Physical Biology

Understanding how cells migrate through confined environments is crucial for elucidating fundamental biological processes, including cancer invasion, immune surveillance, and tissue morphogenesis. The nucleus, as the largest and stiffest cellular organelle, often limits cellular deformability, making it a key factor in migration through narrow pores or highly constrained spaces. In this work, we introduce a geometric surface partial differential equation (GS-PDE) model in which the cell plasma membrane and nuclear envelope are described as evolving energetic closed surfaces governed by force-balance equations. We replicate the results of a biophysical experiment, where a microfluidic device is used to impose compressive stresses on cells by driving them through narrow microchannels under a controlled pressure gradient. The model is validated by reproducing cell entry into the microchannels. A parametric sensitivity analysis highlights the dominant influence of specific parameters, whose accurate estimation is essential for faithfully capturing the experimental setup. We found that surface tension and confinement geometry emerge as key determinants of translocation efficiency. Although tailored to this specific setup for validation purposes, the framework is sufficiently general to be applied to a broad range of cell mechanics scenarios, providing a robust and flexible tool for investigating the interplay between cell mechanics and confinement. It also offers a solid foundation for future extensions integrating more complex biochemical processes such as active confined migration.

A morphospace is an abstract space of theoretically possible biological traits, shapes, or property values. It is interesting to explore which parts of a morphospace life occupies, as compared to those parts which could be occupied, but are not. Comparing random and natural non-coding (nc) RNA secondary structures is an established approach to studying morphospace occupation for RNA structures. Most earlier studies have focused on the minimum free energy (MFE) structure, while relatively few have looked at the Boltzmann distribution, describing the ensemble of energetically suboptimal RNA folds. These suboptimal structures may have important roles and functions, and hence should be examined carefully. Here we compare random and natural ncRNA in terms of their Boltzmann distributions, finding that natural RNA tend to have very similar profiles to random RNA, with the main difference being that natural RNA are slightly more energetically stable, except for very short sequences (20 to 30 nucleotides) which tend to be slightly less stable. We infer that natural ncRNA occupy similar parts of the morphospace that random RNA do, indicating that the biophysics of the genotype-phenotype map largely determines the ensemble properties of ncRNA.

Cells must maintain stable protein levels despite the inherently stochastic nature of gene expression, as excessive fluctuations can disrupt cellular function and impair the reliability of decision-making. Regulatory mechanisms, such as negative feedback, buffer protein fluctuations. Yet, it remains unclear how fluctuations are affected by delays that depend on a molecules specific state. Here, we develop a stochastic model in which proteins are produced in bursts as inactive molecules and pass through a series of intermediate steps before becoming active. The duration of such activation delays depends on the current level of active protein, creating a state-dependent feedback loop. Our model provides explicit analytical expressions relating the delay structure and feedback strength to the variability of active protein levels, quantified using the Fano factor, and shows that state-dependent delays can reduce fluctuations below the baseline expected from simple bursty production. Stochastic simulations confirm these predictions, and incorporating negative feedback in burst production further decreases variability while keeping system behavior predictable. These results reveal how temporal and state-dependent regulation stabilizes protein expression, offering guidance for understanding natural cellular control and designing robust synthetic gene circuits.

Mass-conserved reaction-diffusion systems are used as mathematical models for various phenomena such as cell polarity. Numerical simulations of this system present transient dynamics in which multiple stripe patterns converge to spatially monotonic patterns. Previous studies indicated that the transient dynamics are driven by a mass conservation law and by variations in the amount of substance contained in each pattern, which we refer to as "pattern flux". However, it is challenging to mathematically investigate these pattern dynamics. In this study, we introduce a reaction-diffusion compartment model to investigate the pattern dynamics in view of the conservation law and the pattern flux. This model is defined on multiple intervals (compartments), and diffusive couplings are imposed on each boundary of the compartments. Corresponding to the transient dynamics in the original system, we consider the dynamics around stripe patterns in the compartment model. We derive ordinary differential equations describing the pattern dynamics of the compartment model and analyze the existence and stability of equilibria for the reduced ODE with respect to the boundary parameters. For a specific parameter setting, we obtained results consistent with previous studies. Moreover, we present that the stripe patterns in the compartment model are potentially stabilized by changing the parameter, which is not observed in the original system. We expect that the methodology developed in this paper is extendable to various directions, such as membrane-induced pattern control.

