Robustness of Boolean dynamics under knockouts
Gunnar Boldhaus,1 Nils Bertschinger,2 Johannes Rauh,2 Eckehard Olbrich,2 and Konstantin Klemm1
1Bioinformatics Group, Institute for Computer Science, University of Leipzig, Härtelstraße 16-18, D-04107 Leipzig, Germany
2Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, D-04103 Leipzig, Germany

Received 5 May 2010; revised manuscript received 8 July 2010; published 18 August 2010
The response to a knockout of a node is a characteristic feature of a networked dynamical system. Knockout
resilience in the dynamics of the remaining nodes is a sign of robustness. Here we study the effect of knockouts
for binary state sequences and their implementations in terms of Boolean threshold networks. Besides random
sequences with biologically plausible constraints, we analyze the cell cycle sequence of the species Saccha-
romyces cerevisiae and the Boolean networks implementing it. Comparing with an appropriate null model we
do not find evidence that the yeast wildtype network is optimized for high knockout resilience. Our notion of
knockout resilience weakly correlates with the size of the basin of attraction, which has also been considered
a measure of robustness.
DOI: 10.1103/PhysRevE.82.021916 PACS number
s: 87.16.Yc, 05.10.Ln
I. INTRODUCTION
Living systems show a ubiquitous robustness against mu-
tations, environmental changes and intrinsic nondeterminism
1–3. In particular, each single cell must control its growth
and eventual division by regulating concentrations of pro-
teins in a precise temporal pattern. Using a Boolean state
dynamics 4, this cell cycle network has been argued to be
robustly designed for budding yeast as a model organism 5.
The robustness has been pinpointed as reproducibility of the
dynamics in the presence of stochastic perturbations 5–7.
Resilience against mutations, i.e., changes of the interactions
among the proteins, has been studied as well 5,8,9.
A particular type of mutation, either intrinsic or by inter-
vention, is a complete knockout of a single protein. A gene is
made dysfunctional such that it is no longer transcribed and
its product is effectively removed from the cell. Some knock-
outs can be tolerated or compensated by the cell, whereas
others are lethal. Knockouts are often used to infer the func-
tion of specific proteins. This is only appropriate, if the
knockout is neither lethal nor fully compensated, but disables
a specific function of the cell. Importantly in real experi-
ments, knockout resilient systems are more difficult to ana-
lyze and identify because knockout mutants do not exhibit a
measurable difference.
In this contribution we study the resilience against knock-
outs in two scenarios. First, the system under consideration is
defined only by a sequence of activation patterns regardless
of the specific mechanism producing them
“black box”.
Here, resilience with respect to knockout of a node means
that the information contained in the activation patterns of all
other nodes is still sufficient to unambiguously produce the
original sequence. In the second scenario, we consider the
sequence together with a given implementation by a Boolean
threshold network. Knocking out node j means that we re-
move its interactions with the other nodes. The network is
resilient against this knockout if all other nodes still perform
the original sequence of activation patterns.
After defining these notions of knockouts and resilience
we apply them to the yeast cell cycle sequence and its net-
work implementations. Significance of the results is assessed
by comparison of random sequences as null models with
various constraints.
II. KNOCKOUTS AND ROBUSTNESS OF FUNCTION
A. Definitions
Molecular processes within cells are frequently modeled
as dynamics with Boolean states 4,10,11. Components of a
Boolean state vector correspond to different molecules which
can be present in either high or low concentrations. The dy-
namics is thus described as a temporal sequence of activation
patterns x
0 ,x
1 , . . ., where each activation pattern x has n
binary components xi, i=1, . . . ,n. Low and high concentra-
tion of molecule i at time t are denoted by xi
t=0 and
xi
t=1, respectively. Each component i computes a Boolean
function f i mapping the present concentration pattern to its
activation at the next time step xi
t+1= f i
x1
t , . . . ,xn
t.
As the interaction between the components may be given in
terms of a network
see Sec. III, the components are hence-
forth called nodes.
Let us define what we mean by robustness of a single
node i against knockout of another node j. We assume that
the input nodes follow a certain dynamics x1
t , . . . ,xn
t,
where t
0. The set of possible activation patterns which the
input takes is called the input support. It can be represented
as a set of binary strings of length n.
Node i is robust against knockout of node j if the state
information of node j is either irrelevant for the correct func-
tioning of i or if this information is already contained in the
other available inputs.
Definition. A node i with mapping f i : 0,1n→ 0,1 is
robust against knockout of node j if xi is independent of xj
given x1 , . . . ,xj−1 ,xj+1 , . . .xn. With this we mean that
f i
x1, . . . ,xj−1,0,xj+1, . . . xn = f i
x1, . . . ,xj−1,1,xj+1, . . . xn ,

