Random Boolean Network Models and the Yeast Transcriptional Network

Abstract

The recently measured yeast transcriptional network is analyzed in terms of simplified Boolean network models, with the aim of determining feasible rule structures, given the requirement of stable solutions of the generated Boolean networks. We find that, for ensembles of generated models, those with canalyzing Boolean rules are remarkably stable, whereas those with random Boolean rules are only marginally stable. Furthermore, substantial parts of the generated networks are frozen, in the sense that they reach the same state, regardless of initial state. Thus, our ensemble approach suggests that the yeast network shows highly ordered dynamics.

Full text

Random Boolean network models and the yeast
transcriptional network
Stuart Kauffman*, Carsten Peterson
†‡
, Bjo
¨
rn Samuelsson
†
, and Carl Troein
†
*Department of Cell Biology and Physiology, University of New Mexico Health Sciences Center, Albuquerque, NM 87131; and
†
Complex Systems Division,
Department of Theoretical Physics, Lund University, So¨ lvegatan 14A, S-223 62 Lund, Sweden
Communicated by Philip W. Anderson, Princeton University, Princeton, NJ, October 6, 2003 (received for review June 30, 2003)
The recently measured yeast transcriptional network is analyzed in
terms of simplified Boolean network models, with the aim of
determining feasible rule structures, given the requirement of
stable solutions of the generated Boolean networks. We find that,
for ensembles of generated models, those with canalyzing Boolean
rules are remarkably stable, whereas those with random Boolean
rules are only marginally stable. Furthermore, substantial parts of
the generated networks are frozen, in the sense that they reach the
same state, regardless of initial state. Thus, our ensemble approach
suggests that the yeast network shows highly ordered dynamics.
genetic networks 兩 dynamical systems
T
he regulatory network for Saccharomyces cerevisiae was
recently measured (1) for 106 of the 141 known transcription
factors by determining the bindings of transcription factor
proteins to promoter regions on the DNA. Associating the
promoter regions with genes yields a network of directed gene–
gene interactions. As described in refs. 1 and 2, the significance
of measured bindings with regard to inferring putative interac-
tions are quantified in terms of P values. Lee et al. (1) did not
infer interactions having P values above a threshold value, P
th
⫽
0.001, for most of their analysis. Small threshold values, P
th
,
correspond to a small number of inferred interactions with high
quality, whereas larger values correspond to more inferred
connections, but of lower quality. It was found that for the P
th
⫽
0.001 network, the fan-out from each transcription factor to its
regulated targets is substantial, on the average 38 (1). From the
underlying data (http:兾兾web.wi.mit.edu兾young兾regulatory㛭
network), one finds that fairly few signals feed into each of them;
on the average 1.9. The experiments yield the regulatory network
architecture but yield neither the interaction rules at the nodes,
nor the dynamics of the system, nor its final states.
With no direct experimental results on the states of the system,
there is, of course, no systematic method to pin down the
interaction rules, not even within the framework of simplified
and coarse-grained genetic network models; e.g., ones where the
rules are Boolean. One can nevertheless attempt to investigate
to what extent the measured architecture can, based on criteria
of stability, select between classes of Boolean models (3).
We generate ensembles of different model networks on the
given architecture and analyze their behavior with respect to
stability. In a stable system, small initial perturbations should not
grow in time. This aspect is investigated by monitoring how the
Hamming distances between different initial states evolve in a
Derrida plot (4). If small Hamming distances diverge in time, the
system is unstable and vice versa. Based on this criterion, we find
that synchronously updated random Boolean networks (with a
flat rule distribution) are marginally stable on the transcriptional
network of yeast.
By using a subset of Boolean rules, nested canalyzing functions
(see Methods and Models), the ensemble of networks exhibits
remarkable stability. The notion of nested canalyzing functions
is introduced to provide a natural way of generating canalyzing
rules, which are abundant in biology (5). Furthermore, it turns
out that for these networks, there exists a fair amount of forcing
structures (3), where nonnegligible parts of the networks are
frozen to fixed final states regardless of the initial conditions.
Also, we investigate the consequences of rewiring the network
while retaining the local properties; the number of inputs and
outputs for each node (6).
To accomplish the above, some tools and techniques were
developed and used. To include more interactions besides those
in the P
th
⫽ 0.001 network (1), we investigated how network
properties, local and global, change as P
th
is increased. We found
a transition slightly above P
th
⫽ 0.005, indicating the onset of
noise in the form of biologically irrelevant inferred connections.
In ref. 5, extensive literature studies revealed that, for eu-
karyotes, the rules seem to be canalyzing. We developed a
convenient method to generate a distribution of canalyzing rules,
that fit well with the list of rules presented by Harris et al. (5).
Methods and Models
Choosing Network Architecture. Lee et al. (1) calculated P values
as measures of confidence in the presence of an interaction. With
further elucidation of noise levels, one might increase the
threshold for P values from the value 0.001 used in Lee et al. (1).
To this end, we computed various network properties, to inves-
tigate whether there is any value of P
th
for which these properties
exhibit a transition that can be interpreted as the onset of noise.
