Combinatorial control is prevalent in transcriptional regulatory networks. However, whether there are specific logic patterns over- or under-represented in real networks remains uninvestigated. Using a theoretic model and _in-silico_ simulations, we systematically study how the relative abundance of distinct regulatory logic patterns influences the network&#x2019;s global dynamics. We find that global dynamic characteristics are sensitive to several specific logic patterns regardless of the detailed network topology. We show it is possible to infer logic motifs based on the sensitivity profile and the biological interpretations of these global characteristics

Rui Jiang

Xuebing Wu

Zhirong Sun

English

Nature Precedings

 1 
Logic motif of combinatorial control in transcriptional networks 
 
Xuebing Wu,
 1
 Zhirong Sun 
2,*
, Rui Jiang 
1 
 
1
MOE Key Laboratory of Bioinformatics and Bioinformatics Division, TNLIST / 
Department of Automation  
2
MOE Key Laboratory of Bioinformatics, State Key Laboratory of Biomembrane and 
Membrane Biotechnology, Department of Biological Science and Technology，
Tsinghua University, Beijing 100084, China 
* 
Corresponding author 
 
Combinatorial control is prevalent in transcriptional regulatory networks. 
However, whether there are specific logic patterns over- or under-represented in 
real networks remains uninvestigated. Using a theoretic model and in-silico 
simulations, we systematically study how the relative abundance of distinct 
regulatory logic patterns influences the network’s global dynamics. We find that 
global dynamic characteristics are sensitive to several specific logic patterns 
regardless of the detailed network topology. We show it is possible to infer logic 
motifs based on the sensitivity profile and the biological interpretations of these 
global characteristics.  
 
Introduction 
Gene regulatory networks, or more specifically, transcriptional regulatory networks, 
are vital for many important biological processes, such as development and the 
response to environmental changes. The study of the static architecture and dynamic 
behaviors of gene regulatory networks has long been one of the central problems in 
genetics. Recent works have shown that transcriptional regulatory networks are 
composed of basic building blocks — network motifs 1, 2. Network motifs are 
enriched small sub-network patterns that are more frequently observed than random. 
Some motifs, such as feed-back loops and feed-forward loops, are found to be highly 
enriched in real transcriptional networks and are supposed to be important for the 
dynamics and robustness of the gene network. Essentially, a network motif dissects 
the network at the static topology level, while the network dynamics depends more on 
a higher level: combinatorial control logic. In real biological networks, the behavior 
of genes are controlled by complex combinatorial regulations of multiple inputs, and 
it has been shown that many complex combinatorial control logics could be easily 
implemented even in simple organisms 
3
. Similar to the concept of network motif, we 
would ask: are there any “motifs” of combinatorial logic patterns? i.e., are there 
specific logic patterns favored by nature? Are all theoretically available logic patterns 
uniformly distributed in real biological networks? To answer these questions, one 
needs to know the exact combinatorial control logics for relatively large numbers of 
genes in specific organism. However, currently we do not have sufficient data on real 
networks.  
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 As a first step towards the logic motif problem, we turn to theoretic models and in 
silico simulations. We propose a framework targeting this problem in the reverse 
direction: we study how the relative abundance of distinct regulatory logic patterns 
influences the network‟s global dynamic characteristics, and by forcing these 
characteristics to resemble those of real organisms, we may infer whether a logic 
pattern is over-represented or under-represented in real networks. 
 
Methods 
NK model 
We use Kauffman‟s NK model 4 to model the transcriptional network. NK model is the 
first attempt in deciphering the integrated dynamic behavior of the complex genetic 
network. Genes are taken as boolean variables (1-active, 0-inactive) and the state of 
which is determined by the state of the inputs through some Boolean function, or logic 
patterns here, such as AND, OR and XOR when K=2. In NK models, there are N 
genes, and each gene has K inputs, or regulated by K genes. NK model is much 
simplified, yet is quite powerful in exploring the dynamics of gene regulatory 
networks 
5
. In this report we set K=2, i.e. each gene receives regulation from 2 genes. 
Previous research shows that networks with K=2 are ordered and critical, whose 
dynamics are always stable 
4
, which is an important characteristics for real regulatory 
networks. In addition, large scale experimental data in yeast 
6
 shows that the average 
number of regulators for each gene is 1.9. Therefore it is reasonable to study our 
problem with K=2 networks. Another benefit is, for K=2, there are only 16 (= 22
K
) 
potential logic patterns, which are feasible to be studied all at one time. 
 
