Antimicrobial resistance (AMR) is a critical global health challenge that is increasingly being addressed through quantitative and systems-level approaches. Beyond evolutionary genetics and mutational adaptation, bacterial survival under antibiotics also reflects physiological state transitions and metabolic constraints. We review mathematical models across scales, from population-level approaches to dose optimization, multidrug therapy, community effects, and global epistasis to cellular frameworks based on proteome partitioning and resource allocation. These coarse-grained models reveal how metabolic constraints shape antibiotic action, genetically encoded resistance, and non-genetic tolerance or persistence, particularly for ribosome inhibitors and increasingly for other bacteriostatic and bactericidal drugs. They also identify nonlinear behaviors, including bistability, threshold effects, and regime-dependent resistance strategies. Despite the substantial gap between laboratory models and clinical application, this multiscale framework clarifies the interplay among bacterial metabolism, population dynamics, and antibiotic action. We discuss the achievements and limitations of current approaches and the challenges that must be overcome for clinical translation.
During early development, embryos establish a coordinate system along the head-to-tail, belly-to-back, and left-to-right axes. This process – gastrulation – transforms a uniform group of cells into an elongated, multi-layered body-plan. Although gastrulation differs dramatically between species, remarkable overlaps exist in the overarching principles as well as the underlying mechanisms. Here we review recent studies that use a combined experimental and in silico approach to identify and quantify the mechanisms orchestrating gastrulation across amniotes. From this literature, spanning diverse experimental model systems and in silico frameworks, three self-organising principles recur: (1) initial symmetry breaking; (2) refinement of this broken symmetry into a full head-tail axis; and (3) patterning of pluripotent cells into three distinct germ layers positioned relative to this axis. With the focus on interdisciplinary concepts, we provide an overview of how combining modelling with experiments recently has expanded our understanding of gastrulation and point at potential new directions.
Recent years have seen a convergence between metabolic and whole-cell models, driven by advances that incorporate intracellular component concentrations into metabolic modeling frameworks. Constraint-based models have been extended to include enzyme levels and gene expression machinery, while kinetic models explicitly account for enzyme and metabolite concentrations through nonlinear rate laws. In parallel, coarse-grained self-replicator models have unified these elements within nonlinear optimization frameworks that also capture dilution by growth. Different streams of metabolic modeling are thus converging toward a new class of nonlinear resource allocation models, integrating kinetic rate laws, protein synthesis, and dilution by growth of all model components. These next-generation models are potentially closer to whole-cell models in predictive capacity, while remaining substantially simpler and therefore more readily applicable in research and engineering because of their lower curation demands. In this review, we summarize recent developments in metabolic modeling and outline the prospects for a next generation of metabolic models that integrates these advances.
Current drug discovery and development approaches rely heavily on cell line models, which often limit the translation of discoveries into novel therapeutics. Recently, patient-derived organoids (PDOs) have been developed that more closely resemble disease tissues. To facilitate drug discovery, in vitro expanded patient-derived organoids can be exposed to hundreds of compounds to assess drug effects. These organoids can be analyzed in detail using high-content imaging. Imaging screening generates multi-parametric data at the single cell or organoid level and can provide insights into patient-specific drug effects and drug mode of action. Additionally, image-based approaches benefit co-culture assays by distinguishing perturbation effects on different cell types. Challenges, such as quality control, automation, and data analysis, will increase as organoid models become more complex to reconstitute the in vivo tissue microenvironment, for example, when combining tumor and stromal cells to model drug action. In this review, we discuss the state of the art in using image-based profiling of organoids to drive innovation in drug discovery.
Impairment of adenosine triphosphate (ATP) production and dysregulation of bioenergetic pathways endanger cellular viability and are linked to various diseases. Pyruvate is a critical branching point of the central carbon metabolism and enters the mitochondrial energy production pathway via pyruvate dehydrogenase (PDH). This enzyme is regulated by calcium (Ca 2+ ), a key signaling messenger that modulates cellular function and energy production. However, the mechanisms connecting the Ca 2+ regulation of PDH to metabolic remodeling and disease progression remain poorly understood. Systems biology offers a powerful framework to model dynamical processes of cell bioenergetics and to understand mechanistic adaptations that may lead to a cellular ATP deficit. Here, we review modeling approaches for PDH activation and analyze the potential of Ca 2+ to maintain cellular homeostasis by metabolic control through PDH.