1
whenever the two states
x1 , . . . ,xj−1 ,0 ,xj+1 , . . .xn and

x1 , . . . ,xj−1 ,1 ,xj+1 , . . .xn both lie in the input support. Ro-
bustness against simultaneous knockout of multiple nodes is
defined in a similar fashion.
It is important to note that this definition depends on the
input support. If the input x1 , . . . ,xn takes each possible
value, then robustness against knockout of node j implies
PHYSICAL REVIEW E 82, 021916
2010
1539-3755/2010/82
2/021916
6 ©2010 The American Physical Society021916-1

that f i does not depend on xj. However, if the inputs are
highly correlated, then nontrivial robustness may appear: It
may happen that xi can compensate the knockout of a single
input by reconstructing the missing information from another
input with similar dynamics, but the simultaneous knockout
of both inputs leads to a failure. For example, in the se-
quence of the yeast cell cycle
Table I it is easy to see that
all nodes are robust against knockout of either MBF or SBF,
but in general a simultaneous knockout of both MBF and
SBF cannot be compensated.
Our notion of knockout robustness is based on ideas of Ay
and Krakauer 12. Within the framework defined in 12,
resilience means that the exclusion dependence of the system
with respect to certain knockouts vanishes. Combinatorial
conditions characterizing this situation have been found by
Herzog et al. 13. The theory becomes much simpler in the
present setting restricted to deterministic dynamics.
Our definition of robustness can be applied to each node
of the cell cycle network shown in Fig. 2. In our studies of
the yeast cell cycle the input support will be the set of acti-
vation patterns which appear in this cycle. We then study
each node mapping and ask, which inputs are “essential” for
the functioning of this node and which inputs can be com-
pensated.
0 1 2 3 4 5 6 7 8 9 10
pr
ob
ab
ili
ty
knockout resilience
10-1
10-2
10-3
10-4
10-5
10-6
100(a) (b)
0 1 2 3 4 5 6 7 8 9 10
knockout resilience
100
10-1
10-2
10-3
10-4
0 1 2 3 4 5 6 7 8 9 10
pr
ob
ab
ili
ty
system robustness
100
10-1
10-2
(c) (d)
0 1 2 3 4 5 6 7 8 9 10
system robustness
10-2
10-1
10-3
10-4
100
FIG. 1.
Color online
a,b The distribution of the number of
inputs which can be compensated by all other nodes are shown for
the first null model
a and the more realistic null model
b. The
blue
light gray in print bars represent the distribution of the ran-
dom cell cycles and the single green
dark gray in print bar repre-
sents the knockout resilience of the wildtype network.
c,d The
distribution of the number of nodes which are robust with respect to
all single node knockouts for the first null model
c and the more
realistic null model
d. As above, the blue
light gray in print bars
represent the random cell cycle sequences while the single green

