Additive Functions in Boolean Models of Gene Regulatory Network Modules

Gene-on-gene regulations are key components of every living organism. Dynamical abstract models of genetic regulatory networks help explain the genome’s evolvability and robustness. These properties can be attributed to the structural topology of the graph formed by genes, as vertices, and regulatory interactions, as edges. Moreover, the actual gene interaction of each gene is believed to play a key role in the stability of the structure. With advances in biology, some effort was deployed to develop update functions in Boolean models that include recent knowledge. We combine real-life gene interaction networks with novel update functions in a Boolean model. We use two sub-networks of biological organisms, the yeast cell-cycle and the mouse embryonic stem cell, as topological support for our system. On these structures, we substitute the original random update functions by a novel threshold-based dynamic function in which the promoting and repressing effect of each interaction is considered. We use a third real-life regulatory network, along with its inferred Boolean update functions to validate the proposed update function. Results of this validation hint to increased biological plausibility of the threshold-based function. To investigate the dynamical behavior of this new model, we visualized the phase transition between order and chaos into the critical regime using Derrida plots. We complement the qualitative nature of Derrida plots with an alternative measure, the criticality distance, that also allows to discriminate between regimes in a quantitative way. Simulation on both real-life genetic regulatory networks show that there exists a set of parameters that allows the systems to operate in the critical region. This new model includes experimentally derived biological information and recent discoveries, which makes it potentially useful to guide experimental research. The update function confers additional realism to the model, while reducing the complexity and solution space, thus making it easier to investigate

Genetic Regulatory Networks.

<p>A representation of (a) the transcriptional regulatory network in ES cells and (b) the yeast cell-cycle regulatory network. Arrows point from transcription factor to the target gene. Signs (respectively ) represent activating (respectively repressing) links.</p

Derrida plots of RUFs for ES cell.

<p>(a) (curves for are not reported as RUF rules are symmetrical), and (b) only values close to the critical gene expression value are investigated with refinement steps of . Please note the two different scales in the axes.</p

Derrida plots of RUFs for yeast cell.

<p>(a) (curves for are not reported as RUF rules are symmetrical), and (b) only values close to the critical gene expression value are investigated with refinement steps of . Please note the two different axes scales in the figures.</p

Critical threshold.

<p>Derrida plots and Criticality Distances of Activator Driven Additive functions for the yeast cell-cycle regulatory network.</p

Attractor space analysis.

<p>Numerical simulation results for the ES model. (a) Attractors (average) number, (b) attractors average lengths, and (c) the average basin of attraction size. The statistics are computed on samples of RUF systems, hence the standard error bars, and exhaustively on ADA systems.</p

Regime of the ABA model.

<p>Derrida plot of the simplified ABA model with the original real-life update functions.</p

Real-life networks critical values.

<p>For systems under RUF, we show the function's gene expression probability values for all three regimes, for both ES cells and Yeast cell-cycle. In the case of ADA, we give threshold values also for all three regimes and both studied networks.</p

Derrida plot.

<p>Derrida plot of the original RBN model (see text).</p