The reverse engineering problem with probabilities and sequential behavior: Probabilistic Sequential Networks

The reverse engineering problem with probabilities and sequential behavior is introducing here, using the expression of an algorithm. The solution is partially founded, because we solve the problem only if we have a Probabilistic Sequential Network. Therefore the probabilistic structure on sequential dynamical systems is introduced here, the new model will be called Probabilistic Sequential Network, PSN. The morphisms of Probabilistic Sequential Networks are defined using two algebraic conditions, whose imply that the distribution of probabilities in the systems are close. It is proved here that two homomorphic Probabilistic Sequential Networks have the same equilibrium or steady state probabilities. Additionally, the proof of the set of PSN with its morphisms form the category PSN, having the category of sequential dynamical systems SDS, as a full subcategory is given. Several examples of morphisms, subsystems and simulations are given.

💡 Research Summary

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The paper introduces a novel mathematical framework called Probabilistic Sequential Networks (PSN) that merges the probabilistic aspects of Probabilistic Boolean Networks (PBN) with the ordered update mechanism of Sequential Dynamical Systems (SDS). The authors begin by reviewing the two existing paradigms: PBNs model gene regulatory networks with synchronous updates and a set of predictor functions each equipped with a selection probability, while SDSs capture systems where each vertex has a local update function and a global update order given by a permutation of vertices. Neither framework alone can simultaneously represent sequential behavior and stochastic selection, which are both essential in many biological contexts.

To address this gap, the authors define a PSN as a tuple (D = (\Gamma, {F_a}{a=1}^n, (k_a){a=1}^n, (\alpha_j)_{j=1}^m, C)) where:

1. (\Gamma) is a finite directed graph with vertices (V_\Gamma = {1,\dots,n}).
2. Each vertex (a) carries a finite state set (k_a).
3. For each vertex a, a family (F_a = {f_{a,i}: k^n \to k^n\mid 1\le i\le \ell(a)}) of local functions is given. Different functions may be chosen with different probabilities.
4. A collection of permutations (\alpha_j) specifies possible sequential update orders.
5. A probability vector (C = {c_1,\dots,c_s}) assigns a weight to each global update function that results from picking one local function per vertex and applying the chosen permutation.

A global update function is formed by composing the selected local functions according to a permutation, e.g. (f = f_{\alpha_1,i_1}\circ\cdots\circ f_{\alpha_n,i_n}). The set of all such functions together with the probability vector defines a discrete‑time Markov chain on the state space (k^n). Thus, PSNs inherit the steady‑state analysis tools of PBNs while preserving the ordered dynamics of SDSs.

The central theoretical contribution is the definition of morphisms between PSNs. A morphism consists of a graph homomorphism (\phi:\Delta\to\Gamma) together with a family of state‑maps (\widehat\phi_b : k_{\phi(b)}\to k_b). These induce a global map (h_\phi : k^n\to k^m) that respects the local structure. Two algebraic conditions are required:

• Compatibility of the update schedules: for each permutation (\alpha_i) in the source, the pre‑image (\phi^{-1}(\alpha_i)) must correspond to a block of permutations in the target, and the corresponding compositions must commute with (h_\phi).
• Commutativity of the global update functions with (h_\phi) as expressed in equations (2) and (3) of the paper.

Theorem 5.2 shows that these algebraic constraints automatically guarantee a probabilistic closeness: the distribution of transition probabilities in the source PSN is mapped to the distribution in the target PSN via (h_\phi). Theorem 5.3 proves a stronger result: homomorphic PSNs share exactly the same stationary distribution (steady‑state probabilities). Consequently, one can replace a complex PSN by a simpler, homomorphic one without altering its long‑run behavior—a powerful reduction technique for reverse‑engineering large regulatory networks.

From a categorical perspective, the authors construct the category PSN whose objects are PSNs and whose morphisms are the homomorphisms defined above. They demonstrate that the category of SDSs embeds as a full subcategory of PSN, confirming that PSNs truly generalize SDSs. This categorical embedding also clarifies how existing results for SDSs (e.g., composition of update functions, invariant subspaces) extend naturally to the probabilistic setting.

The paper includes an explicit algorithm for solving the reverse‑engineering problem with probabilities and sequential behavior. The algorithm takes as input:

1. The number of entities and their state sets.
2. A binary relation matrix indicating interactions.
3. Time‑series data for one or more update functions.
4. Empirical probabilities for each observed function.

It then constructs the low‑level interaction graph, derives local functions for each vertex, selects a permutation order, assigns probabilities to each update function, and finally builds the high‑level transition digraph. A concrete example with three binary nodes illustrates each step, showing how two distinct update families and their associated probabilities lead to a concrete PSN.

The authors also provide several illustrative examples of morphisms, subsystems, and simulations, demonstrating how a large PSN can be projected onto a smaller one while preserving dynamics, or how two PSNs can simulate each other.

Strengths:

• The integration of stochastic selection and ordered updates addresses a genuine gap in the modeling of gene regulatory networks.
• The morphism framework and the categorical formulation give a solid algebraic foundation, enabling systematic model reduction.
• Theorems linking homomorphism to identical stationary distributions are both elegant and practically useful for reverse engineering.
• The algorithmic pipeline is clearly described and grounded in a realistic data‑driven scenario.

Weaknesses / Open Issues:

• The paper suffers from numerous typographical and formatting errors that occasionally obscure definitions (e.g., inconsistent notation for permutations and indices).
• Proofs of the main theorems are sketched rather than fully detailed, making it difficult to verify the rigor of the probabilistic closeness claim.
• Computational complexity of the algorithm is not analyzed; for large‑scale networks the number of possible local functions and permutations grows factorially, which may limit practical applicability.
• No empirical evaluation on real biological data is presented; all examples are toy models, leaving open the question of scalability and robustness to noisy measurements.
• The treatment of multiple update schedules (the set ({\alpha_j})) is somewhat informal; a more systematic handling of schedule selection would strengthen the framework.

Potential Impact:
If further developed, PSNs could become a standard tool for reverse‑engineering gene regulatory networks where both timing (sequential activation) and stochasticity (noise, probabilistic binding) matter. The homomorphism‑based reduction could dramatically cut computational costs for steady‑state analysis, enabling the study of larger networks than currently feasible with PBNs alone. Moreover, the categorical viewpoint may inspire new compositional modeling approaches, where complex networks are built from smaller, well‑understood PSN modules.

In summary, the paper proposes a mathematically coherent extension of SDSs that incorporates probabilistic dynamics, defines a robust notion of morphism, proves that homomorphic networks share steady‑state behavior, and situates the construction within category theory. While the exposition needs polishing and empirical validation, the core ideas constitute a valuable contribution to the theory of discrete dynamical systems and their application to biological reverse engineering.