Dynamics and MetaDynamics in Biological and Chemical Networks

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Dynamics and MetaDynamics in
Biological and Chemical Networks

Hugues Bersini

IRIDIA

Universite Libre de Bruxelles

Two examples




One brief example: Hopfield network


A second longer example: Chemical
Network

1. Network ??


Homogeneous units a
i

(t) (the same time evolution
-

the
same differential or difference equations)


da
i
/dt = F(a
j
, W
ij
)


A connectivity matrix: W
ij


A large family of biological networks:


Idiotypic immune network


Hopfield network


Coupled Map Lattice


Boolean network


Ecological network (Lokta
-
Volterra)


Genetic network


Chemical network ?


a + b
--
> c


c + d
--
> e


…..


d[a]/dt =
-
k
abc
[a][b]


d[c]/dt = k
abc
[a][b]


Quadratic form of network


Fixed point dynamics

Dynamics


ai(t)


Time

MetaDynamics


A second level of change


Change in the structure of the network

-

add or remove units

-

add or remove connections

-

modify connection values

Studied examples


Learning in Hopfield Network


Adding or removing antibody types in
idiotypic network


Adding or removing molecules in chemical
network

A key interdependency


First Example: Hopfield Network

da
i
dt


a
i


f
(

w
ij
j

1
n

a
j
)

-


1

tanh
(
x
)
2
f(x) =

A 6
-
neurons Hopfield net

0

-
1

0

0

-
1

-
1
0

0

0

-
1

-
1

-
1
-1

-
1

0

0

-
1

0
-1

-
1

-
1

0

0

0
-1

-
1

0

-
1

0

0
0

-
1

-
1

-
1

0

0














W
ij

=


The dynamics

-2
-1
0
1
2
-1.5
-1
-0.5
0
0.5
1
1.5
figure 1
-2
-1
0
1
2
-1.5
-1
-0.5
0
0.5
1
1.5
figure 2
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
figure 3
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
figure 4
The frustrated chaos: the
idiotypic network
-

the origin

0
50
100
150
200
250
Ab conc
0
50
100
150
200
250
time [d]
300
350
400

2-clone case
0

1
1

0







0
50
100
150
200
250
Ab concentrations
0
50
100
150
200
250
time [d]
300
350
400
0
100
200
300
400
500
0
Ab concentrations
100
200
300
400
500
600
time [d]
700
800
3-clone open chain
0

1

0
1

0

1
0

1

0










3-clone closed c
hain
0

1

1
1

0

1
1

1

0










Properties of this chaos


Typical intermittent chaos: critical bifurcation,
length of cycles increasing…


type 1 or type 2 bifurcation


T(w
ij
) = 1/(m
ij

-

m
ij
T
)
a


Where the intermittent cycles are the relaxing
cycles


= Kaneko’s chaotic itinerancy


Present in immune, CML, Hopfield Net.


Not enough studied


Bifurcation Diagram



MetaDynamics = Learning


Hebbian Learning


d(w
ij
)/dt = ka
i
.a
j


Control the chaos by stabilizing one of the
frustrated cycle.


Learning travels on the bifurcation
diagram


Still to “engineerize” or to “cognitivize”….

Second example: Chemical Network


OO CHEMISTRY

A

ARTIFICIAL

CHEMISTRY

OO COMPUTATION

Artificial Chemistry


a + b
--
> c + d


A set of molecules:


abstract symbols, numbers, lambda expressions, strings, proofs


A set of reaction rules:


string matching, concatenation,lambda calculus, finite state
automata, Turing machines,matrix multiplication, arithmetic,
boolean…


A dynamics:


ODE, difference equations, explicit collision, cellular automata,
reactor, 3
-
D Euclidean space, ..

Example


Molecules: {1,2, …}


Reaction rules:


a + b
--
> a + c with c =



Dynamics = random choice of molecules



Dittrich in Dortmund .


a/b if a mod b = 0

b otherwise

Kaufmann

-

autocatalytic self
-
maintaining




network


Fontana

-

emergence of self
-
maintaining and self
-
producing chaining reactions


Fontana (2)

The three main


“raison d'être” of Alife or Achemistry


Offers biologists or chemists software platforms
to be easily parameterize to allow simulation of
real biology or chemistry
---
>design patterns


Allow the discovery of laws describing universal
emergent behaviors of complex systems.


Like Kauffman’s laws of Boolean networks


Fontana’s emergence of hypercycles


etc….



