Event-Driven Metamorphoses of P Systems - uni-jena.dehinze/slides-wmc2008.pdfMotivation Mass-Action Kinetics ΠPMA Transitions Partially Self-ReproducibleRAM Artiﬁcial Network Evolution

Motivation Mass-Action Kinetics ΠPMA Transitions Partially Self-Reproducible RAM Artificial Network Evolution Outlook

Event-Driven Metamorphoses of P Systems

T. Hinze R. Faßler T. Lenser N. Matsumaru P. Dittrich

{hinze,raf,thlenser,naoki,dittrich}@minet.uni-jena.de

Bio Systems Analysis GroupFriedrich Schiller University Jena

www.minet.uni-jena.de/csb

Ninth Workshop onMembrane Computing (WMC9)

Event-Driven Metamorphoses of P Systems T. Hinze, R. Faßler, T. Lenser, N. Matsumaru, P. Dittrich


OutlineEvent-Driven Metamorphoses of P Systems

• Motivation

• Mass-action kinetics

• P systems ΠPMA

• Transitions between P systems ΠPMA

• Example 1:Partially self-reproducible register machines

• Example 2:Artificial network evolution

• Outlook and acknowledgement


1

10

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11

1

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00

011

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0 001 1

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time


Plasticity = Structural Dynamics

Some biological examples

• Metamorphosis

• Mutationalself-replication

• Population dynamics

• Synaptic plasticity

• Photosynthesis

Structure includes: set of reactions or behavioural rules





• Metamorphosis




• Photosynthesis



model 1 model 2

model 3 model 4

observed data

observed data

observed data

observed data




• Metamorphosis




• Photosynthesis



event

event

conditions

set of speciesreaction rulescompartments

conditions


conditions


conditions


P system 1 P system 2

P system 4P system 3


Mass-Action Kinetics: BackgroundModelling Temporal Behaviour of Chemical Reaction Networks

Assumption: number of effective reactantcollisions Z proportional toreactant concentrations(Guldberg 1867)

A + B k̂−→ C . . . . ZC ∼ [A] and ZC ∼ [B], so

ZC ∼ [A] · [B]

Production rate generating C:vprod([C]) = k̂ · [A] · [B]

Consumption rate of C: . . . . . .vcons([C]) = 0d [C]d t = vprod ([C]) − vcons([C])

d [C]d t = k̂ · [A] · [B]

Initial conditions: [C](0), [A](0), [B](0)to be set






ZC ∼ [A] · [B]



d [C]d t = k̂ · [A] · [B]







ZC ∼ [A] · [B]



d [C]d t = k̂ · [A] · [B]







ZC ∼ [A] · [B]



d [C]d t = k̂ · [A] · [B]







ZC ∼ [A] · [B]



d [C]d t = k̂ · [A] · [B]







ZC ∼ [A] · [B]



d [C]d t = k̂ · [A] · [B]




Mass-Action Kinetics: General ODE ModelChemical reaction system

a1,1S1 + a2,1S2 + . . . + an,1Snk̂1−→ b1,1S1 + b2,1S2 + . . . + bn,1Sn

a1,2S1 + a2,2S2 + . . . + an,2Snk̂2−→ b1,2S1 + b2,2S2 + . . . + bn,2Sn

...

a1,hS1 + a2,hS2 + . . . + an,hSnk̂h−→ b1,hS1 + b2,hS2 + . . . + bn,hSn,

results in ordinary differential equations

d [Si ]

d t=

h∑

ν=1

(

k̂ν · (bi ,ν − ai ,ν) ·

n∏

l=1

[Sl ]al,ν

)

with i = 1, . . . , n.



Mass-Action Kinetics: A Simple Example

2A + 0Bk̂1−→ 0A + 1B

ODE system

d [A]

d t= −2 · k̂1 · [A]2

d [B]

d t= k̂1 · [A]2

Analytic solution

[A](t) =

(

2k̂1t +1

[A](0)

)−1

iff [A](0) > 0 else [A](t) = 0

[B](t) =

(

−2(

2k̂1t +1

[A](0)

))−1

+[A](0)

2+ [B](0)


AB

= 0,02RStoffkonzentrationStoffkonzentrationk

time

spec

ies

conc

entr

atio

n

A][B][B[

A][]

(0) = 24(0) = 0


P Systems ΠPMAWhy?