In Fig. 1, the number of nodes, mean connectivity, mean pairwise
distance (radius), and fraction of node pairs connected are
shown. As can be seen, there appears to be a transition slightly
above P
th
⫽ 0.005. In what follows, we therefore focus on the
network defined by P
th
⫽ 0.005. Furthermore, we (recursively)
remove genes that have no outputs to other genes, because these
are not relevant for the network dynamics. The resulting network
is shown in Fig. 2.
Generating Rules. Lee et al. (1) determined the architecture of the
network but not the specific rules for the interactions. To
investigate the dynamics on the measured architecture, we
repeatedly assign a random Boolean rule to each node in the
network. We use two rule distributions; one null hypothesis and
one distribution that agrees with rules compiled from the
literature (ref. 5; see also Supporting Text, which is published as
supporting information on the PNAS web site). In both cases, we
ensure that every rule depends on all of its inputs because the
dependence should be consistent with the network architecture.
As a null hypothesis, we use a flat distribution among all
Boolean functions that depend on all inputs. For rules with a few
inputs, this will create rules that can be expressed with normal
Boolean functions in a convenient way. In the case of many
inputs, most rules are unstructured and the result of toggling one
input value will appear random.
In biological systems, the distribution of rules is likely to be
structured. Indeed, all of the rules compiled by Harris et al. (5)
are canalyzing (3); a canalyzing Boolean function (3) has at least
one input, such that for at least one input value, the output value
‡
To whom correspondence should be addressed. E-mail: carsten@thep.lu.se.
© 2003 by The National Academy of Sciences of the USA
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is fixed. It is not straightforward to generate biologically relevant
canalyzing functions. A canalyzing rule implies some structure,
but the function of the noncanalyzing inputs (when the canalyz-
ing inputs are clamped to their noncanalyzing values) could be
as disordered as the full set of random Boolean rules. However,
the canalyzing structure is repeated in a nested fashion for almost
all rules in the study by Harris et al. (5). Hence, we introduce the
concept of nested canalyzing functions (see Appendix), which can
be used to generate distributions of canalyzing rules. Actually, of
the 139 rules of Harris et al. (5), only 6 are not nested canalyzing
functions (see Tables 1 and 2, which are published as supporting
information on the PNAS web site).
A special case of nested canalyzing functions is the recently
introduced notion of chain functions (ref. 7; see Appendix).
Chain functions are the most abundant form of nested canalyzing
functions, although 32 of the 139 rules in the study by Harris et al.
(5) fall outside this class.
It turns out that the rule distribution of nested canalyzing
functions in the study by Harris et al. (5) can be well described
by a model with only one parameter (see Appendix). Hence, we
use this model to mimic the compiled rule distribution. The free
parameter determines the degree of asymmetry between active
and inactive states and its value reflects the fact that most genes
are inactive at any given time in a gene regulatory system.
Analyzing the Dynamics. A biological system is subject to a
substantial amount of noise, making robustness a necessary
feature of any model. We expect a transcriptional network to be
stable, in that a random disturbance cannot be allowed to grow
uncontrollably. Gene expression levels can be approximated as
Boolean, because genes tend to be either active or inactive. This
approximation for genetic networks is presumably easier to
handle for stability issues than for general dynamical properties.
Using synchronous updates is computationally and conceptually
convenient, although it may at first sight appear unrealistic.
However, in instances of strong stability, the update order should
not be very important.
To study the time development of small fluctuations in this
discrete model with synchronous updating, we investigate how
the Hamming distance between two states evolves with time. In
a Derrida plot (4), pairs of initial states are sampled at defined
initial distances, H(0), from the entire state space, and their
mean Hamming distance, H(t), after a fixed time, t, is plotted
against the initial distance, H(0). The slope in the low H region
indicates the fate of a small disturbance. If the curve is above兾
below the line, H(t) ⫽ H(0), it reflects instability兾stability in the
sense that a small disturbance tend to increase兾decrease during
the next, t, time steps (see Fig. 3).
It is not uncommon that transcription factors control their own
expression. In some cases, genes up-regulate themselves, with
the effect that their behavior becomes less linear and more
switch-like. This action is readily mimicked in a Boolean net-
work. However, in the other case, where a transcription factor
down-regulates itself, the system will be stabilized in a model
with continuous variables, provided that the time delay of the
self-interaction is not too large. Boolean networks can only
model the limit of large time delays, which gives rise to nodes that
in a nonbiological manner repeatedly flip between no activity
and full activity without requiring any external input. Thus, the
self-interactions need to be treated as a special case in the
Boolean approximation. To this end, we consider three different
alternatives: (i) view the self-interactions as internal parts of the
rules (all self-interactions are removed); (ii) remove the possi-
bility for self-interactions to be down-regulating; and (iii)no
special treatment of self-interactions.
It is natural to use alternative i as a reference point to understand
the effect of the self-interactions in alternatives ii and iii.
