Building synthetic networks of the budding yeast cell-cycle using swarm intelligence
Gonzalo A. Ruz, Tania Timmermann and Eric Goles
Facultad de Ingenierı́a y Ciencias
Universidad Adolfo Ibáñez
Av. Diagonal Las Torres 2640, Santiago, Chile
Email: {gonzalo.ruz, tania.timmermann, eric.chacc}@uai.cl
Abstract—A swarm intelligence technique called the bees
algorithm is formulated to build synthetic networks of the
budding yeast cell-cycle. The resulting networks contain the
original fixed points of the budding yeast cell-cycle network
plus additional fixed points to reduce the basin size of the
fixed point associated to the G1 phase of the cell-cycle, with
the purpose of promoting cell proliferation for biotechnological
applications. One thousand synthetic networks were found
using the bees algorithm, 84.5% had basins size for the G1
fixed point less or equal to 10, whereas the original model
has a basin size for that fixed point of 1764. One of the
synthetic networks was analyzed by a biologist concluding that
the resulting model was quite consistent from a biological point
of view, supporting the proposed method as a tool for biologist
to construct synthetic networks with desired characteristics.
Keywords-Boolean networks; bees algorithm; cell-cycle; gene
regulatory networks; computational biology.
I. I NTRODUCTION
Boolean networks are common models for gene regulatory
networks (GRN), they were introduced by Stuart Kauffman
[1]. In this network, each node represents a gene, that can be
either active (node value 1) or inactive (node value 0), and
the edges represent direct influences or relations between
the genes. The dynamics of the network is computed by
a set of Boolean functions for each node and an updating
scheme. In the original model, the updating scheme was
considered to be synchronous or parallel, which means that
at each discrete time step, all the nodes update their values at
the same time. For a Boolean network with 𝑛 nodes, there
are 2𝑛 possible states, and given the deterministic nature
of this model, the network will converge to steady states,
also known as attractors. There are two types of attractors,
fixed point, where once the network reaches that state it can
never leave it. The other is the limit cycle, where the network
returns to a previous state with a certain periodicity. The set
of states that can lead the network to a specific attractor is
called the basin of attraction. The attractors are of interest in
the context of GRN since they represent different cell types.
Due to the simplicity of the Boolean network model,
other mathematical model for GRN have been proposed,
such as: Probabilistic Boolean networks [2], Petri nets [3],
Bayesian networks [4], recurrent neural networks [5], differential equations [6], linear regression [7], etc. Nevertheless,
Boolean networks are still in demand and new theoretical
and algorithmic results are constantly been reported [8]–
[12].
The mechanisms governing cell division have been well
characterized over the years, in particular, in Saccharomyces
cerevisiae. Furthermore, the genes and pathways for cell
-cycle control seem to be very similar in all eukaryotes,
including mammals. Therefore, studying the cell-cycle in
yeast is of interest.
The cell-cycle involves four phases, G1 (Gap 1), S (Synthesis), G2 (Gap 2) and M (Mitosis). In the G1 phase,
cells increase in size. Furthermore, inside this phase there is
an important checkpoint, named ”Start point” in the yeast.
G1 checkpoint makes the key decision whether the cell
should divide (enter to S phase), delay division, or enter a
resting stage. This decision will depend on the environmental
conditions. In the S phase, DNA replication occurs, in
order to duplicate the genetic material. During this phase,
synthesis is completed as quickly as possible due to the
exposed base pairs being sensitive to external factors such as
any drugs taken or any mutagens. During the gap between
DNA synthesis and mitosis, G2 phase, the cell will continue
to grow. The G2 checkpoint control mechanism ensures
that everything is ready to enter the M (Mitosis) phase and
divide. Finally, when the cell enters into the M phase, the
growth of the cell stops and cellular energy is focused on the
orderly division into two daughter cells. A checkpoint in the
middle of the mitosis (Metaphase checkpoint) ensures that
the cell is ready to complete the cell division. After the M
phase, the cell goes back to the stationary G1 phase, waiting
for a signal for another round of division.