Methods to visualize and quantify the molecular responses of cells to local forces exerted at adhesions are crucial to elucidate how physical forces control cellular behavior. Of the many proteins involved in focal adhesions, vinculin plays a key role in mediating force-sensitive processes. Here, we combined optical tweezers and Forster resonance energy transfer (FRET) microscopy to measure the intensity and FRET efficiency of the vinculin tension sensor, VinTS, in response to a force. Fibroblasts expressing VinTS formed adhesions on fibronectin-coated, 3m-diameter, polystyrene beads. As the beads were displaced by the cell, we applied an optical trap to counteract this movement and increase the traction force required by the cell to maintain the bead displacement. The optical trap stiffness varied from zero (no laser) up to 0.26 pN/nm. In this range, the median bead displacement after 5 min was ~200nm in all trapping conditions inducing counteracting forces in the 10-100pN range. To maintain this displacement, vinculin recruitment increased (up to 35% in relative intensity at high stiffness) while tension increased but more moderately (1-2% decrease in absolute FRET efficiency). For higher trap stiffness, the main response was an increase in vinculin recruitment, while the tension did not increase significantly. The increase in vinculin intensity was correlated with the decrease in FRET efficiency at 0.26 pN/nm but not at lower stiffness. Thus, the presence of the high stiffness optical trap over 5 min appears to induce a positive correlation between vinculin recruitment and vinculin tension. In a few instances, vinculin puncta migrated a few microns away from the bead exceeding the bead movement speed while experiencing an increase in both vinculin intensity and tension. Taken together, the results suggest that combining an optical trap with vinculin tension measurements uncovers novel vinculin dynamics in the presence of a force.

Phase separated condensates are recognized as a ubiquitous mechanism of spatial organization in cell biology. Biophysical modeling of condensates provides critical insights into the dynamics and functions of these subcellular structures that are difficult to extract via experiments. Here we present an efficient computational pipeline, CASPULE (Condensate Analysis of Sticker Spacer Polymers Using the LAMMPS Engine), to simulate and analyze the biological condensates made of sticker-spacer polymers. CASPULE implements a unique force field that combines traditional Langevin dynamics with a "detailed balance proof" protocol for single-valent bond formation between stickers. This framework allows us to study the non-trivial biophysics that emerge out of the single-valent sticker interactions coupled with the effect of separation in energetic contribution by stickers and spacers. We provide detailed documentation on how to setup the simulation environment, perform simulations and analyze the results. Through case studies, we highlight the utility and efficacy of our pipeline. Importantly, we provide statistical parameters to characterize the cluster size distribution often observed in biological systems. We envision this tool to be broadly useful in decoding the interplay of kinetics and thermodynamics underlying the formation and function of biological condensates.

Cytokine-mediated communication is a central mechanism by which immune cells coordinate activation, differentiation and proliferation. While mechanistic reaction-diffusion models provide detailed descriptions of cytokine secretion and uptake at the cellular scale, their computational cost limits their applicability to large and densely packed cell populations. Previously employed approximations of cytokine diffusion fields rely on assumptions that neglect the influence of cellular geometry and volume exclusion. In this work, we study a macroscopic description of cytokine diffusion and reaction dynamics based on homogenization techniques, rigorously linking microscopic reaction-diffusion formulations to effective continuum models. The resulting homogenized equations replace discrete responder cells with a continuous density, while retaining essential features of cellular uptake and excluded-volume effects. Further, we show that in regimes with approximate radial symmetry, classical Yukawa-type solutions emerge as limiting cases of the homogenized model, provided appropriate correction factors are included. Overall, our approach allows efficient multiscale modeling of cytokine signaling in complex immune-cell environments.