dark gray in print bar shows the system robustness for the wild-
type network. Note the logarithmic scale of the four diagrams.
CLN1,2 CLB5,6
CLN3
SIC1
MBF
CLB1,2
CDH1
SBF
CDC20 SWI5
MCM1
Cell Size
Activating Edge
Inhibiting Edge
Inhibiting
Self-Coupling
FIG. 2.
Color online The wildtype network of the cell cycle of
the yeast species Saccharomyces cerevisiae 5. The edges of the
network are directed and can be activating
dashed green arrow or
inhibiting
solid red arrow. All self-couplings
solid yellow are
inhibiting. The network comprises 34 different interactions, 15 of
which are activating and 19 are inhibiting.
TABLE I. The cell cycle sequence of the yeast species Saccharomyces cerevisiae 5. The last phase stationary G1 is a fixed point of the
dynamics, i.e., once the system has reached this state, it stays in this state, until some external event flips the first node from 0 to 1 and
triggers another cell cycle.
Time CLN3 MBF SBF CLN1,2 CLB5,6 CLB1,2 MCM1 CDC20 SWI5 SIC1 CDH1 Phase
1 1 0 0 0 0 0 0 0 0 1 1 START
2 0 1 1 0 0 0 0 0 0 1 1 G1
3 0 1 1 1 0 0 0 0 0 1 1 G1
4 0 1 1 1 0 0 0 0 0 0 0 G1
5 0 1 1 1 1 0 0 0 0 0 0 S
6 0 1 1 1 1 1 1 0 0 0 0 G2
7 0 0 0 1 1 1 1 1 0 0 0 M
8 0 0 0 0 0 1 1 1 1 0 0 M
9 0 0 0 0 0 1 1 1 1 1 0 M
10 0 0 0 0 0 0 1 1 1 1 0 M
11 0 0 0 0 0 0 0 1 1 1 1 M
12 0 0 0 0 0 0 0 0 1 1 1 G1
13 0 0 0 0 0 0 0 0 0 1 1 Stat. G1
BOLDHAUS et al. PHYSICAL REVIEW E 82, 021916
2010
021916-2

In order to study robustness of the system as a whole
there are multiple possibilities: as a measure of system ro-
bustness we count the number of nodes which are robust
with respect to all single node knockouts, i.e., how many
nodes would remain functional if any one input would be
knocked out. Another possibility is to find the knockouts
under which the behavior of the whole system is robust. In
this case we find the single node knockouts which can be
compensated by all
other nodes of the network. The set of
these knockouts is called the resilience combination. The
cardinality of the resilience combination is called the knock-
out resilience.
B. Null models of state sequence
System robustness and knockout resilience of a specific
system may be judged as significantly high or low only in
comparison with a null hypothesis from a model. Here we
define two null models of state sequences under constraints
derived from properties of a cell cycle sequence.
For the first null model, the activation states are drawn
from 0,1 with probabilities 1/2, independently across time
steps and nodes. As an exception, the last activation pattern
is taken as the first activation pattern and then flipping the
state of one randomly chosen node. Thus the first and last
activation patterns differ at a single node. This constraint is
to reflect the fact that the cell cycle is triggered by a single
signal protein.
The second null model is further constrained as follows:

i CLN3 acts as an input node. It is active in the first
activation pattern and inactive at all later time steps.

ii The other nodes are activated and inactivated exactly
once during the cycle. Their activity
or inactivity is there-
fore constrained to a block of successive time steps.
These constraints are motivated by the observation that
switching events
rising and falling edges of activity are
much rarer in the cell cycle than would be expected for un-
correlated random state sequences.
As a natural additional constraint for both models, activa-
tion patterns at different steps along the sequence must be
different. Sequences violating this condition are discarded.
C. Results
We now apply the ideas of Sec. II to the yeast cell cycle

Table I and the null models. Starting with the system ro-
bustness we find that only the four nodes CLN3, CLN1,2,
CLB5,6, and CDC20 are robust against all single node
knockouts. For CLN3 this is trivial since it corresponds to the
constant map fCLN3=0. Regarding the knockout resilience,
the cell cycle still functions correctly if any one of the five
nodes MBF, SBF, MCM1/SFF, SWI5, or CDH1 is knocked
out.
Summing up these observations, the yeast cell cycle has
knockout resilience 5 and system robustness 4. We can now
ask whether these values are a special property of the cell
cycle or rather expected for a support containing 13 out of
211=2048 possible states. Figure 1 shows that all null models
produce a considerable fraction of instances whose resilience
and robustness values are above those of the yeast cell cycle
sequence.
Formulating this insight in terms of a statistical test, our
null hypothesis is that the yeast cell cycle is a random se-
quence as obtained from the null model considered. The
p-value is the probability that the null model generates an
instance with an empirical value greater or equal to the ob-
served value. The first null model yields p-values P1
kr
=0.99999 and P1
sr
=0.702 for knockout resilience and system
robustness, respectively. For the second null model we obtain






