In [13] a Boolean network is used to model the budding
yeast cell-cycle network. It was found that the model was
extremely stable and robust, with seven stationary states or
fixed points and amongst the seven fixed points, there is
one fixed point attracting 1764 states (basin size) which
is approximately 86% of the protein states. Remarkably,
this stable state is the biological G1 stationary state. The
advantage for a cell’s stationary state to be an attractor
with a large basin size is that the stability of the cell state
is guaranteed and the energy cost is lower. Under normal
conditions, the cell will be sitting at this fixed point, waiting
for a signal to begin another round of division.
In this paper, we are interested in finding synthetic net-
works of the budding yeast cell-cycle with a reduced basin
size of the G1 fixed point, with the purpose of promoting cell
proliferation for biotechnological applications. For example,
improvements of the biomass of the yeast would be very
beneficial for the processes involved in the making of bread,
beer and wine, amongst others. All these applications are
aimed at reducing the costs and the production time.
To search for alternative Boolean network models for
the budding yeast cell-cycle network, a swarm intelligence
approach is presented in this paper to reconstruct Boolean
networks that contain the seven original fixed points but with
different basins of attraction, in particular we are searching
for networks with a small basin size for the fixed point
associated to G1.
For the machine learning community, the reconstruction of
GRN from data has become an interesting application. Several approaches using evolutionary computation techniques
have been proposed, such as, genetic algorithms (GA) with
GRN modeled by an S-system [14], genetic programming
(GP) with GRN modeled by a combination of kinetic equations and recurrent neural networks [5]. A comparison of
evolutionary algorithms (GA, GP, evolution strategy, evolutionary programming, and differential evolution) with GRN
modeled by the S-system and neural networks was carried
out in [15]. Other meta-heuristic methods such as simulated
annealing (SA) have been used as well. In [16], SA is
used to learn GRN modeled by Bayesian networks, and
GRN modeled by combinations of kinetic parameters that
produce a desired behavior is presented in [17]. Threshold
Boolean networks constructed by SA is reported in [12],
results from this work showed that there is a power law
between the frequency of the networks that the SA found
and the number of the updating sequences which the network
could preserve the cycle attractor. Swarm intelligence has
also been used for inference of GRN from data. Particle
swarm optimization (PSO) is used for the reconstruction of
gene networks modeled by recurrent neural networks (RNN)
in [18] and [19]. Also, a hybrid of differential evolution and
PSO (DEPSO) for training RNN is investigated in [20]. Less
work has been reported on the use of ants and bees. In [21],
an ant system is implemented to generate candidate network
structures. Recently, a comparison of the bees algorithm
with SA for learning threshold Boolean networks has been
carried out in [22]. The results show that the bees algorithm
outperforms the SA, obtaining more feasible solutions using
less edges than the SA.
II. BACKGROUND
A. Boolean networks
Let x be a finite set of 𝑛 variables, x = {𝑥1 , . . . , 𝑥𝑛 },
with 𝑥𝑖 ∈ {0, 1} for 𝑖 = 1, . . . , 𝑛. A Boolean network is a
pair (𝐺, 𝐹 ), where 𝐺 = (V, E) is a finite directed graph;
V is the set of 𝑛 nodes and E the set of edges. 𝐹 is a
Boolean function, 𝐹 : {0, 1}𝑛 → {0, 1}𝑛 composed of 𝑛
local functions 𝑓𝑖 : {0, 1}𝑛 → {0, 1}. Furthermore, each
local function 𝑓𝑖 depends only on variables belonging to
the neighborhood 𝑉𝑖 = {𝑗 ∈ V∣(𝑗, 𝑖) ∈ E}. The indegree,
𝐾, of vertex 𝑖 is ∣𝑉𝑖 ∣. The updating schemes are repeated
periodically, and since the hypercube is a finite set, the
dynamics of the network converges to attractors which are
fixed points or limit cycles, defined by
∙ Fixed point: 𝑥𝑖 (𝑡 + 1) = 𝑥𝑖 (𝑡) for 𝑖 = {1, . . . , 𝑛}.
