Finding All Steady States in Biological Regulatory Networks
ABSTRACT Motivation. Computing the long term behavior of regulatory and signaling networks is critical in understanding how biological functions take place in organisms. Steady states of these networks determine the activity levels of individual entities in the long run. Identifying all the steady states of these networks is difficult as it suffers from the state space explosion problem. Results. In this paper, we propose a method for identifying all the steady states of regulatory and signaling networks accurately and efficiently. We build a mathematical model that allows pruning a large portion of the state space quickly without causing any false dismissals. For the remaining state space, which is typically very small compared to the whole state space, we develop a randomized algorithm that extracts the steady states. This algorithm estimates the number of steady states, and the expected behaviors of individual genes and gene pairs in steady states in an online fashion. Also, we formulate a stopping criteria that terminates the randomized algorithm as soon as user supplied percentage of the results are returned with high confidence. Finally, in order to maintain the scalability of our algorithm to very large networks, we develop a partitioningbased estimation strategy. We show that our algorithm can identify all the steady states accurately. Furthermore, our experiments demonstrate that our method is scalable to virtually any large real biological network.
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ABSTRACT: Expression of the Drosophila segment polarity genes is initiated by a prepattern of pairrule gene products and maintained by a network of regulatory interactions throughout several stages of embryonic development. Analysis of a model of gene interactions based on differential equations showed that wildtype expression patterns of these genes can be obtained for a wide range of kinetic parameters, which suggests that the steady states are determined by the topology of the network and the type of regulatory interactions between components, not the detailed form of the rate laws. To investigate this, we propose and analyse a Boolean model of this network which is based on a binary ON/OFF representation of mRNA and protein levels, and in which the interactions are formulated as logical functions. In this model the spatial and temporal patterns of gene expression are determined by the topology of the network and whether components are present or absent, rather than the absolute levels of the mRNAs and proteins and the functional details of their interactions. The model is able to reproduce the wildtype gene expression patterns, as well as the ectopic expression patterns observed in overexpression experiments and various mutants. Furthermore, we compute explicitly all steady states of the network and identify the basin of attraction of each steady state. The model gives important insights into the functioning of the segment polarity gene network, such as the crucial role of the wingless and sloppy paired genes, and the network's ability to correct errors in the prepattern.Journal of Theoretical Biology 08/2003; 223(1):118. · 2.30 Impact Factor  SourceAvailable from: David R Smyth[Show abstract] [Hide abstract]
ABSTRACT: Mutations in the APETALA1 gene disturb two phases of flower development, flower meristem specification and floral organ specification. These effects become manifest as a partial conversion of flowers into inflorescence shoots and a disruption of sepal and petal development. We describe the changes in an allelic series of nine apetala1 mutants and show that the two functions of APETALA1 are separable. We have also studied the interaction between APETALA1 and other floral genes by examining the phenotypes of multiply mutant plants and by in situ hybridization using probes for several floral control genes. The results suggest that the products of APETALA1 and another gene, LEAFY, are required to ensure that primordia arising on the flanks of the inflorescence apex adopt a floral fate, as opposed to becoming an inflorescence shoot. APETALA1 and LEAFY have distinct as well as overlapping functions and they appear to reinforce each other's action. CAULIFLOWER is a newly discovered gene which positively regulates both APETALA1 and LEAFY expression. All functions of CAULIFLOWER are redundant with those of APETALA1. APETALA2 also has an early function in reinforcing the action of APETALA1 and LEAFY, especially if the activity of either is compromised by mutation. After the identity of a flower primordium is specified, APETALA1 interacts with APETALA2 in controlling the development of the outer two whorls of floral organs.Development 11/1991; 119(3):721743. · 6.27 Impact Factor  SourceAvailable from: Bela Novak[Show abstract] [Hide abstract]
ABSTRACT: p53 is activated in response to events compromising the genetic integrity of a cell. Recent data show that p53 activity does not increase steadily with genetic damage but rather fluctuates in an oscillatory fashion. Theoretical studies suggest that oscillations can arise from a combination of positive and negative feedbacks or from a long negative feedback loop alone. Both negative and positive feedbacks are present in the p53/Mdm2 network, but it is not known what roles they play in the oscillatory response to DNA damage. We developed a mathematical model of p53 oscillations based on positive and negative feedbacks in the p53/Mdm2 network. According to the model, the system reacts to DNA damage by moving from a stable steady state into a region of stable limit cycles. Oscillations in the model are born with large amplitude, which guarantees an allornone response to damage. As p53 oscillates, damage is repaired and the system moves back to a stable steady state with low p53 activity. The model reproduces experimental data in quantitative detail. We suggest new experiments for dissecting the contributions of negative and positive feedbacks to the generation of oscillations.Cell cycle (Georgetown, Tex.) 04/2005; 4(3):48893. · 5.24 Impact Factor
Page 1
Vol. 00 no. 00 2009
Pages 1–11
Finding All Steady States in Biological Regulatory Networks
Ferhat Ay,Fei Xu,and Tamer Kahveci
Computer and Information Science and Engineering, University of Florida, Gainesville, FL 32611
{fay, feixu, tamer}@cise.ufl.edu
ABSTRACT
Motivation. Computing the long term behavior of regulatory and
signaling networks is critical in understanding how biological functions
take place in organisms. Steady states of these networks determine
the activity levels of individual entities in the long run. Identifying all
the steady states of these networks is difficult as it suffers from the
state space explosion problem.
Results. In this paper, we propose a method for identifying all the
steady states of regulatory and signaling networks accurately and
efficiently. We build a mathematical model that allows pruning a
large portion of the state space quickly without causing any false
dismissals. For the remaining state space, which is typically very
small compared to the whole state space, we develop a randomized
algorithm that extracts the steady states. This algorithm estimates
the number of steady states, and the expected behaviors of individual
genes and gene pairs in steady states in an online fashion. Also, we
formulate a stopping criteria that terminates the randomized algorithm
as soon as user supplied percentage of the results are returned with
high confidence. Finally, in order to maintain the scalability of our
algorithm to very large networks, we develop a partitioningbased
estimation strategy. We show that our algorithm can identify all the
steady states accurately. Furthermore, our experiments demonstrate
that our method is scalable to virtually any large real biological
network.
