Spatial rule-based modeling: a method and its application to the human mitotic kinetochore

Dynamics of spindle assembly and position checkpoints: Integrating molecular mechanisms with computational models

Mitotic checkpoints orchestrate cell division through intricate molecular networks that ensure genomic stability. While experimental research has uncovered key aspects of checkpoint function, the complexity of protein interactions and spatial dynamics necessitates computational modeling for a deeper, system-level understanding. This review explores mathematical frameworks-from ordinary differential equations to stochastic simulations, which reveal checkpoint dynamics across multiple scales, encompassing models ranging from simple protein interactions to whole-system simulations with thousands of parameters. These approaches have elucidated fundamental properties, including bistable switches driving spindle assembly checkpoint (SAC) activation, spatial organization principles underlying spindle position checkpoint (SPOC) signaling, and critical system-level features ensuring checkpoint robustness. This study evaluates diverse modeling approaches, from rule-based models to chemical organization theory, highlighting their successful application in predicting protein localization patterns and checkpoint response dynamics validated through live-cell imaging. Contemporary challenges persist in integrating spatial and temporal scales, refining parameter estimation, and enhancing spatial modeling fidelity. However, recent advances in single-molecule imaging, data-driven algorithms, and machine learning techniques, particularly deep learning for parameter optimization, present transformative opportunities for improving model accuracy and predictive power. By bridging molecular mechanisms with system-level behaviors through validated computational frameworks, this review offers a comprehensive perspective on the mathematical modeling of cell cycle control, with practical implications for cancer research and therapeutic development.

Cell Cycle Complexity: Exploring the Structure of Persistent Subsystems in 414 Models

Background: The regulation of cellular proliferation and genomic integrity is controlled by complex surveillance mechanisms known as cell cycle checkpoints. Disruptions in these checkpoints can lead to developmental defects and tumorigenesis. Methods: To better understand these mechanisms, computational modeling has been employed, resulting in a dataset of 414 mathematical models in the BioModels database. These models vary significantly in detail and simulated processes, necessitating a robust analytical approach. Results: In this study, we apply the chemical organization theory (COT) to these models to gain insights into their dynamic behaviors. COT, which handles both ordinary and partial differential equations (ODEs and PDEs), is utilized to analyze the compartmentalized structures of these models. COT's framework allows for the examination of persistent subsystems within these models, even when detailed kinetic parameters are unavailable. By computing and analyzing the lattice of organizations, we can compare and rank models based on their structural features and dynamic behavior. Conclusions: Our application of the COT reveals that models with compartmentalized organizations exhibit distinctive structural features that facilitate the understanding of phenomena such as periodicity in the cell cycle. This approach provides valuable insights into the dynamics of cell cycle control mechanisms, refining existing models and potentially guiding future research in this area.

MCell4 with BioNetGen: A Monte Carlo simulator of rule-based reaction-diffusion systems with Python interface

Biochemical signaling pathways in living cells are often highly organized into spatially segregated volumes, membranes, scaffolds, subcellular compartments, and organelles comprising small numbers of interacting molecules. At this level of granularity stochastic behavior dominates, well-mixed continuum approximations based on concentrations break down and a particle-based approach is more accurate and more efficient. We describe and validate a new version of the open-source MCell simulation program (MCell4), which supports generalized 3D Monte Carlo modeling of diffusion and chemical reaction of discrete molecules and macromolecular complexes in solution, on surfaces representing membranes, and combinations thereof. The main improvements in MCell4 compared to the previous versions, MCell3 and MCell3-R, include a Python interface and native BioNetGen reaction language (BNGL) support. MCell4's Python interface opens up completely new possibilities for interfacing with external simulators to allow creation of sophisticated event-driven multiscale/multiphysics simulations. The native BNGL support, implemented through a new open-source library libBNG (also introduced in this paper), provides the capability to run a given BNGL model spatially resolved in MCell4 and, with appropriate simplifying assumptions, also in the BioNetGen simulation environment, greatly accelerating and simplifying model validation and comparison.

