Generalized Stochastic SimulationAlgorithm for Artificial Chemistry

Gillespie in artificial chemistry

Hedi Soula

Sorbonne University

September 25 2018

Bacterial reaction graphs

(a) Step 1: From cDNA fasta file, get gene labels.

(b) Step 2: From KEGG gene entries, get ECcodes.

(c) Step 3: From KEGG enzyme entries, getsubstrates and products.

(d) Step 4: Build graph and filter ubiqui-tous metabolites. Ubiquitous metabolites :H2O, ATP, ADP, NAD+, NADH, NADPH,NADP+, CO2, ammonia, sulfate, thiore-doxin, phosphate, PPi, H+.

(e) Final graph.

1

1Hedi Soula Staarc

Artificial Chemistry

MotivationI (Real) Chemistry is ... difficult

Maybe artificial chemistry is easiercan model interesting properties (for e.g. space/diffusion)

I (Real) Life is ... chemistryArtificial Life should have ... artificial chemistry (AC)

What is done (usually)I chemistry is prescribed : small dimension (small # of reactions)I straightforward : chemistry graph is simpleI (somewhat) unrealistic

transition energies are ignored (easier)mass conservation is sloppyA+ B 7→ C and C 7→ A

Hedi Soula Staarc

AC : what is expected

1 large : should have a huge number of reactions2 energy : all reactions are not possible3 mass : reversibility should not be hacked4 open : we don’t know all reactions/molecules

Hedi Soula Staarc

Hutton Artificial Chemistry (1)

Hutton J.I several papers on his AC schemeI only one follow upI straightforward : chemistry is simpleI very complex : chemistry reactions network can be extremely

complexI however (somewhat) unrealistic

I Hutton, Tim J. Evolvable self-replicating molecules in an artificial chemistry. Artificial life 8.4(2002) : 341-356.

I Hutton, Tim J. Evolvable self-reproducing cells in a two-dimensional artificial chemistry. Artificiallife 13.1 (2007) : 11-30.

I Hutton, Tim J. A functional self-reproducing cell in a two-dimensional artificial chemistry.ALIFE9. 2004.

I Hutton, Tim J. The organic builder : A public experiment in artificial chemistries andself-replication. Artificial life 15.1 (2009) : 21-28.

Hedi Soula Staarc

Hutton Artificial Chemistry (2)

Main properties

I molecules are graphsI nodes are domainsI edges are chemical bondingI domains have a fixed type and a

changing statedescribed by a pair (t|s)usually a letter and a number a0,b1 ...

I any domain can have any numbersof links

I the chemistry is composed of fullyconnected subgraphs

Hedi Soula Staarc

Hutton Artificial Chemistry (3)

Geometry and PhysicsI HAC is 2DI spatially resolvedI each domain has an id/positionI links are springs (however weird coding)I (vaguely) Brownian in viscous medium

Hedi Soula Staarc

Hutton Artificial Chemistry (3)

ReactionsReactions are of the form :

(t1|s1)(.|+)(t2|s2) 7→ (t1|s3)(.|+)(t2|s4)

I (ti |sj) is a domain (within a molecule)I (.|+) : is either linked . or no (collision) +I note that type is unchangedI modification are only local (other links unmodified)I conformation . is performed instantaneouslyI reaction + is performed instantaneously upon collisionI when conflict : chose at random

Hedi Soula Staarc

Hutton Artificial Chemistry (4)

ExamplesStarting a reactor with several a0s and 1 (one) a1 with

a0+ a1 7→ a1.a2 yields

�� �� �� �� ��

whereas a0+ a1 7→ a2.a1 yields

��

�� �� �� ��

Hedi Soula Staarc

Hutton Artificial Chemistry (5) : replication

Main propertiesUsing wildcards ...R9 : e8+ e0 −→ e4e3R10 : x4y1 −→ x2y5R11 : x5+ x0 −→ x7x6R12 : x3+ y6 −→ x2y3R13 : x7y3 −→ x4y3R14 : f 4f 3 −→ f 8+ f 8R15 : x2y8 −→ x9y1R16 : x9y9 −→ x8+ y8

e8

a1

f1

e4

a1

f1

e0 e3 e2

a5

f1

e3e2

a7

f1

e3

a0 a6

e2

a7

f1

e2

a3

e2

a4

f1

e2

a3

e2

a2

f5

e2

a3

f0

e2

a2

f7

e2

a3

f6

e2

a2

f7

e2

a2

f3

e2

a2

f4

e2

a2

f3

e2

a2

f8

e2

a2

f8

e2

a9

f1

e2

a2

f8

e2

a8

f1

e2

a8

f8

e9

a1

f1

e2

a8

f8

e9

a1

f1

e9

a1

f8

e2

a9

f1

e2

a9

f1

e8

a1

f1

e8

a1

f1

R9 R10 R11

R12

R13 R10R11

R12 R14 R15

R15 R16

R15

R15

R16

Hedi Soula Staarc

Hutton Artificial Chemistry (6)

Pros and ConsI Pros :

very general2D and nicemass conservation

I Cons :2D and niceabsurdly longno reaction ratesintelligently designed

Hedi Soula Staarc

STAARC : STochastic Atom-based ARtificialChemistry

1 Getting rid of space2 Same data structure (without space)3 reactions with rates4 SSA formalism (Gillespie)5 slightly involved (but not too much)6 github.com/hsoula/staarc

