Refining Boolean models with the partial most permissive scheme

TL;DR

Boolean models may miss finer dynamical properties that multivalued refinements can capture. MRBM introduces a partial most-permissive updating scheme to locate minimal multivalued components whose refinement yields the desired reachability under asynchronous dynamics, with verification by model checking and basin analysis. The method is demonstrated on a toy model and two stem cell differentiation BMs, uncovering refinements that restore reachabilities and match basin sizes predicted by more permissive dynamics. This approach provides a systematic, data-guided way to enrich Boolean networks with multivalued detail, enabling more accurate qualitative predictions when experimental parameters are limited.

Abstract

Motivation: In systems biology, modelling strategies aim to decode how molecular components interact to generate dynamical behaviour. Boolean modelling is more and more used, but the description of the dynamics from two-levels components may be too limited to capture certain dynamical properties. %However, in Boolean models, the description of the dynamics may be too limited to capture certain dynamical properties. Multivalued logical models can overcome this limitation by allowing more than two levels for each component. However, multivaluing a Boolean model is challenging. Results: We present MRBM, a method for efficiently identifying the components of a Boolean model to be multivalued in order to capture specific fixed-point reachabilities in the asynchronous dynamics. To this goal, we defined a new updating scheme locating reachability properties in the most permissive dynamics. MRBM is supported by mathematical demonstrations and illustrated on a toy model and on two models of stem cell differentiation.

Figures (6)

• Figure 1: Toy Boolean Model (BM). ( A) Regulatory graph associated with a BM $f$ composed of three components $g_1, g_2, g_3$, five regulatory interactions (red edges for inhibition, green edges for activation), and logical regulatory functions (operators $"\&", "|","!"$ stand for and, or, not respectively). The values of the logical regulatory function $f_i(x) \in \mathbb{B}$ provide the target levels of the $i$-th component. ( B) State Transition Graph (STG) representing the Toy model's dynamics under the asynchronous scheme. Each node represents a state $x = (x_1, x_2, x_3)$ of the model and each edge represents a transition between two consecutive states. The attractors (fixed points) are colored in blue. ( C) Most permissive and ( D) Partial most permissive applied on gene $g_1$ ( $^{g_1}$m.p. ) STG. For the sake of visualization, we do not show the m.p. states. The trajectories from the state $010$ to the state $011$ are highlighted in red.
• Figure 2: Multivalued Refinement of the Toy model. (A) Regulatory graph of the multivalued refinement of the BM of Figure \ref{['fig:Toy']}A, and associated logical functions. The rectangle represents the multivalued component $g_1$ ($m_1 = 2$); outgoing edges from $g_1$ have labels that specify the threshold levels from which the regulations occur. (B) Asynchronous STG of the multivalued refinement $h$. Each node represents a state $(x_1, x_2, x_3)$. Trajectories representing the reachability $010 \mathrel{\stackunder[2pt]{\stackon[4pt]{$$}{$h$}}{$asyn.$}}011$ are highlighted in red.
• Figure 3: Boolean Model of Early Hematopoïetic Stem Cell Differentiation. (A) Regulatory graph of the BM (edges in red represent inhibitions and edges in green activations). The logical rules are detailed in herault_novel_2023. (B) Configuration (i.e. activity level of each component) of the 5 fixed points and the 5 transient key states (in column). Each cell of the table represents the activity level of a component (0=inactive, in white, 1 = active in dark green, * = 0 or 1, in light green). (C) Reachabilities of the fixed points (in column) from transient states (in row) in m.p. and asynchronous ("asyn) dynamics. ✓ stands for "reachable", and ✗ for "not reachable". The cells are divided if there is a difference between the m.p. and the asynchronous dynamics. (D) Reachabilities of the fixed points from transient states (in row) in the partial $^J$m.p. dynamics. The accessibility of transient states at fixed points in the partial dynamics of $^J$m.p. is satisfied for the $J$ sets specified in the table cell.
• Figure 4: Subgraph of a Multivalued Refinement of the BM of Early HSC Aging. Multivalued components are represented as rectangles, while other components are depicted as ovals. The label on arrows indicates the threshold level from which the regulation by a multivalued component occurs.
• Figure 5: Boolean model of the Asymmetric Stem Cell Division in Arabidopsis Thaliana Root. (A) Regulatory graph of the BM; the logical functions are available in garcia-gomez_system-level_2020. (B) Summary of the 6 fixed points. Each row represents a component, each column corresponds to a fixed point, and each cells to the activity level ($1$ for active, in dark green, $0$ for inactive, in white). (C) Sizes (in percentage) of the basins of attraction of the BM under different updating schemes. The rows represent basins of attraction and the columns correspond to the updating schemes: asynchronous ("Asyn"), m.p. and partial m.p. . For each partial $^J$m.p. dynamics, the contributing set $J$ is indicated in the column headers. We grouped in the same column the partial $^J$m.p. dynamics leading to same sizes of basin of attraction. We only report the smallest set(s) of m.p. components resulting in a specific size of basin of attraction. Values in red correspond to increases in the sizes of the basin of attraction as compared to the sizes obtained in the asynchronous dynamics.

Theorems & Definitions (10)

• Proposition 1
• proof
• Remark 1
• Proposition 2
• proof
• Remark 2
• Theorem 1
• proof
• Theorem 2
• proof