The influence of different kinetic rates on the dynamics of a simple model of catalytic reaction

networks Alessandro Filisetti1,3, Alex Graudenzi2

1Energy and Environment, interdepartmental centre of industrial research (C.I.R.I), University of Bologna.

2 Department of Computer Science, Systems and Communications, University of Milan Bicocca 3 European Centre for Living Technology

WIVACE12 (Italian Workshop for Artificial Intelligence and Evolutionary Computation)

Outline

•  Introduction •  Autocatalytic sets of molecules •  Description of the model •  Previous Results •  Fragility of ACSs •  Simple deterministic model •  Conclusions

2  

Introduction

•  Framework – Complex Systems Biology

•  Topic – Emergence of autocatalytic sets of molecules in

catalytic reaction networks •  Methods – Consider simple models able to capture general

behaviors

3  

Introduction

•  Life that is the result of an evolutionary process���

•  Whatever it is the right scenario (RNA world, metabolism first, …) a minimal level of complexity was surely necessary���

•  Since Eigen (1979) the importance of cycles is a well known property of the living systems

4  

Introduction

•  Our work is an extension of the original work made by Stuart Kauffman (1986).

•  Although autocatalytic sets are frequently seen in models of catalytic networks, they are rarely observed in laboratory.

•  Our aim is to relax some limitations and develop a new model attempting to create a bridge between the theoretical and the experimental approach.

5  

Autocatalytic Sets of Molecules

•  An autocatalytic sets of molecules is achieved when each member is the product of at least one reaction catalyzed by at least one other member, Kauffman (1986)

6  

Description of the model

- Species are linear chains composed of a certain alphabet

- Two types of reaction: -  Condensation -  Cleavage

- A particular species is assumed to catalyse a particular reaction with independent probability P

- Question: Under which conditions does a self consistent autocatalytic subset of molecules emerge?

Kauffman main result

•  As the maximum length of species M increases, the number of species increases exponentially but the number of reactions by which these polymers might interconvert increases yet faster, such that the ratio of reactions to species grows linearly.

Condensations e.g. ABABA -  A + BABA -  AB + ABA -  ABA + BA -  ABAB + A

   

All possible cleavage reactions

, Z

2 3 4 5 6 7 8 9 1011121314151617181920100

102

104

106

108

Maximum Length M

log

# of

pol

ymer

s and

reac

tions

Number of polymersNumber of reactions

A bridge between theory and lab

•  WHY IS IT SO DIFFICULT TO REPRODUCE THAT IN LABORATORY?

These  mathema.cal  models  are  toys  

Increase  molecules  diversity  

System Dynamics

•  Gillespie Stochastic algorithm – Gillespie algorithm allows to

compute which reaction occurs and the time of the reaction

– Open flow reactor

10  

Reaction Graphs •  “Actual” Reaction Graph: Each time that a reaction

occurs, it is drawn on the graph. •  Time window: In order to neglect the presence of very

rare reactions, if the reactions do not occur again within the decay time, they are removed from the graph

11  

Reac.on  1   Reac.on  2   Reac.on  3   Reac.on  4  

Time  

Influence of the reaction probability���i.e. the network connectivity

Filisetti, A., Graudenzi, A., Serra, R., Villani, M., De Lucrezia, D., Fuchslin, R. M., Kauffman, S. A., et al. (2011). A stochastic model of the emergence of autocatalytic cycles. Journal of Systems Chemistry, 2(1), 2. doi:10.1186/1759-2208-2-2

Influence of the incoming flux

•  Increasing the diversity in the composition of the incoming flux the probability of having an ACS increases as well.

Influence of the average residence time

•  Increasing the average residence time of the molecules in the reactor the activity of the system is enhanced.

Average Residence Time

Filisetti, A., Graudenzi, A., Serra, R., Villani, M., Fuchslin, R. M., Packard, N., Kauffman, S. A., et al. (2011). A stochastic model of autocatalytic reaction networks. Theory in biosciences = Theorie in den Biowissenschaften, 1-9. Springer Berlin / Heidelberg. doi:10.1007/s12064-011-0136-x

Introduction of energy constraints

•  Endergonic and Exergonic reactions. •  Introduction of an energy carrier molecule. •  Presence of an incoming flux of energy carriers and

a species energization rate.