• Allow coupling of systems in terms of system transitions

• Adopt systems description by reaction networks

• Discretise mass-action kinetics

Aspects considered for single system

• Suitability for small amounts of reactingparticles (e.g. cell signalling)

• Compliance with mass conservance forundersatisfied reaction scenarios

• Determinism by strict prioritisation ofrewriting rules

• Obtaining simple computational units

• Symbol objects

• Spatial globality in single well-stirred vessel


2 A + B ➜ CD

A

B

B

A

A

D

A

B

B

A

A2 A + B ➜ C2 A + D ➜ E

B

C

A

D

?


P Systems ΠPMA: DefinitionΠPMA = (V ,Σ, [1]1, L0, R)

V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . system alphabetΣ ⊆ V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . terminal alphabet[1]1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . compartmental structureL0 ⊂ V × (N ∪ {∞}) . . . . . . . .multiset for initial configurationR = {r1, . . . , rh} . . . . . . . . . . . . . . . . . . . . . . . set of reaction rules

Each reaction rule ri consists of two multisets and rate constant(reactants Ei , products Pi , ki ) such that

ri = ({(A1, a1), . . . , (An, an)}, {(B1, b1), . . . , (Bn, bn)}, ki ).

We write in chemical denotation:

ri : a1 A1 + . . . + an Anki−→ b1 B1 + . . . + bn Bn

=⇒ Index i specifies priority of ri : r1 > r2 > . . . > rh.Event-Driven Metamorphoses of P Systems T. Hinze, R. Faßler, T. Lenser, N. Matsumaru, P. Dittrich


P Systems ΠPMA: DefinitionΠPMA = (V ,Σ, [1]1, L0, R)

V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . system alphabetΣ ⊆ V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . terminal alphabet[1]1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . compartmental structureL0 ⊂ V × (N ∪ {∞}) . . . . . . . .multiset for initial configurationR = {r1, . . . , rh} . . . . . . . . . . . . . . . . . . . . . . . set of reaction rules

Each reaction rule ri consists of two multisets and rate constant(reactants Ei , products Pi , ki ) such that

ri = ({(A1, a1), . . . , (An, an)}, {(B1, b1), . . . , (Bn, bn)}, ki ).

We write in chemical denotation:

ri : a1 A1 + . . . + an Anki−→ b1 B1 + . . . + bn Bn

=⇒ Index i specifies priority of ri : r1 > r2 > . . . > rh.Event-Driven Metamorphoses of P Systems T. Hinze, R. Faßler, T. Lenser, N. Matsumaru, P. Dittrich


P Systems ΠPMA: Discretised Mass-Action KineticsMapping of rate constant k̂i from ODE model

ki =k̂i

V|Ei |· ∆t

with V: volume of reaction vessel, ∆t: time discretisation interval

Considering reactions r1, . . . , rh consecutivelyIteration scheme for reaction ri = (Ei , Pi , ki)

Lt,0 = Lt

Lt,i = Lt,i−1 ⊖ {multiplicity of reactant particles iff available}

⊎ {multiplicity of generated product particles}

Lt+1 = Lt,h

Multiplicities depend on Lt,i−1, Ei , Pi , ki



Transitions between P Systems ΠPMA

State transition system A

• Each P system ΠPMA represents a state in A

• State transitions in A initiated by triggering events withregard to time (t = τ) or species concentration as ([a] = κ)

A = (Q, T , I,∆, F )

Q = {Π(j)PMA | (j ∈ A) ∧ (A ⊆ N)} . . . . . . . . . . . . . . . . . . . . .states

T ⊆ {(t = τ) | (τ ∈ B) ∧ (B ⊆ N)}∪ . . . . . . . . . input alphabet{([a] cmp κ) | (κ ∈ N) ∧ (a ∈ V (j))∧ . . cmp: =, <,≤, . . .