The apparent pKa of ionizable lipids in lipid nanoparticles (LNPs) is a key determinant of RNA encapsulation during formulation and endosomal release after cellular uptake. However, it is difficult to predict the effective pKa of a given ionizable lipid solely from its solution pKa, because it is sensitive to the membranes composition, as well as solution conditions such as the salt concentration. We developed a simple continuum electrostatics model, based on Gouy-Chapman theory, to predict the shift in effective pKa for ionizable lipids in lipid bilayers as a function of salt concentration and membrane composition. We derive equations for the surface potential and fraction of lipids charged, which are solved self-consistently as a function of solution pH to extract the titration curve and effective pKa. The model shows that the shift in effective pKa is largest when the concentration of titratable lipid is high, and the effect is diminished by increasing salt concentration. We provide a python implementation of the model and an interactive notebook that will allow users to further easily explore the predicted pKa shifts as a function of formulation variables.

Cancer therapeutics are increasingly incorporating engineered receptors due to their ability to detect extracellular ligands and initiate intracellular responses that regulate gene expression. By redesigning these natural signaling systems, synthetic receptors hold great potential for use in novel cell-based therapies. One particularly promising direction is modifying the Notch receptor, a transmembrane protein that naturally mediates ligand-dependent signaling at the cell surface to regulate cell proliferation and differentiation in neurogenesis. Both the intracellular and extracellular domains of Notch can be replaced with alternative domains, creating the family of modified Notch receptors known as synthetic Notch (synNotch). In existing synNotch-activated chimeric antigen receptor (CAR) T-cells, the extracellular domain can be engineered to adjust binding affinity for a specific cancer antigen, enabling precise tuning of therapeutic activity while minimizing off-target effects. To quantify and inform such tuning, we develop differential equations models of synNotch receptor signaling and subsequent gene expression. The mathematical models couple activation dynamics on fast timescales (characteristic of receptor-ligand interactions) and on slow timescales (characteristic of downstream gene expression dynamics). Global Sobol sensitivity analysis of the proposed models highlights parameters that yield the greatest variability in synNotch signal transduction and gene expression, indicating their potential to be engineered for different functions in future cancer therapeutics. For the receptor-ligand interactions in the synNotch model, we find that ligand association and ligand-independent activation are the most sensitive parameters. In the downstream gene expression model, promoter strength and degradation rates of mRNA and gene product are found to be most amenable to engineering.

The COVID-19 pandemic saw successive emergence and global spread of novel viral variants, exhibiting enhanced transmissibility or evasion of immunity. While the genotypic and phenotypic basis of SARS-CoV-2 variants have been extensively characterized, the evolutionary factors governing their patterns of emergence are less well understood. In this study we systematically investigated how the invasion dynamics of viral variants depend on variant phenotype (increased transmissibility or immune evasion), source (local evolution vs importation), the timing of introduction, the distribution of population susceptibility, and the contact network structure. Using a stochastic multi-strain epidemic model, we find that strains with only a transmission advantage are more likely to emerge earlier in the epidemic, and rapidly and predictably dominate the viral population. In contrast, immune-escape variants tend to linger at low prevalence for extended time periods after emergence, avoiding detection, until a critical amount of immunity has built up in the population and they begin to rapidly outcompete existing strains. We find that two common features of realistic human contact networks---heterogeneity in contacts (overdispersion) and clustering---lead to more punctuated evolutionary dynamics. This work provides insight into past dynamics of SARS-CoV-2 variants and can help define planning scenarios for future epidemic modeling efforts.

