Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Robustness of Boolean networks to update modes: an application to hereditary angioedema

Jacques Demongeot, Eric Goles, Houssem ben Khalfallah, Marco Montalva-Medel, Sylvain Sené

TL;DR

The paper analyzes how update schedules shape Boolean regulatory networks modeling hereditary angioedema, introducing intricate update modes to extend block-parallel updates. By leveraging update-digraph theory, it reduces the analysis to representative modes and demonstrates that the simplified GRN $\Psi$ has no fixed points across all block-sequential dynamics, with attractors restricted to cycles of lengths $2,3,4,$ and $6$. An illustrative intricate update mode yields a unique length-$4$ cycle, showing that such modes can capture the dominant cyclic behavior and potentially model clock-driven gene updating (Zeitgebers). These results support the structural robustness of the disease-relevant network under diverse update schedules and provide a framework for exploring timing mechanisms in genetic regulation. The findings have practical relevance for understanding periodic gene expression in hereditary angioedema and for modeling regulatory dynamics when the updating clock is uncertain.

Abstract

Many familial diseases are caused by genetic accidents, which affect both the genome and its epigenetic environment, expressed as an interaction graph between the genes as that involved in one familial disease we shall study, the hereditary angioedema. The update of the gene states at the vertices of this graph (1 if a gene is activated, 0 if it is inhibited) can be done in multiple ways, well studied over the last two decades: parallel, sequential, block-sequential, block-parallel, random, etc. We will study a particular graph, related to the familial disease proposed as an example, which has subgraphs which activate in an intricate manner (\emph{i.e.}, in an alternating block-parallel mode, with one core constantly updated and two complementary subsets of genes alternating their updating), of which we will study the structural aspects, robust or unstable, in relation to some classical periodic update modes.

Robustness of Boolean networks to update modes: an application to hereditary angioedema

TL;DR

The paper analyzes how update schedules shape Boolean regulatory networks modeling hereditary angioedema, introducing intricate update modes to extend block-parallel updates. By leveraging update-digraph theory, it reduces the analysis to representative modes and demonstrates that the simplified GRN has no fixed points across all block-sequential dynamics, with attractors restricted to cycles of lengths and . An illustrative intricate update mode yields a unique length- cycle, showing that such modes can capture the dominant cyclic behavior and potentially model clock-driven gene updating (Zeitgebers). These results support the structural robustness of the disease-relevant network under diverse update schedules and provide a framework for exploring timing mechanisms in genetic regulation. The findings have practical relevance for understanding periodic gene expression in hereditary angioedema and for modeling regulatory dynamics when the updating clock is uncertain.

Abstract

Many familial diseases are caused by genetic accidents, which affect both the genome and its epigenetic environment, expressed as an interaction graph between the genes as that involved in one familial disease we shall study, the hereditary angioedema. The update of the gene states at the vertices of this graph (1 if a gene is activated, 0 if it is inhibited) can be done in multiple ways, well studied over the last two decades: parallel, sequential, block-sequential, block-parallel, random, etc. We will study a particular graph, related to the familial disease proposed as an example, which has subgraphs which activate in an intricate manner (\emph{i.e.}, in an alternating block-parallel mode, with one core constantly updated and two complementary subsets of genes alternating their updating), of which we will study the structural aspects, robust or unstable, in relation to some classical periodic update modes.

Paper Structure

This paper contains 18 sections, 4 theorems, 14 equations, 11 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Mathematical context
  3. Definitions
  4. Basic notations
  5. Boolean networks
  6. Block-sequential update modes
  7. Block-parallel update modes
  8. Intricate update modes
  9. Threshold Boolean automata network
  10. Attractors
  11. Robustness
  12. Network dynamics
  13. Results and discussion
  14. Genetic regulatory network of hereditary angioedema
  15. Robustness to block-sequential update modes
  16. ...and 3 more sections

Key Result

Theorem 1

A labeled digraph is an update digraph if and only if changing the direction of only its arcs with $<$-labels results in a new digraph (possibly a multidigraph) that does not contain any cycle with a $<$-edge.

Figures (11)

  • Figure 1: Boolean automata network $f$ of Example \ref{['ex:3pos-cycle']} on page \ref{['ex:3pos-cycle']}; (a) its definition by means of functions; (b) its associated interaction graph.
  • Figure 2: Different periodic dynamics of the Boolean automata network of Example \ref{['ex:3pos-cycle']} on page \ref{['ex:3pos-cycle']} defined in Figure \ref{['fig:3pos-cycle']}; (a) its parallel dynamics according to $\mu_\textsc{par} = (\{1, 2, 3\})$; (b) its block-sequential dynamics according to $\mu_\textsc{bs} = (\{1, 2\}, \{3\})$; (c) its block-parallel dynamics according to $\mu_\textsc{bp} = \{(1), (3, 2)\} \equiv (\{1, 2\}, \{1, 3\})$; (d) its intricate dynamics according to $\mu_\textsc{in} = (\{2, 3\}, \{1, 3\}, \{1,2\})$.
  • Figure 3: (a) Boolean automata network $f$ of Example \ref{['ex:UD']}; (b) its associated interaction graph $G = (V,E)$, where the set of vertices is $V = \{1, 2, 3\}$ and the set of edges is $E = \{(1,2), (2,1), (2,3)\}$.
  • Figure 4: All the update digraphs associated to the digraph $G$ of Figure \ref{['fig:3digraph']} at stake in Example \ref{['ex:UD']}, where (a) is obtained with $s_1$ and $s_7$, (b) is obtained with $s_2$, (c) is obtained with $s_3$, $s_8$, $s_9$ and $s_{10}$, (d) is obtained with $s_4$ and $s_{11}$, (e) is obtained with $s_5$, and (f) is obtained with $s_6$, $s_{12}$ and $s_{13}$.
  • Figure 5: The six distinct possible block-sequential dynamics of the Boolean automata network of Example \ref{['ex:UD']}, where (a), (b), (c), (d), (e) and (f) respectively derive from update modes equivalence classes $[s_1]_G$, $[s_2]_G$, $[s_4]_G$, $[s_5]_G$, $[s_6]_G$ and $[s_8]_G$ defined in Table \ref{['tab:UDeq']}.
  • ...and 6 more figures

Theorems & Definitions (8)

  • Example 1
  • Theorem 1
  • Theorem 2
  • Example 2
  • Example 3
  • Proposition 1
  • proof
  • Proposition 2