We want to examine how the geometry of networks influence
the dynamics. It is known (3) that the distributions of in- and
out-connectivities of the nodes strongly affect the dynamics in
Boolean networks, but how important is the overall architec-
ture? If for each node, we preserve the connectivities, but
otherwise rewire the network randomly (6), how is the dynamics
Fig. 1. Topological properties of the yeast regulatory network described by
Lee et al. (1) for different P value thresholds excluding nodes with no outputs:
number of nodes (solid line), mean connectivity (dotted line), mean pairwise
distance (radius) (dotted–solid line), and fraction of node pairs that are
connected (dashed line). The right y axis corresponds to the number of nodes,
whereas the other quantities are indicated on the left y axis. Self-couplings
were excluded, but the figure looks similar when they are included. The
dashed vertical line marks the threshold, P
th
⫽ 0.005.
Fig. 2. The P
th
⫽ 0.005 network excluding nodes with no outputs to other
nodes. The filled areas in the arrowheads are proportional to the probability
of each coupling to be in a forcing structure when the nested canalyzing rules
are used on the network without self-interactions. This probability ranges
from approximately one-fourth, for the inputs to YAP6, to one, for the inputs
to one-input nodes. Nodes that will reach a frozen state (on or off) in the
absence of down-regulating self-interactions, regardless of the choice of
rules, are shown as dashes. For the other nodes, the grayscale indicates the
probability of being frozen in the absence of self-interactions, ranging from
⬇97% (bold black) to ⬎99.9% (light gray).
Kauffman et al. PNAS
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affected? For a Derrida plot with t ⫽ 1, there is no change. If we
only take a single time step from a random state, the outputs will
not have time to be used as inputs. There will be correlations
between nodes, but the measured quantity H(1) is a mean over
all nodes, and this is not affected by these correlations. Hence,
H(1) is not changed by the rewiring. To obtain a better picture
of the dynamics, we need to increase t. However, if we go high
enough in t to probe larger structures in the networks, we lose
sight of the transient effects of a perturbation.
To remedy this situation, we opt to select a fixed initial
Hamming distance, H(0), and examine the expectation value of
the distance as a function of time, by using the nested canalyzing
rules. As noise entering the biological network would act on the
current state of the system rather than on an entirely random
one, we select one of the states to be a fixed point of the
dynamics, and let the probability of any given fixed point be
proportional to the size of its attractor basin. A graph of H(t)
shows the relaxation behavior of the perturbed system where the
self-interactions have been removed (see Fig. 4a). We investigate
the role of the self-interactions both in terms of relaxation of a
perturbed fixed point (see Fig. 4b) and in terms of probabilities
for random trajectories to arrive at distinct fixed points and
cycles.
The assumption that the typical state of these networks is a
fixed point can be motivated. A forcing connection (3) is a pair
of connected nodes, such that control over a single input to one
node is sufficient to force the output of the other node to one of
the Boolean values. With canalyzing rules, this outcome is
fulfilled when the canalyzed output of the first node is a
canalyzing input to the second. The condition of forcing struc-
tures implies stability, because a (forcing) signal traveling
through such a structure will block out other inputs and is
thereby likely to cause information loss. Abundant forcing
structures should tend to favor fixed points.
Results and Discussion
Despite the absence of knowledge about initial and final states,
we have been able to get a hint about possible interaction rules
within a Boolean network framework for the yeast transcrip-
tional network. Our findings are as follows: (i) Canalyzing
Boolean rules confer far more stability than rules drawn from a
flat distribution as is clear from the Derrida plots in Fig. 3. Yet,
even a flat distribution of Boolean functions yields marginal
stability; (ii) The dynamical behavior around fixed points is more
stable for the measured network than for the rewired ones,
although only in the early time evolution (two to three time
steps) of the systems (see Fig. 4a). The behavior at this time scale
can be expected to depend largely on small network motifs,
whose numbers are systematically changed by the rewiring (6);
(iii) The removal of self-couplings increases the stability in these
networks. However, the relaxation is only changed significantly
if we allow the toggling of self-interacting nodes (see Fig. 4b).
This finding means that a node with a switch-like self-interaction
is not likely to be toggled by its inputs during the relaxation, nor
do the down-regulating self-interactions alter the relaxation.
This result means that the overall properties of relaxation to
fixed points can be investigated regardless of how the self-
interactions should be modeled; (iv) The number of attractors
and their length distribution are strongly dependent on how the
self-interactions are modeled. The average numbers of distinct
fixed points per rule assignments found in 1,000 trials of different
trajectories are 1.02, 4.33, and 3.79, respectively, for the three
self-interaction models. The numbers of two-cycles are 0.02,
0.09, and 0.38, respectively. Longer cycles are less common; in
total they sum up to 0.03, 0.11, and 0.11, respectively; and (v)
Fig. 3. Evolution of different Hamming distances, H(0) with one time step to
H(1) [Derrida plots (4)] for random rules (dark gray) and nested canalyzing
rules (light gray) with and without self-couplings (dashed borders), respec-
tively. (Down-regulating self-couplings are allowed.) The bands correspond to
1