Lead to new engineering tools


OO Computation


OO reconnects programming and simulation


the program objets are “real” objects


Using UML diagram helps to visualize the
program. Visualizing allows better
understanding


Objects have state and behaviour


Objects mutually interact by sending messages
(orders)

OO Chemistry

Component
concentration
modifyConcentration( )
0..*
ChemicalComponent
reactivi ty
0..*
SourceComponent
0..*
0..*
1
1..*
0..*
1
myIdentity
Atom
identity
keys
locks
valence
energyReceptors
1..*
myConnectedAtoms
1
*
1
Link
nbrOfBounds
energy
exchangeLink( )
headAtom
1
1
AtomInMolecule
aCopy_AtomInMolecule
dupl icate( )
compare( )
1..*
1
*
1
CrossOver
OpenBound
Molecule
numberOfInstances[]
1
1..*
0..*
1
1
1
Reaction
1
1..*
1..*
1
2.5 Molecule


Atoms aggregation


attributes : which atom and how many
instances of each


methods: constructors :


from two atoms


from one atom and one molecule


from two molecules


by splitting one molecule


One front door: the headAtom =
AtomInMolecule for the structure of the complex


2.6 AtomInMolecule


As soon as an atom get into a molecule


they have identity related with atom


they code the tree or the graph structures


they have pointers called myConnectedAtoms


the well
-
known computational trick to handle
tree and graphs.


What molecules do, atomInMolecule have to do:
test affinity, duplicate, be compared.

Basic atoms


1
-

valence 4


2
-

valence 2


3
-

valence 1


4
-

valence 1


Basic diatomic molecules: 1(1), 2(2), 3(3),
4(4)

1(1(4 4 4) 2 (1 (3 3 3) 2 (2 (3)) 2 (4))

4
4
4
1
3
3
3
1
2
3
2
2
4
2
1
A MOLECULE = A COMPUTATIONAL TREE

Not far from the SMILES notation

2.7 Link


A link between two atomsInMolecule: poleA and
poleB


Two capital attributes:


the nbr of bounds


the energy


One key method in the crossover type of
reactions:


aLink.exchangeLink(anotherLink)

if (the identity of n < the identity of m) { the smaller is n }
else
if (the identity of n = the identity of m)
{ if ( n has no connected atom and m has no connected atom) {the smaller is n}
else
if (the number of connected atoms of n > the number of connected atoms of m) {the smaller is n}
else
if ( the number of connected atoms of n = the number of connected atoms of m)
{for all j connected atoms of n and m
{if (the identity of the jth connected atom of n < the identity of the jth connected of m) {the smaller is n , break-
the-loop}
else
{ for all connected atoms of n and m
{ redo recursively the same testing procedure}}}}
Table 1: Which is the smaller between the
atomInMolecule n and m.
THE CANONICALISATION:

ONE TREE = ONE MOLECULE

Still miss:


Isomerism


Merging molecules: aromaticity,…


Cristals


…….

The different reaction mechanisms


Chemical CrossOver:


HCL + NaOH
--
> NaCL + H
2
O


N
2

+ 3H
2

--
> 2NH
3


C
2
H
5
OH+CH
3
COOH
--
> CH
3
COOC
2
H
5

+ H
2
O


multiple
-
link CrossOver:


CH
4

+ 2O
2

--
> CO
2

+ 2H
2
O




OpenBound Reaction:


C
2
H
2

+ 2H
2

--
> C
2
H
6


CloseBound Reaction:


2Na
2
Cl
--
> 2NO
2

+ Cl
2


Reorganisation:


CH
3
CHO
--
> CH
4

+ CO

One simple crossover

Figure 3: a single-link-crossover
1
2
3
4
2
3
1
4
+
=

The single
-
link crossover:

1(1) +
[4]

2 (3 4)


1(3 3 3 3) +

1(2(4) 2(4) 2(4) 2(4)



The multiple
-
link crossover:




1
(
4

4

4

4
)

+

[
2
]

2
(
2
)



1
(
2

2
)

+

[
2
]

2
(
4

4
)

One open
-
bond reaction

1

1

+

4

4

=

1

1

4

4

4

4

4

4

Difference with the GA crossover


Xover occurs between trees = genetic
programming


valence plays an important role (no
engineering needs)


one or more links can be involved


CANONICALISATION (discussed in the
following)


FITNESS (discussed in the following)

CANONICALISATION


Not necessary with GP, only the result of
the tree is important not its structure


Don’t care about similar fonctionnal trees
in the population because no explicit need
of the concentration or the diversity.

FITNESS


Reactions lowering the fitness are much
more probable.


So fitness must be implicitly distributed on
the links


Molecule presents weak epistasis


Similar to (Baluja and Caruana, 1995)


Where the fitness is explicilty distributed
on the schema


The random simulation loop




Take randomly one molecule


Take randomly another molecule


Make them react according to either:


-

the Crossover


-

the Open
-
Bond reaction


In each reaction the link which breaks is the weakest link.


Generate the new molecule in its canonical form only if they don’t
exist already in the system.


Calculate the rate of the reaction.

The determistic simulation


Ad

infinitum

do

{

-

time

=

time

+

1

-

For

all

molecules

i

of

the

system


For

all

molecules

j

(going

from

1

to

i)

of

the

system



{

-

Make

the

reaction

(i,j)

according

to

a

specific




reaction

mechanism




-

Put

the

products

in

the

canonical

form


-

If

the

products

of

the

reaction

already

exist,

increase

their

concentration,

if

not


add

them

in

the

system

with

their

specific

concentration
.



