Dynamical properties and path dependence in a gene-network model of cell differentiation

Abstract

In this work, we explore the properties of a control mechanism exerted on random Boolean networks that takes inspiration from the methylation mechanisms in cell differentiation and consists in progressively freezing (i.e. clamping to 0) some nodes of the network. We study the main dynamical properties of this mechanism both theoretically and in simulation. In particular, we show that when applied to random Boolean networks, it makes it possible to attain dynamics and path dependence typical of biological cells undergoing differentiation.

Full text

Soft Computing (2021) 25:6775–6787
https://doi.org/10.1007/s00500-020-05354-0
FOCUS
Dynamical properties and path dependence in a gene-network model
of cell differentiation
Michele Braccini1·Andrea Roli1,4 ·Marco Villani2,4 ·Roberto Serra2,3,4
Published online: 2 November 2020
© The Author(s) 2020
Abstract
In this work, we explore the properties of a control mechanism exerted on random Boolean networks that takes inspiration
from the methylation mechanisms in cell differentiation and consists in progressively freezing (i.e. clamping to 0) some nodes
of the network. We study the main dynamical properties of this mechanism both theoretically and in simulation. In particular,
we show that when applied to random Boolean networks, it makes it possible to attain dynamics and path dependence typical
of biological cells undergoing differentiation.
Keywords Boolean networks ·Cell differentiation ·Path dependence ·Epigenetics ·Methylation
1 Introduction
The co-existence of different cell types, which share the same
genome, in a multicellular organism, and the processes of
progressive cell differentiation raise challenging theoretical
issues. The possibility that the same set of genes, with the
same type of reciprocal influences, gives rise to very different
phenotypes is associated to the fact that not all the genes are
active in every cell type, i.e. to the existence of different stable
activation patterns of the same set of genes. An important
question which naturally raises concerns how these stable
patterns are determined, among the huge amount of states
which are possible in principle. It is well-known that active
genes interfere with the activation of other genes, through
proteins or other gene products like, e.g. miRNA. Therefore,
it is natural to associate the stable patterns, i.e. the cell types,
to the attractors of the dynamical system which describes the
Communicated by Tomas Veloz.
BMichele Braccini
m.braccini@unibo.it
1Department of Computer Science and Engineering, Campus
of Cesena - Alma Mater Studiorum Università di Bologna,
Cesena, Italy
2Department of Physics, Informatics and Mathematics,
Università di Modena e Reggio Emilia, Modena, Italy
3Institute for Advanced Study (IAS), University of
Amsterdam, Amsterdam, The Netherlands
4European Centre for Living Technology, Venice, Italy
complex web of interactions among genes, RNA, proteins,
other gene products and other molecules in the cell.
There are beautiful models of various parts of this system;
for example, some describe in detail the steps which lead
from DNA to mRNA to ribosomes, and some describe the
details of the regulation processes. They are very well suited
to describe in-depth the translation and transcription pro-
cesses, but here we are rather interested in global activation
patterns of thousands of heterogeneous entities. Therefore,
it is mandatory to introduce some simplifications to keep the
model manageable and meaningful. One interesting possi-
bility (pioneered by Stuart Kauffman about 50 years ago) is
that of simplifying the picture to that of a network of inter-
acting genes only, without explicitly taking into account the
other entities—whose influence is subsumed by the interac-
tion rules among the genes.
Following this approach, we will consider a dynamical
system of n“genes”, which can take different activation val-
ues. The time change of the activation of a gene is ruled by a
differential (or difference) equation; the rules can be differ-
ent for different genes. As time passes, the system will tend
to one of its stable steady states—to be associated to a cell
type.
This framework is attractive, due to its simplicity, yet it
has to be reconciled with the crucial process of cell differ-
entiation, i.e. the development of pluripotent types into fully
differentiated ones, through various intermediate stages. If
we associate a pluripotent type with a stable or metastable
attractor, then it is necessary to understand what can make this
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6776 M. Braccini et al.
attractor unstable. One might be tempted to identify the pro-
cess of cell maturation with transients, while attractors would
be associated to mature cell types only. However, pluripotent
intermediate types may be long lived, so it is difficult to devise
satisfactory models along this line. It seems more appropri-
ate to associate attractors also to the long-lived intermediate
states, but in this case it is necessary to identify mechanisms
which can actually modify the dynamical system, making
these attractors unstable and driving the system towards the
other attractors.
This is possible since the cell is not a fully autonomous
system, but it is coupled with the environment in which it
lives. Various molecules can indeed interfere with the gene
expression mechanisms, switching on or off single genes or
groups of genes. However, one has to resist the temptation of
simplistic explanations based only upon detailed properties
of external influences, because the gene network (or rather,
the whole dynamical system it represents) is always present.
Switching one gene on or off does not only lead to the synthe-
sis of a specific protein or gene product, but it may also affect
the expression of other genes. The activation patterns must
be coherent with the whole dynamics of the network Huang
and Kauffman (2013).
In this paper, we explore the behaviour of a control mech-
anism, which affects the attractors of a gene network and
their stability. This mechanism, which is simple enough to
be amenable to large-scale simulation, is directly inspired by
biological observations and experimental knowledge about
the effects of DNA methylation; refer to Sect. 2for a brief
introduction on biological methylation and its role in the gene
expression process.
While the model is described in detail in Sect. 2, let us
mention here that we will make the simplifying assumption
that nodes are either off or on, i.e. that they take Boolean
values, neglecting intermediate values. Moreover, let us sup-
pose that the time evolution of the network is synchronous
and deterministic in discrete steps. The attractors of any finite
network are therefore either cycles or fixed points (which
can be regarded as period-1 cycles). The silencing effects
of methylation will therefore be modelled by keeping con-
stantly equal to zero the activation of some nodes (which
will be called “externally frozen”, briefly e-frozen, or some-
times “blocked”). The network is assumed to be in one of its
attractors at time t<0, while at time t=0 some nodes are
externally frozen. The clamping to zero of a node may affect
other nodes, inducing in time a propagation of freezing (i.e.
an increase in the number of constant nodes) and the attractor
landscape will be modified.