Availability. All the code developed for this paper is available at
http://bioinformatics.cise.ufl.edu/palSteady.html
1
Analyzing biological networks is essential in understanding the
machinery of living organisms which has been a main goal for
scientists. We use the term biological regulatory networks (BRN)
to combine gene regulatory networks and signal transduction
pathways. To capture the biological meaning of BRNs, it is
necessary to characterize their long term behavior. A common way
to achieve this is to identify the steady states of the dynamic system
defined by BRNs. A state shows the activity levels of genes at a
certain time point. The state of a BRN can change over the time due
to internal regulations or external stimulants. We say that a state of a
BRN is steady if that state will periodically be revisited forever once
the BRN reaches to that state (see Figures 2 (a) and (b)). In other
words, the steady states represent the possible long term outcomes
of the corresponding BRN. Identification of steady states is a crucial
problem and is useful in many applications such as the treatment
of various human cancers [12] (e.g. leukemia, glioblastoma) and
INTRODUCTION
genetic engineering [21]. In this paper, we consider the problem of
identification of all steady states in BRNs.
The naive approach to this problem is to exhaustively search
the state space for the steady states. However, the number of
possible states of a BRN is exponential in the number of its
genes. For a BRN with n genes, the state space size is 2n
for the boolean state model. Therefore, exhaustive methods are
computationally infeasible for even moderate sized BRNs. To
address this problem, some existing methods use finitestate Markov
chains [10], binary decision diagrams (BDD) [9], constraint
programming [6], probabilistic boolean networks [24] and relational
programming [22]. Orthogonal to the selection of the above method,
there are two options for modeling the state transitions. These
are synchronous and asynchronous models and both are used in
the literature. Synchronous model assumes that the activity levels
of all the genes change simultaneously. Hence, the next state
is deterministically decided by the current state. On the other
hand, asynchronous model assumes that the time is divided into
small enough intervals, such that only one gene can change its
state at a time and state change is equally likely for all genes.
The advantages/disadvantages of these models together with their
effect on running time of steady state identification algorithms are
discussed in several papers in the literature [1, 2, 8]. Here, we use
the asynchronous model in our algorithm discussion since it allows
theidentificationofanimportanttypeofsteadystates, namelycyclic
steady states. However, it is important to point out that our method
is independent of the choice of state transition model.
Garg et al. [9] proposed one of
the most recent methods to identify
the steady states of BRNs. They
use the boolean network model for
BRNs.They represent the state
space using the BDD data structure.
Their algorithm, however, fails to
classify a significant percentage of
the states correctly. Here, we present
a brief example on a real subnetwork
given in Figure 1 to illustrate the
problemsofthismethodofGargetal.
(Detailed evaluation of this method
is in Section 4.2.) For this 6 gene
subnetwork of human p53 signaling
pathway, Garg et al.’s method finds
four steady states.The sets of
active genes in these steady states
????
??????
????
????
???
???
Fig. 1: A portion of p53
signalingpathway.
pathway plays an important
role in the origination of
differentcancer
Pointed arrow
represent positive regulation
(activation) and arrow heads
with bars represent negative
regulation (inhibition).
p53
types.
heads
c ? Oxford University Press 2009.
1
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Ay et al
are: {MDM2, MDMX}, {MDM2},
{MDMX}andØ. Inordertoevaluate
the accuracy of their results, we surveyed the literature on p53
network and extracted its steady states. We observed that the first
three states reported above by their algorithm are not steady (i.e.
they are false negatives). Also, their method fails to find the steady
states corresponding to oscillation(s) between p53 and MDM2
levels (false negatives) that are observed in the literature [4, 14].
Finally, the running time of their method grows quickly as the
complexity or the size of the underlying BRN increases.
Our Contributions: In this paper, we develop an algorithm that
identifies all the steady states of BRNs accurately and efficiently.
We use the boolean network model proposed by Kauffman et al [13]
due to its parallelism with the level of currently available data and
its successful applications on real data sets [7, 16]. We build a
hypothetical state transition graph using the interactions in a BRN.
We develop a mathematical model that classifies each state into one
of the three classes, namely Type 0, Type 1 and Type 2. Type 0
and Type 2 states are guaranteed to be steady and transient (i.e.
not steady), respectively. Type 1 states can be either one. We use
BDDs to partition the states into these classes efficiently. To further
classify the Type 1 states, we develop a randomized algorithm.
While sampling, we calculate the estimators for the number of
steady states, expected steady state distribution of individual genes
and jointsteady state distributions of gene pairs. We calculate a
stopping criteria from the statistical information of explored states.
This criteria allows early termination of sampling when the user
defined percentage of steady states are found with high confidence.
To make our algorithm scalable to large BRNs, we incorporate
a partitioning strategy. Our partitioning strategy works for both
disconnected and weakly connected subnetworks without losing
steady state information. In summary, our technical contributions
are:
• We build a mathematical model for pruning a very large portion
of state space quickly without any information loss.
• We develop a randomized algorithm that computes estimators
for the number of steady states and the fraction of individual genes
and gene pairs being active in these states in an online fashion.
Our algorithm guarantees to find all the steady states after sufficient
number of iterations.
• We develop a partitioningbased strategy to utilize the weak
connectivity of BRNs.
The organization of the rest of this paper is as follows: Section 2
presents the background information. Section 3 discusses our
methods. Section 4 presents the experimental results. Section 5
concludes the paper.
2BACKGROUND
In order to find the long term behavior of BRNs, a dynamic model
should be used to express the state changes. Here, we describe how
the dynamic behavior of BRNs are modeled. First, we define a state
and a steady state of a BRN. Then, we discuss the existing methods
and models for identification of steady states in dynamic biological
networks.
Assume that a BRN
has n genes. A state
???
???
???
???
???
???
???
???
???
???
???
???
???
???
???
???
???
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of this BRN consists
of activity levels of
individual genes. We
represent the state of
the ith gene by Xi =
1 (active) or Xi = 0
(inhibited). The state
of that BRN is denoted
as [X1X2···Xn]. A
set of genes alters the
state of another gene
set if the first set has
activatorsorthesuppressors
of the genes in the
second set. Figure 2
shows how the state
of a BRN changes for
threehypotheticalBRNs.
