as by the National Cancer Institute [grant number
R01CA125109]. The content is solely the responsibility of
the authors and does not necessarily represent the official
views of the National Cancer Institute or the National
Institutes of Health.
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