∙ Limit cycle: 𝑥𝑖 (𝑡 + 𝑝) = 𝑥𝑖 (𝑡) for 𝑖 = {1, . . . , 𝑛}.
where 𝑝 > 1 is a positive integer called the limit cycle
length. The set of states that can lead the network to a
specific attractor is called the basin of attraction. There are
many ways of updating (deterministically) the values of a
Boolean network, some examples are [23]:
∙ Parallel or synchronous mode: where every node is
updated at the same time.
∙ Sequential updating mode: where in every time step,
every node is updated in a defined sequence.
∙ Block-sequential: the set of nodes, for a given sequence,
is partitioned into blocks. The nodes in a same block
are updated in parallel, but blocks follow each other
sequentially.
∙ Asynchronous deterministic: where in every time step,
only one node is updated following a defined sequence.
B. Budding yeast cell-cycle network
The budding yeast cell-cycle network is modeled by a
threshold Boolean network in [13]. It is composed by eleven
nodes (see Fig. 1). The gene updates are computed by:
∑𝑛
⎧
𝑖𝑓
𝜔𝑖,𝑗 𝑥𝑗 (𝑡) − 𝜃𝑖 < 0
⎨ 0
∑𝑗=1
𝑛
𝜔𝑖,𝑗 𝑥𝑗 (𝑡) − 𝜃𝑖 > 0
1
𝑖𝑓
𝑥𝑖 (𝑡 + 1) =
∑𝑗=1
⎩
𝑛
𝑖𝑓
𝑥𝑖 (𝑡)
𝑗=1 𝜔𝑖,𝑗 𝑥𝑗 (𝑡) − 𝜃𝑖 = 0
(1)
with 𝜔𝑖,𝑗 the weight of the edge coming from node 𝑗 into
the node 𝑖, and 𝜃𝑖 the activation threshold of node 𝑖. The
weight matrix and the threshold vector used in (1) are
shown in Fig. 2. This network has seven fixed points and
no limit cycles. The fixed points are (following the node
order in Fig. 2):[00001000100, 00110000000, 01001000100,
00000000100, 01000000100, 00000000000, 00001000000].
The first fixed point is the state associated to the biological
G1 stationary state.
C. Bees algorithm
The bees algorithm (BA) is a population based search
algorithm for function optimization and combinatorial optimization problems. It was first introduced in [24], and
comparisons to other meta-heuristic algorithms appear in
[25]. The algorithm is based on the honeybees’ food foraging
process. In the BA each bee represents a candidate solution,
the flower patch represents a local search area, and the
amount of food the bee collects from the flower patch is
the fitness value. The parameters of the BA that must be
⎛
⎜ 𝐶𝑙𝑛3
⎜ 𝑀 𝐵𝐹
⎜
⎜ 𝑆𝐵𝐹
⎜
⎜𝐶𝑙𝑛1, 2
⎜
⎜ 𝐶𝑑ℎ1
𝑊 =⎜
⎜ 𝑆𝑤𝑖5
⎜
⎜ 𝐶𝑑𝑐20
⎜
⎜ 𝐶𝑙𝑏5, 6
⎜
⎜ 𝑆𝑖𝑐1
⎝
𝐶𝑙𝑏1, 2
𝑀 𝑐𝑚1
𝐶𝑙𝑛3
−1
1
1
0
0
0
0
0
0
0
0
Figure 2.
