Frontiers in Physics

The COVID-19 pandemic triggered an unprecedented volume of real-time discourse on social media platforms, with Twitter serving as a global forum for public reactions, fears, and evolving narratives. Traditional sentiment analysis approaches treat tweets as independent, static samples, failing to capture the temporal evolution and geographic heterogeneity of public opinion. This paper presents a comprehensive spatio-temporal framework that integrates fine-grained sentiment classification using COVID-Twitter-BERT with dynamic topic modeling via BERTopic to automatically discover and track evolving narratives. Using a corpus of 2.4 million geolocated tweets collected between January 2020 and June 2022, our analysis reveals distinct pandemic phases: early fear-driven narratives about mask shortages (Q1 2020), vaccine optimism followed by polarization (2021), and pandemic fatigue (2022). Regional comparisons show significant differences, with US discourse dominated by freedom-versus-mandate debates while European discussions emphasized collective solidarity. Our framework achieved 76% F1-score in sentiment classification and successfully identified 50 distinct narratives with high coherence scores. This work provides a powerful methodology for real-time epidemiological narrative surveillance and crisis communication monitoring.

A fundamental biological process of transcription occurs in the cell nucleus, which is a complex medium that also contains multiple heterogeneous structures known as biomolecular condensates. Interestingly, some of these condensates contain mRNA molecules in addition to proteins, suggesting an important cellular role in transcription that is not yet well understood. In this work, we develop a minimal theoretical framework for quantitative investigation of the role of reversible mRNA condensation in transcription. Our discrete-state stochastic approach accounts for the most relevant processes, allowing us to explicitly evaluate the properties of the system and clarify the effects of condensation. Analytical calculations supported by computer simulations suggest that reversible mRNA condensation influences the transcription processes by maintaining a constant level of free mRNA in the nucleoplasm while lowering the degree of stochastic noise and increasing the robustness against external perturbations. Physicochemical arguments are presented to explain these observations. The proposed theoretical framework elucidates important microscopic aspects of transcription, providing a convenient quantitative tool for investigating complex biological phenomena.

Mechanical properties of biological tissues, driven by passive and active forces, play a vital role in several processes ranging from development to cancer metastasis. However, the dynamical responses of cells in tissues, subject to mechanical deformations such as shear and the associated rheological properties, are not well characterized. Here, we use three-dimensional agent-based models for normal and cancer tissues to investigate their responses to simple shear as a function of cell stiffness and stochastic active forces. In the normal epithelium, with uniform strength of active force, the yield stress as a function of shear rate follows the Herschel-Bulkley form over a range of cell volume fraction. Strikingly, the shear rate dependence and the elasticity-dependent changes in the yield stress fall on master curves upon suitable scaling. To model cancer-like behavior, a certain fraction (Np) of cells was chosen to have enhanced activity and decreased stiffness. As Np increases, the extent of collective cell movement decreases, transitioning from affine (collective) to non-affine (individualistic) movement, a finding that is in accord with imaging experiments. Simulations of a model of a stiff solid tumor, with radius Rs embedded in normal tissue, show that as Rs increases, the yield stress increases. Interestingly, the cells migrate collectively as Rs increases. A Gaussian Mixture Model (GMM) and a mean field theory quantitatively account for the simulation as well as experimental results on cancerous, non-cancerous, and a mixture of these two types. The combined theoretical and experimental study establishes that heterogeneity in stiffness and activity determines non-affine movements in normal and cancer tissues.

Cardiac blood flow is a regulator of several important developmental and remodelling processes in the heart, including through fluid shear forces sensed by the endothelial cells lining the heart. However, optically mapping these flow fields in the complex 3D geometry of the heart is challenging even in transparent animal models such as the zebrafish. One of the main challenges is the difficulty in measuring the out-of-plane (axial) velocity component, preventing accurate mapping of the complete 3-component-3-dimension (3C-3D) blood flow velocity field; image-based techniques such as microscopic particle image velocimetry ({micro}PIV) traditionally only provide the in-plane flow components. Here we present a computational approach to achieve full time-varying 3C-3D blood flow vector mapping using a standard selective plane illumination microscope (SPIM), based on robust cardiac phase assignment, precise measurement-driven registration of sequentially acquired z-stacks, and PIV data fusion from multiple sample orientations. Our approach holds the key to understanding the complex dynamic flow fields within the developing heart, and their role in shaping cardiac development.