The basic reproduction number of an infectious disease is known to depend on the structure of contacts between individuals in a population. This relationship has been explored mathematically through two well-known models: one which depends on a matrix of contact rates between different demographic groups, and another which depends on the variability of contact rates over the population. Here we introduce a model that combines and generalises these two approaches. We derive a formula for the basic reproduction number and validate it through comparisons to simulated outbreaks. Applying this method to contact survey data collected in Belgium between 2020 and 2022, we find that our model produces higher estimates of the basic reproduction number and larger relative changes over periods when social contact behaviour was changing during the COVID-19 pandemic. Our analysis suggests some practical considerations when using contact data in models of infectious disease transmission.

Cerebrospinal fluid (CSF) is a Newtonian fluid that bathes the brain and spinal cord and oscillates in response to the physiological periodic changes in brain volume, of which the cardiac cycle is a major driver. Understanding this motion is essential for clarifying its contribution to solute transport, waste clearance, and drug delivery. In this work, we study oscillatory and steady streaming flow in the cranial subarachnoid space using a lubrication-based theoretical framework. The model represents the cranial CSF compartment as a thin fluid layer bounded internally by the brain surface and externally by the dura, driven by time-dependent brain surface displacements. We first derive simplified governing equations for flow over an arbitrary smooth sphere-like brain surface and obtain analytical solutions for an idealised spherical geometry with uniform displacements. We then incorporate realistic displacement fields reconstructed from MRI measurements in healthy subjects and solve the reduced equations numerically. The results show that oscillatory forcing produces a steady streaming component that may enhance solute transport compared with diffusion alone. This work provides a mechanistic description of the flow generated by physiological brain motion and highlights the potential presence of steady streaming in cranial subarachnoid fluid dynamics.

The interactions of droplets and filaments can lead to mutual deformations and complex combined behavior. Such interactions also occur within the cell, where biomolecular condensates, distinct liquid phases often composed of proteins, have been observed to structure and affect the organization of the cytoskeleton. In particular, biomolecular condensates have been shown to undergo characteristic deformations when cytoskeletal filaments are fully embedded within them. However, a full understanding of the underlying physical mechanisms is still missing. Here, we combine experiments with coarse-grained molecular dynamics simulations and analytical models to uncover the physical mechanisms that define emerging shapes of droplets containing filaments. We find that the surface tension of the liquid phase and the bending energy of the filament(s) suffice to accurately capture emerging shapes if the length of the filament is small compared to the liquid volume. As the volume fraction of filament(s) increases, wetting effects become increasingly important, setting physical constraints within which surface and bending energies compete to define the droplet shapes. We find that mutual deformations of condensate and filament extend accessible shapes beyond classical stability considerations, leading to structuring and entrapment of contained filaments. Shape deformations may further affect ripening dynamics that favor certain geometries. Our findings provide a physical framework for a better understanding of the possible roles of biomolecular condensates in cytoskeletal organization.

BackgroundTo control leishmaniasis, chemotherapy drugs are currently under development. However, these drugs often exhibit poor efficacy and are associated with toxicity, adverse effects, and drug resistance. At present, no specific drug is available for the treatment of leishmaniasis. Meanwhile, vaccine research is ongoing. Recent studies have analysed some experimental vaccines using mathematical models. AimIn previous work, drug targeting was focused on the entire human body rather than specifically addressing infected macrophages and parasites. In our current approach, we aim to eliminate infected macrophages and parasites through nano-drug design. Specifically, we utilise two types of nanoparticles: iron oxide and citric acid-coated iron oxide. Moving forward, we plan to advance this strategy using mathematical modelling of macrophage-parasite interactions. MethodsWe design PDE-based models of macrophages and parasites, incorporating cytokine dynamics, to support nano-drug development. Drug efficacy is estimated using posterior distributions to analyse phenotypic fluctuations of macrophages and parasites during the design phase. We investigate implicit and semi-implicit treatment schemes, focusing on energy decay properties. To model drug flow during treatment, we introduce a three-phase moving boundary problem. Comparative analyses are conducted to evaluate macrophage and parasite behaviour with and without treatment. Finally, the entire framework is implemented within a virtual lab environment. ResultsThe results show that the nano-drug exhibits better efficacy compared to combined drug doses. We analysed and compared two types of nano-drug particles: iron oxide and citric acid-coated iron oxide. We discuss how the drug effectively targets and eliminates infected macrophages and parasites. ConclusionOur models results and simulations will support researchers conducting further studies in nano-drug design for leishmaniasis. These simulations are performed within a virtual lab environment.