A steady state in the
state transition graph is
a state in which the
system stabilizes. This stability is achieved at a state either if none
of the genes issue a change in their activity levels or if there is only
one possible chain of state transitions from the current state which
deterministically leads to the same state after a certain number of
state transitions. In other words, once the state transitions lead to a
steady state the probability of revisiting this state through a fixed
sequence of states is 1. Thus, the steady states reflect the long
term behavior of the biological system. We call all the remaining
states transient. For example, consider the states of the BRN in
Figure 2(a). Here, the state [110] is steady as the BRN remains in
that state forever once it reaches to that state. The BRN shown in
Figure 2(b) has four steady states, namely [011], [111], [110] and
[010]. This is because the BRN deterministically visits these states
in cycle forever once it reaches to one of them. We call these type of
steady states “cyclic steady states”. On the other hand, all the states
in Figure 2(c) are transient. The reason is that once the state [110]
is reached, it is impossible to know which path will be taken before
revisiting [110].
Existing work on steady state identification: The trivial way
to identify all the steady states is to analyze the whole state
transition graph exhaustively. However, the size of this graph grows
exponentially with the number of genes in the BRN. Therefore,
enumeration of all the states is not practical and efficient methods
are necessary.
A number of methods have been developed to find the steady
states of BRNs. Devloo et al. [6] used generalized logical
analysis [29] together with constraint programming for this purpose.
They introduced an image function to traverse the state transition
graph by utilizing the logical parameters described in Snoussi et
al.[26]. However, their algorithm does not work for networks
containing multiple arcs of same weight from the same origin.
Since networks with multiple arcs of same weight from the same
origin are common [17], this limitation restricts the applicability of
2
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Finding All Steady States in Biological Regulatory Networks
their algorithm. Using the Probabilistic Boolean Network Model,
Shmulevich et al.[25] calculated the steady state probabilities
of genes via random gene perturbations. However, they don’t
distinguish between different interaction types such as activations
and inhibitions. Ignoring the type of interactions in the modeling
phase degrades the accuracy of their algorithm from the very
beginning. Shlomi et al.[23] used linear programming to extract the
steady states after gene perturbations. However, their method cannot
find the cyclic steady states in BRNs. Recently, Schaub et al.[22]
proposed an extension of boolean network model called Qualitative
Networks and analyzed signaling networks by symbolic methods.
They identified the infinitely visited states of multicellular systems
by using relational computation tools. However, their idea of using
partitioned transition relations to speed up the algorithm is not valid
for single cell model.
To the best of our knowledge, the most recent algorithm for
identification of all steady states of biological regulatory networks
is proposed by Garg et al.[9]. They used boolean network model to
represent the regulatory networks and BDD data structure for the
representation of its state space. This method, however, incurs both
false positives and false negatives (see the example in Section 1).
Furthermore, its running time increases quickly as the complexity
and the size of the BRN increases. We evaluate this algorithm in
detail in Section 4.2.
3
This section discusses our algorithm for identifying all the steady
states of BRNs. Section 3.1 describes the mathematical model for
expressing the states of a BRN. Section 3.2 discusses the strategy
we use to split the search space into three subspaces. Section 3.3
presents our randomized algorithm for extracting steady states
from the third subset above. Section 3.5 discusses our partitioning
strategy to accelerate our algorithm.
3.1 State Transition Model
In order to identify the steady states of a BRN, we first need to build
a mathematical model that explains its states and how the network
moves from one state to another.
Let Xi(t) = true/false denote the state of the ith gene at time
t. Here true denotes that ith gene is “active” and false denotes that
it is “inactive”. We use Xiinstead of Xi(t) for simplicity wherever
appropriate.
We summarize the interactions that determine the next state of
the ith gene from the activity values at time t as follows. The ith
gene will be inhibited if at least one of its suppressors is active. If
all the suppressors of the ith gene are inactive and at least one of
its activators is active, then it becomes active in the next time step.
The assumption that inhibitors are stronger is due to the binding
mechanism of inhibitors. In all other situations the state of the ith
gene remains unchanged. The following equation summarizes this
process:
Xi(t + 1) : (Xi(t) ∨ pA(t)) ∧ ¬pS(t)
In this equation, the symbols ∨ and ∧ denote the logical “AND”
and “OR” operators, pA(t) and pS(t) represent predicates for the
activators and the suppressors of the ith gene at time t, respectively.
METHODS
(1)
We compute these predicates as pA(t) =?
and the suppressors of the ith gene.
EXAMPLE 1. Consider the BRN in Figure 1. Assume that E2F1
and ATR are the only active genes at some point in time. E2F1 will
activate p14ARF and ATR will activate p53. When p53 becomes
active, it will suppress p14ARF even when E2F1 remains active. p53
will also activate MDM2, whose activity will then suppress p53.
It is worth mentioning that our model uses boolean values for the
states of the genes since it is successfully used in the literature for
BRNs [7, 16]. The method we develop in this paper is independent
of this choice. It can be used for categorical state model, where
the state of a gene can have more than two activity levels (e.g.
low, medium, high activity) and more complex predicates for
interactions.
j∈AXj(t) and pS(t) =
?
j∈SXj(t), where A and S are the sets of indices for activators
f
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
??
??
??
steadytransient
observed
not observed
Type 0 Type 1
Type 2
a
b
c
d
e
?
?
?
?
Fig. 3: Summary of the traversal process for a randomly picked state
from unobserved Type 1 states. The unobserved state is either a
steady state as shown with f or a transient state as shown with a. In
the latter case, the traversal may take different paths before labeling
a as transient. In both cases, all the states on traversed paths are also
labeled as steady or transient.
An important observation is that, even though the next state of the
ith gene is deterministically calculated, there can be multiple next
states for the whole network since we use asynchronous model. A
state of a given BRN is defined by the states of individual genes.
Let u = [X1 ··· Xn] denote a state of the network. The network
can move from state u to state v = [X1 ··· Xi−1 ¬Xi Xi+1 ···
Xn] only if the ith gene is one of the genes that can have a state
change. Individual genes that can issue a state change at a given
state determines the possible next states of the network.
We model the changes in the states of a BRN using an abstract
graph representation. In this graph, each vertex corresponds to a
possible state of the BRN. Thus, if there are n genes in a BRN,
3
Page 4
Ay et al
then the corresponding graph contains 2nvertices. There is an edge
from vertex u to vertex v, if it is possible to change the state of the
BRN from the state represented by u to the state represented by v
by only changing the state of a single gene. There can be up to n2n
edges between these states. This graph is hypothetical as we use it
only for building our mathematical model. We never materialize this
exponential sized graph in our method.
We classify the vertices of this graph into three classes based
on the number of their outgoing edges. Figure 2 provides visual
examples for all three state types listed below:
• Type 0: The vertices that do not have any outgoing edges (except
self cycles). These vertices correspond to steady states as the
state of the network cannot change once one of them is visited.