variation among the different rule assignments generated on the archi-
tecture in Fig. 2. Statistics were gathered from 1,000 starts on each of 1,000
rule assignments.
Fig. 4. The average time evolution of perturbed fixed points for nested canalyzing rules, starting from Hamming distance, H(0) ⫽ 5; impact of the network
architecture (a) and impact of the self-interactions (b). The lines marked with circles in both figures correspond to the network in Fig. 2 without self-interactions.
The gray lines in a show the relaxation for 26 different rewired architectures with no self-interactions, with 1

errors of the calculated means indicated by the
line widths. The black lines in b correspond to the network in Fig. 2 with self-interactions. The upper line shows the case when it is allowed to toggle nodes with
self-interactions as a state at H(0) ⫽ 5 is picked, whereas the lower line shows the relaxation if this toggling is not allowed. The widths of these lines show the
difference between allowing self-interactions to be repressive or not repressive.
14798
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Forcing structures (3) are prevalent for this architecture with
canalyzing rules, as is evident from Fig. 2. On average, 56% of
the couplings belong to forcing structures. As a consequence,
most nodes will be forced to a fixed state regardless of the initial
state of the network. Even the highly connected nodes (in the
center of the network) will be forced to a fixed state for a vast
majority of the random rule assignments. In most cases, the
whole network will be forced to a specific fixed state. At first
glance, this might seem nonbiological. However, in the real
world, there are more inputs to the system than the measured
transcription factors, and to study a process such as the cell cycle,
one may need to consider additional components of the system.
With more inputs, such a strong stability, of the measured part
of the network, may be necessary for robustness of the entire
system.
Future reverse engineering projects in transcriptional net-
works may be based on the restricted pool of nested canalyzing
rules, which have been shown to generate very robust networks
in this case. It should be pointed out that the notion of nested
canalyzing functions is not intrinsically Boolean. For instance,
the same concept can be applied to nested sigmoids.
Appendix: Nested Canalyzing Functions
The notion of nested canalyzing functions is a natural extension
of canalyzing functions. Consider a K input Boolean rule, R, with
inputs i
1
,...,i
K
and output o. R is canalyzing on the input i
m
if
there are Boolean values, I
m
and O
m
, such that i
m
⫽ I
m
f o ⫽
O
m
. I
m
is the canalyzing value, and O
m
is the canalyzed value for
the output.
For each canalyzing rule, R, renumber the inputs in a way such
that R is canalyzing on i
1
. Then, there are Boolean values, I
1
and
O
1
, such that i
1
⫽ I
1
f o ⫽ O
1
. To investigate the case i
1
⫽ not
I
1
, fix i
1
to this value. This defines a new rule R
1
with K ⫺ 1 inputs;
i
2
,...,i
K
. In most cases, when picking R from compiled data, R
1
is also canalyzing. Then, renumber the inputs in order for R
1
to
be canalyzing on i
2
. Fixing i
2
⫽ not I
2
renders a rule R
2
with the
inputs i
3
,...,i
K
. As long as the rules R, R
1
, R
2
, . . . are canalyzing,
we can repeat this procedure until we find R
K⫺1
, which has only
one input, i
K
, and, hence, is trivially canalyzing. Such a rule R is
a nested canalyzing function and can be described by the
canalyzing input values, I
1
,...,I
K
, together with their respective
canalyzed output values, O
1
,...,O
K
, and an additional value,
O
default
. The output is given by
o 
冦
O
1
if i
1
 I
1
O
2
if i
1
 I
1
and i
2
 I
2
O
3
if i
1
 I
1
and i
2
 I
2
and i
3
 I
3
·
·
·
O
K
if i
1
 I
1
and 䡠䡠䡠and i
K⫺1
 I
K⫺1
and i
K
 I
K
O
default
if i
1
 I
1
and 䡠䡠䡠and i
K
 I
K
.
The notion of chain functions in Gat-Viks and Shamir (7) is
equivalent to nested canalyzing functions that can be written on
the form I
1
⫽ 䡠䡠䡠 ⫽ I
K⫺1
⫽ false.
We want to generate a distribution of rules with K inputs, such
that all rules depend on every input. The dependency require-
ment is fulfilled if and only if O
default
⫽ not O
K
. Then, it remains
to choose values for I
1
,...,I
K
and O
1
,...,O
K
. These values are
independently and randomly chosen with the probabilities
P共I
m
 true兲  P共O
m
 true兲 
exp共⫺2
⫺m