-

To

do

so

calculate

the

rate


-

Decrease

the

concentration

of

i

and

j



}

}




How is the rate calculated


K = exp(
-
Ea/T)


if (
S

E
r
links
>
S

E
p
links
) Ea =
D



else








Ea =
S

E
p
links
-

S

E
r
links

+
D

E
r
links

E
p
links

D

Departure of the reactions


Four molecules:


1
(
1
)


2
(
2
)


3(3)


4(4)

After several steps of the
simulation


1

(

3

3

3

3

)

,

1

(

2

(

2

(

3

)

)

3

3

3

)

,

1

(

1

(

3

3

3

)

3

3

3
)

,

1

(

4

4

4

4

)

,

1

(

3

4

4

4

)

,

2

(

2

(

4

)

3

)

,

1

(

1

(

4

4

4

)

3

3

4

)

,

1

(

1

(

3

3

3

)

1

(

3

3

3

)

1

(

3

3

3

)

1

(

3

3

3

)

)

,

1
(
2

(
1

(

3

3

3

)

)

3

3

3

),

1

(

2

(

4

)

3

3

4

)

,

1

(

2

(

1

(

4

4

4

)

)

4

4

4

)

,

1

(

3

3

4

4

)

,

2

(

2

(

4

)

4

)

,

1

(

1

(

3

3

3

)

1

(

4

4

4

)

1

(

4

4

4

)

1

(

4

4

4

)

)

,

1

(

1

(

3

3

3

)

1

(

3

3

3

)

1

(

3

3

3

)

2

(

2

(

1

(

3

3

3

)

)

)

)

,

1

(

2

(

2

(

2

(

1

(

3

3

3

)

)

)

)

3

3

3

)

,

1

(

1

(

4

4

4

)

1

(

4

4

4

)

1

(

4

4

4

)

2

(

1

(

3

3

3

)

)

)

,

1

(

1

(

2

(

4

)

2

(

4

)

2

(

4

)

)

2

(

3

)

2

(

3

)

2

(

3

)

)

,

1

(

1

(

1

(

4

4

4

)

2

(

1

(

3

3

3

)

)

2

(

3

)

)

1

(

1

(

4

4

4

)

2

(

1

(

3

3

3

)

)

2

(

3

)

)

1

(

1

(

4

4

4

)

2

(

1

(

3

3

3

)

)

2

(

3

)

)

1

(

1

(

4

4

4

)

2

(

1

(

3

3

3

)

)

2

(

3

)

)

)

.


A 93 atoms molecule


1

(

1

(

1

(

4

4

4

)

1

(

4

4

4

)

2

(

1

(

1

(

4

4

4

)

1

(

4

4

4

)

1

(

4

4

4

)

)

)

)

1

(

1

(

4

4

4

)

1

(

4

4

4

)

2

(

1

(

1

(

4

4

4

)

1

(

4

4

4

)

1

(

4

4

4

)

)

)

)

1

(

1

(

4

4

4

)

1

(

4

4

4

)

2

(

1

(

1

(

4

4

4

)

1

(

4

4

4

)

1

(

4

4

4

)

)

)

)

1

(

1

(

4

4

4

)

1

(

4

4

4

)

2

(

1

(

1

(

4

4

4

)

1

(

4

4

4

)

1

(

4

4

4

)

)

)

)

)


The dynamics


First order reaction:


a + b
--
> c


[c] = [c] + k [a][b]


[a] = [a]
-

k[a][b]


[b] = [b]
-

k[a][b]

First Simple results


A chemical reactor only containing:






And simple Crossover reaction



3


3

And


4


4

Irreversible
-

simulation deterministe

0
50
100
150
200
250
1
5
9
13
17
21
25
29
33
37
41
45
49
53
57
61
Series1
Series2
Series3
reversible

0
10
20
30
40
50
60
70
80
90
100
1
4
7
10
13
16
19
22
25
28
31
34
37
40
Series1
Series2
Series3

More general simulations departing with
1(1), 2(2), 3(3) and 4(4).


To avoid exponential explosion:


only make the nth first molecules interact
with the nth first molecules




OR


only make the molecules with concentration
above a certain threshold to interact


0
10
20
30
40
50
60
70
80
90
100
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Series1
Series2
Series3
Series4
Series5
Series6
Series7
Series8
Series9
Series10
Series11
Series12
Series13
Series14
Results


Emergence of survival network


Which network, which molecule, is hard to
predict ??


Depending on the dynamics and
metadynamics


Very sensitive in an intricate way to a lot
of factors

Conclusions


Very general abstract scheme studying how
metadynamics and dynamics interact in natural
networks


Mainly computer experiments


For Immune nets: tolerance, homeostasis, memory ...


For NN
--
> possible connection with learning and the
current new wave NN (chaos, oscillation and
synchronicity)


For chemistry: how and which surviving networks
emerge in an unpredictable way