A Study of Degeneracy in Random BooleanNetworks

Roberto Gutierrez1, David A. Rosenblueth1, James Whitacre2

& Carlos Gersenson1

1Instituto de Investigaciones en Matematicas Aplicadas y en SistemasUniversidad Nacional Autonoma de Mexico

2Centre of Excellence for Research in Computational Intelligence and ApplicationsUniversity of Birmingham

Eurepean Conference on Complex Systems, 2012

Introduction Experiments Discussion

Contents

1 Introduction

2 Experiments

3 Discussion

Introduction Experiments Discussion

Random Boolean Networks

Figure: RBN with connectivity K = 2.

Figure: Dynamics of RBNs in a) ordered,b) critical & c) chaotic phase, respectively.

n(t) p(t) o(t + 1)0 0 10 1 01 0 01 1 1

Table: Lookup table for net o.

Introduction Experiments Discussion

Robustness vs. Evolvability

Robustness

“A system is robust if it continues to function in the face ofperturbations” (A. Wagner, 2005).

Evolvability

The capacity to discover beneficial, heritable adaptations. (Wagnerand Altenberg, 1996).

Introduction Experiments Discussion

Robustness vs. Evolvability

Robustness

“A system is robust if it continues to function in the face ofperturbations” (A. Wagner, 2005).

Evolvability

The capacity to discover beneficial, heritable adaptations. (Wagnerand Altenberg, 1996).

Introduction Experiments Discussion

Degeneracy

definition

Describes the coexistence of structurally distinct components thatcan perform similar roles or are interchangeable under certainconditions, yet have distinct roles under other conditions (Edelmanand Gally, 2001).

Figure: Robustness and evolvability. 1

1Image from “Degeneracy: a design principle for achieving robustness

and evolvability”, James M. Whitacre, 2010.

Introduction Experiments Discussion

Simple RBN (core)

Figure: RBN with N = 3, K = 2.

Introduction Experiments Discussion

Adding redundancy (RBN with redundancy)

Figure: RBN with N = 4, K = 2.2

2“The Role of Redundancy in the Robustness of Random Boolean

Network” Carlos Gershenson, Stuart A. Kauffman, Ilya Shmulevich, 2006.

Introduction Experiments Discussion

RBN with function degeneracy

Figure: RBN with N = 4, K = 2.

Introduction Experiments Discussion

RBN with input degeneracy

Figure: RBN with N = 4, K = 2.

Introduction Experiments Discussion

RBN with output degeneracy

Figure: RBN with N = 4, K = 2.

Introduction Experiments Discussion

Experiments

Simple RBN (core).

Function Degeneracy.

Input Degeneracy.

Output Degegeracy.

Redundancy.

Introduction Experiments Discussion

Number of Attractors (A)

A Reflects how many distinct sets of states can “capture” thedynamics of the RBN.

Figure: Dynamics of an RBN. 3

3Image taken from http://www.sussex.ac.uk/Users/andywu/

Introduction Experiments Discussion

Number of Attractors (A)

Figure: RBN with NTotal = 20,K = 5,Ndeg = 5.

Introduction Experiments Discussion

States in Attractors (SIA)

SIA is dependent on the number and length of attractors.

Figure: RBN with NTotal = 20,K = 5,Ndeg = 5.

Introduction Experiments Discussion

Sensitivity to Initial Conditions I

Calculated with the normalized Hamming distance:

∆H = H(Sf ,S′f ) − H(Si ,S

′i ).

Figure: RBN with NTotal = 20,K = 5,Ndeg = 5.

Introduction Experiments Discussion

Sensitivity to Initial Conditions II

Figure: RBN with NTotal = 200,K = 5,Ndeg = 20.

Introduction Experiments Discussion

Discussion I

Figure: Propagation of mutations.

Introduction Experiments Discussion

Discussion II

Figure: Propagation of mutations.

Introduction Experiments Discussion

Discussion III

These results suggest that degeneracy, as well as redundancy,can facilitate robustness and evolvability, allowing newfunctionalities to arise from nodes with small variations offunction or structure without changing too much thedynamics (phenotype).

Thanks for your attention

Introduction Experiments Discussion

References I

Carlos Gershenson. Introduction to Random BooleanNetworks. Centrum Leo Apostel, Vrije Universiteit Brussel.

Tononi G, Sporns O, Edelman GM Measures of degeneracyand redundancy in biological networks 0027-8424 1999,96:3257-3262.

James M Whitacre. Degeneracy: a design principle forachieving robustness and evolvability School of ComputerScience, University of Birmingham, Edgbaston, UK

James M Whitacre. Degeneracy: a link betweenevolvability, robustness and complexity in biologicalsystems School of Computer Science, University ofBirmingham, Edgbaston, UK

Introduction Experiments Discussion

References II

James M Whitacre. The Role of Redndancy in theRobustness of Random Boolean Networks School ofComputer Science, University of Birmingham, Edgbaston, UK

A. Wagner Distributed robustness versus redundancy ascauses of mutational robustness BioEssays, vol. 27, pp.176-188, 2005

Fernandez P., Sole R.(2004). The role of computation incomplex regulatory networks In Koonin, E. V., Wolf, Y. I.,and Karev, G. P., editors, Power Laws, Scale-Free Networksand Genome Biology. Landes Bioscience.

Introduction Experiments Discussion

References III

Carlos Gershenson, Stuart A. Kauffman, Ilya Shmulevich(2006). The Role of Redundancy in the Robustness ofRandom Boolean Networks Artificial Life X, Proceedings ofthe Tenth International Conference on the Simulation andSynthesis of Living Systems.