(Figure 2(a))
• Type 2: The vertices that have two or more outgoing edges. These
vertices represent transient states. (Figure 2(c))
• Type 1: The vertices that have exactly one outgoing edge. These
are the tricky vertices. The states for these vertices can be steady
or transient. (Figure 2(b) and 2(c))
In the following section, we describe the method we use for the
segregation of states into the above three types.
3.2Segregation of States using BDDs
As we discussed in the previous section, we never generate the state
transition graph of the input network. A simple observation on our
state transition model allows us to segregate the states without this
materialization. This segregation results in not only the immediate
identification of all noncyclic steady states, but also eliminates a
huge portion of states by classifying them as transient. For instance,
for the pyk2 network with 24 genes and 16,777,216 (224) possible
states, our segregation method classifies 114,300 (≈ 217) states
as Type 0 (i.e. steady) and 16,086,712(≈ 224) states as Type 2
(i.e. transient) in only 0.08 seconds. Remaining 576,204 states are
labeled as Type 1. Thus, we need to explore only a small fraction of
the state space (which still can be large for practical purposes).
Here, we describe how we construct the BDDs for all Type 0
states and all Type 1 states, namely Z0 and Z1. We first define a
predicate that will be handy in this discussion.
Ci : Xi(t + 1) ⊕ Xi(t)
(2)
Here, ⊕ denotes the logical “XOR” operator. Ci evaluating to true
at time t means that gene i will change its state from Xito ¬Xiat
time t + 1. Otherwise, it preserves its current state. The following
equations, show the formulas of BDDs representing Type 0 and
Type 1 states:
?
Z0 = “True” represents the states that do not satisfy any of the
Ci conditions (i.e. none of the genes change state). The states in
Z1 = “True” satisfy exactly one of the Ci conditions (i.e. exactly
one gene changes state). The states which are not included in the
Z0 :
i
¬Ci
and
Z1 :
?
i
(Ci∧ (
?
j?=i
¬Cj).
two BDDs above are called Type 2 and they are all transient states
since they have more than one possible successor states. The BDD
for these states can be constructed similarly. However, we eliminate
these states since they do not reflect the long term behavior of the
system. By doing this without materialization, we quickly reduce
the state space of the problem to a significantly smaller one. In the
next section, we describe how we decide the steadiness of Type 1
states.
3.3
We have shown that we can identify steady states corresponding to
Type 0 vertices and transient states corresponding to Type 2 vertices
efficiently. In this section, we develop a randomized algorithm that
identifies the steady states of Type 1. We call these states “cyclic
steady”. AnexampleforthisisthecycleoffourstatesinFigure2(b).
Our randomized algorithm traverses the Type 1 vertices and
reports the ones corresponding to steady states. After traversing
a portion of the vertices, we estimate the total number of steady
states, the probability of each gene being active and the joint
probability of gene pairs being coexpressed in steady states. This
is desirable because the user can decide to stop the traversal
when sufficient number of steady states are discovered with high
confidence (e.g. when 90% of the steady states are reported with
95% confidence). This kind of online querying is useful especially
when the underlying BRN is very large and the space of Type 1
states is too big to completely traverse.
Algorithm 1 briefly describes how we traverse the Type 1 states.
Next, we elaborate on different steps of this algorithm.
Extracting Cyclic Steady States
Algorithm 1 RANDOMIZED TRAVERSAL OF TYPE 1 STATES
1. Randomly get an unobserved vertex from the Type 1 set.
2. Follow the out going edge to traverse the graph until seeing one
of the following vertices
(i) A vertex that is labeled as transient or steady in previous
iterations.
(ii) A vertex that is traversed in this iteration.
3. Label all the traversed vertices as transient or steady and update
the estimators.
4. Stop if number of steady states observed so far is sufficient.
Step 1. We obtain a random seed state among the untraversed
satisfying assignments of the BDD for Type 1 states. We do this by
traversing the BDD from root node to the leaf level. At each step of
the traversal, we randomly pick a child node of the currently visited
node. The probability of moving to a child is equal to the number of
states under the subtree of that child divided by the number of states
under the subtree of the parent node. Each level of the BDD decides
the state of one gene. Thus, by randomly branching to a child
node we choose the state of a gene randomly from the satisfying
assignments of the BDD. When we reach the leaf level of the BDD,
4
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Finding All Steady States in Biological Regulatory Networks
the states of all the genes are determined and hence, our seed state
for the whole BRN.
Step 2. Once we choose an unobserved Type 1 state, the next step is
tounderstandwhetherornotwecan reachtoanewsteadystatefrom
this state. To do this, we traverse the state transition graph starting
from this vertex by following the edges.
Since the chosen state is of Type 1, by definition, it has only
one outgoing edge. Thus, we can easily find the next state as the
state that satisfies the transition condition. We continue traversal
by applying the same principle. Figure 1 of the Supplementary
Materials summarizes the possible cases that can occur during this
traversal. Totally there are four different cases. Starting from an
unobserved state if we traverse one of the following three paths then
all the states visited on this path are transient:
• A path ending in a Type 0 state
• A path ending in a Type 2 state
• A path ending in a state that is observed in previous iterations
Notice that all three cases correspond to Step 2(i) of our traversal
method. The next case produces both cyclic steady and transient
states.
• A path leading to a cycle of states visited in current observation
In this case, we label all the states on the cycle as steady and the
other states on the path as transient.
Step 3. At each iteration of the while loop, we traverse a path in the
state transition graph and label each state on this path as transient
or steady. We name the set of vertices visited in each such traversal
as an observation. Using these observations, we develop estimators
for the total number and the “profile” of steady states. The profile of
the steady state is the vector where the ith entry is the expected
fraction of the steady states at which the ith gene is active. For
example, if the second entry of the profile is 0.95, it means that we
expect that the second gene is active in 95% of the steady states. We
also compute the estimators for the joint expression (coexpression)
fractions of gene pairs. Computing these estimates is important as
they can lead to early prediction of the steady state profile and early
termination of the algorithm.
Here, we describe in detail the calculation and the analysis of the
estimator for the total number of cyclic steady states. First of all,
we prove that it is an unbiased estimator. Then, we discuss how to
minimize the variance of this estimator. For the other estimators we
only give the formulations.
First, let us introduce some notation we use throughout this
section:
• N0, N1: Number of Type 0 and Type 1 states, respectively.