Hedi Soula Staarc

Gillespie in AC

ModificationsI Reaction have a rate :

(t1|s1)(.|+)(t2|s2) 7→ (t1|s3)(.|+)(t2|s4) : λI For each + reaction :

We count the number of n1 = (t1|s1) and n2 = (t2|s2)minus the number of n12 = (t1|s1).(t2|s2) already linkedthe propensity is a = λ (n1 ∗ n2 − n12)

I For each . reaction :we count the number of n12 = (t1|s1).(t2|s2) already linkedthe propensity is a = λn12

I then classical Gillespie algorithmI once a reaction is selected we apply the modification to a

given pair (selected at random uniformly)

Hedi Soula Staarc

Gillespie

First stepcompute all propensities ai Compute the combined rates of allreactions

a0 =R∑i=1

ai

To compute the time of the next reaction, draw a random number

τ = − log(rn)/a0

This answers the when

Hedi Soula Staarc

Gillespie

What reactionTo compute the what, we choose randomly the equation weightedby their relative weight i.e

Pr(next reaction isi) =aia0

in practice, shoot a random number uniformrn in [0, 1] and find rsuch as :

r−1∑i=1

ai ≤ a0rn <r∑

i=1

ai

Hedi Soula Staarc

STAARC : STochastic Atom-based ARtificialChemistry

Properties1 now all reactions have rates : real chemistry2 simulated 3D well mixed medium3 almost all HAC properties conserved4 we can simulate diffusion (reaction rate for collision)5 way faster computation

This turn a local resolution to a global oneI well mixed mediumI with infinite crowding

not intuitive ...

Hedi Soula Staarc

Replication : molecules size

Simple experiment : the replication according to the initial numberof particles (n ∈ {30, 60, 600, 6000}) and λ ratio (between collisionreaction + and conformation reaction .)

-6 -5 -4 -3 -2 -1 0 1rate ratio

0

5

10

15Av

erag

e C

onne

cted

Com

pone

nts

30606006000

Hedi Soula Staarc

STAARC : replication

PropertiesI Not very resilient - original sequence quickly lostI Scales very badly

Not that intelligently designed :)

Race conditionsI Race condition – diffusion vs conformationI Race condition – concurrent replications

Hedi Soula Staarc

STAARC : replication

e2

a7

f1

e3

a6

e2

a2

f4

e2

a2

f3

e2

a7

f1

e3

a6

R12

e2

a7

f1

e2

a6

e2

a7

f1

e3

a3

A

e2

a7

f1

e3

a6 R12

e2

a2

f4

e2

a2

f2

e2

a7

f1

e3

a3

B

Hedi Soula Staarc

STAARC : long replication

Long replication experimentI Starting with size 6 replication seedI Production and degradation

(x |0) −→ ∅∅ −→ (x |0)

I Let the simulation for 450,000 reactionsI Compute average size and number of divisions

Hedi Soula Staarc

Long replication : replication event

0 2 4 6 8 10 12 14Time �109

0

2000

4000

6000

8000

10000

12000

# of

repl

icat

ions

Hedi Soula Staarc

Long replication : average size

0 2 4 6 8 10 12 14Time �109

0

20

40

60

80

100

120

140

Aver

age

Mol

ecul

e Si

ze

Hedi Soula Staarc

Long replication : standard deviation

0 2 4 6 8 10 12 14Time �109

0

100

200

300

400

500

600

700

Stan

dard

Dev

iatio

n of

Mol

ecul

e Si

ze

Hedi Soula Staarc

STAARC : long replication

Long replication experimentI Replication is extremely stableI Division occurs in bumpsI The limit size is ... 6I Big compounds created transiently

Hedi Soula Staarc

Random Chemical World

Random generation of reactionI available types : {a, b, c}I maximum number of state is 5.I compute all the possible reactions

with no production nor degradation.keeping only a fraction p ∈ [0, 1] randomly

I starting N = 10, 000 particles (t|s) with t ∈ {a, b, c} and0 ≤ s ≤ 4

ExperimentI Maximum of 2,000 reactions.I Compute the time neededI The ratio of number of molecules

Hedi Soula Staarc

Random Chemical World

-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5Log Fraction

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

Com

pone

nts

Rat

io-6.0-5.0-4.0

Hedi Soula Staarc

Random Chemical World

-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5Log Fraction

-10

-8

-6

-4

-2

0

2

4

Log

Tim

e-6.0-5.0-4.0

Hedi Soula Staarc

Random Chemical World

Phase transitionI Huge variability in the middleI in molecules createdI in the time to the end of reactionsI this middle is for a low number of reactions

DrawbacksI Results not conclusiveI Better random selection of reactionsI Common elements

Hedi Soula Staarc

STAARC

github.com/hsoula/staarc

Hedi Soula Staarc

Perspectives

CodeI Open-Endedness procedural enzymeI Tri-molecular reactionsI Graph-based random chemistryI ’Metabolic network’-like graph

ProjectsI Explore Random ChemistryI Use it in an evolutionnary set-upI Artificial ’bacterial world’

Hedi Soula Staarc

thanks

hedi.soula@upmc.fr // sites.google.com/site/hsoula/

Soula HA, Generalized Stochastic simulation algorithm for ArtificialChemistry Proceedings of ALIFE XV, 2016 Cancun, Mexico