Cancer-associated fibroblasts (CAFs) are a central component of the tumor microenvironment that facilitate a supportive niche for cancer progression and metastasis. Experimental evidence suggests that CAFs can facilitate estrogen-independent tumor growth, thereby reducing the efficacy of anti-hormonal therapies. Understanding and quantifying the complex interactions between tumor cells, hormonal signalling, and the microenvironment are crucial for designing more effective and individualized treatment strategies. We propose a mathematical framework to explore the influence of CAFs on ER+ breast cancer progression and to evaluate strategies to mitigate their impact. We develop a system of nonlinear ordinary differential equations that substantiates the experimental observations by providing a mechanistic basis for the role of CAFs in regulating estrogen-independent growth dynamics. We then employ optimal control theory to evaluate distinct therapeutic approaches involving monotherapy or combinations of: (i) inhibition of tumor-to-CAF signaling, (ii) inhibition of CAF-to-tumor proliferative signaling, and (iii) endocrine therapy. Taken together, our results demonstrate that CAF-targeted strategies can enhance treatment efficacy across various estrogen dosing regimens. Our study provides new insights into the potential of CAF as a therapeutic target that could help to improve existing approaches for endocrine therapies.

Biomolecular condensates formed by complex coacervation of highly charged proteins provide a powerful framework to understand how microscopic interactions give rise to macroscopic material properties. Atomistic molecular dynamics simulations provide detailed insights but remain limited in accesing the spatio-temporal scales relevant for condensate behavior. Here, we use the residue-level coarse-grained Mpipi-Recharged model to investigate condensates formed by ProT and positively charged partners, including histone H1, protamine, poly-lysine, and poly-arginine. Material properties, in this context, provide a stringent experimental benchamark for coarse-grained models. Our model reproduces salt-dependent phase behavior, protein binding affinities, and sequence-specific stability trends in agreement with in vitro experiments, despite the fact that material properties were not included in the model parametrization. We then establish a direct link between protein dynamics and macroscopic material properties by quantifying monomeric diffusion, conformational reconfiguration, and translational mobility within the dense phase, and relating these to condensate viscosity. By comparing dynamics across dense and dilute phases, we uncover a pronounced length scale-dependent behavior. While residue-level binding and unbinding events remain equally fast in both phases, protein reconfiguration time and self-diffusion are significantly slowed down within the condensates. This decoupling reveals how fast intermolecular interactions coexist with slow mesoscale condensate dynamics depending on the molecular length scale. Together, our results establish a predictive framework that links encoded sequence intermolecular forces and multiscale dynamics to the emergent material properties of complex biomolecular condensates.

Nucleus shape is a sensitive indicator of cell state, influenced by numerous bio-chemical and physiological factors. While prior work has cataloged how perturbations alter nucleus morphology, we address the inverse: inferring underlying molecular changes from nucleus shape alone. We previously developed a mechanical model yielding two nondimensional parameters: flatness index and scale factor, which are surrogate measures for cortical actin tension and nuclear envelope compliance respectively. In this study, we apply these parameters to investigate the dynamics in cellular mechanics during confined migration. We fabricated polydimethylsiloxane (PDMS) microchannels with widths of 3 {micro}m (high confinement) and 10 {micro}m (low confinement) and tracked cells migrating through them. We captured high-frequency 3D nucleus shapes via double fluorescence exclusion microscopy and custom image analysis. Fitting the model and estimating flatness index and scale factor to time-resolved shapes revealed dynamic regulation in 3 {micro}m channels: actin tension decreased and nucleus compliance increased immediately before nucleus entry into the constriction, with rapid restoration to baseline upon exit. No such changes occurred in 10 {micro}m channels, indicating active, confinement-dependent cytoskeletal adaptation. Immunostaining for YAP and lamin-A,C confirmed these model inferences. Our results uncover mechanostasis, active mechanical homeostasis, during confined migration and establish the combination of double fluorescence exclusion microscopy and nondimensional nucleus shape parameters as a powerful, non-invasive tool for single-cell mechanobiology studies.