Global dynamic characteristics 
Four global characteristics are defined and studied in the attractor state space S of 
each NK network: distinct attractor number (DAN), average attractor cycle length 
(ACL), gene expression rate (GER), and gene expression variance (GEV). A state is a 
string of „0‟ and „1‟, representing the expression level (active or inactive) of all genes 
in the network. The attractor state space S consists of all possible states for all 
attractors. It is well recognized that the number of distinct attractors can be interpreted 
as distinct cell types and the attractor cycle length represents the cell differentiation 
period 
4
. The expression rate of a state is the proportion of genes activated (taking on 
the value 1), and the expression rate of the network (GER) is the average of 
expression rates of all states in the attractor space. The global characteristic GER can 
be interpreted as the average level of how the network is activated. For simple species 
(with relative small networks), this value should be high, for example 4664 out of 
about 6000 yeast genes are active under general conditions, with GER>0.75 
7
. The 
fourth characteristic gene expression variance (GEV) is defined by 
2
1
0.5
0.5
1 N
i
i
P
GE
N
V

 
  
 
  
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Where N is the number of genes in the network and Pi measures how likely gene i is 
active in a given states and equals to the ratio of the number of states where gene i is 
active to the total number of states. GEV describes how far away P is from 0.5, the 
random case. Considering the fact that housekeeping gene tends to express in all cell 
types (thus Pi →1) while tissue-specific genes only express in a few special cell types 
(Pi →0), GEV is supposed to be high in real organisms.  
 
Multi-parameter sensitivity analysis (MPSA) 
 In this report, we want to find out if some of the defined global characteristics are 
sensitive to specific logic patterns, and if so, how the relative abundance of the logic 
patterns influence the global characteristics. In traditional sensitivity analysis, if one 
wants to find out whether the output is sensitive to specific parameters (or inputs), all 
other parameters are fixed, then one checks if the output changes significantly when 
the parameter understudy varies. However, the relative abundance of logic patterns is 
inherently correlated. If the relative abundance (proportion) for one logic pattern 
changes, the numbers of other logic patterns also changes, as their sum is fixed. At 
this point, we need a tool that simultaneously tests all parameters at one time.     
 The tool we use is multi-parameter sensitivity analysis (MPSA), a method 
recently introduced in finding the relative important factor of a complex system 
8
. We 
first generate a random network with N genes and 2N edges. This is done by randomly 
assign K (=2) genes as inputs to each gene in the network. For each network, we 
randomly sampled 1000 N-by-1 vectors by Latin Hypercube Sampling method. Each 
number in the vector indicates the number of a certain pattern in the network. Then 
logic patterns are randomly assigned to each gene, so that the number of each logic 
pattern is equal to that in the specified vector. Finally, the NK model is run, attractors 
are identified, and the four global characteristics are calculated. The vector specifying 
the relative abundance of all logic patterns is taken as the multivariate input and the 
global characteristics as the multivariate output. Then a threshold is selected for each 
characteristic and each vector is assigned acceptable or unacceptable depending on 
whether the corresponding output passes the threshold or not. Finally the cumulative 
frequencies of acceptable and unacceptable vectors are calculated for each pattern, 
and we define the maximum vertical difference of the cumulative frequency curve as 
the KS value, which indicates the relative importance of the pattern to the global 
characteristic. Thus in each network, for each characteristic, each pattern gets a KS 
value. A higher KS value indicates that the characteristic is more sensitively affected 
by changes in the number of the corresponding pattern. We define the vector of the KS 
values for all patterns as the sensitivity profile. A sensitivity profile shows the 
sensitivity of all logic patterns with respect to a certain characteristic in a network 
with specified topology.  
 
Results & Discussion 
As a proof-of-concept analysis, we study network with N = 20 and K = 2. In total 10 
networks with distinct topology are generated, and we find that the sensitivity profile 
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shows high consistency across different networks. The sensitivity profiles and their 
consistency between different networks can be visualized by colored map (Fig. 1). We 
list all 16 logic patterns on the left side in Fig. 1. Note that there is symmetry in 
sensitivity profile between patterns and their inverse. 
 To determine whether the sensitive patterns positively or negatively influence the 
characteristics, we use the simulation data to show how the average value of each 
characteristic varies when the abundance of logic patterns increases. In Fig. 2, for a 
fixed point (x, y) on the curve, x is the proportion of a pattern in the network, and y is 
the characteristic values averaged on the output of 10000 abundance vectors where 
the pattern‟s proportion is x. The slope of the curve also indicates the sensitivity of the 
characteristic to the pattern and is consistent with results of Fig. 1. 
 Now we consider the problem of identifying over- or under-represented logic 
motifs in gene regulatory networks. If real organisms favor larger values of a global 
characteristic, then logic patterns positively correlated with this characteristic will be 
over-represented, while those patterns negatively correlated will be under-represented. 
For example, it can be seen from Fig. 1 that pattern 7 and pattern 10 are sensitive ones 
for GEV. From Fig. 2 we see they both negatively correlated with GEV. If GEV tends 
to be high in real organisms, then pattern 7 and pattern 10 should be 
under-represented. Interestingly, this conclusion is supported with some evidences. 
Pattern 7 and pattern 10 are both “exclusive or” (XOR) type, which are also the only 
two patterns that do not belong to the class of canalyzing functions. It has been shown 
that network constructed from canalyzing functions exhibit a tendency toward ordered 
behavior and are widely observed in real genetic networks 
9
. Some explanations have 
been proposed for the rarity of XOR patterns, such as the difficulty in constructing 
such functions in reality, or its contribution to the instability of the network. Here we 
propose another explanation based on our sensitivity analysis. Fig.2 shows that both 
XOR type patterns have negative impact on the GEV characteristics. The more XOR 
patterns exist in the network, the lower GEV is. As long as housekeeping genes take a 
large part of the genome, the GEV should be high. Networks with low GEV would 
have lots of housekeeping genes inactivated; which means that some of the 
fundamental processes remain silent. Thus networks with such configuration are not 
favored by nature, therefore such patterns will eventually become under-represented.  
The current knowledge on gene interactions of genome wide is not sufficient to test 
our conclusions. Nevertheless, our proof-of-concept analysis shows that it is 
beneficial to explore the logic motif problem by theoretical analysis and simulation 
studies. Further work is needed to discuss the situation with larger N and K.  
 
