=Paper=
{{Paper
|id=None
|storemode=property
|title=Dynamical models of genetic regulatory networks
|pdfUrl=https://ceur-ws.org/Vol-741/INV03_Serra.pdf
|volume=Vol-741
|dblpUrl=https://dblp.org/rec/conf/woa/Serra11
}}
==Dynamical models of genetic regulatory networks==
https://ceur-ws.org/Vol-741/INV03_Serra.pdf
On the dynamics of gene regulatory networks
Roberto Serra
Modena and Reggio Emilia University
European Centre for Living Technology, Venice
roberto.serra@unimore.it
Abstract— It is shown that a simplified model of genetic shown that it can actually describe some quantitative features
regulatory networks, aimed at the study of their generic of real genetic regulatory networks. The possibility to verify
properties, can shed light on some important biological the appropriateness of such a model has been opened by the
phenomena. Two cases are analyzed, namely perturbations in availability of DNA microarrays, which allow genome-wide
gene expression induced in an organism by the knock-out of monitoring of the changes in gene expression levels. In this
selected genes, and cell differentiation. The role of simplified paper, after reviewing the RBN model in section 2, I will
models in biology is discussed. briefly discuss in section 3 its application to the study of
perturbations induced by single gene knock-out in the yeast S.
I. INTRODUCTION Cerevisiae, which was the first application of RBNs to the
The wealth of data nowadays available in genomics and simulation of quantitative properties of real genes in a cell.
other -omics has superseded our understanding of the key Interestingly enough, the same study opens a way to test one of
biological processes which they refer to, so there is a strong the strongest claims which have been put forth in complex
need for theories and models in order to make sense of the data systems biology, i.e. that evolution has led organisms to critical
themselves. dynamical states, intermediate between order and chaos [3],
which are sometimes referred to as "the edge of chaos". This
The movement of systems biology has contributed to shift
aspect will also be discussed in section 3.
the focus of research from a naive genocentric viewpoint to a
more sophisticated approach, which takes properly into account In section 4 I will describe another application of RBNs to
system-level interactions. In this field, most models are quite real biological systems. In this case the quantitative data which
specific, as they refer to the behaviour of a particular organ can validate or disprove the model are not yet available,
(like e.g. the heart in mammals) or to a particular genetic- however the model is interesting in principle, as it shows that it
metabolic subsystem. is possible to describe several different phenomena involved in
cell differentiation in a unified way by supposing that
A useful complementary approach (which has been called
differentiation is an emergent phenomenon of a genetic
complex systems biology by Kaneko [1]) is centered instead on
regulatory network, without postulating particular gene
the search for generic properties, common to many different
circuits. It is also possible to devise experiments to test the
biological systems. An example of this kind of properties is the
hypotheses which lie at the basis of the model, as well as some
widespread fat-tailed distributions of several biological
of their consequences.
variables, and an even more striking example is provided by
the scaling law which relates power consumption to body mass A final remark concerns the reference list, which might
of different living species, irrespective of the differences have become very long, had all the relevant original papers
among their anatomical and physiological properties [2]. Data been mentioned. I prefer to keep the list short, quoting only a
indicate that the power consumption grows as mass3/4. It seems few papers where all the other relevant references can easily be
indeed that the scaling law applies to very different sizes, from found.
blue whales to small mammals and birds to mithocondria, thus
spanning an impressive range of 20 orders of magnitude. II. RANDOM BOOLEAN NETWORKS
Interestingly enough, the smartest explanation of this regularity The model is fairly well-known and the reader is referred to
and of the value of the exponent is related to the number of [4,5] for a more detailed description. A RBN is a dynamical
(relevant) spatial dimensions, i.e. 3, and to the (generic) system whose N variables take values {1,0} which can change
hypothesis that biological evolution has tuned the features of in time according to a well-defined function (called transition
living organisms in order to optimize power efficiency (under function) of their inputs. It is convenient to think of it as a
suitable constraints). directed graph, with nodes associated to variables. If variable i
The search for generic properties of living beings had been depends upon variable j then there is a link from node j to node
pioneered by Stuart Kauffman, who introduced his model of i. In the case considered here, all the nodes have exactly k input
random boolean networks (RBNs) more than 40 years ago in links, and the updating is synchronous. The network is random
an attempt at exploring the properties of genetic regulatory in that both the connections are drawn at random (choosing the
networks. The model later became popular, in particular in the k inputs to a node with uniform probability among the other N-
complex systems community, but it has only recently been 1) and the Boolean function associated to a node is chosen at
random. Usually one either chooses the Boolean function with
uniform probability from a predefined set, or generates each set concerns 6325 genes and 227 experiments and the
Boolean function by associating with a certain probability the comparison with the experimental distribution of avalanches
output 1 or 0 to each of the 2k input possible input vectors. turned out to be good.