Fluorescent reporters such as fluorescent proteins or chemigenetic indicators are indispensable tools for studying biological processes using light microscopy. Choosing an appropriate fluorescent tag is a crucial step in experimental design not only for imaging but also for quantitative measurements such as fluorescence fluctuation spectroscopy. Two key parameters should be considered: Fluorescent brightness and photo-bleaching. Change to fluorescence intensity due to photobleaching is relatively easy to assess in different biological environments, while brightness is more elusive. Here, we develop and employ a fluorescence correlation spectroscopy (FCS) based excitation scan assay that determines fluorescent protein performance and validate it in tissue culture and zebrafish embryos. We employ our FCS pipeline to compare a set of 10 established fluorescent proteins as well as HALO and SNAP tags for both cellular imaging and measurements of diffusion dynamics with FCS. We show that mNeonGreen outperforms mEGFP in tissue culture and zebrafish embryos. We also compare StayGold variants against other green fluorescent proteins and chemigenetic reporters in tissue culture. Overall, we present a broadly applicable approach for determining fluorescent reporter brightness in the living system of interest.

1Long QT syndrome Type 2 (LQT2) is a genetic disorder caused by missense mutations in the KCNH2 gene that encodes the potassium channel KV11.1. Previous studies have shown that most KV11.1 missense mutations with loss-of-function phenotypes result from impaired trafficking from the endoplasmic reticulum to the plasma membrane. To investigate the molecular basis of these defects, we used molecular dynamics simulations to analyze two sets of disease-associated missense mutations: those that suppress and those that maintain normal channel trafficking. We focused initially on the conformational and dynamics differences between wild-type and several mutants of KV11.1 via molecular dynamics simulations when two K+ were placed in the selectivity filter (SF). Our study reveals that missense mutations in the S4 helix allosterically disrupt the selectivity filter, a critical determinant for proper channel trafficking. Trafficking-competent variants largely retained a wild-type selectivity filter structure, whereas trafficking-deficient mutants exhibited pronounced structural perturbations in this region. These findings suggest that certain LQT2-associated missense mutations in KCNH2 impair channel trafficking by compromising the structural integrity of the selectivity filter. We additionally found that second-site variants Y652C in the drug binding vestibule can correct structural defects associated with some mistrafficking variants.

The information encoded in DNA sequences can be rigorously quantified using Shannon entropy and related measures. When placed in an evolutionary context, this quantification offers a principled yet underexplored route to constructing gene regulatory networks (GRNs) directly from sequence data. While most GRN inference methods rely exclusively on gene expression profiles, the regulatory code is ultimately written in the DNA sequence itself. Here we review the mathematical foundations of information theory as applied to gene sequences, survey existing computational methods for GRN inference--with emphasis on information-theoretic and sequence-based approaches--and examine how evolutionary conservation constrains sequence entropy to preserve biological function. We then propose a four-layer integrative framework that combines per-position Shannon entropy profiles, evolutionary conservation scoring via Jensen- Shannon divergence, expression-based mutual information and transfer entropy, and DNA foundation model embeddings to construct GRNs from sequence. Through worked examples on the Escherichia coli SOS regulatory sub-network, we demonstrate how conservation-weighted mutual information improves edge discrimination and how transfer entropy resolves regulatory directionality. The framework generates testable predictions: edges supported by low-entropy regulatory regions should show higher experimental validation rates, and cross-species entropy profile conservation should predict GRN topology conservation. This work bridges three scales of biological information--nucleotide-level entropy, evolutionary constraint patterns, and network-level regulatory logic--establishing information entropy as the natural mathematical language for sequence-to-network regulatory inference.