Computing all persistent subspaces of a reaction-diffusion system

An algorithm is presented for computing a reaction-diffusion partial differential equation (PDE) system for all possible subspaces that can hold a persistent solution of the equation. For this, all possible sub-networks of the underlying reaction network that are distributed organizations (DOs) are identified. Recently it has been shown that a persistent subspace must be a DO. The algorithm computes the hierarchy of DOs starting from the largest by a linear programming approach using integer cuts. The underlying constraints use elementary reaction closures as minimal building blocks to guarantee local closedness and global self-maintenance, required for a DO. Additionally, the algorithm delivers for each subspace an affiliated set of organizational reactions and minimal compartmentalization that is necessary for this subspace to persist. It is proved that all sets of organizational reactions of a reaction network, as already DOs, form a lattice. This lattice contains all potentially persistent sets of reactions of all constrained solutions of reaction-diffusion PDEs. This provides a hierarchical structure of all persistent subspaces with regard to the species and also to the reactions of the reaction-diffusion PDE system. Here, the algorithm is described and the corresponding Python source code is provided. Furthermore, an analysis of its run time is performed and all models from the BioModels database as well as further examples are examined. Apart from the practical implications of the algorithm the results also give insights into the complexity of solving reaction-diffusion PDEs.

Performance Analysis of a Solar-Powered Multi-Purpose Supply Container

In this article, the performance of a solar-powered multi-purpose supply container used as a service module for first-aid, showering, freezing, refrigeration and water generation purposes in areas of social emergency is analyzed. The average daily energy production of the solar panel is compared to the average daily energy demands of the above-mentioned types of service modules. The comparison refers to five different locations based on the Köppen–Geiger classification of climatic zones with the data for energy demand being taken from another publication. It is shown that in locations up to mid-latitudes, the supply container is not only able to power all types of modules all year round but also to provide up to 15 m3 of desalinated water per day for drinking, domestic use and irrigation purposes. This proves and quantifies the possibility of combining basic supply with efficient transport and self-sufficiency by using suitably equipped shipping containers. Thus, flexible solutions are provided to some of the most challenging problems humans will face in the future, such as natural disasters, water scarcity, starvation and homelessness.

Structure and Hierarchy of SARS-CoV-2 Infection Dynamics Models Revealed by Reaction Network Analysis

This work provides a mathematical technique for analyzing and comparing infection dynamics models with respect to their potential long-term behavior, resulting in a hierarchy integrating all models. We apply our technique to coupled ordinary and partial differential equation models of SARS-CoV-2 infection dynamics operating on different scales, that is, within a single organism and between several hosts. The structure of a model is assessed by the theory of chemical organizations, not requiring quantitative kinetic information. We present the Hasse diagrams of organizations for the twelve virus models analyzed within this study. For comparing models, each organization is characterized by the types of species it contains. For this, each species is mapped to one out of four types, representing uninfected, infected, immune system, and bacterial species, respectively. Subsequently, we can integrate these results with those of our former work on Influenza-A virus resulting in a single joint hierarchy of 24 models. It appears that the SARS-CoV-2 models are simpler with respect to their long term behavior and thus display a simpler hierarchy with little dependencies compared to the Influenza-A models. Our results can support further development towards more complex SARS-CoV-2 models targeting the higher levels of the hierarchy.

FluoSim: simulator of single molecule dynamics for fluorescence live-cell and super-resolution imaging of membrane proteins

Fluorescence live-cell and super-resolution microscopy methods have considerably advanced our understanding of the dynamics and mesoscale organization of macro-molecular complexes that drive cellular functions. However, different imaging techniques can provide quite disparate information about protein motion and organization, owing to their respective experimental ranges and limitations. To address these issues, we present here a robust computer program, called FluoSim, which is an interactive simulator of membrane protein dynamics for live-cell imaging methods including SPT, FRAP, PAF, and FCS, and super-resolution imaging techniques such as PALM, dSTORM, and uPAINT. FluoSim integrates diffusion coefficients, binding rates, and fluorophore photo-physics to calculate in real time the localization and intensity of thousands of independent molecules in 2D cellular geometries, providing simulated data directly comparable to actual experiments. FluoSim was thoroughly validated against experimental data obtained on the canonical neurexin-neuroligin adhesion complex at cell-cell contacts. This unified software allows one to model and predict membrane protein dynamics and localization at the ensemble and single molecule level, so as to reconcile imaging paradigms and quantitatively characterize protein behavior in complex cellular environments.

Trends in mathematical modeling of host-pathogen interactions

Pathogenic microorganisms entail enormous problems for humans, livestock, and crop plants. A better understanding of the different infection strategies of the pathogens enables us to derive optimal treatments to mitigate infectious diseases or develop vaccinations preventing the occurrence of infections altogether. In this review, we highlight the current trends in mathematical modeling approaches and related methods used for understanding host-pathogen interactions. Since these interactions can be described on vastly different temporal and spatial scales as well as abstraction levels, a variety of computational and mathematical approaches are presented. Particular emphasis is placed on dynamic optimization, game theory, and spatial modeling, as they are attracting more and more interest in systems biology. Furthermore, these approaches are often combined to illuminate the complexities of the interactions between pathogens and their host. We also discuss the phenomena of molecular mimicry and crypsis as well as the interplay between defense and counter defense. As a conclusion, we provide an overview of method characteristics to assist non-experts in their decision for modeling approaches and interdisciplinary understanding.