𝑀 𝐵𝐹
0
0
0
0
0
0
0
1
0
0
0
𝑆𝐵𝐹
0
0
0
1
0
0
0
0
0
0
0
𝐶𝑙𝑛1, 2
0
0
0
−1
−1
0
0
0
−1
0
0
𝐶𝑑ℎ1
0
0
0
0
0
0
0
0
0
−1
0
𝑆𝑤𝑖5
0
0
0
0
0
−1
0
0
1
0
0
𝐶𝑑𝑐20
0
0
0
0
1
1
−1
−1
1
−1
0
𝐶𝑙𝑏5, 6
0
0
0
0
−1
0
0
0
−1
1
1
𝑆𝑖𝑐1
0
0
0
0
0
0
0
−1
0
−1
0
𝐶𝑙𝑏1, 2
0
−1
−1
0
−1
−1
1
0
−1
0
1
Weight matrix and threshold vector of the budding yeast cell-cycle network in [13]
Cell Size
initialize a random population
fitness
evaluation
Cln3
MBF
SBF
𝑀 𝑐𝑚1⎞
⎛0⎞
0 ⎟
⎜0⎟
⎟
0 ⎟
⎜0⎟
⎜ ⎟
0 ⎟
⎜0⎟
⎟
⎜ ⎟
0 ⎟
⎜0⎟
⎟
⎜ ⎟
⎟
0 ⎟
⎟
Θ=⎜
⎜0⎟
⎟
1 ⎟
⎜0⎟
⎜ ⎟
1 ⎟
⎜0⎟
⎟
⎜ ⎟
0 ⎟
⎜0⎟
⎟
⎝ ⎠
0 ⎟
⎠
0
1
0
−1
local
search:
elite sites
+ best sites
global
search
Sic1
Clb5,6
Cln1,2
new
population
Mcm1/SFF
Clb1,2
Cdh1
Stop?
yes
no
solution
Cdc20&Cdc14
Swi5
Figure 1.
The budding yeast cell-cycle threshold Boolean network
representation [13] . (Color online) The green/solid edges represent positive
Pajek
weights (activations), the red/dashed edges represent negative weights
(inhibitory).
Table I
T HE BEES ALGORITHM PARAMETERS
ns
ne
nb
nre
nrb
maxi
number of scout bees
number of elite sites
number of best sites
recruited bees for elite sites
recruited bees for best sites
maximum number of iterations
specified by the user are shown in Table I and the flowchart
of the algorithm appears in Fig. 3.
III. U SING THE BEES ALGORITHM TO BUILD SYNTHETIC
NETWORKS OF THE BUDDING YEAST CELL - CYCLE
The problem consists in finding new weight matrices, such
that the resulting network, must contain at least the seven
original fixed points, and the basin size of the fixed point
associated to G1 must be reduced. The threshold vector will
remain the same, that is, 𝜃𝑖 = 0 for all 𝑖. For this, we propose
to use the BA. When implementing the BA, the following
three parts must be specified.
Figure 3.
The bees algorithm flowchart.
A. Coding of solutions
The solutions, networks (weight values), are coded using
an adjacency matrix Ω, as in Fig. 2, where all the non-zeros
in each row represent the incoming edges to the node of that
row. The initial Ω𝑔 , 𝑔 = 1, . . . , 𝑛𝑠 in the BA are generated
randomly, with an indegree 𝐾 ≤ 3.
B. Definition of the fitness function
The fitness function for the Boolean network 𝐵, is given
by the deviation of the network output, defined by 𝑜𝑖 for each
node 𝑖, and the target value 𝑎𝑖 (attractor) for each node 𝑖,
which is computed by
7
𝑓 𝑖𝑡𝑛𝑒𝑠𝑠(𝐵) =
𝑛
1 ∑∑
2
(𝑜𝑖 (𝑡) − 𝑎𝑖 (𝑡))
7𝑛 𝑡=1 𝑖=1
(2)
where 𝑛 is the number of nodes in the network, and 7 is
the number of fixed points that the network must contain.
Also the fitness function was penalized when the number of
edges of the network was greater than 50. This restriction
was introduced in order to obtain synthetic networks that
could have some biological meaning.
Notice that since we are only imposing on the network
to contain the original seven fixed point, the network can
contain more fixed points, this will yield that some states
that were part of the basin of attraction of the G1 fixed
point (and the other six) will have to change to the basins
of attraction of the new fixed points, thus, achieving our
objective of reducing the basin size of the G1 fixed point.