■


■









■















■












■
■








■




■




■



■





■



■



■






■






■


■


■





■







■

■





■
■


■




■
■






■
■



■■






■

■■




■
■

■■


■





■



■
■



■■










■


■
■



■■




■



■■


■
■


N
or
m
al
iz
ed
H
ist
og
ra
m
Resilience Combination
100
10-2
10-4
10-6
10-8
10-10
10-12
10-14
Cln3
MBF
SBF
Cln1,2
Clb5,6
Clb1,2
Mcm1
Cdc20
Swi5
Sic1
Cdh1
FIG. 3. The distribution of the specific knockout schemes is
shown. The filled bars represent the wildtype component. The open
bars represent all components and the single dashed bar represents
the resilience combination of the wildtype itself. Each column rep-
resents one of the 24 different resilience combinations which occur
among the functional networks. The nodes of the networks are
shown as squares
see legend for names. If the knockout of a
particular node is functional the square is filled, otherwise it is
open. The resilience combinations are ordered such that the average
basin size increases.






















■


■









■















■












■
■








■




■




■



■





■



■



■






■






■


■


■





■







■

■





■
■


■




■
■






■
■



■■






■

■■




■
■

■■


■





■



■
■



■■










■


■
■



■■




■



■■


■
■


Av
er
ag
eN
um
be
rO
fE
dg
es
Resilience Combination
70
Cln3
MBF
SBF
Cln1,2
Clb5,6
Clb1,2
Mcm1
Cdc20
Swi5
Sic1
Cdh1
50
60
30
40
20
FIG. 4.
Color online The distribution of positive
dashed
lines, negative
dotted lines, and all edges
straight lines for each
of the corresponding resilience combinations are shown. The lighter
lines
light gray represent networks from all components, while the
darker lines
dark gray with squares represent networks only reach-
able by mutations from the wildtype. The resilience combinations
are depicted as in Fig. 3.
ROBUSTNESS OF BOOLEAN DYNAMICS UNDER KNOCKOUTS PHYSICAL REVIEW E 82, 021916
2010
021916-3

P2
kr
=0.808 and P2
sr
=0.226. It is common practice to reject the
hypothesis only for p-values less than 0.05 14. Here all
p-values are greater than 0.2. Therefore the null hypothesis,
stating that the yeast cell cycle sequence is not more opti-
mized for tolerating knockouts than a null model sequence,
cannot be rejected.
Another result seen in Fig. 1 is that both knockout resil-
ience and system robustness in the second null model are
lower than those of the first null model. This is due to the
correlations between activation patterns in a sequence gener-
ated according to the second null model. Fewer state changes
lead to more similar patterns along the sequence. Then a
knockout is more likely to make activation patterns at differ-
ent times indistinguishable so the sequence can no longer be
performed.
III. ROBUSTNESS OF IMPLEMENTATION
A. Linear threshold networks and neutral graph
The concepts defined up to now apply to the sequence of
activation patterns. They do not take into account the inter-
action network implementing this sequence. In order to in-
vestigate the effects of specific network structures we focus
on linear threshold networks. Such a network is given as a
directed graph on n nodes with weighted edges. We allow
loops, i.e., edges starting and ending at the same node. For
simplicity we only allow weights wij=1
activating or posi-
tive edges and wij=−1
inhibiting or negative edges for
each directed edge i← j. If there is no edge from node j to
node i we write wij=0. To each node i we associate a time-
dependent variable xi
t with dynamics given by
xi
t + 1 =

1 if 
j
wijxj
t 0,
0 if 
j
wijxj
t 0,
xi
t else.


2
It is possible to refine this model by allowing more values for
the edge weights. Furthermore it is customary to assign an
activation threshold individually to each node i. Then the
dynamics is determined by comparing  jwijxj
t to this
threshold. In this work we do not make use of these possi-
bilities and restrict ourselves to the simple model.
TABLE II. The number of networks which allow certain resilience combinations among all networks which implement the cell cycle
sequence
column all components. Among these networks the number of networks in the wildtype components is shown in the column
wildtype component.
Number
Wildtype
component
All
components CLN3 MBF SBF CLN1,2 CLB5,6 CLB1,2 MCM1 CDC20 SWI5 SIC1 CDH1 # Knockouts
1 5.131025 5.071034










