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.