C. Neighborhood search strategy
During the local search, each recruited bee from the elite
or best sites must generate a candidate solution, Ω𝑛𝑒𝑤 , based
on the current solution, Ω𝑜𝑙𝑑 , corresponding to the scout bee
which found that site. For this, a simple neighborhood search
strategy is used. The pseudocode appears in Algorithm 1.
Algorithm 1 Neighborhood search
1: procedure N ET S EARCH (𝑛, Ω𝑜𝑙𝑑 )
2:
Ω𝑛𝑒𝑤 ← Ω𝑜𝑙𝑑
3:
𝑥1 ← 1 + round((𝑛 − 1) ∗ rand)
4:
𝑥2 ← 1 + round((𝑛 − 1) ∗ rand)
5:
𝑦1 ← 1 + round((𝑛 − 1) ∗ rand)
6:
𝑦2 ← 1 + round((𝑛 − 1) ∗ rand)
7:
if rand > 0.5 then
8:
Ω𝑛𝑒𝑤 (𝑥1, 𝑦1) ← 1
9:
else
10:
Ω𝑛𝑒𝑤 (𝑥1, 𝑦1) ← −1
11:
end if
12:
Ω𝑛𝑒𝑤 (𝑥2, 𝑦2) ← 0
13:
return Ω𝑛𝑒𝑤
14: end procedure
Using the above definitions, the general bees algorithm is
shown in Algorithm 2.
Algorithm 2 The Bees Algorithm
1: procedure BA(𝑛𝑠, 𝑛𝑒, 𝑛𝑏, 𝑛𝑟𝑒, 𝑛𝑟𝑏, 𝑚𝑎𝑥𝑖)
2:
Initialize a random population Ω𝑔 , for 𝑔 = 1, . . . , 𝑛𝑠
3:
Evaluate each candidate solution Ω𝑔 in the fitness
function
4:
𝑖𝑡𝑒𝑟 ← 0
5:
while 𝑖𝑡𝑒𝑟 < 𝑚𝑎𝑥𝑖 do
6:
Search for 𝑛𝑟𝑒 candidates in the neighborhood
of the first 𝑛𝑒 fittest solutions
7:
Search for 𝑛𝑟𝑏 candidates in the neighborhood
of the [𝑛𝑒 + 1, 𝑛𝑒 + 𝑛𝑏] fittest solutions
8:
Search for 𝑛𝑠 − (𝑛𝑒 + 𝑛𝑏) candidates randomly
9:
Evaluate each new candidate solution in the
fitness function
10:
if the fittest solution =0 then
11:
Break
12:
end if
13:
𝑖𝑡𝑒𝑟 ← 𝑖𝑡𝑒𝑟 + 1
14:
end while
15: end procedure
IV. E XPERIMENT SETUP
The BA algorithm was used to find 1000 networks that
contained the seven fixed points. For each resulting network,
the basin size of the fixed point associated to G1, the
number of edges, and the number of fixed points was
recorded. In the simulations, the following BA parameters
where used 𝑛𝑠 = 20, 𝑛𝑒 = 5, 𝑛𝑏 = 10, 𝑛𝑟𝑒 = 20,
𝑛𝑟𝑏 = 10, and 𝑚𝑎𝑥𝑖 = 2000. These parameters were found
empirically after several runs based on the effectiveness
of learning the networks weights. There are some interactions in the network that must be present for the model
to be biologically consistent. The following nine weights
were fixed to the value of one throughout the simulations:
𝜔2,1 ; 𝜔3,1 ; 𝜔4,3 ; 𝜔8,2 ; 𝜔9,6 ; 𝜔9,7 ; 𝜔11,6 ; 𝜔11,7 ; 𝜔11,10 .