Spatial Rule-Based Simulations: The SRSim Software

SRSim combines rule-based reaction network models with spatial particle simulations allowing to simulate the dynamics of large molecular complexes changing according to a set of chemical reaction rules. As the rule can contain patterns of molecular complexes and specific states of certain binding sites, a combinatorially complex or even infinitely sized reaction network can be defined. Particles move in a three-dimensional space according to molecular dynamics implemented by LAMMPS, while the BioNetGen language is used to formulate reaction rules. Geometric information is added in a specific XML format. The simulation protocol is exemplified by two different variants of polymerization as well as a toy model of DNA helix formation. SRSim is open source and available for download.

Mathematical modeling and numerical simulation of the mitotic spindle orientation system

The mitotic spindle orientation and position is crucial for the fidelity of chromosome segregation during asymmetric cell division to generate daughter cells with different sizes or fates. This mechanism is best understood in the budding yeast Saccharomyces cerevisiae, named the spindle position checkpoint (SPOC). The SPOC inhibits cells from exiting mitosis until the mitotic spindle is properly oriented along the mother-daughter polarity axis. Despite many experimental studies, the mechanisms underlying SPOC regulation remains elusive and unexplored theoretically. Here, a minimal mathematical is developed to describe SPOC activation and silencing having autocatalytic feedback-loop. Numerical simulations of the nonlinear ordinary differential equations (ODEs) model accurately reproduce the phenotype of SPOC mechanism. Bifurcation analysis of the nonlinear ODEs reveals the orientation dependency on spindle pole bodies, and how this dependence is altered by parameter values. Partial differential equation (PDEs) model as well as linear stability analysis indicate that diffusion play no major role using experimental high diffusion values. These results provide for systems understanding on the molecular organization of spindle orientation system via mathematical modeling. The presented mathematical model is easy to understand and, within the above mentioned context, can be used as a base for further development of quantitative models in asymmetric cell-division.

Mathematical analysis and modeling of DNA segregation mechanisms

The precise regulation of cell life division is indispensable to the reliable inheritance of genetic material, i.e. DNA, in successive generations of cells. This is governed by dedicated biochemical networks which ensure that all requirements are met before transition from one phase to the next. The Spindle Assembly Checkpoint (SAC) is an evolutionarily mechanism that delays mitotic progression until all chromosomes are properly linked to the mitotic spindle. During some asymmetric cell divisions, such as those observed in budding yeast, an additional mechanism, the Spindle Position Checkpoint (SPOC), is required to delay exit from mitosis until the mitotic spindle is correctly aligned. These checkpoints are complex and their elaborate spatiotemporal dynamics are challenging to understand intuitively. In this study, bistable mathematical models for both activation and silencing of mitotic checkpoints were constructed and analyzed. A one-parameter bifurcation was computed to show the realistic biochemical switches considering all signals. Numerical simulations involving systems of ODEs and PDEs were performed over various parameters, to investigate the effect of the diffusion coefficient. The results provide systems-level insights into mitotic transition and demonstrate that mathematical analysis constitutes a powerful tool for investigation of the dynamic properties of complex biomedical systems.

A Mathematical Framework for Kinetochore-Driven Activation Feedback in the Mitotic Checkpoint

Proliferating cells properly divide into their daughter cells through a process that is mediated by kinetochores, protein-complexes that assemble at the centromere of each sister chromatid. Each kinetochore has to establish a tight bipolar attachment to the spindle apparatus before sister chromatid separation is initiated. The spindle assembly checkpoint (SAC) links the biophysical attachment status of the kinetochores to mitotic progression and ensures that even a single misaligned kinetochore keeps the checkpoint active. The mechanism by which this is achieved is still elusive. Current computational models of the human SAC disregard important biochemical properties by omitting any kind of feedback loop, proper kinetochore signals, and other spatial properties such as the stability of the system and diffusion effects. To allow for more realistic in silico study of the dynamics of the SAC model, a minimal mathematical framework for SAC activation and silencing is introduced. A nonlinear ordinary differential equation model successfully reproduces bifurcation signaling switches with attachment of all 92 kinetochores and activation of APC/C by kinetochore-driven feedback. A partial differential equation model and mathematical linear stability analyses indicate the influence of diffusion and system stability. The conclusion is that quantitative models of the human SAC should account for the positive feedback on APC/C activation driven by the kinetochores which is essential for SAC silencing. Experimental diffusion coefficients for MCC subcomplexes are found to be insufficient for rapid APC/C inhibition. The presented analysis allows for systems-level understanding of mitotic control, and the minimal new model can function as a basis for developing further quantitative-integrative models of the cell division cycle.