(Π(j)PMA = (V (j),Σ(j), [1]1, L(j)

0 , R(j)) ∈ Q) ∧ (j ∈ A)}I ⊆ Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . initial states∆ ⊆ Q × T × Q . . . . . . . . . . . . . . . . . . . . . . . . . transition relationF ⊆ Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . final states



Transitions between P Systems ΠPMA

P system transition Π(j)PMA

c7→ Π

(m)PMA ∈ ∆ in detail

From Π(j)PMA = (V (j),Σ(j), [1]1, L(j)

0 , R(j))

To Π(m)PMA = (V (m),Σ(m), [1]1, L(m)

0 , R(m))

Triggered by c ∈ T :

V (m) = V (j) ∪ AdditionalSpeciesV (j ,m) \ VanishedSpeciesV (j ,m)

Σ(m) = Σ(j) ∪ AdditionalSpeciesΣ(j ,m) \ VanishedSpeciesΣ(j ,m)

L(m)0 = L(j)

t ⊎ {(a, 0) | a ∈ AdditionalSpeciesV (j ,m)}

R(m) = R(j) ⊎ AdditionalReactions(j ,m) ⊖ VanishedReactions(j ,m)

Re-prioritisation of reaction rules if necessary



Example 1Chemical register machine (RAM) withself-reproducible components

• Construction of chemical reaction networks for booleanlogic gates

• Introduction of a chemical clock by oscillating reactions

• Specification of a chemical master-slave flip-flop (MSFF)

• Utilise chemical master-slave flip-flop as 1-bit storage unit(initial register)

• Extend registers if needed by integration of further 1-bitstorage units (self-replicable components)

• Transform register machine program into chemicalprogram control (INC, DEC, IFZ, HALT)

• Sequential as well as parallelised register machinechemistry



Chemical Implementation of Boolean Variables andLogic Gates

Chemical reaction network for NAND

Boolean variable z represented bytwo correlated species Z T and Z F

x1 x2 y

0 0 10 1 11 0 1

1 01

F2xF

1xFy Ty

Fy TyF1x F

2x F1x F

2x+ + + +

T2xT

1xTy Fy

Ty FyT1x T

2x T1x T

2x+ + + +

T2xF

1xFy Ty

F2xT

1xFy Ty

Fy TyF1x T

2x F1x T

2x

Fy TyT1x F

2x F2xT

1x

+ + + +

+ + + +



A Chemical Clock• Based on Belousov-Zhabotinsky reactions• Cascade of auxiliary reactions for fast-switching behaviour• Two offset oscillators provide clock signals [C1] and [C2]



Master-Slave Flip-FlopReliable 1-bit storage unit, well-studied

master slave

C

S

R

M

Q

M



Chemical MSFF ImplementationTwo-stage switching from FALSE to TRUE usingtrigger species and offset clocks C1 and C2

C 1

C 2

species MF , MT : master bit valuespecies SF , ST : slave bit value




C 2

C 1





C 2

C 1





C 2

C 1




From MSFF to Register

• Four network motifs (all switching scenarios) form MSFF

• Chaining of MSFFs to build register of arbitrary length

• Assumption of MSFF as self-replicable modular unit



From MSFF to Register

• Four network motifs (all switching scenarios) form MSFF

• Chaining of MSFFs to build register of arbitrary length

• Assumption of MSFF as self-replicable modular unit



Chemical Program ControlSimple example for sequential instruction flow:#0 : IFZ R1 #2 #1

#1 : DEC R1 #0

#2 : HALT



Simulation: Adding Binary Numbers

• Denotation of register machine by P systems ΠPMA

• Dynamical network behaviour emulates computation

• Stepwise extension of registers: system transitions

• Simulation carried out using CellDesigner (SBML) 0

0.2

0.4

0.6

0.8 1

1.2

0 100 200 300 400 500 600 700 800

ConcentrationT

ime scale

> 0> 0

0

0.2

0.4

0.6

0.8 1

1.2

0 100 200 300 400 500 600 700 800

Concentration

Tim

e scale

Π Π Π 0

0.2

0.4

0.6

0.8 1

1.2

0 100 200 300 400 500 600 700 800

Concentration

Tim

e scale

01101010010101000000

0000000101011010101111

00

0

0.2

0.4

0.6

0.8 1

1.2

0 100 200 300 400 500 600 700 800

Concentration

Tim

e scale

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 100 200 300 400 500 600 700 800

Concentration

Tim

e scale

C 2M 1,1

T[ ]b 1b 2 M 1,2T[ ] M 2,1

T[ ]b 1b 2 M 2,2T[ ]

