Percolation in bipartite Boolean networks and its role in sustaining life

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Boolean networks are popular models for gene regulation, where
genes are regarded as binary units, that can be either expressed or not, each
updated at regular time intervals according to a random Boolean function of
its neighbouring genes. Stable gene expression profiles, corresponding to cell
types, are regarded as attractors of the network dynamics. However, the random
character of gene updates does not allow to link explicitly the existence of
attractors to the biological mechanism with which genes interact. We propose
a bipartite Boolean network approach which integrates genes and regulatory
proteins (i.e. transcription factors) into a single network, where interactions
incorporate two fundamental aspects of cellular biology, i.e. gene expression
and gene regulation, and the resulting dynamics is highly non-linear. Since any
finite stochastic system is ergodic, the emergence of an attractor structure, stable
under noisy conditions, requires a giant component in the bipartite graph. By
adapting graph percolation techniques to directed bipartite graphs, we are able
to calculate exactly the region, in the network parameters space, where a cell can
sustain steady-state gene expression profiles, in the absence of inhibitors, and we
quantify numerically the effect of inhibitors. Results show that for cells to sustain
a steady-state gene expression profile, transcription factors should typically be
small protein complexes that regulate many genes. This condition is crucial for
cell reprogramming and remarkably well in line with biological facts.
Original languageEnglish
Article number334002
JournalJournal Of Physics A-Mathematical And Theoretical
Issue number33
Publication statusPublished - 24 Jul 2019


  • Asymmetric neural networks
  • Bipartite graphs
  • Boolean networks
  • Bootstrap
  • Cavity method
  • Gene-regulatory networks
  • Percolation


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