The system is deterministic and synchronous, therefore if N The reason why such a simple model worked so well has
is finite its asymptotic states are cycles (a fixed point being a been uncovered by analytical methods which have proven that
cycle with period one). Depending upon its structural features the distribution of avalanches depends only upon the outdegree
(i.e. topology and boolean functions), a family of networks distribution, while the indegree distribution plays no role [7].
shows a typical time behaviour, although single network Moreover, in the case of classical random Boolean networks,
realizations can behave in a way different from that typical of where the distribution of outgoing connections is Poissonian, it
their family. Indeed, some combinations of structural can be also proven that the distribution of small avalanches
parameters give rise to behaviours which have been termed depends only upon a single parameter, the so-called Derrida
ordered, other combinations lead to disordered or "chaotic" exponent which is given by the equation:
states. In the case of ordered systems the typical length of the
asymptotic cycles increases slowly with the system size;
moreover, it often happens that two nearby initial conditions
where A is the average connectivity of the network and q is the
evolve to the same final attractor. In the case of disordered
probability that a chosen node does not change its value when
networks the length of the cycles increases sharply with N, and
one (and only one) of its inputs has changed (note that q
it often happens that close initial conditions lead to different
attractors. Critical networks have been defined as those whose depends on the choice of the set of Boolean functions). had
structural parameters take values which are intermediate been introduced in the past in order to distinguish between
between those of ordered and those of disordered networks. ordered and disordered dynamical regimes (1 being the critical
value), and it turns out that it also rules the distribution of
A bold theoretical ansatz which has been proposed [3] is avalanches. Therefore it is possible to estimate its value from
that critical RBNs are endowed with features which make them the distribution of avalanches, so these analyses provide a
particularly well-suited to perform complex tasks in a changing general way to test the criticality hypothesis and, within the
environment; therefore it has been argued that biological limitations of the data set presently available, they also provide
evolution should have driven biological organisms in, or close support to it.
to this region in parameter space.
IV. CELL DIFFERENTIATION
III. PERTURBATIONS IN GENE REGULATORY NETWORKS
One of the major challenges in complex systems biology is
It has recently been possible to study the expression levels that of providing a general theoretical framework to describe
of all the genes of an organism, and to compare their global the phenomena involved in cell differentiation, i.e. the process
properties with those of genetic network models. A more whereby stem cells, which can develop into different types,
detailed description of the results summarized below can be become progressively more specialized. The model described
found in [6-8] and further references quoted therein. below (for more details see 9-10 and further references quoted
In an important series of experiments a single gene of S. therein) is an abstract one (it does not refer to a specific
cerevisiae was knocked-out, and the expression levels of all organism or cell type) and it aims at reproducing the most
the genes, in cells with a knocked-out gene, was compared with relevant features of the process: (i) the existence of different
those in normal, wild type cells. In order to make precise degrees of differentiation, that span from totipotent stem cells
statements about the number of genes perturbed in a given to fully differentiated cells; (ii) stochastic differentiation, where
experiment, and to compare them with Boolean models, it is populations of identical multipotent cells stochastically
required that a threshold be defined, such that the difference is generate different cell types; (iii) deterministic differentiation,
regarded as "meaningful" if the ratio of the expression of gene i where signals trigger the progress of multipotent cells into
in experiment j to the expression of gene i in the wild type cell more differentiated types, in well defined lineages; (iv) limited
is greater than the threshold (or smaller than its reciprocal). In reversibility: differentiation is almost always irreversible, but
order to describe the global features of these experiments it is there are limited exceptions under the action of appropriate
convenient to introduce the notion of avalanche, which is the signals; (v) induced pluripotency: fully differentiated cells can
number of genes affected by the perturbation induced by a come back to a pluripotent state by modifying the expression of
particular knock-out experiment. some genes and (vi) induced change of cell type: modification
of the expression of few genes can directly convert one
The knock-out experiment can be simulated in silico by differentiated cell type into another.
comparing the evolution of two identical RBNs which start
from the same state of an attractor, the only difference being The key hypotheses are that the differentiation process is an
that one gene is clamped permanently to the value 0 in the emerging property due to the interactions of very many genes
network which simulates knock-out. A gene belongs to the (so its main features should be shared by a variety of different
avalanche associated to a particular knock-out if it differs in the organisms) and that cellular noise plays a crucial role. To check
final states of the two networks at least once in the attractor these hypotheses a noisy version of the RBN model can be
cycle. The initial simulations were performed using a classical used (briefly refereed to as NRBN).
RBN with 2 input connections per node, restricting the set of Noise is modelled as a transient flip of a single node,
Boolean functions to the so-called canalyzing ones. The data chosen at random. Attractors of deterministic RBNs are
unstable with respect to noise even at these low levels, and if a most stringent outcome of the model, and could be amenable to
node is flipped for a single time step in an attactor state one experimental test.
sometimes observes transitions from that attractor to another
one. Therefore, by flipping all the states belonging to the This hypothesis explains in a straightforward way the fact
that there are different degrees of differentiation (i.e. property
attractors of a RBN, it is possible to create a complete map of
the transitions among the attractors. In these conditions single i), corresponding to different threshold values. It is also
straightforward to describe stochastic differentiation (i.e.