In Sect. 3, we will present a simplified, mean-field descrip-
tion of this process of propagation in time of freezing. In
Sect. 4, we will describe the results of several simulations
of the effects of freezing on the attractor landscape, investi-
gating the effects of the choice of some key parameters. A
particularly important question concerns the importance of
the order (in time) of gene silencing: if gene Ais silenced at
a certain time step, and gene Bis silenced at a later stage,
will the attractors and their basins be the same as in the case
where silencing of Bprecedes that of A? Interestingly, one
can prove that the answer is (sometimes) no, so the system
described here can show a strong form of path dependence.
This is described in detail in Sect. 5. The final section is
devoted to a discussion of the results and of their biological
implications.
2 Methylation model
Epigenetic mechanisms significantly contribute to the deter-
mination and maintenance of cell fates in biological organ-
isms. Methylation, in particular, can occur to both DNA
regions and proteins. DNA methylation typically occurs at
CpG sites1changing the activity of DNA sequences, e.g.
blocking gene promoter activities and hence their ability
to induce transcription, without requiring mutations. Differ-
ently, histone methylations change the degree of compactness
of the chromatin by adding methyl groups to histone pro-
teins: proteins on which DNA of eukaryotic cells wrap
around to form structural units called nucleosome. Changes
the degree of compactness of chromatin have implications
on the expression of genes belonging to the DNA regions
involved: condensed DNA (heterochromatin condition) pre-
vent transcription by polymerases, while loosely packed
regions (euchromatin condition) allow transcription. Differ-
ential methylation is therefore a mechanism exploited by
biological cells to modulate gene regulation and expression
during development and differentiation.
Taking into account histone methylation, we observe that
although its effects depend on the particular positions on his-
tones on which it acts, it most often leads to heterochromatin
conditions (Gilbert and Barresi 2016; Perino and Veenstra
2016; Schuettengruber and Cavalli 2009). In addition, along
lineages, the attained configurations of DNA methylation are
inherited and progressively extended as cells become more
specialised Kim and Costello (2017). Therefore, methylation
has a prominent role in the maintenance and stabilisation of
the attained gene expressions that ultimately characterise the
identities of the various cell states. It is also worth mentioning
that methylation is tightly regulated by complex interactions,
and that its dysregulation can be the causa prima of a lot of
disorders, from cognitive, neurological and chronic diseases
to cancer. It is precisely due to the complexity of these mech-
anisms that the adoption of models can support the analysis
1Sites in which a cytosine nucleotide is followed by a guanine
nucleotide.
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Dynamical properties and path dependence in a gene-network… 6777
of the role of methylation, and more in general of epigenetics,
in (patho)physiological processes.
Several works making use of mathematical models of
specific aspects of epigenetic processes are present in the
literature (Bull 2014; Miyamoto et al. 2015; Turner et al.
2017,2013); however, to the best of our knowledge, there is
no systematic attempt to study the capabilities of reproducing
differentiation dynamics by methylation-like mechanisms in
modelling. A noteworthy model of differentiation, based on
an entirely different mechanism (i.e. intracellular noise), had
been presented elsewhere (Serra et al. 2010; Villani et al.
2011; Villani and Serra 2013). It is likely that the mecha-
nism described here might flank the previous one in giving
rise to the complex phenomena of differentiation. In a previ-
ous work conducted by us Braccini et al. (2019), we began to
investigate the effects of a simplified model of methylation in
the dynamics of Boolean networks. Boolean networks (BNs)
are a well-known model of biological genetic regulatory net-
work introduced by Kauffman (1969), frequently employed
for the investigation of the causes of the rich dynamics arising
in complex systems, such as biological cells. Indeed, despite
their simplifications, they proved to be suitable systems to
represent the dynamics of biological GRNs to many level of
abstractions Graudenzi et al. (2011), Serra et al. (2006), Serra
et al. (2007), Shmulevich et al. (2005).
Formally, BNs are discrete-time and discrete-state dynam-
ical systems. Following their original formulation, they can
be represented by a directed graph with nnodes each having
associated a Boolean variable xi,i=1,...,nand a Boolean
function gi=(xi1,...,xik)which depends on kother nodes.
One among the most prominent classes of BNs is that of ran-
dom BNs (RBNs), in which functions and connections are
chosen according to pre-defined distributions. A special case
is the one in which nodes receive exactly kdistinct inputs
chosen at random (avoiding self-loops) and Boolean func-
tions are defined by choosing for each of the 2kentries of the
truth table value 1 with probability p, being pcalled the bias.
RBNs exhibit a phase transition between order and chaos
depending on the values of kand pDerrida and Pomeau
(1986), Bastolla and Parisi (1997). For 2 p(1−p)k<1,
RBNs have on average an ordered behaviour, whilst for
2p(1−p)k>1 the networks show extreme sensitivity
to initial conditions and very long cyclic attractors, which
denote a chaotic behaviour. A critical regime is attained for
2p(1−p)k=1. Critical RBNs have proven to show prop-
erties typical of real cells Nykter et al. (2008), Roli et al.
(2018), Serra et al. (2007), Serra et al. (2004), Villani et al.
(2018).
The abstract methylation model to which we will refer
has been introduced in our work mentioned above and it
will be summarised in its main concepts in the following.
It is inspired by the idea of progressive methylation of chro-
matin along the development and differentiation of biological
Fig. 1 Schematic representation of the methylation-inspired model.
Grey nodes represent frozen nodes (blocked to value 0, regardless of
the actual values of their inputs). The specific expression patterns of
nodes in the ennuples that represent the state of the BN over time have
no other meaning than to exemplify the increase in methylated nodes,
characteristic of the methylation process introduced
cells. So, by analogy with the heterochromatin status, the
methylation process in Boolean networks has been modelled
by blocking the expression of some BN nodes to value 0;
these nodes will be referred to as e-frozen in the follow-
ing (this process is sketched in Fig. 1). This model is based
on the hypothesis that—even though it is not the only phe-
nomenon in place—the progression of frozen nodes imposes
the arrow of time of the differentiation process. A differ-
entiation model, and therefore also this methylation-based
one, should accommodate, at least some, properties related to
the differentiation phenomenology. We have already shown
that the model in question can give rise, progressively, to
reduced alternatives and an increment in stability along the
differentiation process—representing cell types by means of
the system’s attractors Huang et al. (2005), Huang et al.