INC# 0 R 1 # 1INC# 1 R 1 # 2INC# 2 R 2 # 3

# 6 # 4IFZ# 3 R 1

# 6 # 4IFZ# 3 R 1

# 6 # 4IFZ# 3 R 1HALT# 6

INC# 5 R 2 # 3

DEC# 4 R 1 # 5

INC# 5 R 2 # 3

DEC# 4 R 1 # 5

PMA(0)

registerR2 b 1

registerR1 b 1

PMA(1)

registerR2 b 1

registerR1 b 2b 1

F 2,1I[ ] PMA

(2)

registerR2 b 2b 1

registerR1 b 2b 1

F 1,1I[ ]

clock register register R2R1



Example 2: Artificial Network Evolution

Task: addition of two positive real numbers

R0

output1

X1

input1

R1

X2

R2

input2

R0

X1 output1

input2

R2 R1

input1

snapshots of artificial network evolution

• R0, R1, R2 identify reactions• input1 , input2 , output1 :

distinguished species• X1, X2: auxiliary species• Stepwise modification of network

structure and kinetic parameters


R0.htm

output1.htm

X1.htm

input1.htm

R1.htm

X2.htm

R2.htm

input2.htm

R0.htm

X1.htm

output1.htm

input2.htm

R2.htm

R1.htm

input1.htm


Two-Level Evolutionary Algorithm• Separation of structural evolution from parameter fitting• Idea: parameters can adapt to mutated network structure

R R

R

0 0

0

i i

i

n n

n

p p

p

u u

u

t t

t

o o

o

u u

u

t t

t

p p

p

u u

u

t t

t

R R

R

1 1

1

R

R

2

2

r r

r

e e

e

s s

s

o o

o

u u

u

r r

r

c c

c

e e

e

X

X

4

4

Mutation of Network Structure

mutate parameterscreate offspring &

selection

para

met

er s

ets

Parameter Fitting & Fitness Evaluation

Selection&

OffspringCreation

Population

• Upper level: network structure, analogue to graph-GP• Lower level: parameter fitting using standard Evolution

Strategy

=⇒ All networks handled as SBML modelsEvent-Driven Metamorphoses of P Systems T. Hinze, R. Faßler, T. Lenser, N. Matsumaru, P. Dittrich


Evolutionary Operators and ParameterisationEA used here employs eight different mutations

Operators for structural evolution• Addition/deletion of a species

• Addition/deletion of a reaction

• Connection/removal of an existingspecies to/from a reaction

• Duplication of a species with all itsreactions (discussed in detail later)

Operator for parameter evolution• Mutation of a randomly selected

kinetic parameter by additionof a Gaussian variable

Further information on SBMLevolver software: www.esignet.net


addition of a species

deletion of a species

addition of a reaction

deletion of a reaction

disconnection of species

connection of species

species duplication


Outlook

Take home message• Coordination of temporally local subsystems into common

framework requires homogeneous approach

• P systems suit here: discreteness, combine different levelsof abstraction

• Exploring structural dynamics in Systems Biology

• Understand/predict functionality of complex dynamicalsystems as a whole beyond molecular computing

Further work• Comprise P systems of (selected) different classes and

with compartmental structures into common transitionframework



Acknowledgement: ESIGNET Project Funded by EUEvolving Cell Signalling Networks in silico

European interdisciplinary research project

• University of Birmingham (Computer Science)

• TU Eindhoven (Biomedical Engineering)

• Dublin City University (Artificial Life Lab)

• University of Jena (Bio Systems Analysis)

Objectives

• Study the computational properties of bionetworks

• Develop new ways to model and predict real bionetworks

• Gain new theoretical perspectives on real bionetworks

Computing facilities

• Cluster of 33 workstations(two Dual Core AMD OpteronTM 270 processors)



Our Team for Bio Systems Analysis in Jena

Peter Dittrich (PI)

Thomas Hinze (PostDoc)

Gerd Grünert (PhD student)Bashar Ibrahim (PhD student)Thorsten Lenser (PhD student)Naoki Matsumaru (PhD student)Stefan Peter (PhD student)

Franz Carlsen (research assistant)Raffael Faßler (research assistant)Christoph Kaleta (research assist.)Stephan Richter (research assist.)

www.minet.uni-jena.de/csb

Thank you for your attention. Questions? Remarks?


Documents

Event-Driven Metamorphoses of P Systems - uni-jena.dehinze/slides-wmc2008.pdfMotivation Mass-Action Kinetics ΠPMA Transitions Partially Self-ReproducibleRAM Artiﬁcial Network Evolution