Interleukin-13 (IL-13) is an immunomodulatory cell signaling cytokine that has been implicated in neurodegenerative disease and chronic inflammation. IL-13 binds to its low and high affinity receptors, IL-13 receptor 1 (IL-13R1) and IL-13 receptor 2 (IL-13R2), respectively, with residence times that vary accordingly. As the binding kinetics of the cytokine-receptor complexes influence cellular responses, we employed the molecular dynamics (MD) simulation-based{tau} -random acceleration molecular dynamics method ({tau}RAMD) to compute relative residence times for wild-type (WT) IL-13 and 19 IL-13 mutants to the two receptors. Comparison with experimental kinetic data shows that the{tau} RAMD computations capture the trends in residence times. Analysis of simulated dissociation trajectories of the cytokine-receptor complexes reveals two distinct dissociation pathways of IL-13 from each of the receptors. This study thus pinpoints key determinants of the interaction of IL-13 with its receptors which could be targeted for therapeutic design. Statement of SignificanceCytokines are regulatory proteins that bind to cell surface receptors and thereby send signals to the cellular interior. Interleukin-13 (IL-13) is a cytokine that has a low and a high affinity receptor. It has important physiological roles, and its deregulation is involved in diseases such as atopic dermatitis and asthma. We employed a molecular dynamics simulation-based method to compute the effects of changes in the sequence of IL-13 on the lifetimes of complexes of IL-13 and its receptors. Comparison with experiments supports the validity of the computational approach and analysis of the simulations reveals two distinct ways in which IL-13 dissociates from each receptor. These results thus provide a map for targeting IL-13 - receptor interactions for the design of therapeutics.

Wound repair is an important mechanism to preserve tissue integrity in organisms after injury. However, why different tissues exhibit different mechanisms to repair wounds is a long-standing question that remains unanswered. In this work, we theoretically explore the role of the purse string, an actomyosin contractile cable used by tissues to close small wounds. Does the tissue 3D geometry influence the efficiency of the purse string in driving wound closure? Using a 3D biophysical model, we study in silico tissues with the same cell volumes but different aspect ratios, ranging from squamous to thick and tall tissues. The model predicts that taller cells are easily deformed by the purse string. In contrast, very squamous cells require a very strong purse string that might demand additional cellular mechanisms to close the gap. These findings establish a theoretical framework to predict the optimal biophysical mechanisms of wound healing in different tissues. Graphical abstractCells of different aspect ratios can be observed in a range of organisms with different function and mechanics. The wound healing efficiency of the purse string increases with the cell aspect ratio in our theoretical exploration. O_FIG O_LINKSMALLFIG WIDTH=200 HEIGHT=135 SRC="FIGDIR/small/743165v1_ufig1.gif" ALT="Figure 1"> View larger version (23K): org.highwire.dtl.DTLVardef@d44ab0org.highwire.dtl.DTLVardef@1737cbaorg.highwire.dtl.DTLVardef@101b5d4org.highwire.dtl.DTLVardef@1487f26_HPS_FORMAT_FIGEXP M_FIG C_FIG

In the active loop extrusion model, the cohesin protein complex creates chromatin loops in eukaryotic cells. Extrusion maintains topologically associated domains (TADs), which are contiguous segments of chromatin that preferentially colocalize in space and are typically bounded by CTCF proteins that pause cohesin translocation. Here, we model active loop extrusion with hybrid molecular dynamics - Monte Carlo simulations in entangled flexible linear polymer melts. Intra-chain contact probabilities of polymers with active loop extrusion are enhanced compared to their equilibrium, passive counterparts. Extrusion causes the size of chain segments to be much smaller than in passive melts. While the overlap parameter in passive melts without extrusion monotonically increases with segment length, it is nonmonotonic in active melts and on the order of unity within the parameters of this study. Active loop extrusion suppresses contacts between TADs in favor of intra-TAD contacts. Reduction of overlaps between chain segments dilutes entanglements in active melts. Depending on parameters, active extrusion without TADs may induce more compact conformations than with TADs, due in part to fractal loopy globule-like dynamics. This work suggests that active loop extrusion reduces overlaps between TADs, contributing to effective gene regulation by cis-regulatory elements.