(2009), Huang and Ingber (2000). Moreover, we assessed
its ability to maintain diversity in terms of possible asymp-
totic states originating from different combinations of frozen
nodes, during all the differentiation process described by the
progressive freezing itself.
In this work, we provide a theoretical prediction—
supported by a mean-field model—on the progression of the
number of fixed genes as a consequence of perturbations in
terms of externally frozen genes. The model predictions will
be validated through simulations of ensembles of RBNs. This
step highlights the twofold role of the model: its applicabil-
ity in the determination of prediction in finite-size model’s
instances and its role as a means to generalise the obtained
results.
Furthermore, we experimentally investigate, by employ-
ing ensembles of RBNs, the presence of a stronger version of
the classical path dependence, i.e. the possibility of reaching
different asymptotic states (cell types) by freezing the same
set of nodes (genes), but in different temporal orders.
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6778 M. Braccini et al.
2.1 Related models
Since the seminal works by Kauffman (1969), Kauffman
(1993), Boolean networks have become one among the most
used models of GRNs Albert and Thakar (2014), as also
proven by the availability of Boolean models in the Cell
Collective repository Helikar et al. (2012), hence our first
motivation for studying the effectof a methylation-like mech-
anism in RBNs. Besides this, BNs provide a suitable level
of description for cellular dynamics, as they both make it
possible to reify the notion of gene activity, and enable to
apply the ensemble approach Kauffman (2004) by in silico
simulations of BN models with given characteristics.
However, several variants of BNs exist and alternative or
complementary models are available De Jong (2002). An
overview of these models is out of the scope of this contribu-
tion; anyway, here we provide a succinct list of the ones we
may suggest as a starting point for the interested reader. Apart
from BNs subject to asynchronous dynamics (Harvey and
Bossomaier 1997;DiPaolo2000) and mixed dynamics Dara-
bos et al. (2007), a prominent variant of classical BNs is
that of Probabilistic BNs Shmulevich and Dougherty (2010),
which extend the original model by allowing for multiple
functions per node, each associated to a probability of being
chosen for the update. Multi-valued logic networks have also
been proposed Thomas et al. (1995). Nonlinearity can be still
preserved also in models including continuous variables, as in
piecewise linear differential equations (Glass and Pasternack
1978; Kappler et al. 2003;Rolietal.2010). We conclude this
summary by mentioning a couple of recent BN extensions
that allow for formal verification of some dynamical proper-
ties by means of modal logics. The first is that of Reactive
BNs Figueiredo and Barbosa (2018), which introduces the
notion of reactive frames Gabbay and Marcelino (2009a)into
BNs. The second one builds upon Abstract BNs Yordanov
et al. (2016)—whereby update functions might be partially
known—and provides a model checking tool for the verifica-
tion of network dynamical properties Goldfeder and Kugler
(2018).
3 A mean-field description of the spreading
of freezing
The external freezing of some nodes can modify the dynam-
ics of the network, since the values of other nodes may be
affected. Moreover, at successive time steps, more nodes will
be frozen in cascade (locked at either 0 or 1 as a consequence
of the external freezing), so there will be a spreading of freez-
ing in the system. In order to understand the main features of
this process a simple, mean-field model can be studied. Of
course, specific networks can behave in a way very different
from the average one.
Note that we will consider here only RBNs with two inputs
per node, which suffice to provide information about the
most relevant properties of the model. Generalisations are
formally straightforward but calculations can become com-
plicated. We will also assume that all the Boolean functions
are allowed, with the same probability, so there is a symmetry
between 0’s and 1’s. As discussed above, these assumptions
imply that the network starts in a critical state.
Let us suppose to start in attractor A at time t <0: There
is an initial set Viof nodes which can take different values
in A, and a set Fiof fixed nodes in A. The union of these
two sets is of course the whole set of nodes Vi∪Fi=U.
Note that here we do not care about the value of a fixed node,
but we focus on the distinction between fixed and oscillating
nodes. Let fiand vibe the fractions of fixed and variable
nodes in A, with
fi+vi=1(1)
At time t=0 some nodes are externally blocked to 0. If they
are constant nodes, the balance of fixed versus variable nodes
does not change, so we will consider only those nodes which
were previously variable. So at time t=0 let a fraction b0of
nodes (which were in V) be externally blocked to 0, and let
B0be their set. At time t=0, the total fraction of constant
nodes jumps to f0=fi+b0, while the fraction of variable
nodes shrinks to v0=vi−b0. Summarising, at time 0:
v0=vi−b0
f0=fi+b0
(2)
In the evolution from t=0 new nodes can become con-
stant due to spreading of freezing. Now we consider how
many nodes will be frozen at the following time step t=1
(excluding from this count those which are already constant
at time 0). Each node has two inputs, therefore a randomly
chosen node in Vcan become frozen iff one of the following
conditions apply:
(i) its two inputs were in Vand are now both in B, due to
novel freezing; or
(ii) only one input is in B due to novel freezing, while the
other was already in F due to the network dynamics; or
(iii) only one input is in B due to novel freezing, while the
other one is variable, and the Boolean functions are such
that the daughter node which was in V now becomes
frozen (i.e., when the function of the node is canalizing2
in that particular value assumed by the frozen parent).
2A Boolean function is called canalizing if the output value is deter-
mined by only one input value—e.g. in the OR function one input equal
to 1 is enough to have 1 as output.
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Dynamical properties and path dependence in a gene-network… 6779
Note that it is also possible that the daughter of two nodes
in Vbe in F, and that this might be altered by the blocking
of one node at t=0. The probability of this event depends
quadratically upon vi, which decreases along the freezing
process. Therefore, we decided to neglect this term.