Acknowledgements 
We thank Dr. R Wang, Dr. J.S. Zheng, Dr. L Li, Mr. H Chen and Mr. Y Zhou for 
useful discussions. This work was supported by the National Key Foundation 
Research Grant in China (973-2003CB715903; 863-2002AA234041; 
NSFC-90303017). 
 
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Figures 
 
DAN
1 2 3 4 5 6 7 8 910
      0
     AB
    AB^
      A
    A^B
      B
A^B+AB^
    A+B
   A~B~
AB+A^B^
     B^
   A+B^
     A^
   A^+B
  A^+B^
      1
ACL GER GEV
0 500 1000
0
0.5
1
Cumulative Frequency Curve
P1-GEV KS=0.3001
 
 
0 500 1000
0
0.5
1
P9-GEV KS=0.0434 
 
SP
acpt.
unacpt.
0.05
0.1
0.15
0.2
0.25
0.3
(a) (b) (c)
colorbar
 
Fig. 1. Sensitivity Profiles. (a) the sensitivity profiles for all the four global characteristics studied. 
Each column represents the SP of one network, such as (b), the first column of GEV. Each row 
represents one logic pattern labeled on the left side, in which „^‟ means the inverse. The color encodes 
the KS value, red for higher KS value thus higher sensitivity, blue ones on the contrary. KS value is the 
maximum vertical distance between the cumulative frequency curves for acceptable and unacceptable 
abundance vectors, see illustration in (c).  
 
Fig. 2. Relationship between pattern abundance and global characteristics. Each row shows the curves 
of all 16 patterns for one characteristic. Each subplot in one row illustrates how the expectation of the 
characteristics varies with the abundance of a certain pattern (labeled by the number on the bottom). 
The horizontal axes all vary between [0, 0.5], representing the proportion of a pattern in a network. 
 
References 
1. Shen-Orr, S.S., Milo, R., Mangan, S. & Alon, U. Network motifs in the transcriptional 
regulation network of Escherichia coli. Nat Genet 31, 64-68 (2002). 
2. Milo, R. et al. Network motifs: Simple building blocks of complex networks. Science 298, 
824-827 (2002). 
3. Buchler, N.E., Gerland, U. & Hwa, T. On schemes of combinatorial transcription logic. P Natl 
Acad Sci USA 100, 5136-5141 (2003). 
4. Kauffman, S.A. Metabolic stability and epigenesis in randomly constructed nets. J Theor Biol 
22, 437-467 (1969). 
N
at
ur
e 
Pr
ec
ed
in
gs
 : 
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01
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/n
pr
e.
20
08
.2
22
7.
1 
: P
os
te
d 
25
 A
ug
 2
00
8
 6 
5. Kauffman, S., Peterson, C., Samuelsson, B. & Troein, C. Random Boolean network models 
and the yeast transcriptional network. P Natl Acad Sci USA 100, 14796-14799 (2003). 
6. Lee, T.I. et al. Transcriptional Regulatory Networks in Saccharomyces cerevisiae. Science 298, 
799-804 (2002). 
7. Lewin, B. Gene VIII. (Pearson Education, Inc., 2004). 
8. Zi, Z.K. et al. In silico identification of the key components and steps in IFN-gamma induced 
JAK-STAT signaling pathway. Febs Lett 579, 1101-1108 (2005). 
9. Kauffman, S., Peterson, C., Samuelsson, B. & Troein, C. Genetic networks with canalyzing 
Boolean rules are always stable. P Natl Acad Sci USA 101, 17102-17107 (2004). 
 
 
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Logic motif of combinatorial control in transcriptional networks

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