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

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Two examples

• One brief example Hopfield network
• A second longer example Chemical Network

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1. Network ??

• Homogeneous units ai (t) (the same time evolution
  - the same differential or difference equations)
• dai/dt F(aj, Wij)
• A connectivity matrix Wij
• A large family of biological networks
• Idiotypic immune network
• Hopfield network
• Coupled Map Lattice
• Boolean network
• Ecological network (Lokta-Volterra)
• Genetic network

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Chemical network ?

• a b --gt c
• c d --gt e
• ..
• da/dt -kabcab
• dc/dt kabcab
• Quadratic form of network
• Fixed point dynamics

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Dynamics

• ai(t)

Time
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MetaDynamics

• A second level of change
• Change in the structure of the network

- add or remove units - add or remove
connections - modify connection values
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Studied examples

• Learning in Hopfield Network
• Adding or removing antibody types in idiotypic
  network
• Adding or removing molecules in chemical network

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A key interdependency
Dynamics
MetaDynamics
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First Example Hopfield Network
f(x)
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A 6-neurons Hopfield net
Wij
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The dynamics
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The frustrated chaos the idiotypic network - the
origin
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Properties of this chaos

• Typical intermittent chaos critical bifurcation,
  length of cycles increasing
• type 1 or type 2 bifurcation
• T(wij) 1/(mij - mijT)a
• Where the intermittent cycles are the relaxing
  cycles
• Kanekos chaotic itinerancy
• Present in immune, CML, Hopfield Net.
• Not enough studied

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Bifurcation Diagram
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MetaDynamics Learning

• Hebbian Learning
• d(wij)/dt kai.aj
• Control the chaos by stabilizing one of the
  frustrated cycle.
• Learning travels on the bifurcation diagram
• Still to engineerize or to cognitivize.

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Second example Chemical Network
OO COMPUTATION
OO CHEMISTRY
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Artificial Chemistry

• a b --gt 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,
  ..

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Example

• Molecules 1,2,
• Reaction rules
• a b --gt a c with c
• Dynamics random choice of molecules
• Dittrich in Dortmund .

a/b if a mod b 0
b otherwise
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Kaufmann - autocatalytic self-maintaining
network
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Fontana - emergence of self-maintaining and
self-producing chaining reactions
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Fontana (2)
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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---gtdesign patterns
• Allow the discovery of laws describing universal
  emergent behaviors of complex systems.
• Like Kauffmans laws of Boolean networks
• Fontanas emergence of hypercycles
• etc.
• Lead to new engineering tools

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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)

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OO Chemistry
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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

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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.

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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)

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A MOLECULE A COMPUTATIONAL TREE
1(1(4 4 4) 2 (1 (3 3 3) 2 (2 (3)) 2 (4))
Not far from the SMILES notation
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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)

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THE CANONICALISATION ONE TREE ONE MOLECULE
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Still miss

• Isomerism
• Merging molecules aromaticity,
• Cristals
• .

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The different reaction mechanisms

• Chemical CrossOver
• HCL NaOH --gt NaCL H2O
• N2 3H2 --gt 2NH3
• C2H5OHCH3COOH --gt CH3COOC2H5 H2O
• multiple-link CrossOver
• CH4 2O2 --gt CO2 2H2O

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• OpenBound Reaction
• C2H2 2H2 --gt C2H6
• CloseBound Reaction
• 2Na2Cl --gt 2NO2 Cl2
• Reorganisation
• CH3CHO --gt CH4 CO

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One simple crossover
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• 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)

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One open-bond reaction
1
1
4


4
4
4
4
1
1
4
4
4
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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)

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CANONICALISATION

• Not necessary with GP, only the result of the
  tree is important not its structure
• Dont care about similar fonctionnal trees in the
  population because no explicit need of the
  concentration or the diversity.

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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

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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 dont exist already in the system.
• Calculate the rate of the reaction.

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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

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How is the rate calculated

• K exp(-Ea/T)
• if ( S Erlinksgt S Eplinks) Ea
  D else Ea S Eplinks- S Erlinks D

D
Erlinks
Eplinks
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Departure of the reactions

• Four molecules
• 1(1)
• 2(2)
• 3(3)
• 4(4)

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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 ) ) ) .

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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 ) ) ) ) )

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The dynamics

• First order reaction
• a b --gt c
• c c k ab
• a a - kab
• b b - kab

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First Simple results

• A chemical reactor only containing
• And simple Crossover reaction

And
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Irreversible - simulation deterministe
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reversible
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• 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

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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

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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 --gt 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