The probabilities associated to the previous situations are
the following
(i) two parents move to B:b2
0;
(ii) one moves to B, the other was in F:2b0fi=2b0(f0−
b0). The factor 2 is due to the fact that the node origi-
nally in Vcan be either parent;
(iii) one moves to B, the other was in V:2ηb0v0=b0v0
where the spreading constant ηis the probability that a
randomly chosen node becomes frozen (at time t+1
etc.) if one and only one of its parent nodes becomes
frozen (at time t) while the other parent node was not
frozen at time t. In the case k=2, all the Boolean
functions allowed with equal probability it turns out
that η=1/2. Therefore 2ηb0v0=b0v0.
Of course, in order to have a change in the fraction of fixed
nodes it is necessary that the target node was a variable one
(an event whose probability is v0), so the above values must
be multiplied by v0. Therefore, summing the contributions
from (i) to (iii), the fraction of newly frozen nodes at time
t=1is
b1=v0(b2
0+2b0f0−2b2
0+b0v0)
=b0v0(1+f0−b0)(3)
where one has taken into account that f0+v0=1. Sum-
marising, at time t=1
⎧
⎪
⎨
⎪
⎩
b1=b0v0(1+f0−b0)
v1=v0−b1
f1=f0+b1
(4)
At the following time step, the situation is basically the same,
except that now the system starts from the values of fand v
given by Eq. 4, and the newly frozen nodes are b1. Applying
the same procedure one gets
⎧
⎪
⎨
⎪
⎩
b2=b1v1(1+f1−b1)
v2=v1−b2
f2=f1+b2
(5)
This is true in general, so
⎧
⎪
⎨
⎪
⎩
bk+1=bkvk(1+fk−bk)
vk+1=vk−bk+1
fk+1=fk+bk+1
(6)
●●●●●●●●●
0.6 0.7 0.8 0.9 1.0
steps
Fraction of fixed nodes
012345678
●
b0
0.2
0.1
0.05
0.03
0.01
Fig. 2 Typical progression of fixed nodes ( fk) as predicted by the
model. We suppose that the network has initially 67% of blocked
nodes (which is the average value found experimentally for 100
nodes networks) and we externally block to 0 a fraction b0∈
{0.01,0.03,0.05,0.1,0.2}of nodes. We can observe that the values
tends to finite asymptotic values strictly lower than 1
This formula can be applied at the following time steps. How-
ever, dealing with finite networks it is necessary to observe
that nothing changes if less than one node has to be added, so
the freezing necessarily halts when bk<1/N. It is interest-
ing to note that the freezing dynamics described above tends
to finite asymptotic values, as shown in Fig. 2.
4 Simulation results against model
predictions
We set up the following in silico experimental setting to
compare the RBNs dynamics against the model predictions
concerning the spreading of freezing induced by external
perturbations—in terms of nodes clamped to 0 values—to
system’s attractors.
For assessing the freezing progression in RBNs, we
considered different experimental conditions: ensembles of
RBNs with 100 nodes in critical dynamical regime—k=2,
bias= 0.5—and with b0∈{1,3,5,10}. In accordance with
the methylation-inspired model previously formulated, we
used a synchronous updating scheme for state update of
BNs. For each parameter configuration, we drew 100 sam-
ples (RBNs) and for each of them we started from one of
its attractor (let’s call it A0), chosen at random; with refer-
ence to the mean-field model, this represents the situation at
t<0. Then, we clamp to value 0 a number b0of nodes;
otherwise, if the attractor does not present a number of vari-
able nodes greater than or equal to b0we replace the network
with another one that satisfies this requirement. After having
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6780 M. Braccini et al.
1 2 5 10 20 50 100
0 20 40 60 80 100
steps of trajectories (log scale)
no. of fixed nodes over trajectory (100 networks)
Fig. 3 Progression of fixed nodes in trajectories of 100 RBNs in the
case of no. 3externally frozen nodes. Blue dots denote the number of
fixed nodes in the starting attractor (t<0), while the red triangles
show the same for the final attractor: the fixed nodes count includes the
e-frozen nodes
perturbed the attractor A03we follow its trajectory record-
ing the number of nodes that are fixed on a value (regardless
of whether they are 0 or 1) and which remain so until the
dynamic relaxes to its new attractor. In this way, we were
able to keep track of the trend of the fixed nodes along all
100 trajectories, as shown in Fig. 3(see also Figs. 15,16,17
in Appendix): with the blue dots we denote the fixed nodes at
t<0, while with the red triangles we denote the final number
of frozen nodes in the new stable dynamic condition.
The figures regarding the progression of fixed nodes
show considerable variability. As predicted by the mean-field
model, in each replica we observe the tendency to con-
verge to a value smaller than the maximum, i.e. 100 nodes
for these experiments. However, to perform a comparison
between the experimental and theoretical results, we must
take into consideration the progression of the number of the
mean fixed nodes predicted by the model and compare it—
step by step—with the average of the experimental values
of fixed nodes. This comparison has been done for all the
experimental conditions reported above. The steps reported,
including standard deviation and standard error of the mean,
are presented for the number of steps for which at least 30
samples have been found. The results for b0=3areshown
in Fig. 4(see also Figs. 12,13,14 in Appendix for the other
comparisons). As starting condition for computing the model
predictions, we have used an initial value of fixed nodes equal
to the average value of the fraction of fixed nodes found in
3Wechoose the last state in ascending lexicographic order, to be almost
sure to perturb the attractor’s state also in cyclic attractors effectively.
steps
Fraction of fixed nodes
●
●●●
●●●●
0.0 0.2 0.4 0.6 0.8 1.0
01234567
●
●
●
●
●
●
●
●
●
●
Experiments' mean
Model's mean
Sample standard deviation
Standard error of the mean
Fig. 4 Comparison between the mean-field model predictions of fixed
nodes (model’s mean) and the mean value of the fraction of fixed nodes
found in the experiments (Experiments’ mean) for the case of b0=3.