0
2 3.981024 2.591031









 1
3 6.221023 1.501031









1
4 6.221023 1.501031









1
5 1.771031









1
6 2.241029








 2
7 3.151032









1
8 9.671028








2
9 9.671028








2
10 3.521027








2
11 4.631022 9.621027








 2
12 3.511027








2
13 4.631022 9.621027








 2
14 8.871025







 3
15 8.871025







 3
16 1.281029








2
17 9.121020 3.221024
 






 3
18 2.441022
 





 4
19 6.251023
 






3
20 2.791025







3
21 1.741022 8.041027
 







2
22 2.791025







3
23 4.821025
 






3
24 4.951021
 





4
Sum 5.661025 5.111034
BOLDHAUS et al. PHYSICAL REVIEW E 82, 021916
2010
021916-4

We focus on networks that perform the sequence of acti-
vation patterns of the yeast cell cycle
Table I when initial-
ized with the START pattern at time 1. One of these func-
tional networks called the wildtype is shown in Fig. 2. The
interactions of the wildtype are based on empirical evidence
5. Note that here the term wildtype does not denote the
actual yeast wildtype organism. It is used only to distinguish
this network
which is believed to model the actual yeast
organism most accurately from other functional networks on
the level of linear threshold models.
As a first null model in the assessment of robustness of an
implementation
a particular network we use a flat distribu-
tion on the set of all functional networks, containing 5.11
1034 elements 8.
As a second null model we consider a restriction to those
networks that may be reached from the wildtype by an evo-
lutionary path. To make this precise, we define the neutral
graph. Its node set is the set of the functional networks. Two
networks A and B are connected by an undirected edge if A
can be obtained from B by adding or deleting a single inter-
action 9,15. In our case the neutral graph is disconnected
and falls into a large number of connected components 9.
The 5.661025 networks in the connected component con-
taining the wildtype
called wildtype component serve as
another null model, again weighted with a flat distribution.
B. Robustness definition
In the context of linear threshold models we can define a
related notion of robustness, which we will call robustness of
implementation: a knockout of a node j is modeled by re-
moving a node from the network, together with all of the
edges involving this node. We can then analyze the dynamics
of the changed network and compare it to the dynamics of
the unperturbed network. If the dynamics of the remaining
n−1 nodes is not changed, then we say that the implemen-
tation is robust against the knockout of node j. Mathemati-
cally this means that the sign of k
jwikxk
t equals the sign
of kwikxk
t. The resilience combination is the set of nodes
against whose knockout the network is robust. Now the
knockout resilience of the network is the cardinality of the
resilience combination. This is a property of the implemen-
tation and thus depends on the connection structure of the
network implementing the cell cycle pattern. Its value cannot
be greater than the knockout resilience of the function, as
defined above.
C. Results
There are 24 different occurring resilience combinations.
Out of these, eight are found in the component containing the
wildtype
see Table II.
The majority of networks, i.e.,
99% of all the functional
networks of the neutral graph and
90% of the networks in
the wildtype component, cannot cope with any single node
knockout. However, a maximum of four independent single
node knockouts is found. The wildtype itself stays functional
after knockout of node CDH1, see Fig. 3. Networks reach-
able by a mutational path from the wildtype can manage at
most three independent knockouts.
One might speculate that high knockout resilience re-
quires redundant wiring of the network which would be ob-
servable as an increased edge density. In Fig. 4, we look at
the distribution of the number of positive, negative and all
edges for the different resilience combinations. There is no
clear correlation between the average number of edges and
the knockout resilience.
As suggested by Li et al. 5, the basin of attraction of the
G1 fixed point
stationary state is a measure of robustness.
The basin consists of all activation patterns from which the
G1 state is eventually reached by following the dynamics
Eq.
2. The average basin size of networks with a given
resilience combination is shown in Fig. 5. In Table III the
average basin size for the different knockout resiliences is
shown. With an increasing capability to cope with more
single node knockouts the average basin size increases. Ad-
ditionally, average basins sizes in the wildtype component
are larger than their corresponding basin sizes in all compo-
nents. However, all network implementations in the wildtype
component have a network resilience of at most three.
IV. DISCUSSION
We have studied network dynamics with Boolean state
vectors
activation patterns under knockout of single nodes






