兲
1  exp共⫺2
⫺m

兲
for m ⫽ 1,...,K. For all generated distributions, we let

⫽ 7.
The described scheme is sufficient to generate a well defined
rule distribution, but each rule has more than one representation
in I
1
,...,I
K
and O
1
,...,O
K
.InSupporting Text, we describe how
to obtain a unique representation, which is applied to the rules
compiled in Harris et al. (5). This result enables us to present a
firm comparison between the generated distribution and the list
of rules in Harris et al. (5). (See Fig. 5, which is published as
supporting information on the PNAS web site.)
We thank Stephen Harris for providing details underlying ref. 5. C.T.
thanks the Swedish National Research School in Genomics and Bioin-
formatics for support. This work was initiated at the Kavli Institute for
Theoretical Physics (Santa Barbara, CA) (C.P. and S.K.) and was
supported in part by National Science Foundation Grant PHY99-07949.
1. Lee, T. I., Rinaldi, N. J., Robert, F., Odom, D. T., Bar-Joseph, Z., Gerber,
G. K., Hannett, N. M., Harbison, C. T., Thompson, C. M., Simon, I., et al. (2002)
Science 298, 799–804.
2. Hughes, T. R., Marton, M. J., Jones, A. R., Roberts, C. J., Stoughton, R.,
Armour, C. D., Bennett, H. A., Coffey, E., Dai, H., He, Y. D., et al. (2000) Cell
102, 109–126.
3. Kauffman, S. A. (1993) Origins of Order: Self-Organization and Selection
in Evolution (Oxford Univ. Press, Oxford).
4. Derrida, B. & Weisbuch, G. (1986) J. Physique 47, 1297–1303.
5. Harris, S. E., Sawhill, B. K., Wuensche, A. & Kauffman, S. (2002) Complexity
7, 23–40.
6. Maslov, S. & Sneppen, K. (2002) Science 296, 910–913.
7. Gat-Viks, I. & Shamir, R. (2003) Bioinformatics 19, Suppl. 1, 1108–
1117.
Kauffman et al. PNAS
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vol. 100
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no. 25
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