To disentangle complex fiber configurations that remain challenging for diffusion MRI tractography, insights might be gained from microscopy tractography. Indeed, by precisely following small white matter (WM) fascicles invisible at the resolution of diffusion MRI, microscopy tractography can help explain how fiber populations are organized at the finest scales. Serial optical coherence tomography (S-OCT) is an imaging modality relying on the intrinsic contrast of a sample. When applied to brain tissues, the S-OCT contrast is primarily driven by the myelin reflectivity. Due to its high resolution, on the order of microns, and its 3D nature, S-OCT offers promise for studying WM connections at the microscale. However, while other microscopy imaging modalities have been shown to enable tractography, whether the reflectivity contrast from S-OCT supports the reconstruction of long-range WM fascicles at the microscale remains unknown. Furthermore, there is a gap in the literature regarding how an ideal microscopy tractography algorithm should behave with respect to the choice of tractography algorithm, tracking maps definition and microscale orientation distribution functions (ODF) estimation. In this work, we describe a tailored approach to reconstruct WM fascicles at the microscale from S-OCT acquisitions. We improve microscale orientation distribution functions (ODF) estimation by implementing a sliding-window formulation allowing the estimation of ODF at S-OCT resolution, and use apodized Dirac delta functions for reducing unwanted interference. We validate our approach on a simulated microscopy-like FiberCup dataset, and show that using multiscale Frangi filters for estimating ODF outperforms structure tensor analysis. We also show that particle filtering tractography with anatomical constraints enables targetted, region-to-region tractography, and outperforms standard deterministic or probabilistic tracking approaches. We further demonstrate our method on a whole mouse brain S-OCT reconstruction at 10 m by reconstructing the thalamocortical white-matter projections. Overall, our results show that S-OCT tractography recovers fine white matter fascicles visible at the microscale, and that these connections are supported by viral tracing experiments from the Allen Mouse Brain Connectivity Atlas. Moreover, this work shows the first ODF estimation and fully-3D probabilistic particle filtering tractography of the mouse brain from S-OCT reconstructions at 10 m isotropic resolution.

Living cells are highly dynamic and densely crowded environments in which organelles such as vesicles undergo continuous motion that is essential for cellular processes. Therefore, accurate tracking of individual organelles is crucial for understanding intercellular dynamics and functions. However, precise tracking of individual organelles in living cells remains challenging due to high organelle densities, frequent particle overlap, and the coexistence of stationary and motile organelles. In particular, stationary organelles can obscure the trajectories of moving organelles, leading to tracking errors and fragmented tracks. To overcome these challenges, we developed Multiple Particle Tracking via Velocity Filtering (MPT-vVF), an unbiased, semi-automated tracking framework that incorporates a mathematically derived velocity-filtering algorithm to selectively identify and track moving organelles with high accuracy in crowded intracellular environments. MPT-vVF integrates denoising, background subtraction, and a velocity-matching detection step that discriminates true particle motion from noise based on spatiotemporal continuity, followed by robust trajectory linking. We demonstrate that MPT-vVF can accurately resolve nanometer-scale displacements of immobilized beads, highlighting its high tracking precision. We also validate the robustness of MPT-vVF by quantifying the transport of brain-derived neurotrophic factor (BDNF)-mRFP-containing vesicles in living hippocampal neurons. Furthermore, MPT-vVF reveals that exposure to 50-nm nanoplastics impairs vesicular transport, reducing both travel length and speed of BDNF-containing vesicles in living neurons. These findings establish MPT-vVF as a powerful method for quantitative analysis of intracellular organelles in crowded living cells and suggest its broad application to biophysics, cell biology, and soft matter research.

While geometric constraints shape tissue development, quantifying the resulting growth dynamics remains a central challenge in tissue engineering. Conventional methods often struggle to capture multi-scale kinetics without complex labeling or difficult single-cell tracking. Here, we analyze geometrically controlled growth of microtissues derived from human dermal fibroblasts using time-resolved, label-free brightfield microscopy, combined with optical flow and semi-automated deep learning mitosis detection. By extracting multi-scale flow fields and integrating them with tissue segmentation, we quantify directional tissue dynamics, separating flow into parallel and normal components relative to the local tissue contour. Applying this framework, we contrast the quiescent tissue interior with the advancing growth front where localized dynamics and cell proliferation drive expansion. Our results demonstrate that, compared to the bulk, the growth front exhibits higher fluctuations parallel to the tissue contour, positive mean normal flow, and significantly increased mitotic activity. Furthermore, evaluating flow divergence around mitotic events reveals distinct spatial behaviors: with the onset of mitosis, a contraction and subsequent expansion occurs in the vicinity of the dividing cells. Beyond the immediate cellular neighborhood, the broader regional dynamics remain consistent before and after mitosis onset, with net tissue expansion in proximity to the growth front and contraction within the tissue interior. By extracting continuous kinetic data from easily accessible, label-free brightfield imaging, this approach serves as a non-invasive, complementary tool for evaluating in vitro tissue morphogenesis and growth dynamics. This analytical framework can be expanded to study locally resolved tissue morphogenesis and growth kinetics in other microsystems, ranging from embryos to organoids. Statement of SignificanceUnderstanding how localized cellular forces drive tissue growth is critical for mechanobiology. However, mapping these dynamics traditionally requires complex, invasive fluorescent labeling. We present an accessible, label-free computational framework combining optical flow and deep learning-based mitosis detection to quantify continuous tissue kinematics directly from standard brightfield microscopy. Applying this to 3D microtissues, we reveal a distinct spatial coupling between cell division, local mechanical fluctuations, and directed tissue expansion at the active growth front. This non-invasive approach bridges the gap between single-cell mechanics and macroscopic morphogenesis, offering a versatile tool to monitor complex in vitro model systems-like organoids and bioengineered tissues-without disrupting their native state.