■


■









■















■












■
■








■




■




■



■





■



■



■






■






■


■


■





■







■

■





■
■


■




■
■






■
■



■■






■

■■




■
■

■■


■





■



■
■



■■










■


■
■



■■




■



■■


■
■


Av
er
ag
eB
as
in
Si
ze
Resilience Combination
Cln3
MBF
SBF
Cln1,2
Clb5,6
Clb1,2
Mcm1
Cdc20
Swi5
Sic1
Cdh1
2200
2000
1800
1600
1400
1200
1000
FIG. 5.
Color online The averages of the basin sizes corre-
sponding to the specific resilience combination are shown. The
straight line
dark gray represents networks from all components,
while the line with squares
light gray represents networks only
from the wildtype component. Additionally, the area between the
10% and 90% quantiles is inked. The resilience combinations are
depicted as in Fig. 3. For each resilience combination 105 networks
were sampled.
TABLE III. The average basin sizes for the different knockout
resiliences. Again the wildtype component is considered separately.
Knockout resilience Wildtype component All components
0 1663.44 1447.42
1 1629.86 1485.06
2 1675.99 1513.21
3 1779.22 1588.93
4 1594.82
ROBUSTNESS OF BOOLEAN DYNAMICS UNDER KNOCKOUTS PHYSICAL REVIEW E 82, 021916
2010
021916-5

using the example of the yeast cell cycle. The yeast wildtype
network is not optimized for knockout resilience, given the
sequence of activation patterns. There are networks with sig-
nificantly larger knockout resilience implementing the same
sequence.
The model of the yeast cell cycle studied in this work is
far from being a complete description of cell function. For
example, a more realistic model could include further nodes
as well as graded activations
i.e., the concentrations of the
molecules at the nodes are considered as real valued activa-
tions. An increasing number of nodes would generically in-
crease the robustness of the null models since additional state
information is available to compensate node knockouts.
Thereby, our conclusion that the yeast cell cycle is not espe-
cially robust would not change under this generalization. The
analysis of graded concentrations instead of binary activa-
tions would require a notion of state distance taking into
account which disturbances are considered and how they af-
fect the functioning of the cell. This is beyond reach of the
present idealized model.
Finally we stress that our definition of knockout resilience
checks whether a node is dispensable for integrating and
transmitting information only in the context of the regulatory
network we consider. In reality, the considered node may be
involved in other functions indispensable for survival. This
would further reduce the knockout resilience.
ACKNOWLEDGMENTS
This work was supported by the Volkswagenstiftung. We
are grateful to Nihat Ay for useful comments on the paper.
1 M. B. Elowitz, A. J. Levine, E. D. Siggia, and P. S. Swain,
Science 297, 1183
2002.
2 H. Kitano, Nat. Rev. Genet. 5, 826
2004.
3 A. Wagner, Robustness and Evolvability in Living Systems

Princeton University Press, Princeton, NJ, 2005.
4 S. Kauffman, J. Theor. Biol. 22, 437
1969.
5 F. Li, T. Long, Y. Lu, Q. Ouyang, and C. Tang, Proc. Natl.
Acad. Sci. U.S.A. 101, 4781
2004.
6 S. Braunewell and S. Bornholdt, J. Theor. Biol. 245, 638

2007.
7 W.-B. Lee and J.-Y. Huang, FEBS Lett. 583, 927
2009.
8 K.-Y. Lau, S. Ganguli, and C. Tang, Phys. Rev. E 75, 051907

2007.
9 G. Boldhaus and K. Klemm, Eur. Phys. J. B
to be published,
doi: 10.1140/epjb/e2010-00176-4
10 M. I. Davidich and S. Bornholdt, PLoS ONE 3, e1672
2008.
11 I. Albert, J. Thakar, S. Li, R. Zhang, and R. Albert, Source
Code for Biology and Medicine 3, 16
2008.
12 N. Ay and D. C. Krakauer, Theory Biosci. 125, 93
2007.
13 J. Herzog, T. Hibi, F. Hreinsdóttir, T. Kahle, and J. Rauh, Adv.
Appl. Math. 45, 317
2010.
14 R. A. Fisher, Statistical Methods, Experimental Design, and
Scientific Inference
Oxford University Press, New York,
1990.
15 P. Schuster, W. Fontana, P. F. Stadler, and I. L. Hofacker, Proc.
R. Soc. London, Ser. B 255, 279
1994.
BOLDHAUS et al. PHYSICAL REVIEW E 82, 021916
2010
021916-6