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Boolean nets
THEORETICAL BIOLOGY
Theoretical biology is a science of general
laws and principles governing life. There is still no commonly
accepted paradigm for the theory of life, since according to
the prevailing Neo-Darwinian and molecular-reductionistic views
on life (see A
successful form of reductionism ), life has no autonomous
laws or principles - it is governed by the laws of physics
and chemistry as well as the law of natural selection. However,
the latter operates very well also in non-living systems.
According to this reducionistic view, there
are no autonomous biological laws and therefore any theoretical
biology is limited only to the theory of evolution and mathematical
models of some living processes (biochemical, physiological,
ecological, etc.). However, there are many different scientific
views on life - and each yields a quite different theoretical
biology.
Although at the Bion Institute, we strongly
sympathise with contemporary organistic (structuralistic) aspect
(see here)
we focused our research efforts on the concepts that are
mostly covered by the complexity
theory. It studies complex systems, their capability
of self-organisation and self-ordering (see Complexity
Papers Online)
and other general principles. Mathematical models of such
sytems are dynamic Boolean networks, first studied by a
great theoretical biologist Stuart
Kauffman.
They may be a robust representation of some life processes
and organizational principles, which include so called
order-for-free (mathematical order that does not necessitate
any income of free energy). This order is therefore implicit
in organisms.
BOOLEAN NETWORKS
Boolean networks are abstract mathematical
models for an extensive study of complex dynamical systems
with multitude of coupled variables [Kauffman,
93].
Such systems also include biological systems on various levels
of organization, like genetic networks, metabolic networks,
immune system, neural networks, organisms, ecosystems, biosphere
etc.
A simple dynamic Boolean network
A Boolean network consists of N binary
elements; each of them has randomly assigned a Boolean or logical
function that determines the value of an element according
to the permutation of its input values. Interactions between
the elements are defined with randomly assigned inputs to elements.
The number of inputs per element (K) varies (this
is called variable K).
An element can have many inputs, few of them or even none.
Boolean networks had primarily constant K (all elements
had the same number of inputs), but they are not as realistic
as variable K networks.
Once the connectivity and Boolean functions
are assigned, they remain constant for a given network. We
randomly choose the initial values of elements and start the
computation of the network. The values of elements are synchronously
updated, according to the previous state of the network. Therefore,
the network follows a deterministic succession of discrete
states, which is represented with a curve (a trajectory) in
the state space. State space is a mathematical tool for representation
of complex system's behaviour. It is an abstract n-dimensional
space (the number of dimensions depends on the number of variables
in the system). A point in the state space represents a state
of the system, which is uniformly determined by certain values
of variables. Each state lies on one of trajectories. Eventually
all trajectories run into attractors. Attractors are end states
of the network, and are either point attractors (static end
states) or cyclic attractors (periodic end states). All trajectories
that run into the same attractor constitute a basin of attraction.
An external perturbation (switching the value of one or more
elements) causes the system to leap from one trajectory to
another that can run into the same or into some other attractor.
Attractors determine the network's dynamics that can be either
ordered or disordered. Point attractors and short and moderately
long cyclic attractors are characteristic for ordered dynamics.
Long and extremely long cyclic attractors denote disordered
dynamics, because in real time we cannot observe any periodic
patterns. The degree of order can be estimated from the number
of attractors, their length, their similarity, their basin
size and the fraction of frozen elements.
(See also a
Boolean Network Model).
An example of a periodic attractor
An example, where the dynamics
of a network with 100 elements settled in
a periodic attractor
with four states. Blue elements are frozen
to the value 0,
violet to 1, whereas yellow (value 0) and red (value 1) are
active.
OUR RESEARCH
Basic (variable K)
Independently from a similar research [Fox
and Hill 01]
we introduced a variable K model and first analysed
the degree of order and its origin, and second the perturbations
of attractors. We studied networks with average number of inputs
(K)
= 2; and apart from a research of Fox and Hill, we
also allowed elements with no inputs (source or constitutive
elements) and elements with no outputs (sinks). The distribution
of inputs we used was skewed binomial. We were interested in
how ordered are variable K networks, compared to constant K networks,
and what are the origins of this order. We also analysed the
influence of source elements (no input elements) on network's
dynamics. We presumed that variable K networks are
even more ordered than constant K networks, because
the distribution of inputs alone creates frozen areas that
introduce an additional amount of order. Fox and Hill independently
demonstrated the effect of various K distributions on the orderliness
and supported our presumption. They also determined the lower
limit for the size of frozen component originating from canalising
Boolean functions and propagation of their effect through the
network.
We determined the entire effective K distribution
(the shift of distribution when only effective inputs are considered),
and thoroughly analysed the sources of order, one being the distribution
of inputs and canalising properties of Boolean functions, and
the other one the complex dynamics itself. We also presumed that
no-input elements markedly influence network's dynamics, and
confirmed this to be true, since increasing number of no-input
elements strongly increases dynamic diversity of the network
(referring to the number and the similarity of attractors). Most
of our work is based on numerical simulations, while the effective K distribution
in a part, which depends on the canalising properties of Boolean
functions and spreading of their influence through the network,
was determined analytically. Our contributions are also two originally
developed methods for analysing the similarity and the relatedness
of attractors. For more details see [Skarja
et al. 03].