Pattern formation is a widespread phenomenon in many biological processes; yet, some of the principles governing pattern emergence and maintenance remain elusive. Here, we outline key mechanisms governing pattern formation of cell fates, where the intrinsic discreteness of cells requires treating them as distinct, distinguishable units. We first highlight that such patterns involve mechanisms and properties unique to cellular discreteness, posing challenges that differ fundamentally from those in continuum descriptions. We then address common aspects of pattern formation in both continuous and discrete tissues, discussing the emerging view of considering patterns as transient, dynamic processes rather than stationary states. Finally, we introduce the notion of self-organized multistability, a phenomenon in which a system spontaneously organizes in the vicinity of a multistable pattern regime and suggest it can confer plasticity and adaptability to patterns, with potential significance for understanding the fundamental principles of pattern formation.
Digital twins (DTs) represent the logical extension of systems biology modelling toward clinical application; however, current implementations are predominantly static, organ specific, and limited to single-instance personalisation. This review identifies five primary barriers impeding the development of adaptive, body-scale DTs: (1) parameter explosion driven by complexity, (2) weak structural identifiability, (3) lack of bidirectional multiscale coupling, (4) challenges in integrating heterogeneous asynchronous data, and (5) computational intractability for real-time applications. While each barrier has partial solutions, none is fully resolved. Hybrid mechanistic learning frameworks, such as neural ordinary differential equations, can uncover unknown couplings while maintaining interpretability. Modular architectures help manage complexity and identifiability, and uncertainty quantification supports riskaware clinical reasoning. Agentic systems offer a promising, though still exploratory, avenue that requires further development before clinical deployment. Progress in this field requires three key shifts: prioritising modular, identifiable models over encyclopaedic coverage; integrating mechanistic modelling with data-driven methods; and designing DTs as adaptive reasoning tools that support, rather than replace, clinical evaluation.
Alzheimer's disease is characterized by the accumulation of (3amyloid (A(3) peptides, with oligomeric forms exerting particularly toxic effects through disruption of neuronal Ca2+ homeostasis. The reciprocal interactions between A(3 and Ca2+ create a complex feedback loop with wide-ranging consequences for neuronal signaling, metabolism, and survival. Computational modeling offers a powerful means to disentangle these nonlinear processes yet its impact on experimental research has remained limited. This review synthesizes and highlights the contribution of current modeling approaches addressing A(3/ Ca2+ dysregulation. The aim is to clarify the hypotheses and objectives of each of them to underscore their relevance for guiding experimental investigations.
Cell fate decisions can often be understood as bifurcations in gene regulatory networks where cells cross critical thresholds to commit to distinct developmental trajectories. Comparable retinal injury stimuli trigger regeneration in zebrafish but quiescence in mammalian M & uuml;ller glia despite conserved orthologous genetic components. We propose that this species difference reflects network topology rather than gene sequence alone, with blocking regulators preventing bifurcation crossing in mammals. The core regenerative circuit is a bistable switch: Ascl1 activates Lin28a, which suppresses let-7 microRNAs; since let-7 represses both Ascl1 and c-Myc, its removal locks the system irreversibly in the regenerative state. In zebrafish, four conditions enable sustained Ascl1 elevation sufficient to cross the bifurcation threshold: constitutive absence of Prox1 in M & uuml;ller glia, partial chromatin accessibility at regenerative loci, post-injury STAT3 dampening and NFI suppression. In mammals, four corresponding blocking regulators prevent threshold crossing despite conserved orthologous genetic components: sustained STAT3 activity, NFImediated quiescence enforcement, elevated Prox1, and epigenetic constraints on chromatin accessibility. Because blocking regulators form a mutually reinforcing system, coordinated perturbation of multiple regulatory nodes is likely required for sustained regeneration, with the specific combination depending on which constraints dominate in a given cellular context. Dynamical Network Biomarker (DNB) theory formalises the early warning signals that precede threshold crossing, enabling prospective identification of cells approaching a bifurcation threshold and informing targeted intervention strategies. This framework reshapes therapeutic strategy and generates specific, testable experimental predictions for regenerative medicine.