Contact inhibition of proliferation (CIP) enables epithelial tissues to self-regulate growth and maintain tissue homeostasis. However, how cell-level mechanical contact, tissue-scale structural order, and proliferation kinetics interplay remains a fundamental open question in living matter physics. Here, we present a particle-based model of a confluent epithelial monolayer governed by overdamped dynamics, where individual cells interact via a two-dimensional hard core- soft shoulder potential. By comparing structural evolution during quasistatic densification with previously reported experimental division kinetics, we find that the dynamics of proliferation arrest mimics the onset of direct steric contacts between the hard cores of the shell. Identifying hard core contacts as the physical driver of CIP, we couple our mechanical model with a stochastic Monte Carlo division scheme in which the instantaneous division rate decreases to zero from an intrinsic value as the number of hard core contact increases to six from zero. We demonstrate that for high intrinsic division rates, the cellular densification outpaces mechanical relaxation. This kinetic mismatch drives premature hard-core contact formation, shifts the onset of jamming and contact inhibition to lower packing fractions, and induces increasingly disordered transient configurations before the tissue universally converges to a hexagonal close-packed limit. Our model's predicted division kinetics and structural order evolution are consistent with epithelial monolayer experiments, both reported and our own. This minimal physical framework links single-cell steric contact mechanics directly to tissue-scale growth regulation and structural evolution.

Structural network representations of metastatic dissemination typically focus on static topology without resolving transport dynamics, relaxation timescales, or steady-state behaviour. Here, we formulate a discrete Markovian transport model on a directed higher-order network with transition rates derived from qualitative clinical affinity classes. By constructing a non-Hermitian row-stochastic transfer operator, we characterise the relaxation dynamics through its spectral decomposition. The system exhibits a fast-mixing regime characterised by a spectral gap of {gamma} {approx} 0.67, corresponding to a characteristic relaxation timescale of {tau} {approx} 1.49 discrete steps, with the influence of the primary tumour origin progressively attenuated during dissemination. Convergence towards a non-equilibrium steady state (NESS) is accompanied by a reduction in Shannon entropy, concentrating probability mass within specific topological sinks. This spectral relaxation delineates two distinct dynamical regimes: early transient dissemination (n < {tau}), dominated by local organ-specific transition probabilities (organotropism), and the asymptotic regime (n > {tau}), determined increasingly by the global transport architecture of the network. Comparison with independent clinical and autopsy observations across 21 primary tumours and 23 target organs indicates that the predicted stationary distribution is consistent with the observed hierarchy of metastatic organ involvement.

Violent conflicts increasingly involve multiple armed actors competing for influence over shared civilian populations, creating complex dynamics that challenge conventional security analysis and policy design. We present a framework that adapts epidemiological methods informed by the conflict landscape in Nigeria to model multi-actor violent conflict as an epidemic process. We derive a basic insecurity reproduction number ($R_0$), identify violence-free and persistent-violence equilibria, and introduce a novel Civilian Harm Index (CHI) to quantify humanitarian impact. Sensitivity analyses identify recruitment, ideological support from civilian populations, and abduction as the key drivers of conflict persistence and civilian harm. The framework reveals several counterintuitive findings. Interventions that most effectively suppress violence transmission are not necessarily those that minimise civilian harm, demonstrating that epidemic control and humanitarian protection may require distinct optimisation criteria. Likewise, interventions effective against one armed actor may be ineffective, or even counterproductive, when applied uniformly across groups. In addition, prisoner exchange and ransom payments increase violence persistence and civilian harm. Although developed as an illustrative rather than predictive framework, our results show that epidemiological methods provide quantitative metrics for evaluating intervention priorities and trade-offs in complex multi-actor conflicts.