In the plot, only the steps in which there was a number of samples equal
to or greater than 30 have been reported. For these steps also the sample
standard deviation (grey area) and the standard error of the mean (green
area) have been computed and reported
random attractors chosen as initial condition in our ensem-
bles of RBNs at time t<0 (i.e. in their wild condition). In
particular, we used f0∈{0.6866,0.6903,0.6419,0.6424}
for values of b0∈{1,3,5,10}, respectively.
The comparison shows, in general, a high agreement
between the values predicted by the model and the experi-
mental results. On the one hand, as pointed out in the previous
sections, this supports the validity of the mean-field model
for the prediction of fixed nodes spreading and allows us to
generalise the results obtained even for larger networks in
size, which would be otherwise computationally too expen-
sive to simulate. On the other hand, these results provide
further support to the validity of this specific methylation-
inspired mechanism as a driving mechanism for reproducing
differentiation phenomenology in Boolean networks. Indeed,
a bounded spreading of fixed nodes in response to externally
frozen genes is a necessary condition for the existence of a
progression of different (meta)stable activation patterns of
gene expressions, and so for the existence of differentiation
lineages.
5 Path dependence
The notion of path dependence is expressed as the prop-
erty of attaining different outcomes from an initial common
condition, depending on the path taken by the process Des-
jardins (2011). The role of path dependence in biological
systems has been thoroughly discussed (see, e.g. Longo
(2018), Szathmáry (2006)) and is a core ingredient in cell
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Dynamical properties and path dependence in a gene-network… 6781
Fig. 5 A graphical representation of the specific formulation of the path
dependence taken into account throughout this work. σ1and σ2labels
represent different sets of freezing genes. We ascertain the presence
of path dependence if the attractors A4and A5, reached by different
temporal sequences of the same freezing sets of genes, are different
differentiation Huang (2009). In this work, we focus on a
specific definition of path dependence that refers to the capa-
bility of reaching different attractors under different freezing
sequences of the same genes, i.e. the attractor reached by
freezing first gene σ1and then gene σ2can be different
than the attractor reached by first freezing σ2and then σ1.
In a sense, this is a stronger version of the classical path
dependence, as it refers to the possibility of reaching a dif-
ferent outcome by acting on the same set of control variables,
but in different temporal order. This property is particularly
important because it makes it possible to manoeuver a lim-
ited number of control variables to lead the system towards
target asymptotic states.
To assess to what extent RBNs exhibit path dependence as
a function of the order of gene freezing, we run experiments
in which two genes (or two small groups of genes) σ1and σ2
are successively frozen in both orders starting from the same
state, and the final attractor reached is compared. A graphi-
cal representation of this process is depicted in Fig. 5: let us
suppose to select two sets σ1and σ2of nodes to be externally
frozen; starting from a state in attractor A14we freeze the
nodes in σ1(resp. σ2) and let the network relax to the asymp-
4For both the freezing sequences, we take the same state in the attractor,
so as to avoid the variance caused by starting from different attractor
phases. In particular, we choose the first state in ascending lexicographic
order.
Table 1 Parameters of the experiments for assessing path dependence
n=100 , k=2, bias = 0.5
|σi|∈{1,3},i=1,2
100 randomly sampled pairs σ1,σ
2
m∈{10,40,70}
totic condition represented by attractor A2(resp. A3); once
in A2(resp. A3), the other group of nodes is frozen and the
final attractor A4(resp. A5) is stored. If the two final asymp-
totic states are different, then we can say that the network
shows path dependence under the condition that the starting
attractor is A1and the pair of sets to be frozen is σ1,σ
2.
By replicating this comparison with random samples of σ1
and σ2, we can estimate the tendency of a RBN to exhibit
path dependence. The comparison between two asymptotic
states reached after freezing two different groups of nodes
requires some care, because one cannot simply compare the
two attractors tout court, i.e. on all the nodes, for they are any-
way very likely to differ in the frozen nodes by construction
(they might coincide only in the rare case in which in both
the attractors A2and A3, reached by, respectively, freezing
the two different groups of nodes σ1and σ2, all the nodes in
σ1are constant at 0 in A3and all the nodes in σ2are con-
stant at 0 in A2). Furthermore, a comparison based on the set
of nodes complementing σ1∪σ2would miss an important
modelling assumption: The observed cell types are a func-
tion of a limited number of coding genes, which are in turn
controlled by the remaining part of the network (Borriello
et al. 2018; Espinosa-Soto et al. 2004; Fumiã and Martins
2013). Therefore, we compared the asymptotic states on the
basis of the values assumed by the nodes in a subset M, that
we name attractor projection of size m=|M|. Referring to
Fig. 5,proj(A4|M)=proj(A5|M)if the network states of
the two attractors restricted to the nodes in Mare the same.
The nodes to be frozen are chosen outside Mand excluding
those nodes always at 0 along the attractor.
The assessment of the general tendency towards path
dependence of RBNs has been undertaken in different
experimental conditions, summarised in Table 1. For each
parameter configuration, 1000 RBNs were generated and for
each of them the experiment has been replicated for 100
random samples of σ1and σ2and from a different initial
number of externally frozen nodes. In this way, it is possi-
ble to ascertain the progressive tendency of exhibiting path
dependence, while the number of externally frozen nodes
is incremented. A representative example of the results we
obtained is depicted in Fig. 6—the results of the other con-
figurations are qualitatively the same and can be found in the
Appendix. For each RBN, the fraction of pairs σ1,σ
2that
led to differing asymptotic states (y-axis) is shown w.r.t. the
number of initially frozen nodes (x-axis). The area below the
123
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6782 M. Braccini et al.
0.00.10.20.30.40.50.6
Initial no. of frozen nodes
Fraction of freezing sequences producing path dependence
0 6 24 42
Legend:
10 samples
100 samples
1000 samples
Fig. 6 Pictorial view of the results concerning path dependence for
the case of 40 nodes chosen for the projection and |σ1|=|σ2|=3.
