Reducing Boolean networks with backward equivalence.

BACKGROUND: Boolean Networks (BNs) are a popular dynamical model in biology where the state of each component is represented by a variable taking binary values that express, for instance, activation/deactivation or high/low concentrations. Unfortunately, these models suffer from the state space explosion, i.e., there are exponentially many states in the number of BN variables, which hampers their analysis. RESULTS: We present Boolean Backward Equivalence (BBE), a novel reduction technique for BNs which collapses system variables that, if initialized with same value, maintain matching values in all states. A large-scale validation on 86 models from two online model repositories reveals that BBE is effective, since it is able to reduce more than 90% of the models. Furthermore, on such models we also show that BBE brings notable analysis speed-ups, both in terms of state space generation and steady-state analysis. In several cases, BBE allowed the analysis of models that were originally intractable due to the complexity. On two selected case studies, we show how one can tune the reduction power of BBE using model-specific information to preserve all dynamics of interest, and selectively exclude behavior that does not have biological relevance. CONCLUSIONS: BBE complements existing reduction methods, preserving properties that other reduction methods fail to reproduce, and vice versa. BBE drops all and only the dynamics, including attractors, originating from states where BBE-equivalent variables have been initialized with different activation values The remaining part of the dynamics is preserved exactly, including the length of the preserved attractors, and their reachability from given initial conditions, without adding any spurious behaviours. Given that BBE is a model-to-model reduction technique, it can be combined with further reduction methods for BNs.

The current status of gene expression profilings in COVID-19 patients.

Background: The global pandemic of coronavirus disease 2019 (COVID-19) caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has swept through every part of the world. Because of its impact, international efforts have been underway to identify the variants of SARS-CoV-2 by genome sequencing and to understand the gene expression changes in COVID-19 patients compared to healthy donors using RNA sequencing (RNA-seq) assay. Within the last two and half years since the emergence of SARS-CoV-2, a large number of OMICS data of COVID-19 patients have accumulated. Yet, we are still far from understanding the disease mechanism. Further, many people suffer from long-term effects of COVID-19; calling for a more systematic way to data mine the generated OMICS data, especially RNA-seq data. Methods: By searching gene expression omnibus (GEO) using the key terms, COVID-19 and RNA-seq, 108 GEO entries were identified. Each of these studies was manually examined to categorize the studies into bulk or single-cell RNA-seq (scRNA-seq) followed by an inspection of their original articles. Results: The currently available RNA-seq data were generated from various types of patients' samples, and COVID-19 related sample materials have been sequenced at the level of RNA, including whole blood, different components of blood [e.g., plasma, peripheral blood mononuclear cells (PBMCs), leukocytes, lymphocytes, monocytes, T cells], nasal swabs, and autopsy samples (e.g., lung, heart, liver, kidney). Of these, RNA-seq studies using whole blood, PBMCs, nasal swabs and autopsy/biopsy samples were reviewed to highlight the major findings from RNA-seq data analysis. Conclusions: Based on the bulk and scRNA-seq data analysis, severe COVID-19 patients display shifts in cell populations, especially those of leukocytes and monocytes, possibly leading to cytokine storms and immune silence. These RNA-seq data form the foundation for further gene expression analysis using samples from individuals suffering from long COVID.

Exact maximal reduction of stochastic reaction networks by species lumping.

MOTIVATION: Stochastic reaction networks are a widespread model to describe biological systems where the presence of noise is relevant, such as in cell regulatory processes. Unfortunately, in all but simplest models the resulting discrete state-space representation hinders analytical tractability and makes numerical simulations expensive. Reduction methods can lower complexity by computing model projections that preserve dynamics of interest to the user. RESULTS: We present an exact lumping method for stochastic reaction networks with mass-action kinetics. It hinges on an equivalence relation between the species, resulting in a reduced network where the dynamics of each macro-species is stochastically equivalent to the sum of the original species in each equivalence class, for any choice of the initial state of the system. Furthermore, by an appropriate encoding of kinetic parameters as additional species, the method can establish equivalences that do not depend on specific values of the parameters. The method is supported by an efficient algorithm to compute the largest species equivalence, thus the maximal lumping. The effectiveness and scalability of our lumping technique, as well as the physical interpretability of resulting reductions, is demonstrated in several models of signaling pathways and epidemic processes on complex networks. AVAILABILITY AND IMPLEMENTATION: The algorithms for species equivalence have been implemented in the software tool ERODE, freely available for download from https://www.erode.eu. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Maximal aggregation of polynomial dynamical systems.

Ordinary differential equations (ODEs) with polynomial derivatives are a fundamental tool for understanding the dynamics of systems across many branches of science, but our ability to gain mechanistic insight and effectively conduct numerical evaluations is critically hindered when dealing with large models. Here we propose an aggregation technique that rests on two notions of equivalence relating ODE variables whenever they have the same solution (backward criterion) or if a self-consistent system can be written for describing the evolution of sums of variables in the same equivalence class (forward criterion). A key feature of our proposal is to encode a polynomial ODE system into a finitary structure akin to a formal chemical reaction network. This enables the development of a discrete algorithm to efficiently compute the largest equivalence, building on approaches rooted in computer science to minimize basic models of computation through iterative partition refinements. The physical interpretability of the aggregation is shown on polynomial ODE systems for biochemical reaction networks, gene regulatory networks, and evolutionary game theory.