In 1969, Stuart Kauffman proposed using Random Boolean Networks (RBNs) as an abstract model
of gene regulatory networks.
Where Each vertex represents a gene and “on” state of a vertex indicates that the gene
is expressed.
An edge from one vertex to another implies the the former gene regulates the later and
“0” and “1” values on edges indicate presence/ absence of activating/repressing
proteins.
Boolean functions assigned to vertices represent rules of regulatory interactions between genes
If we consider a 3 genes system with following rules
X(t+1) = X(t) and Y(t)
Y(t+1) = X(t) or Y(t)
Z(t+1) = X(t) or (not Y(t) and Z(t))
Then we can calculate all possible transient states for the given network.
We can demonstrate all attractors and their basin of attraction using a graph called,
state transition graph or STG.
So we can see that the number of accessible states is finite, 2^N and Cyclic trajectories
are possible.
- Not every state must be approachable from every other state.
The successor state is unique; the predecessor state is not unique.
If the rules for updating states are unknown, we
need to select rules randomly.
So suppose that Boolean functions are assigned to RBN vertices so that they evaluate to 0
with probability p and evaluate to 1 with probability 1-p.
For example, p = 0.5 means that Boolean functions are assigned independently and uniformly at
random from the set of 16 Boolean functions of 2 variables.
An NK automaton is an autonomous random network of N Boolean logic elements.
Each element has K inputs and one output.
The signals at inputs and outputs take binary (0 or 1) values.
The Boolean elements of the network and the connections between elements are chosen in
a random manner.
There are no external inputs to the network.
The number of elements N is assumed large.
An automaton operates in discrete time.
The set of the output signals of the Boolean elements at a given moment of time characterizes
a current state of an automaton.
During an automaton operation, the sequence of states converges to a cyclic attractor.
The states of an attractor can be considered as a "program" of an automaton operation.
The number of attractors M and the typical attractor length L are important characteristics
of NK automata.
With K connections, there is 22K Boolean input functions; these Nets are free of external
inputs.
Once, connections and rules are selected, they remain constant and the time evolution
is deterministic.
What is the relationship between the average connectedness of genes and the ability of
organisms to evolve?
Has fortunate evolutionary history selected only nets of highly ordered circuits that
alone insure metabolic stability; Or are stability and epigenesist, even in nets of randomly
interconnected regulatory circuits, to be expected as the probable consequence of as
yet unknown mathematical laws?
Are living things more akin to precisely programmed automata selected by evolution, or to randomly
assembled automata…?
Given N bulbs and K connections behavior is
1) Chaotic: If K is large, the bulbs keep twinkling chaotically
2) Frozen or periodic: If K is small (K = 1), some flip on and off, most soon stop
3) Complex: If K is around 2, complex patterns appear, in which twinkling islands of stability
develop, changing shape at their borders.
A network that is frozen either solid or chaotic cannot transmit information and thus cannot
adapt.
Gene regulatory networks of living cells are believed to exhibit phase transition behavior,
on the border between the frozen and chaotic phases.
Kaufman has shown that if k = 2 and p = 0.5, then the statistical features of RBNs match
the characteristics of living cells:
number of attractors number of cell types
length of attractors cell cycle time
When we study a system, our motivation is usually a search for causal relations.
Although in everyday life we frequently make causal statements, such as “I couldn’t
get up on time this morning because I was up late last night”, in general we cannot
“see” causal relations but can only infer their existence.
Our current systems theory, including all that is taken from physics or physical science,
deals exclusively with simple systems or mechanisms.
While Complex and simple systems are disjoint categories.
von neuman thought that a critical level of system size would trigger the onset of complexity
but Complexity is more a function of system qualities rather than size.
Complex systems require that all aspects of them be encoded in order to be more completely
understood.
This is not possible only using traditional parameter dependent modelling.
The world of simple mechanisms is a surrogate world created
by traditional science.
The real world is complex and a new view is needed.