The figures concerning the other cases are reported in Appendix. For
each RBN, the fraction of pairs σ1,σ
2that led to differing projected
asymptotic states (y-axis) is shown w.r.t. the number of initially frozen
nodes (x-axis). The area below the segmented line is coloured so as to
have a pictorial view of the density of this property across RBNs: the
darker the colour, the higher the fraction of RBNs with that amount of
path dependence
segmented line is coloured so as to have a pictorial view of the
density of this property across RBNs: the darker the colour
of a point , the higher the fraction of RBNs with that amount
of path dependence. We first observe the darker area, cor-
responding to the majority of replicas: The impact of path
dependence is about the 20% in the case without initially
frozen nodes and decreases linearly to a smaller yet non-
negligible fraction. This trend is qualitatively the same by
considering the distribution of this fraction across the RBNs;
nevertheless, we observe that path dependence is still quite
intense even for about one fourth of initially frozen nodes in
a considerable fraction of replicas.
6 Conclusion
In the previous sections, we have shown that RBNs subject to
(possibly progressive) external clamping of some nodes to a
fixed value exhibit the properties of (i) keeping the avalanche
of frozen nodes bounded—so that on average the asymptotic
fraction of frozen nodes is less than 1—and (ii) allowing
the possibility of reaching two different asymptotic states by
exerting the external freezing of the same genes but in dif-
ferent temporal order. The first property guarantees that in
general the network has still multiple fates after e-freezing,
i.e. the external intervention can constrain and control the
future asymptotic states but does necessarily imply a final
unique asymptotic state. The second property—i.e. strong
path dependence—shows that it is possible to reach differ-
ent fates by simply permuting the sequence of external node
freezing. This overall outcome supports the use of external
freezing in analogy with methylation mechanisms in real
cells, enabling us to address some questions on cell dif-
ferentiation by looking for generic properties in RBNs. For
example, one may ask how many control actions are avail-
able to reach fates with given characteristics, or, conversely,
whether there exist fates that can be reached by specific or
fragile sequences of external freezing interventions. Future
work is indeed planned to assess in more detail the control-
lability of RBNs under e-freezing in terms of control theory.
A question may arise as to whether the choice of externally
freezing the nodes to 0 might impact the results. In RBNs
with k=2 and Boolean functions uniformly distributed, the
average distribution of 0s and 1s is symmetrical, therefore the
choice of either 0 or 1 simply breaks the symmetry towards
one of the two values but the results are the same. Of course,
results might differ if the RBNs considered have a different
number of inputs per node and an asymmetric distribution
of Boolean functions, but they are not expected to undergo
qualitative changes.
In addition, we remark that our results are the outcome
of models and experiments characterised by the hypothesis
that networks are random, therefore one may ask to what
extent these results can be valid also for real cells, which
are supposed to be characterised by genes with non-random
relations. First of all, some questions can be addressed in
terms of generic properties—in the frame of the ensemble
approach Kauffman (2004)—that can be investigated also
in random models. Second, if some properties are generally
found in random networks, even if with a mild tendency,
it means that selection could easily exploit and enhance
those properties during evolution. So ensembles of networks
that are the result of evolutionary processes, or have differ-
ent architectures than the one tested here (different degree
of connectivity, scale-free, modular, etc.), or are subject to
other updating mechanisms (asynchronous), or can accom-
modate some kind of expression noise (extrinsic and intrinsic
ones), could better match the statistical features of real cells.
On the other hand, the simple control mechanism we have
discussed in this work may be also applied in the case
of Boolean models of real gene regulatory networks. Fur-
thermore, other models or formalisms that can be applied
to the study of cellular differentiation can extend our pro-
posed model. For this purpose, reactive graphs Gabbay and
Marcelino (2009b), or more specifically reactive Boolean
networks Figueiredo and Barbosa (2019), can assume the role
of the control module and so represent the causal interactions
that ultimately produces the progression of e-frozen nodes—
currently externally controlled—that drive the differentiation
process. Future work is planned in these directions.
Funding Open access funding provided by Alma Mater Studiorum -
Universitá di Bologna within the CRUI-CARE Agreement.
123
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Dynamical properties and path dependence in a gene-network… 6783
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right holder. To view a copy of this licence, visit http://creativecomm
ons.org/licenses/by/4.0/.
Appendix
Pictorial view of the results concerning path dependence.
For each RBN, the fraction of pairs σ1,σ
2that led to dif-
fering projected asymptotic states (y-axis) is shown w.r.t. the
number of initially frozen nodes (x-axis). The area below
the segmented line is coloured so as to have a pictorial view
of the density of this property across RBNs: the darker the
colour, the higher the fraction of RBNs with that amount of
path dependence (Figs. 7,8,9,10,11,12,13,14,15,16 and
17).
0.0 0.2 0.4 0.6
Initial no. of frozen nodes
Fraction of freezing sequences producing path dependence
0103763
Fig. 7 10 nodes, |σ1|=|σ2|=3
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Initial no. of frozen nodes
Fraction of freezing sequences producing path dependence
0 4 13 22
Fig. 8 70 nodes, |σ1|=|σ2|=3
0.00 0.05 0.10 0.15 0.20 0.25 0.30
Initial no. of frozen nodes
Fraction of freezing sequences producing path dependence
0103763
Fig. 9 10 nodes, |σ1|=|σ2|=1
123
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6784 M. Braccini et al.