Small gene regulatory networks (GRNs) are well-established biological modules that underpin cellular decisions and dynamical function. Their theoretical understanding has largely been shaped by the motif idea, which links simple network wiring patterns to behaviours. This approach has been extremely influential providing a clear and widely used language for regulatory logic, facilitating the understanding of behaviours such as bistability, ultrasensitivity, or oscillations. However, a growing body of theoretical, and experimental work now challenges the idea that circuit behaviour is fully determined by topology alone, revealing that even very small GRNs can exhibit much richer dynamics once molecular implementation, stochasticity, and upstream modulation are taken into account.Recent advances show that the timing, precision, and reversibility of cell-fate decisions depend critically on signal history, noise structure, and molecular context, even in minimal circuits. Furthermore, there is growing evidence that small GRNs support a wide range of non-canonical dynamical behaviours, including mushroom and isola bifurcations, hybrid oscillatory–switching regimes, and pronounced critical slowing down, substantially expanding their functional repertoire without increasing topological complexity. Crucially, these behaviours are highly sensitive to how regulation is implemented at the molecular level: distinct promoter architectures, regulatory logics, and stochastic mechanisms—often hidden by standard Hill-function descriptions—can qualitatively reshape circuit dynamics, requiring an explicit link between abstract network structure and specific biophysical processes.Together, these results expose fundamental limits of inferring function from topology alone, or reconstructing mechanism from expression data. Rather than simplified motifs, Small GRNs still provide a uniquely powerful setting in which to explore these open questions in order to progress in the development mechanistic, nonlinear descriptions of gene regulation.
Understanding how life emerged from non-living matter remains one of the most profound challenges in science. Empirical constraints and the scarcity of ancient evidence make this question particularly suitable for theoretical and computational approaches. Here, we review recent progress toward a systems-level understanding of life’s origins, focusing on how mathematical models describe the progressive emergence of complexity across three interconnected levels: chemical, informational, and ecological. At the chemical level, models of autocatalytic networks and protometabolic organization capture how self-sustaining reaction systems and feedback loops could arise spontaneously under out-of-equilibrium conditions. At the informational level, studies of polymerization and template-assisted replication of biopolymers shed light on how simple molecular systems could give rise to the emergence of catalytic RNA, genetic heritable information and error-prone molecular evolution. Finally, ecological and thermodynamic models illuminate how protocells and subsequent microbial consortia might have diversified, interacted, and self-organized into the first ecosystems. Together, these approaches highlight a sequence of transitions and bifurcations that call for the development of a coherent framework for studying life’s origins and early evolution, grounded in complex systems theory, prebiotic chemistry, and ecological dynamics. We believe that such an integrative modeling effort will be essential for identifying universal principles underlying the emergence of living systems, bridging the current gap between molecular and ecological levels of organization, and guiding future experimental and computational research in origins-of-life studies.
Nonlinear models are fundamental in systems biology, yet their complexity often limits analytical solutions and real-time prediction. We introduce Carleman linearization as a practical method to approximate nonlinear differential systems by transforming them into infinite-dimensional linear systems and truncating for manageable approximate solutions. This approach, rarely applied in biology, enables interpretable formulas that retain essential dynamics without relying solely on numerical simulations. To demonstrate its utility, we apply Carleman linearization to the classical Susceptible-Infected-Removed epidemic model. The resulting approximations provide explicit expressions for infection dynamics and an algebraic formula for the effective reproduction number, requiring only simple averages of prevalence data. Strikingly, the widely used Gompertz growth law emerges naturally from the structure of Carleman approximants, offering a theoretical basis for its empirical success in epidemic modeling. These findings position Carleman linearization as a versatile tool for Systems Biology, delivering controlled approximations that support rapid and reliable estimates in complex dynamical processes.
The generation of robust circadian oscillations in the suprachiasmatic nucleus relies on intercellular coupling and synchronization of cell-autonomous clocks. When uncoupled, individual cells display considerable heterogeneity in amplitude and period, and a large fraction of cells undergo damped oscillations. We review here modeling approaches and summarize theoretical findings regarding synchronization mechanisms and entrainment properties of the multi-cellular circadian network. We also discuss recent experimental efforts aiming to characterize the synchronization dynamics.
Biological systems are generally complicated and/or complex. In the former approach, one sets up a model with a large number of parameters to describe the system in detail. The latter approach focuses on understanding the universal aspects of biological systems. In this case, an appropriate simple model represents a universality class. The extraction of universal properties is supported by evolutionary robustness and the reduction of dimensionality in high-dimensional states. Integrating the data-driven omics approach with the universality approach is an important step in systems biology.