• Oi = (si,ti):
observed steady and transient states traversed in this observation.
• Si, Ti, Ui: Total number of observed steady states, observed
transient states and unobserved states after first i observations,
respectively.
ith observation. si and ti are the number of
From the definitions above, we can calculate Ui = N1−Si−Ti,
Si =?i
assuming at any time one and only one of the Bi’s can be 1. In other
words, E[BiBj] = 0 for any i ?= j. Notice that E[Bi] = E[Bn
si+ti
N1
number of Type 1 steady states at the ith iteration as:
j=1sjand Ti =?i
j=1tj. Now, we introduce a 0/1 random
variable Bi for each observation Oi. We simulate our sampling by
i] =
for observation Oi. We formulate the estimator of the total
Fi =
i
?
k=0
Bksk
N1
sk+ tk
(3)
LEMMA 1. The estimator Fiis an unbiased estimator.
Proof: We prove this by showing the expected value of Fiis equal
to the total number of Type 1 steady states. Taking expectations of
both sides and replacing E[Bk] with
sk+tk
N1
:
E[Fi] = E[
i
?
k=0
Bksk
N1
sk+ tk] =
i
?
k=0
E[Bksk
N1
sk+ tk] =
i
?
k=0
sk
After defining the estimator, the next step is to calculate its
variance.
LEMMA 2. The variance of Fiis
V ar[Fi] =
i
?
j=0
s2
j(
N1
sj+ tj) − [
i
?
j=0
sj]2.
Proof: We know that, V ar[Fi] = E[F2
compute F2
i] − E2[Fi]. We first
i.
F2
i =
i
?
?
j=0
Bjsj
N1
sj+ tj
i
?
N1
sj+ tj)(
k=0
Bksk
N1
sk+ tk
=
j?=k
BjBksjsk(
N1
sk+ tk) +
i
?
j=0
B2
js2
j(
N1
sj+ tj)2
When we take the expected value of F2
E[BjBk] = 0 for any i ?= j. Hence, the variance of Fi can be
computed as:
ithe first term cancels since
V ar[Fi] = E[F2
i] − E2[Fi] = E[
i
?
j=0
B2
js2
j(
N1
sj+ tj)2] − (E[Fi])2
=
i
?
j=0
s2
j(
N1
sj+ tj) − [
i
?
j=0
sj]2
There are many ways to build an estimator from Fjs.
However, it is desirable to build an estimator with a small variance
as it converges to true solution faster. Following Lemma builds the
estimator with minimum variance.
5
Page 6
Ay et al
LEMMA 3. The estimator that has the smallest variance is
T =
?
j
1
ViVjFj
?
i
1
Proof:
F1,F2,...,Fn with variances V1,V2...,Vn to minimize the overall
variance of our estimation. In other words, we want to find the
weight parameters γ1,γ2,...,γn such that
variance of the estimator for total number of steady states of Type
1 is minimized. Let us denote this new estimator as T =?γiFi.
V ar(T) =
Mathematically, our aim is to minimize?γ2
?
Taking derivative of both sides with respect to each γi, we get the
equations:
Now,we discuss how we combine the estimators
?γi = 1 and the
Then,
?
γ2
iVi
iVi given?γi = 1.
We formulate this problem by using Lagrange Multiplier as follows:
L =γ2
iVi− λ(
?
γi− 1)
2γiVi− λ = 0,λ =
1
?
1
2Vi
Solving these equations we get the γi values that minimizes the
V ar(T) as:
γj =
1
1
ViVj
?
Thus, by using the value of γis we find that the estimator with
smallest variance is
T =
?
j
1
1
?
ViVjFj
Next, we give the formulations of the estimators for the fractions
of each gene and each gene pair being active in steady states. First,
we formulate our estimator for the fraction of a gene being active in
cyclic steady states. Assume that the number of steady states at the
ith observation in which the kth gene is active is nk,i. An estimator
for the kth gene after the ith iteration is then :
Gk,i=
i
?
j=1
nk,j/Si
(4)
Let na↔b,i denote the number of steady states in which gene a
and gene b are both active or both inactive after the ith observation.
We calculate the estimator of joint probability of two genes having
the same activity level at a steady state as:
Ja↔b,i=
i
?
j=1
na↔b,j/Si
(5)
The probability of a gene being active in steady states is important
in characterizing the overall steady state behavior of this gene. We
found that some genes can be active in more that 90% of the steady
states, whereas some others show activation in only less than 10% of
these states. The joint probability of two genes being active together
is called coexpression. Coexpression of gene pairs are crucial in
functional annotation of genes [24]. Therefore, the estimators we
calculate above are useful in interpreting the coexpressed genes
in BRNs. We discuss their biological significance in experimental
results section.
Step 4. When our randomized algorithm finishes traversing all Type
1 states (steps 1 to 3), it finds all the steady states. However, in some
applicationsitmightbesufficienttofindapredeterminedpercentage
ofsteadystates. Wedevelopstatisticalcriteriatobeabletoterminate
the algorithm quickly after a sufficient portion of the Type 1 states
areexplored. Ourmethodstillguaranteesthatthedesiredpercentage
of the results are found with high confidence. More precisely, when
the user supplies a parameter α (e.g. 0.9), we compute a confidence
c ∈ [0,1], at each iteration such that “at least α×100 percent of the
steady states are found with probability at least c”. This is desirable
as the user can terminate the loop when c is large enough for the
underlying application.
Now, let us describe how the stopping criteria works. Let A∗
denote the actual number of total Type 1 steady states. If we have
known the value of A∗we could have stopped sampling with a
confidence value of c = 1 when A∗< Si +
satisfied. That is the time when we are sure that α × 100 percent
of the steady states are already reported. Since we do not know A∗
in advance, we use the information gathered from observed portion
of states. We compute Ai which denotes the minimum number of
total steady states that needs to be present for our method to decide
to continue traversal.
(1−α)(N0+Si)
α
is
Ai = Si+ (1 − α)(N0+ Si)/α
(6)
Trivially, if Ai > Ui + Si we just stop sampling with 100%
confidence as even if all the unobserved states were to be steady,
the reported ones constitute at least α × 100 percent of total steady
states. Otherwise, we calculate the confidence value in ith iteration
as the probability that we would have observed at least Si steady
states in our observations so far if there were Aiunobserved steady
states. Formally, we compute the confidence as:
?
k
C(Ai) =
Si+Ti
?
k=Si
[
Si+ Ti
?
qk
i(1 − qi)Si+Ti−k]
(7)
qi in Equation 7 represents the percentage of steady states if there
were Aisteady states in Type 1 states (i.e. qi =
of the summation represents “The probability of getting exactly k
steady states from Si+Ticurrently observed states if the probability
of a state being steady is qi”.