Perturbations of attractors
We were further interested in how much homeostasis do attractors
have, so we investigated the perturbations of attractors. A
perturbation means that the value of one or more elements is
switched from 0 to 1 or vice versa. A perturbation throws the
network out from attractor on a trajectory that can run into
the same or into some other attractor. How far the perturbation
throws the network depends on the number of elements perturbed
(the more, the farther). We expected that with only one element
perturbed, the network mostly returns into the same attractor
and that with more element perturbed this homeostasis decreases
(the network falls into other attractors more frequently).
We also expected that perturbations of no input elements throw
the network more often into other attractors, since they have
stronger influence on network dynamics.
Preliminary results show that homeostasis
in networks when few elements are perturbed is on average quite
high and decreases, as expected, with increasing number of
perturbed elements. With one element perturbed, the network
mostly returns into the same attractor, when the number of
perturbed elements increases, the network falls into other
attractors more frequently. With no-input (or constitutive)
elements perturbed the network mostly falls into other attractors
(the homeostasis is significantly lower). Different attractors
have very diverse homeostasis, that depends on its basin
sizes (attractors with large basins have higher homeostasis,
than attractors with smaller basins). Perturbations of attractors
turned out to be a very efficient method for discovering
new attractors (that were not found with random initial states
method). With further perturbations of newly found attractors,
some new attractors have been found again (this way we found
some orders of new attractors). These new orders of attractors
seem to be closely connected, since new attractors could be
found only from previously discovered ones (they are difficult
to find with random searching, probably because they have
small, irregularly shaped or fragmented basins). In genetic
interpretation of the model, no-input elements represent
constitutive genes or external factors that influence gene
expression, and their perturbation mostly diverts the network
into some other attractor, that represents a different gene
expression pattern. Another way to divert the network into
other attractor is to perturb more elements. These diverting
processes would happen during cell differentiation or some
pathological disturbances of normal gene expression patterns
(carcinogenesis). (For more details, see [Remic
et al. 03, Remic
et al. 02])
Mathematical laws of order in complex systems
This abstract model is devoid of physical
laws (so there are no thermodynamic fluxes that would drive
self-organization of the system). The order originates solely
from interactions between elements that are mathematically
defined. This demonstrates that non-equilibrium thermodynamics
and other physical laws are not the single source of order
in the nature. The mathematical laws are probably as much important.
Besides physical and chemical foundations, in all probability
life has an important mathematical basis as well!
Genetic interpretation of variable K networks
We interpreted our model as a genetic network,
but with smaller modifications, it can also represent a metabolic,
a neural, or an ecological network. In this case, perturbations
are internal or external disturbances that influence gene expression.
In variable K networks, each element has its
own Ki indicating
the number of inputs, and Ko indicating
the number of outputs (they are uniformly determined by the
inputs). According to that, we distinguish four kinds of
elements:
- no input elements ( Ki =
0, Ko >= 1)
(also "sources" or "constitutive" elements)
- connected elements ( Ki >=
1, Ko >= 1)
- no output elements ( Ki >=
1, Ko =
0)
(also "sinks")
- unconnected elements ( Ki =
0, Ko = 0)
No-input elements have strong influence on network's dynamics,
but the network has no feedback influence on them. They represent
constitutive genes or external factors that regulate the expression
of inducible genes. Their value remains constant during computation
(depending on their initial state), except when the element
is perturbed.
Connected elements influence the network's dynamics through
their outputs and the network influences them back through
their inputs. They represent genes inducible by other genes
that can further induce some other genes. The regulatory self-inputs
are also possible.
No-output elements have no direct influence on the network,
but they influence it indirectly, because they reduce the effective
number of connections (increase the order), they represent
inducible genes that do not induce any other genes.
Unconnected elements do not interact with the network. They
are members of the network, but are not dynamically connected
to it. They could be interpreted as non-inducible genes, having
no influence on other genes - but they might still be important
for the organism.
(See also genetic
networks).
An example of a genetic network
References
[Kauffman, 93] S. A. Kauffman, Origins
of Order : Self-Organization and Selection in Evolution,
Oxford University Press, Oxford, 1993.
J. J. Fox and C. C. Hill, "From topology to dynamics in biochemical
networks," Chaos 11, 809-815 (2001)
Our references:
M.
İkarja, B.
Remic, I. Jerman: Boolean
networks with variable number of inputs (K),
submitted to Chaos, 2003. [PDF, 196
kB]
B. Remic, M. İkarja, I. Jerman: High dynamic order in Boolean networks with variable number of connections, in Proceedings of the 5th International Multi-Conference Information Society - Cognitive Sciences, Ljubljana, Slovenia, 2002. [PDF, 542 kB]
B.
Remic, M. İkarja, I. Jerman: Perturbation
stability in variable k Boolean nets,
in Proceedings of the 6th International Multi-Conference
Information Society - Cognitive Sciences, Ljubljana, Slovenia,
2003.
Presentation
from the conference [PDF,
884 kB]
B. Remic: Boolean networks as models of genetic and other biological systems, M.Sc. Thesis, University of Ljubljana, Biotechnical Faculty, Department of Biology, 2004. [PDF, 108 kB]
B. Remic, M. İkarja, I. Jerman: Systemic biology, in Proceedings of the 7th International Multi-Conference Information Society - Cognitive Sciences, Ljubljana, Slovenia, 2004, in press. [PDF, 236 kB]
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