In silico spatial simulations reveal that MCC formation and excess BubR1 are required for tight inhibition of the anaphase-promoting complex

In response to the activation of the mitotic spindle assembly checkpoint (SAC), distinct inhibitory pathways control the activity of the anaphase-promoting complex (APC/C). It remains unclear whether the different regulatory mechanisms function in separate pathways or as part of an integrated signalling system. Here, five variant models of APC/C regulation were constructed and analysed. The simulations showed that all variant models were able to reproduce the wild type behaviour of the APC. However, only one model, which included both the mitotic checkpoint complex (MCC) as well as BubR1 as direct inhibitors of the APC/C, was able to reproduce both wild and mutant type behaviour of APC/C regulation. Interestingly, in this model, the MCC as well as the BubR1 binding rate to the APC/C was comparable to the known Cdc20-Mad2 binding rate and could not be made higher. Mad2 active transport towards the spindle mid-zone accelerated the inhibition speed of the APC/C but not its concentration level. The presented study highlights the principle that a systems biology approach is critical for the SAC mechanism and could also be used for predicting hypotheses to design future experiments. The presented work has successfully distinguished between five potent inhibitors of the APC/C using a systems biology approach. Here, the favoured model contains both BubR1 and MCC as direct inhibitors of the APC/C.

Systems Biology Modeling of Five Pathways for Regulation and Potent Inhibition of the Anaphase-Promoting Complex (APC/C): Pivotal Roles for MCC and BubR1

Correct DNA segregation is a fundamental process that ensures the precise and reliable inheritance of genomic information for the propagation of cell life. Eukaryotic cells have evolved a conserved surveillance control mechanism for DNA segregation named the Spindle Assembly Checkpoint (SAC).The SAC ensures that the sister chromatids of the duplicated genome are not separated and distributed to the spindle poles before all chromosomes have been properly linked to the microtubules of the mitotic spindle. Biochemically, the SAC delays cell cycle progression by preventing activation of the anaphase-promoting complex (APC/C) or cyclosome whose activation by Cdc20 is required for sister-chromatid separation; this marks the transition into anaphase. In response to activation of the checkpoint, various species control the activity of both APC/C and Cdc20. However, the underlying regulatory pathways remain largely elusive. In this study, five possible model variants of APC/C regulation were constructed, namely BubR1, Mad2, MCC, MCF2, and an all-pathways model variant. These models were validated with experimental data from the literature. A wide range of parameter values has been tested to find the critical values of the APC/C binding rate. The results show that all variants are able to capture the wild-type behavior of the APC/C. However, only one model variant, which included both MCC as well as BubR1 as potent inhibitors of the APC/C, was able to reproduce both wild-type and mutant type behavior of APC/C regulation. In conclusion, the presented work informs the regulation of fundamental processes such as SAC and APC/C in cell biology and has successfully distinguished between five competing dynamical models using a systems biology approach. The results attest that systems-level approaches are vital for molecular and cell biology.

Spindle assembly checkpoint is sufficient for complete Cdc20 sequestering in mitotic control

The spindle checkpoint assembly (SAC) ensures genome fidelity by temporarily delaying anaphase onset, until all chromosomes are properly attached to the mitotic spindle. The SAC delays mitotic progression by preventing activation of the ubiquitin ligase anaphase-promoting complex (APC/C) or cyclosome; whose activation by Cdc20 is required for sister-chromatid separation marking the transition into anaphase. The mitotic checkpoint complex (MCC), which contains Cdc20 as a subunit, binds stably to the APC/C. Compelling evidence by Izawa and Pines (Nature 2014; 10.1038/nature13911) indicates that the MCC can inhibit a second Cdc20 that has already bound and activated the APC/C. Whether or not MCC per se is sufficient to fully sequester Cdc20 and inhibit APC/C remains unclear. Here, a dynamic model for SAC regulation in which the MCC binds a second Cdc20 was constructed. This model is compared to the MCC, and the MCC-and-BubR1 (dual inhibition of APC) core model variants and subsequently validated with experimental data from the literature. By using ordinary nonlinear differential equations and spatial simulations, it is shown that the SAC works sufficiently to fully sequester Cdc20 and completely inhibit APC/C activity. This study highlights the principle that a systems biology approach is vital for molecular biology and could also be used for creating hypotheses to design future experiments.