Lemma 4 shows that, the confidence value reported when we stop
sampling is never an over estimation. In other words, the stopping
criteria does not cause any false dismissals.
Ai
N1). The inner term
LEMMA 4. The confidence value given in Equation 7 by using
Aidoes not lead to false dismissal.
6
Page 7
Finding All Steady States in Biological Regulatory Networks
Proof: Here, we have three cases to consider:
• Case1 : (A∗> Ai)
Then, q∗ =
calculated as the area under the right hand side of a binomial
probability distribution function (i.e. inverse CDF), c value will
be larger for a larger value of q. Hence, C(A∗) > C(Ai). That
means whenever we stop sampling the confidence we report is
conservative.
• Case2 : (A∗= Ai)
Trivially, C(A∗) = C(Ai) when we terminate the sampling.
• Case3 : (A∗< Ai)
This case implies that we overestimated the total number of
steady states in confidence value calculation. Only thing that can
happeninsuchacaseisthatourmethodcontinuessamplingwhen
it does not need to. Since the actual number of steady states are
less than what we have guessed, when we decide to stop we have
already sampled at least as many steady states as we needed.
A∗
N1>
Ai
N1= qi. Since the confidence value is
Corollary 1 follows from Lemma 4.
COROLLARY 1. Our algorithm guarantees to find all the steady
states when the confidence value reaches 1.0.
3.4
An interesting case happens when the BRN can be partitioned into
disconnected subnetworks. Finding the steady states in the entire
BRN from the results of the individual parititions is relatively easy
in this case. When there are no activations or inhibitions between
two or more subnetworks of the input network, the steady states of
a partition can be determined independent of the other partitions. In
the case of BRNs, that type of partitioning is usually applicable [11].
Assume that we have k independent partitions for an input BRN.
We know that Type 0 states have no outgoing edges and Type 1
steady states have only one outgoing edge which is a part of a cycle.
Therefore, the combination of two Type 0 steady states from two
partitions results in a Type 0 steady state for the union of these
partitions. This is also true for multiple partitions. On the other
hand, combining a Type 1 steady state of a partition with a Type
0 steady state of another partition we get a Type 1 steady state for
theunionofthesepartitions. However, combiningtwoType1steady
states produces two outgoing edges from combined state. Hence, all
such statesare transient. Clearly, any othertype of state combination
results in transient states.
Let, Θ1,Θ2,··· ,Θkand Ω1,Ω2,··· ,Ωkdenote the number of
Type 0 and Type 1 steady states in each partition. Using the above
observations on combining the steady states, we formulate the total
number of Type 0 and Type 1 steady states (Θ and Ω respectively)
of the actual network as:
Partitioning of independent subnetworks
Θ =
k
?
i=1
Θi
and
Ω =
k
?
i=1
[Ωi
?
i?=j
Θj].
3.5
The initial segregation of states and the use of stopping criteria
for sampling accelerate our algorithm significantly. Since the state
space grows exponentially, further speed up might be necessary for
large networks. For this purpose, we propose a network partitioning
based strategy to increase the scalability of our method. Especially,
when the number of Type 1 states and the number of cyclic steady
states are large, partitioning the network before applying the above
steps can improve the running time significantly. In here, we
discuss how we combine the results gathered from these partitions
to construct overall steady states of the actual network without loss
of any information.
Combining the results from different partitions is rather easy
when the partitions are independent of each other (i.e. when they are
disconnected). We leave that case to the Supplementary Materials
and discuss the harder case when the partitions weakly depend on
each other (i.e. there are a small number of interactions between
differentpartitions). WeuseTheBoostGraphLibraryfunctions[28]
to identify the two components of the network that have the least
number of interactions between each other. When necessary, this
division can be repeated recursively. In here, we explain our
method on a network that is divided into two subnetworks which
are connected by only one interaction of activation type. The
discussion can be generalized to partitions connected by more
than one interactions. However, the complexity of combining
the results of two partitions can increase exponentially with the
number of interactions between them. Therefore, it is important
to use partitioning of this type for only weakly dependent sub
networks. Thegood newsisthat, the weaklydependentsubnetworks
are commonly observed in real BRN data due to the sparsity
of interactions. For instance, the average number of interactions
per gene is only 1.12 in average in the real networks we use in
our experiments. We discuss the applicability of our partitioning
strategy on real data sets in Section 4.2.
Let us turn our attention back to the case with two subnetworks
connected by a single activation. Let P and R denote these
partitions and Ap ∈ P and Br ∈ R be the two genes such that
Ap activates Br (i.e. Ap → Br). Define another partition R?such
that R?= R ∪ {Ap}. We first identify all the steady states of each
of these three partitions using our algorithm. The tricky part is to
combine the steady states of these three partitions to find all the
steady states of the original network without losing any information.
We do this as follows. We first pick one steady state, say uP, from
the solution space of partition P. Depending on the type of uP, we
have three different cases to consider:
• Case1 : uP is of Type 0. In this case, each steady state of
partition R?can be combined with uP to create a combined steady
state. If Apis active in state uP we combine uP with only the steady
states of R?that have gene Apin active state. The case when Apis
inactive is combined similarly.
• Case2 : uP is of Type 1 and the state of Ap is fixed in the
steady state cycle that contains uP. If gene Ap is inactive in uP,
we combine it with only the Type 0 steady states of partition R.
When Apis always zero in a cyclic steady state of P, the activation
Further Partitioning the BRN
7
Page 8
Ay et al
between Ap and Br has no effect on the states of R and two
partitions act independently. In the case when Ap is always active,
we only combine uP with Type 0 steady states of R which have Ap
in active state.
• Case3 : uP is of Type 1 and the state of Ap is not fixed in the
steadystatecyclethatcontainsuP. IfgeneApchangesactivitylevel
(oscillates) in a steady state cycle with state uP, then we can only
combine uP with the Type 0 steady states of partition R in which
gene Aphas no effect on the state of gene Br. The only case when
Ap can have a state changing effect on Br is when all the other
activators and all inhibitors of gene Br are in inactive state. Hence,
we combine uP with only the steady states of R in which at least
one activator or an inhibitor of Bris in active state.