Energy Description

A + B + C + LEC à A* + B + C + UECà C.A* + B +UECà C + AB + UEC(+ *)

A B  

LEC  

A   B  

UEC  

C  AA

C  AA

C  AAA  

B  

UEC  

C  AA

UEC  

B  A*  

*  

*  

Loaded  energy  carrier  Unloaded  energy  carrier  

Results���Production of species

In correspondance of particular values of the energetic parameters the system shows a maximum in the species production. Nevertheless the concentration produced within an ACS remains low.

Filisetti, A., Graudenzi, A., Serra, R., Villani, M., De Lucrezia, D., & Poli, I. (2011). The role of energy in a stochastic model of the emergence of autocatalytic sets. In L. T, G. M, H. Bersini, P. Bourgine, M. Dorigo, & R. Doursat (Eds.), Advances in Artificial Life, ECAL 2011 Proceedings of the Eleventh European Conference on the Synthesis and Simulation of Living Systems (pp. 227-234). MIT Press, Cambridge, MA.

Results���Chance of having an ACS

0,1

10

1000

0

0.1

0.2

0.3

0

1.00E-23 1.00E-22

1.00E-21 1.00E-20

Knr

g

% A

CS

[EC]

0.2-0.3 0.1-0.2 0-0.1

Although it is not high, also the probaility of having an ACS increasis for a particular combination of the parameters

ACS structure fragility

Concentra)on  produced  within  the  ACSs  as  a  func.on  of  the  number  of  species  with  at  least  

one  exemplar    (20  different  simula.ons)  

Bo/leneck  

The  concentra)on  is  mainly  produced  by  a  reac)on  chain  composed  of  the  species  

 11  à45  à  68  

5474

5474

10290

2261

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578

109

56

56

37

9

2

1145

68

94

3

0

95

12

132

1 93

3590

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3323

3323

547454741016

10164944

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1645

1461

1461

148168081930562856283397

1346124596939

19175

19175

5520

2791

3349

5749

5642

9119

7600

4988

10290

3532

5201

4089

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1827

1089

1089

959

1188

1188

2762

2021

4498

2151

3740

3740

2261

2117

2209

3076

1369

1369

2175

2175

12751275887

1248537

537

1313

733

733

1078

12842017

589589

824

570

712

531

615

654

426

461

578

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352

352

189

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6075

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56

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8

10

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9

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9

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66

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9167

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1

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39

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3564

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2797

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17210862516

96

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243

Time  windo

w  =  10  

Time  windo

w  =  100

 

ACS structure fragility (II)

Some observations

•  Which is the right time-frame to observe? •  New layers of species lengths emerge

influx  

The  dots  size  represents  the  concentra.ons  of  the  species    

2°  layer  of  new  species  

1°  layer  of  new  species  

Influence of the kinetics���A very simple Model

Results

K1 = k2 = k3 = k4 = k5 = k6 = 1 K1 = k2 = k3 = 1 k4 = k5 = k6 = 10

The behavior is trivial but the implications could be very important in a protocell context

Autocatalysis Introduction ���D molecules

K1 = k2 = k3 = k4 = k5 = k6 = 1

K1 = k2 = k3 = 1 k4 = k5 = k6 = 10

In this case the original ACS is no longer favored.

Autocatalysis Introduction ���F molecules

K1 = k2 = k3 = k4 = k5 = k6 = 1

K1 = k2 = k3 = 1 k4 = k5 = k6 = 10

Although the first steps of the chain are diluted F and G survives because of the autocatalytic nature of F

Inorganic catalyst Introduction The inorganic catalyst rules the behavior of the system

Conclusions

•  The presence of an autocatalytic set is not sufficient in order to achieve its survival.

•  The introduction of the stochasticity changes completely the dynamics.

•  Big systems composed of several reaction pathways could suppress the ACS.

•  Analytical analysis could be useful in order to decipher the behaviour of the system.

Conclusion •  Compartmentalization could have played an

important role: – The introduction of amphiphilic molecules leads to the

spontaneous formation of micelles. – Some molecules are entrapped within the micelles. – The micelles compete for the same limited resources. – The micelle able to grow and divide synchronizing the

processes of growth and molecules replication is favored in the competition.

– The micelle containing ACSs grow faster than the other.

Protocell Unit

Self-Assembly

Self-Reproduction Self-Maintenance Emergence of the synchronization

between the key processes Catalytic reaction networks and the emergence of autocatalytic cycles

Environment

for free

Cognition

Life as a process

Environment  

Protocell  

Co-­‐emergence  

evolu.on  

Actual Cell Design ���Complex Unit

Thanks for the attention!!!

32  

ACKNOWLEDGEMENTS

alessandro.filisetti@unibo.it