0.00 0.05 0.10 0.15 0.20 0.25 0.30
Initial no. of frozen nodes
Fraction of freezing sequences producing path dependence
0 6 24 42
Fig. 10 40 nodes, |σ1|=|σ2|=1
0.0 0.1 0.2 0.3 0.4 0.5
Initial no. of frozen nodes
Fraction of freezing sequences producing path dependence
0 4 13 22
Fig. 11 70 nodes, |σ1|=|σ2|=1
0.0 0.2 0.4 0.6 0.8 1.0
steps
Fraction of fixed nodes
●
●
●●
●
01234
●
●
●
●
●
Experiments' mean
Model's mean
Sample standard deviation
Standard error of the mean
Fig. 12 Comparison between the mean-field model predictions of fixed
nodes (model’s mean) and the mean value of the fraction of fixed nodes
found in the experiments (Experiments’ mean) for the case of b0=1
steps
Fraction of fixed nodes
●
●●●●●●
●●
0.0 0.2 0.4 0.6 0.8 1.0
01 423 5678
●
●
●
●
●
●
●
●
●
●
Experiments' mean
Model's mean
Sample standard deviation
Standard error of the mean
Fig. 13 Comparison between the mean-field model predictions of fixed
nodes (model’s mean) and the mean value of the fraction of fixed nodes
found in the experiments (Experiments’ mean) for the case of b0=5
123
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Dynamical properties and path dependence in a gene-network… 6785
0.0 0.2 0.4 0.6 0.8 1.0
steps
Fraction of fixed nodes
●
●●
●
●●●
0123456
●
●
●
●
●
●
●
Experiments' mean
Model's mean
Sample standard deviation
Standard error of the mean
Fig. 14 Comparison between the mean-field model predictions of fixed
nodes (model’s mean) and the mean value of the fraction of fixed nodes
found in the experiments (Experiments’ mean) for the case of b0=10
1 2 5 10 20 50 100
0 20 40 60 80 100
steps of trajectories (log scale)
no. of fixed nodes over trajectory (100 networks)
Fig. 15 Progression of the number of fixed nodes in 100 trajectories
after externally freezing 1node. Blue dots indicate the number of fixed
nodes in the starting attractor, while the red dots show the same for
the final attractor: the fixed nodes count includes the externally frozen
nodes
12 51020 50
0 20406080100
steps of trajectories (log scale)
no. of fixed nodes over trajectory (100 networks)
Fig. 16 Progression of the number of fixed nodes in 100 trajectories
after externally freezing 5nodes. Blue dots indicate the number of fixed
nodes in the starting attractor, while the red dots show the same for the
final attractor: the fixed nodes count includes the externally frozen nodes
12 51020
0 20406080100
steps of trajectories (log scale)
no. of fixed nodes over trajectory (100 networks)
Fig. 17 Progression of the number of fixed nodes in 100 trajectories
after externally freezing 10 nodes. Blue dots indicate the number of
fixed nodes in the starting attractor, while the red dots show the same
for the final attractor: the fixed nodes count includes the externally
frozen nodes
123
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6786 M. Braccini et al.
References
Albert R, Thakar J (2014) Boolean modeling: a logic-based dynamic
approach for understanding signaling and regulatory networks and
for making useful predictions. WileyInterdiscip Rev Syst Biol Med
6(5):353–369
Bastolla U, Parisi G (1997) A numerical study of the critical line of
Kauffman networks. J Theor Biol 187(1):117–133
Borriello E, Walker SI, Laubichler MD (2018) A unified, mechanis-
tic framework for developmental and evolutionary change. arXiv
preprint arXiv:1809.02331
Braccini M, Roli A, Villani M, Montagna S, Serra R (2019) A simpli-
fied model of chromatin dynamics drives differentiation process in
Boolean models of GRN. In: The 2019 conference on artificial life,
vol 31, pp 211–217. https://doi.org/10.1162/isal_a_00163.https://
www.mitpressjournals.org/doi/abs/10.1162/isal_a_00163
Bull L (2014) Evolving boolean regulatory networks with epigenetic
control. Biosystems 116:36–42
Darabos C, Giacobini M, Tomassini M (2007) Semi-synchronous acti-
vation in scale-free Boolean networks. In: European conference
on artificial life. Springer, pp 976–985
De Jong H (2002) Modeling and simulation of genetic regulatory sys-
tems: a literature review. J Comput Biol 9(1):67–103
Derrida B, Pomeau Y (1986) Random networks of automata: a simple
annealed approximation. Europhys Lett 1:45–49
Desjardins E (2011) Reflections on path dependence and irreversibility:
lessons from evolutionary biology. Philos Sci 78(5):724–738
Di Paolo E (2000) Searching for rhythms in asynchronous random
Boolean networks. In: Artificial life VII: proceedings of the sev-
enth international conference on artificial life, vol 7. MIT Press, p
73
Espinosa-Soto C, Padilla-Longoria P, Alvarez-Buylla E (2004) A gene
regulatory network model for cell-fate determination during ara-
bidopsis thaliana flower development that is robust and recovers
experimental gene expression profiles. Plant Cell 16(11):2923–
2939
Figueiredo D, Barbosa L (2018) Reactive models for biological regu-
latory networks. In: International symposium on molecular logic
and computational synthetic biology. Springer, pp 74–88
Figueiredo D, Barbosa LS (2019) Reactive models for biological regu-
latory networks. In: Chaves M, Martins MA (eds) Molecular logic
and computational synthetic biology. Lecture notes in computer
science. Springer, Cham, pp 74–88. https://doi.org/10.1007/978-
3-030- 19432-1_5
Fumiã H, Martins M (2013) Boolean network model for cancer path-
ways: predicting carcinogenesis and targeted therapy outcomes.
PLoS ONE 8(7):e69008
Gabbay D, Marcelino S (2009) Modal logics of reactive frames. Stud
Log 93(2–3):405
Gabbay DM, Marcelino S (2009) Modal logics of reactive frames. Stud
Log 93(2):405. https://doi.org/10.1007/s11225-009-9214-1
Gilbert SF, Barresi MJF (2016) Developmental biology, XI edn., chap-
ter 3. Sinauer Associates Inc, pp 50–52
Glass L, Pasternack J (1978) Stable oscillations in mathematical models
of biological control systems. J Math Biol 6(3):207
Goldfeder J, Kugler H (2018) Temporal logic based synthesis of
experimentally constrained interaction networks. In: International
symposium on molecular logic and computational synthetic biol-
ogy. Springer, pp 89–104
Graudenzi A, Serra R, Villani M, Damiani C, Colacci A, Kauffman SA
(2011) Dynamical properties of a boolean model of gene regulatory
network with memory. J Comput Biol 18(10):1291–303
Harvey I, Bossomaier T (1997) Time out of joint: attractors in asyn-
chronous. In: Fourth European conference on artificial life, vol 4.