4
In this section, we evaluate the performance and accuracy of our
algorithm through various experiments on real BRNs. We compare
the significance of the steady states found by our algorithm to
both experimentally observed and computationally predicted steady
states. We measure the number of steady states observed, running
time and the activity levels of individual genes and gene pairs in
steady states.
Code: We implement our algorithm using C++. We use the BDD
implementation of the BuDDy package [15] as our data structure.
For partitioning the BRNs, we use Boost Graph Library [28]. We
obtain the executable of the method proposed by Garg et al.[9] from
the authors.
Dataset: We use the BRNs that are available from the literature.
Also, we use the signaling pathways and gene regulatory networks
available in the KEGG Pathway database [19].
Environment:We run all the experiments on a desktop computer
running Ubuntu 8.04 with a 3.20 GHz processor and 2 GB of RAM.
RESULTS AND DISCUSSION
4.1
To evaluate the accuracy of the results reported by our algorithm,
we compared the steady states that we found with the steady states
that are reported in the literature. For this purpose, we use the
genetic regulatory network that controls flower morphogenesis in
Arabidopsis thaliana which has been studied in detail in several
papers [17, 6].
The network shown in Figure 4 is taken from Mendoza et al. [17].
They analyzed this network using generalized logical analysis. They
focused on two subnetworks, namely the subnetworks that contain
the genes {AP3, PI} and {TFL1, LFY, AP1, AG}. They enumerated
allthepossiblestatesofeachsubnetworkandexhaustivelyidentified
all their steady states. We use the vector notation to represent the
activity levels of an ordered gene set. For instance, for a gene
set of {g1, g2, g3} [010] represents the state when g2 is active
and g1, g3 are inactive. Using this notation, the two steady states
found for {AP3, PI} are [00] and [11]. For the subnetwork of
{TFL1, LFY, AP1, AG} they considered two cases depending
on the activity of flower inhibitor gene EMF1. If EMF1 is active
[1000] is found as the only steady state for {TFL1, LFY, AP1, AG}
subnetwork. For the case when EMF1 is inactive, [0001] and [0010]
are found as the two possible steady states. When identifying these
Biological Significance
steady states, they did not consider the effect of input genes (LUG,
UFO and SUP) and the interactions between two subnetworks.
They assumed independent behavior for two subnetworks and only
transient activity for the three inputs. However, these assumptions
prevent them from combining the steady states correctly and cause
loss of biologically meaningful steady states as we will discuss next.
Assume that the states we give
in vector notation will correspond
to the states of the genes in the
order EMF1, TFL1, LFY, AP1,
AG, LUG, UFO, AP3, PI and SUP.
The six steady states found by
Mendoza et al. are described by
using ABC combinatorial model
proposedforflowermorphogenesis[5]
as follows: [0001000000] ⇒ A
function, [0001000110] ⇒ A and
B functions, [0000100110] ⇒ B
and C functions, {[1100000000],
[1100000110]} ⇒ nonfloral state
and [0000100000] ⇒ C function.
Our algorithm identified the last
four steady states when A function
is not active. The two states with
active genes of {AP1} and {AP1, AP3, PI} are not reported by
our method. Indeed, there is a high parallelism between the activity
levels of AP1 and LFY genes [3]. Additionally, they have a mutually
reinforcing effect which is denoted by the positive feedback loop in
Figure 4. Due to these reasons, the two states with active AP1 and
inactive LFY proposed by Mendoza et al. for describing the states
of genes when A function is active are not realistic. To express
the states when only A function is active and when both A and B
functions are active more realistically, we propose the two steady
states [0011000001] and [0011000110] which are identified by our
algorithm. The first one corresponds to high activity levels of AP1,
LFY and SUP genes. This describes the case when only A function
is active. The reason why SUP is active in this state is that it
suppresses the activator effect of LFY on the genes related to B
function. That way it keeps A as the only active function in this
state. Thesecondstateweproposecorrespondstothesituationwhen
SUP loses its inhibition effect on AP3 and PI. In that case, the active
genes are AP1, LFY, AP3 and PI. The first two genes are responsible
from A function and the last two are responsible from B function.
The two states that we propose above cannot be found by
Mendoza et al.’s method. The reason is, when they separate the
network into two parts they do not consider how LFY activity
can lead to different transitions than the case when the network
is complete. Since our algorithm takes only milliseconds for
identifying all the steady states of this network, we did not partition
the network. However, even if it was the case that we partitioned
the network, our method would have assured that none of the steady
states are lost by considering all the possible cases while combining
the results of partitions as we described in Section 3.5.
????
???
???
???
???
????
???
??
???
??
Fig. 4:
of
in
Pointed arrow heads represent
activation and arrow heads with
bars represent inhibition.
Regulatory network
flower morphogenesis
Arabidopsis Thaliana.
8
Page 9
Finding All Steady States in Biological Regulatory Networks
We provide additional results on coexpression of genes to test
the biological significance of the results in the Supplementary
Materials.
4.2
In this section, we compare the performance of our method to that of
Garg et al. [9]. We compared the number of steady states found by
each algorithm and the running times for a number of real BRNs
taken from KEGG. Table 1 reports the results for two different
versions of their method and our method. We explain what we mean
by different versions in the caption of this table.
When we use the original BRNs as inputs, Garg et al.’s method
crashed for most of the cases. We believe that, one of the reasons
of this is that their method assumes that the input genes without self
activations are always inactive. However, this is not the case in many
real BRNs. Additionally, this assumption literally disconnects input
genes from the network which results in a considerably smaller but
not complete model of the real BRN. As a result, the method of
Garg et al. misclassifies a significant portion of the steady states.
An example for that case is p53 network given in Table 1. For this
network, we report 9.4E+13 steady states, whereas their method
could find only 1 trivial steady state which is when all genes are
inactive. Thus, their assumption in modeling degrades the accuracy
and the applicability of their algorithm significantly.
We tried to fix the above problem in Garg et al. by adding
self activations to all the genes in the networks. Interestingly, the
numbers of steady states for modified networks are significantly
larger than the ones for the original versions. Also, we observed that
their method does not crash for any of the modified input networks.
However, it still incurs false positives and false negatives. The first
row of the Table 1 provides an example when the modified version
of Garg et al.’s method incurs false negatives (i.e. misses some of
the steady states). The second row, THelper network, shows the
case when it has false positives (i.e. transient states are classified as
steady).