Current drug discovery and development approaches rely heavily on cell line models, which often limit the translation of discoveries into novel therapeutics. Recently, patient-derived organoids (PDOs) have been developed that more closely resemble disease tissues. To facilitate drug discovery, in vitro expanded patient-derived organoids can be exposed to hundreds of compounds to assess drug effects. These organoids can be analyzed in detail using high-content imaging. Imaging screening generates multi-parametric data at the single cell or organoid level and can provide insights into patient-specific drug effects and drug mode of action. Additionally, image-based approaches benefit co-culture assays by distinguishing perturbation effects on different cell types. Challenges, such as quality control, automation, and data analysis, will increase as organoid models become more complex to reconstitute the in vivo tissue microenvironment, for example, when combining tumor and stromal cells to model drug action. In this review, we discuss the state of the art in using image-based profiling of organoids to drive innovation in drug discovery.
Lakes undergo multiple types of regime shifts such as nutrient pollution, algal blooms, species invasions, overfishing, or restoration. Mechanisms of some types of regime shifts have been understood by coordinating ecosystem models with experiments on whole intact ecosystems. Emerging challenges, such as greenhouse gas release from aquatic ecosystems and the long-term effects of climate warming and continental drying, may involve novel regime shifts. Understanding of the ongoing change will require coordination of models with approaches such as long-term observations, cross-ecosystem comparisons, cross-scale interactions, and experiments at realistic scales of time and space.
Systems biology continues to face the challenge of uniting causal explanation and interpretability with the drive for greater predictive power and scalability. Mechanistic models based on ordinary differential equations (ODEs) provide interpretability and causal grounding in systems biology, yet they often suffer from parameter uncertainty, limited scalability, and computational costs. Machine learning (ML) approaches offer strong predictive performance by learning from high-dimensional, noisy biological data, but this data-driven strength comes at the cost of limited transparency and limited generalizability. Hybrid approaches that integrate mechanistic modeling with ML are emerging as a powerful new paradigm: data-driven modules reduce dimensionality and noise, encode multimodal and longitudinal data, and serve as surrogates for expensive mechanistic submodels, while mechanistic constraints guide ML toward biologically meaningful solutions. This synergy opens the door to uncertainty-aware, generalizable, and computationally tractable models with enhanced predictive power. Applications in cancer and aging research illustrate the promise of hybrid models in predicting treatment success, charting aging trajectories, and designing preventive strategies. Hybrid mechanistic-ML frameworks are not merely incremental improvements but represent a step towards personalized digital twins of biological systems, adaptive, interpretable, and predictive tools for precision medicine and geroscience.
Living cells achieve spatial regulation through phase-separated biomolecular condensates that compartmentalize reactions without membranes. We review how condensates emerge, emphasizing the underlying physical principles of nucleation, diffusion, and mechanical feedback. We outline the main observables of biomolecular condensates and provide insights into how emerging experimental and theoretical approaches reveal their mechanisms. Finally we connect condensates to their biological roles at genomic loci, where they coordinate DNA repair, transcription, and epigenetic control.
Making decisions is a core task for living matter, and a task that can only be carried out through processes out of equilibrium. While important elements of non-equilibrium processes occur in the spatial distribution of resources and within genetic heterogeneity, temporal dynamics of protein concentrations play a critical yet underappreciated role in cellular decision-making. This review explores our current understanding of how cells use dynamic signaling patterns such as oscillations, to regulate processes and cell-fate decisions across multiple biological systems. We focus on examples including the tumor suppressor p53, immune signaling via NF kappa B, and oscillatory networks in development. We further highlight theoretical frameworks and the current horizon in our understanding of this fundamental interplay.
In animal embryos, cell fate specification coordinates with the spatial organisation of cells into distinct domains, leading to the establishment of embryonic boundaries. Multiple hypotheses have been proposed to explain the mechanisms underlying cell sorting, focussing on differences in tissue adhesiveness, contractility, or cell motility. Here, we summarise the main mathematical models of cell sorting and review recent findings showing how these models yield new insights into the problem. Finally, we briefly comment on the limitations of each approach and the importance of identifying the key biophysical ingredients when building a model.