V. R ESULTS AND D ISCUSIONS
Histograms of the basin size of the G1 fixed point, number
of edges, and number of fixed points are shown in Fig. 4, 5, 6
respectively. Figure 4 only shows the frequency of the
networks found for basins size less or equal to ten, which
were 845 networks out of the 1000. We see an exponential
decay, starting from basin size one which was the basin size
with the largest frequency. Figure 5 shows that the number
of edges of the networks found are approximately normal,
with a mode equal to 40 edges. Figure 6 shows that the
distribution of the number of fixed point of the synthetic
networks found is skewed to the right. The large number of
fixed points reduces the basin size of the G1 fixed point. An
example of a synthetic network found by the BA is shown in
Fig. 7. This network has 15 fixed points, 7 of which are the
original ones. The G1 fixed point has a basin size of one.
The network was analyzed by a biologist concluding that
the resulting model was quite consistent from a biological
point of view, nevertheless, it is important to point out that
this synthetic network (and others) can only be validated
making live experiments with genetically modified yeast
that replicate the conditions of the synthetic network. The
cell will be able to complete the cycle until the M phase,
nevertheless, it will not be able to enter the G1 phase since
the Cdc20 protein will be absent, because the two positive
edges that activate protein Cdc20 are not present in the
synthetic network. Without Cdc20, the cell can not pass
from the M phase to the G1 phase, because the complex
Cdc20/APC will not be formed and, without this complex,
several key activations and degradation will not happen, then
the cell will remain in metaphase (inside M phase), unable to
complete the cycle. If the cell does not enter the G1 phase,
there will be no cell growth that allows the increase of the
concentration of cyclin 3 (Cln3), which triggers the entrance
to the cell-cycle. Therefore, the Cln3 concentration could be
increased to emulate the final stage of G1, where if the final
environmental conditions are appropriate, Cln3 will pass a
concentration threshold, thus entering the S phase, starting
a new cycle. To accomplish this, one approach is to supply
300
450
400
Frequency of synthetic networks found
Frequency of synthetic networks found
250
350
300
250
200
150
100
200
150
100
50
50
0
0
1
2
3
4
5
6
7
8
Basin size of the G1 fixed point
9
10
Figure 4. Frequency of the networks found with different basins size of
the G1 fixed point.
0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Number of fixed points
Figure 6. Frequency of the networks found containing different numbers
of fixed points.
Cell Size
200
Frequency of synthetic networks found
180
160
SFM
Cln3
B FM
140
120
Clnc1,
100
Cl061&
Sidc
80
60
Cl0c1,
40
0
B d4 cw
SMM
C2bc
20
34
35
36
37
38 39 40 41 42
Number of edges
43
44
45
46
47
Figure 5. Frequency of the networks found with different number of edges.
the cell with cAMP and glucose, because the mRNA levels
of Cln3 rise in response to these molecules [26].
C2d, hmC2dc/
S5i6
Figure 7. An example of a synthetic network of the budding yeast cellcycle found by the bees algorithm with a reduced basin size of the G1
fixed point. (Color online) The green/solid edges represent positive weights
(activations), the red/dashed edges represent negative weights (inhibitory).
Pajek
VI. C ONCLUSION
A swarm intelligence approach, by means of the bees
algorithm, was formulated to build synthetic networks of
the budding yeast cell-cycle under the threshold Boolean
network model. The resulting networks contain the original
fixed points of the budding yeast cell-cycle network plus
additional fixed points to reduce the basin size of the fixed
point associated to the G1 phase of the cell cycle, with the
purpose of promoting cell proliferation for biotechnological
applications. The proposed method can help biologist find
synthetic networks with the desired characteristics, in this
case small basin size of the G1 fixed point, which would be
very difficult and time consuming if they were to construct
these networks by hand. The resulting synthetic networks
must be tested later on in the lab by means of experiments
with genetically modified yeast that replicate the conditions
of the synthetic network. Future research will consider the
study of the update robustness of the synthetic networks.
ACKNOWLEDGMENT
The authors would like to thank Conicyt-Chile under grant
Fondecyt 11110088 (G.A.R.), Fondecyt 1100003 (E.G.),
Fondecyt 1110850 (T.T.), Basal(Conicyt)-CMM (E.G.), and
ANILLO ACT-88 (G.A.R., E.G.) for financially supporting
this research.
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