The running times in Table 1 indicate that, even when we do not
use our partitioning strategy, our method runs significantly faster
than the modified version of the method of Garg et al. for all
the cases. It took around 1.5 minutes for our method to identify
the steady states for the largest network (MAPK), whereas their
algorithm could not find all the steady states in 10 days.
The last column in Table 1 reports the running times in bold
for the cases when partitioning into independent subnetworks
resulted in partitions having more than 20 genes. For these cases,
we used our partitioning strategy of separating weaklydependent
components. For instance, for the p38MAPK network with one
connected component (26 genes) our algorithm took 21 minutes.
When we divide the network into two components with 17 and 9
genesthatareconnectedbyonlyoneedge, therunningtimedropped
to 0.2 seconds.
The above results showed that our algorithm is more accurate
and significantly faster than the method of Garg et al. Furthermore,
our nontrivial partitioning strategy makes it scalable for even large
networks. Therefore, we believe that our method can be used for
Comparison with Existing Methods
accurate identification of all the steady states for even the large
scale BRNs.
4.3
To evaluate the quality of our samplingbased estimators, we
measuredtheircorrectnessandconvergencerate. Correctnessmeans
that the estimates will eventually converge to the correct value.
For the convergence rate, a good estimator should approximate the
correct value after a small fraction of the state space is explored.
We use a portion of p53 network in this experiment. We measure
the estimated number of steady states at which a gene is active for
each gene at each iteration of our algorithm. Our algorithm traverses
the entire space of Type 1 states in approximately 2,500 iterations
for this network. Figure 5 shows the results for seven different
genes. We plot these genes as they have different characteristics.
They vary in the fraction of steady states in which they are active
(e.g. CHK3 is active whereas p21 is suppressed in many steady
states). The results show that our estimators converge to the correct
ratio for all genes in less than 500 iterations. The rapid convergence
suggests that our algorithm approximates the correct profile of gene
levels without traversing the whole space of Type 1 states. Hence,
ouralgorithmisalsopracticalforlargeBRNswithhighconnectivity
which is the case when partitioning the network becomes difficult.
Accuracy of Estimators
4.4
As discussed in Section 3.3 of the paper, we calculate the fraction
of steady states in which a two genes are in active state together.
Biologically this fraction corresponds to the coexpression of the
two genes. Revealing coexpressed genes has great significance in
discovery of conserved genetic modules [27] and identification of
differentially expressed genes [20].
In here, we compare the coexpression values for gene pairs
found by our algorithm with the values reported in the gene co
expression database, COXPRESdb [18]. For this purpose, we use
Hedgehog signaling network of Homo Sapiens given in KEGG.
This network consists of 17 genes and hence, 136 possible gene
pairs. We sorted the gene pairs according to their coexpression
values in a decreasing order and compared our ordering with the
one in COXPRESdb. We picked top 20 gene pairs from our list
and searched for the indices of these pairs in the ordering of
COXPRESdb. Here, we report the largest index, l, among these k
indices for different values of k.
For k = 1 we have l = 1, which means that the highest
coexpressed gene pair (GL1SMO) in our ordering is also the
top scoring pair in COXPRESdb. For k = 5 we have l = 6,
meaning that the five gene pairs (GL1SMO, GSK3BFBXW11,
RAB23GAS1, GLI1IHH and SUFUSMO) with the highest ranks
in our ordering are in between the top 6 pairs in the ranking of
COXPRESdb. For the other values of k = 10 and k = 15, the l
values are 16 and 35 respectively. Hence, the gene pairs reported by
our method that are found to be active together in the steady states
suggest that there is a coexpression between these two genes. This
suggests that our algorithm is useful in predicting coexpression of
genes in BRNs.
Coexpression of gene pairs
9
Page 10
Ay et al
Table 1. The comparison of our algorithm with an existing method on eight different BRNs.1The method of Garg et al. when we use BRNs as they are
given in KEGG database.2The method of Garg et al. when we add self activations for all the genes in original BRNs.3Our algorithm without partitioning
the BRNs into weaklyconnected subnetworks.4Our algorithm when we partition the weaklyconnected subnetworks of size larger than 20. “” indicates
that the corresponding method crashed and “’>10 days’ indicates that the method could not finish reporting all the steady states after 10 days.
Network size
Genes
17
23
26
27
35
40
55
114
Number of Steady States
Garg et al.1
Garg et al.2
 616
34,685
23,087
 81,864
40
>665,450
47,520
1
>545,807

>304,131
Total Running Time (seconds or days)
Garg et al.1
Garg et al.2
 4.79s
0.178s 723s
3.6E+4s
 4.8E+5s
0.05s
>10 days
 9.8E+4s
0.2s
>10 days

>10 days
Network
Hedgehog
THelper
p38MAPK
fmlp
Adipocytokine
GnRH
p53
MAPK
Edges
11
35
28
31
35
23
56
55
Our Algo.3,4
11,700
2,152
23,087
81,864
8.3E+6
5.8E+7
9.4E+13
1.17E+30
Our Algo.3
0.02s
48s
1.26s
1.3s
125s
0.09s
144s
91s
Our Algo.4
0.02s
0.9s
0.2s
1.3s
0.4s
0.09s
0.1s
0.13s
5
In this paper, we proposed a computational method that can identify
all the steady states of BRNs accurately and efficiently. We built
a mathematical model that can explain the changes in the state of
BRNs. Using this, we employed BDDs to filter a large percentage
of the state space that has only transient states. The BDDs allowed
us to pick another subset in the state space which consists of all the
Type 0 steady states. For the remaining state space, we developed
a randomized traversal algorithm. This algorithm estimates the
number of steady states and the expected behavior of individual
genes and pairs of genes in the steady state in an online fashion.
We formulated a stopping criteria that terminates the randomized
algorithm as soon as user supplied percentage of the results are
returned with high confidence. We also developed a partitioning
based strategy that makes our algorithm scalable to virtually any
large network. Our experiments showed that, our method finds
many steady states that cannot be found using existing methods.
Furthermore, ouralgorithmrunssignificantlyfasterthantheexisting
methods especially for large networks.
CONCLUSION
0 5001000
Number of Iterations
1500 20002500
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Estimated fraction of steady states in which gene is active
CHK1
ATR
cyclinE1
CDC2
cyclinG
p21
SFN
Fig. 5: Convergence of the estimators of the fraction of steady states that
each gene is active.
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