Structural analysis of in silico mutant experiments of human inner-kinetochore structure

Large multi-molecular complexes like the kinetochore are lacking of suitable methods to determine their spatial structure. Here, we use and evaluate a novel modeling approach that combines rule-bases reaction network models with spatial molecular geometries. In particular, we introduce a method that allows to study in silico the influence of single interactions (e.g. bonds) on the spatial organization of large multi-molecular complexes and apply this method to an extended model of the human inner-kinetochore. Our computational analysis method encompasses determination of bond frequency, geometrical distances, statistical moments, and inter-dependencies between bonds using mutual information. For the analysis we have extend our previously reported human inner-kinetochore model by adding 13 new protein interactions and three protein geometry details. The model is validated by comparing the results of in silico with reported in vitro single protein deletion experiments. Our studies revealed that most simulations mimic the in vitro behavior of the kinetochore complex as expected. To identify the most important bonds in this model, we have created 39 mutants in silico by selectively disabling single protein interactions. In a total of 11,800 simulation runs we have compared the resulting structures to the wild-type. In particular, this allowed us to identify the interaction Cenp-W-H3 and Cenp-S-Cenp-X as having the strongest influence on the inner-kinetochore's structure. We conclude that our approach can become a useful tool for the in silico dynamical study of large, multi-molecular complexes.

Active transport can greatly enhance Cdc20:Mad2 formation

To guarantee genomic integrity and viability, the cell must ensure proper distribution of the replicated chromosomes among the two daughter cells in mitosis. The mitotic spindle assembly checkpoint (SAC) is a central regulatory mechanism to achieve this goal. A dysfunction of this checkpoint may lead to aneuploidy and likely contributes to the development of cancer. Kinetochores of unattached or misaligned chromosomes are thought to generate a diffusible “wait-anaphase” signal, which is the basis for downstream events to inhibit the anaphase promoting complex/cyclosome (APC/C). The rate of Cdc20:C-Mad2 complex formation at the kinetochore is a key regulatory factor in the context of APC/C inhibition. Computer simulations of a quantitative SAC model show that the formation of Cdc20:C-Mad2 is too slow for checkpoint maintenance when cytosolic O-Mad2 has to encounter kinetochores by diffusion alone. Here, we show that an active transport of O-Mad2 towards the spindle mid-zone increases the efficiency of Mad2-activation. Our in-silico data indicate that this mechanism can greatly enhance the formation of Cdc20:Mad2 and furthermore gives an explanation on how the “wait-anaphase” signal can dissolve abruptly within a short time. Our results help to understand parts of the SAC mechanism that remain unclear.

Multi-state modeling of biomolecules

Multi-state modeling of biomolecules refers to a series of techniques used to represent and compute the behavior of biological molecules or complexes that can adopt a large number of possible functional states. Biological signaling systems often rely on complexes of biological macromolecules that can undergo several functionally significant modifications that are mutually compatible. Thus, they can exist in a very large number of functionally different states. Modeling such multi-state systems poses two problems: the problem of how to describe and specify a multi-state system (the “specification problem”) and the problem of how to use a computer to simulate the progress of the system over time (the “computation problem”). To address the specification problem, modelers have in recent years moved away from explicit specification of all possible states and towards rule-based formalisms that allow for implicit model specification, including the κ-calculus [1], BioNetGen [2]–[5], the Allosteric Network Compiler [6], and others [7], [8]. To tackle the computation problem, they have turned to particle-based methods that have in many cases proved more computationally efficient than population-based methods based on ordinary differential equations, partial differential equations, or the Gillespie stochastic simulation algorithm[9], [10]. Given current computing technology, particle-based methods are sometimes the only possible option. Particle-based simulators fall into two further categories: nonspatial simulators, such as StochSim [11], DYNSTOC [12], RuleMonkey [9], [13], and the Network-Free Stochastic Simulator (NFSim) [14], and spatial simulators, including Meredys [15], SRSim [16], [17], and MCell [18]–[20]. Modelers can thus choose from a variety of tools, the best choice depending on the particular problem. Development of faster and more powerful methods is ongoing, promising the ability to simulate ever more complex signaling processes in the future.