MOTIVATION: Analysis of metabolic networks through extreme rays such as extreme pathways and elementary flux modes has been shown to be effective for many applications. However, due to the combinatorial explosion of candidate vectors, their enumeration is currently limited to small- and medium-scale networks (typically <200 reactions). Partial enumeration of the extreme rays is shown to be possible, but either relies on generating them one-by-one or by implementing a sampling step in the enumeration algorithms. Sampling-based enumeration can be achieved through the canonical basis approach (CBA) or the nullspace approach (NSA). Both algorithms are very efficient in medium-scale networks, but struggle with elementarity testing in sampling-based enumeration of larger networks. RESULTS: In this paper, a novel elementarity test is defined and exploited, resulting in significant speedup of the enumeration. Even though NSA is currently considered more effective, the novel elementarity test allows CBA to significantly outpace NSA. This is shown through two case studies, ranging from a medium-scale network to a genome-scale metabolic network with over 600 reactions. In this study, extreme pathways are chosen as the extreme rays, but the novel elementarity test and CBA are equally applicable to the other types. With the increasing complexity of metabolic networks in recent years, CBA with the novel elementarity test shows even more promise as its advantages grows with increased network complexity. Given this scaling aspect, CBA is now the faster method for enumerating extreme rays in genome-scale metabolic networks. AVAILABILITY AND IMPLEMENTATION: All case studies are implemented in Python. The codebase used to generate extreme pathways using the different approaches is available at https://gitlab.kuleuven.be/biotec-plus/pos-def-ep.

Individual-based modeling (IbM) is an instrumental tool for simulating spatial microbial growth, with applications in both microbial ecology and biochemical engineering. Unlike Cellular Automata (CA), which use a fixed grid of cells with predefined rules for interactions, IbMs model the individual behaviors of cells, allowing complex population dynamics to emerge. IbMs require more detailed modeling of individual interactions, which introduces significant computational challenges, particularly in resolving spatial overlaps between cells. Traditionally, this is managed using arrays or kd-trees, which require numerous pairwise comparisons and become inefficient as population size increases. To address this bottleneck, we introduce the Discretized Overlap Resolution Algorithm (DORA), which employs a grid-based framework to efficiently manage overlaps. By discretizing the simulation space further and assigning circular cells to specific grid units, DORA transforms the computationally intensive pairwise comparison process into a more efficient grid-based operation. This approach significantly reduces the computational load, particularly in simulations with large cell populations. Our evaluation of DORA, through simulations of microbial colonies and biofilms under varied nutrient conditions, demonstrates its superior computational efficiency and ability to accurately capture microbial growth dynamics compared to conventional methods. DORA's grid-based strategy enables the modeling of densely populated microbial communities within practical computational timeframes, thereby expanding the scope and applicability of individual-based modeling.

Various machine learning problems rely on kernel-based methods. The power of these methods resides in the ability to solve highly nonlinear problems by reformulating them in a linear context. The dominant eigenspace of a (normalized) kernel matrix is often required. Unfortunately, the computational requirements of the existing kernel methods are such that the applicability is restricted to relatively small data sets. This letter therefore focuses on a kernel-based method for large data sets. More specifically, a numerically stable tracking algorithm for the dominant eigenspace of a normalized kernel matrix is proposed, which proceeds by an updating (the addition of a new data point) followed by a downdating (the exclusion of an old data point) of the kernel matrix. Testing the algorithm on some representative case studies reveals that a very good approximation of the dominant eigenspace is obtained, while only a minimal amount of operations and memory space per iteration step is required.