MIT Press, p 67
Helikar T, Kowal B, McClenathan S, Bruckner M, Rowley T, Madrahi-
mov A, Wicks B, Shrestha M, Limbu K, Rogers JA (2012) The cell
collective: toward an open and collaborative approach to systems
biology. BMC Syst Biol 6(1):96
Huang S (2009) Non-genetic heterogeneity of cells in development:
more than just noise. Development 136(23):3853–3862
Huang S, Eichler G, Bar-Yam Y, Ingber D (2005) Cell fates as high-
dimensional attractor states of a complex gene regulatory network.
Phys Rev Lett 94(128701):1–4
Huang S, Ernberg I, Kauffman S (2009) Cancer attractors: a systems
view of tumors from a gene network dynamics and developmental
perspective. In: Seminars in cell & developmental biology, vol 20.
Elsevier, pp 869–876
Huang S, Ingber DE (2000) Shape-dependent control of cell growth,
differentiation, and apoptosis: switching between attractors in cell
regulatory networks. Exp Cell Res 261(1):91–103
Huang S, Kauffman S (2013) How to escape the cancer attractor: ratio-
nale and limitations of multi-target drugs. In: Seminars in cancer
biology, vol 23. Elsevier, pp 270–278
Kappler K, Edwards R, Glass L (2003) Dynamics in high-dimensional
model gene networks. Sig Process 83(4):789–798
Kauffman S (1969) Metabolic stability and epigenesis in randomly con-
structed genetic nets. J Theor Biol 22(3):437–467
Kauffman S (1993) The origins of order: self-organization and selection
in evolution. Oxford University Press, Oxford
Kauffman S (2004) A proposal for using the ensemble approach to
understand genetic regulatory networks. J Theor Biol 230:581–
590
Kim M, Costello J (2017) Dna methylation: an epigenetic mark of cel-
lular memory. Exp Mol Med 49(4):e322
Longo G (2018) How future depends on past and rare events in systems
of life. Found Sci 23(3):443–474
Miyamoto T, Furusawa C, Kaneko K (2015) Pluripotency, dif-
ferentiation, and reprogramming: a gene expression dynamics
model with epigenetic feedback regulation. PLoS Comput Biol
11(8):e1004476
NykterM,PriceN,AldanaM,RamseyS,KauffmanS,HoodL,Yli-
Harja O, Shmulevich I (2008) Gene expression dynamics in the
macrophage exhibit criticality. In: Proceedings of the national
academy of sciences, USA, vol 105, pp 1897–1900
Perino M, Veenstra G (2016) Chromatin control of developmental
dynamics and plasticity. Dev Cell 38(6):610–620
Roli A, Vernocchi F, Serra R (2010) Continuous network models of
gene expression in knock-out experiments: a preliminary study.
In: Artificial life and evolutionary computation. World Scientific,
pp 293–301
Roli A, Villani M, Filisetti A, Serra R (2018) Dynamical criticality:
overview and open questions. J Syst Sci Complex 31(3):647–663
Schuettengruber B, Cavalli G (2009) Recruitment of polycomb group
complexes and their role in the dynamic regulation of cell fate
choice. Development 136(21):3531–3542
Serra R, Villani M, Barbieri A, Kauffman S, Colacci A (2010) On the
dynamics of random Boolean networks subject to noise: attractors,
ergodic sets and cell types. J Theor Biol 265(2):185–93
Serra R, Villani M, Graudenzi A, Kauffman SA (2006) On the dis-
tribution of small avalanches in random Boolean networks. In:
Ruusovori P et al (eds) Proceedings of the 4th TICSP workshop
on computational systems biology. Juvenes Print, Tampere, pp 93–
96
Serra R, Villani M, Graudenzi A, Kauffman SA (2007) Why a simple
model of genetic regulatory networks describes the distribution of
avalanches in gene expression data. J Theor Biol 246(3):449–460
Serra R, Villani M, Semeria A (2004) Genetic network models and
statistical properties of gene expression data in knock-out experi-
ments. J Theor Biol 227:149–157
123
Content courtesy of Springer Nature, terms of use apply. Rights reserved.

Dynamical properties and path dependence in a gene-network… 6787
Shmulevich I, Dougherty E (2010) Probabilistic Boolean networks: the
modeling and control of gene regulatory networks. SIAM, vol 118
Shmulevich I, Kauffman SA, Aldana M (2005) Eukaryotic cells
are dynamically ordered or critical but not chaotic. PNAS
102(38):13439–13444
Szathmáry E (2006) Path dependence and historical contingency in
biology. In: Understanding change. Springer, pp 140–157
Thomas R, Thieffry D, Kaufman M (1995) Dynamical behaviour of
biological regulatory networks —I. Biological role of feedback
loops and practical use of the concept of the loop-characteristic
state. Bull Math Biol 57(2):247–276
Turner AP, Caves LS, Stepney S, Tyrrell AM, Lones MA (2017) Artifi-
cial epigenetic networks: automatic decomposition of dynamical
control tasks using topological self-modification. IEEE Trans Neu-
ral Netw Learn Syst 28(1):218–230
Turner AP, Lones MA, Fuente LA, Stepney S, Caves LS, Tyrrell AM
(2013) The incorporation of epigenetics in artificial gene regula-
tory networks. BioSystems 112(2):56–62
Villani M, Barbieri A, Serra R (2011) A dynamical model of genetic
networks for cell differentiation. PLoS ONE 6(3):e17703
Villani M, La Rocca L, Kauffman S, Serra R (2018) Dynamical criti-
cality in gene regulatory networks. Complexity 2018
Villani M, Serra R (2013) On the dynamical properties of a model of cell
differentiation. EURASIP J Bioinf Syst Biol 2013(1):4. https://doi.
org/10.1186/1687- 4153-2013- 4
Yordanov B, Dunn SJ, Kugler H, Smith A, Martello G, Emmott S (2016)
A method to identify and analyze biological programs through
automated reasoning. NPJ Syst Biol Appl 2(1):1–16
Publisher’s Note Springer Nature remains neutral with regard to juris-
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