attractors can no longer be associated to cell types, as it is
usually assumed [4]. Ribeiro and Kauffman [11] observed that property ii): in this vision the fate of a cell depends on the
particular attractor where the system is found when the noise
it is possible to identify in the attractors’ landscape subsets of
attractors, which they called Ergodic Sets, which entrap the level changes. The new cell type will be that corresponding to
the new TES to which the attractor belongs at the new
system in the long time limit, so the system continues to jump
between attractors which belong to the set. Unfortunately it threshold level.
turns out that most NRBNs have just one such set: this There exist also several processes, e.g. during the
observation rules out the possibility to associate them to cell embryogenesis, in which cell differentiation is not stochastic
types. but it is driven towards precise, repeatable types by specific
A possible solution to this problem is based on the chemical signals, which activate or silence some genes. These
signals can be simulated by permanently fixing to 1 or 0 the
observation that flips are a kind of noise fairly intense, as they
amount to silencing an expressed gene or to express a gene state of some nodes. However this single action doesn’t
influence the level of noise, and therefore doesn’t enable
which would otherwise be inactive: a particular transition may
well be an event too rare to happen with significant probability differentiation: in order to have deterministic differentiation it
is necessary that so-called "switch" nodes exist, whose
in the cell lifetime, if it can happen only by perturbing a
specific gene, or very few ones. It is possible therefore to permanent perturbation coupled with a change in noise level
always leads the system to the same TES. The existence of
introduce a threshold θ, and to neglect all the transitions having
switch nodes has actually been verified to be a common
an occurrence probability lower than that. In such a way, the
property (found in about 1/3 of the nets), thereby proving the
notion of Ergodic Set has to be modified in that of Threshold
effectiveness of the model (i.e. property iii).
Ergodic Set (briefly, TES), a set of attractors linked only by
jumps having a probability higher than θ, that entrap the Moreover, by simulating the overexpression of a few genes,
system in the long time limit. A TES is therefore a subset of it has been possible to simulate also the other properties
attractors which are directly or indirectly θ-reachable summarized above, and in particular the important processes of
(reachable by means of transition whose probability exceeds induced pluripotency and transitions among different cell
the threshold θ) from at least another member of the set, and types.
from which no transition can allow escaping. The threshold
clearly is related to the level of noise in the cell, and scales V. CONCLUSIONS
with the reciprocal of the frequency of flips [9]. The above examples show that relatively simple generic
models of gene regulatory networks are able to describe the
An ergodic set can be described therefore as a TES with quantitative features of the perturbations induced by gene
θ=0; by increasing the threshold, one usually observes the birth knock-out, and to form the basis of an interesting model of cell
of more and more TESs until, above a certain level, all the differentiation. It goes without saying that more sophisticated
attractors of the deterministic model are also independent models might be necessary to describe other important
TESs. It is therefore possible to associate cell types to TESs, properties.
that represent coherent stable ways of functioning of the same
genome even in the presence of noise. Several authors, on But the point which is worth stressing is that even models
theoretical and experimental bases, associate different levels of which are based on crude approximations may well provide
noise to different levels of differentiation, the noise being insights on complex phenomena. This is well known in
higher the less differentiated the cell is. So the degree of physics, where the aim is often that of finding very general
differentiation appears to be related to the possibility for an properties, and simple models which display these properties
undifferentiated cell to wander in a portion of phase space are considered very useful. On the contrary, researchers in
greater than the corresponding portions covered by more biology and social sciences often overstate the need for detailed
differentiated cells. In the NRBN model a convenient proxy for models, which entrap all the features and the interactions which
the available portion of phase space could be the number of they suppose might be important - a requirement which, if
different attractors belonging to the TES associated to that cell. taken too seriously, might even prevent the development of
A 0-threshold TES could therefore be associated to a totipotent dynamical medelling in those fields. What might be envisaged
cell, while as the threshold is increased smaller TESs appear, is a hierarchy of models, where the simpler ones, which
corresponding to more differentiated biological forms, until at however capture some key properties, are used to understand
high enough threshold values all the attractors are TESs, thus some of the most relevant aspects, and to suggest further
representing the fully differentiated cells. The increase of the experiments. They may well be complemented by more
threshold would correspond to a decrease of noise level, that detailed models able to provide more accurate quantitative (and
could be related to an improvement in the mechanisms sometimes also qualitative) behaviours.
whereby fluctuations are kept under control. This association
of differentiation to changes in the noise level represents the
ACKNOWLEDGMENT [5] Aldana M., Coppersmith S., Kadanoff L.P. (2003) Boolean dynamics
with random couplings. In: Kaplan, E., Marsden, J.E., Sreenivasan, K. R.
Useful discussions with Stuart Kauffman, Annamaria (Eds), Perspectives and Problems in Nonlinear Science, 23–89. Springer
Colacci, Marco Villani, Alex Graudenzi, Alessia Barbieri and Applied Mathematical Sciences Series
Chiara Damiani are gratefully acknowledged. This work has [6] Serra, R. Villani, M., Semeria, A. (2004) Genetic network models and
been supported by the Italian MIUR-FISRproject nr. statistical properties of gene expression data in knock-out experiments. J.
Theor. Biol. 227: 149-157, 2004
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Venezia. simple model of genetic regulatory networks describes the distribution of
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