Within the studies of stochastic gene expression, apart from the variability of copy number of gene products, the problems of threshold crossing of those products are biologically important as they often lead to terminal cellular events. Here, we study the threshold crossing problem of the messenger ribonucleic acid (mRNA) and present an exact probability distribution of first passage times in Laplace space. The function furnishes moments of any order and also predicts the characteristic time of the exponential tail of the distribution, which we match against Gillespie simulations. We find that all the measures of relative fluctuations of the threshold crossing times show U-shapes within this simple model of mRNA, as was found earlier in more mathematically involved models of threshold crossing time statistics of proteins. Furthermore, we extend the exact formula to include the phenomenon of DNA duplication and the corresponding doubling of transcription rate. As expected, the distribution varies considerably depending on the onset of the duplication stage within the cell cycle.

The study of tissue dynamics has been stimulated during the last decades thanks to the use of quantitative descriptions, with the development of several theoretical and computational frameworks, many of them revolving around the notion of reaction-diffusion systems, eventually with additional structure variables beyond time and space. The use of structure variables can accommodate phenotypic traits. In this work, we study a family of competition models, where a given population depends on a resource (e.g. oxygen) and several populations are competing for it. Our quantitative description incorporates phenotypic traits and heterogeneity at the level of cell cycle variations, which influence replication rates via oxygen consumption. This enables us to replicate the fitness of specific subpopulations to environmental conditions (e.g. oxygen shortage or external influences). Using numerical simulations, we show that such models display dynamical pattern formation in the form of coupled travelling wave profiles that expand or retreat at the same wave speed. The full theoretical analysis of such dynamics is quite involved; to circumvent this difficulty, we introduce a quasi-stationary approximation for the resource dynamics. We find that this approximation can reproduce the overall behaviour very accurately, with the additional benefit of allowing theoretical treatment of the reduced model. In this way, we provide estimates on the wave speed which are numerically shown to be robust across a wide range of macroscopic parameters of the full model. The wave speeds are thus found to depend strongly on the proliferation rate of the fittest population, resembling a winner-takes-all dynamics.

Residual dipolar couplings (RDCs) obtained via magnetic field-induced alignment offer a powerful, media-free approach for the structural analysis of biomolecules. However, their detection in short, fast-tumbling nucleic acids remains elusive at conventional magnetic fields due to insufficient alignment and sensitivity. Here, we demonstrate the direct observation of these RDCs at 1.3 GHz in a 14-mer hairpin fragment derived from an HIV-1 Vif-targeting aptamer. The 1JNH scalar couplings of imino protons were measured at fields ranging from 600 MHz to 1.3 GHz. While the coupling constants remained invariant between 600 and 900 MHz, a clear deviation was exclusively captured at 1.3 GHz for all base-paired stem residues, demonstrating the first media-free detection of field-induced RDCs in a short RNA of this size. This breakthrough arises from a synergistic B07/2 scaling, combining enhanced alignment ({propto} B02) and sensitivity ({propto} B03/2). These RDCs showed excellent agreement with the NOE-derived structure. Additionally, the flexible loop residue G8 exhibited no detectable RDC, but displayed a field-dependent TROSY/anti-TROSY intensity inversion at 1.3 GHz, reflecting an unusual 1H chemical shift anisotropy (CSA) tensor that corroborates the local base-packing environment. Our findings highlight 1.3 GHz NMR as an indispensable tool for the structural analysis of short RNAs.

Communities exposed to flooding can experience markedly different mental health outcomes, yet conventional resilience indicators capture only part of the social and contextual conditions that may explain this variation. This study develops a multilevel and predictive framework for examining community resilience and depressive symptoms following flood exposure in Indonesia. Data were drawn from 20,303 respondents aged 15 years and older nested within 312 communities in the Indonesia Family Life Survey (IFLS-5). Depressive symptoms were assessed using the 10-item Centre for Epidemiologic Studies Depression Scale (CES-D-10), with Rasch Partial Credit Model calibration used to examine measurement properties. Bayesian multilevel models quantified between-community heterogeneity and assessed how far observable structural resources accounted for this variation. Community resilience was represented through two complementary constructs: structural resilience, based on observable socioeconomic and social-capital resources, and Latent Community Protective Capacity (LCPC), a model-derived proxy for residual contextual variation in depressive-symptom risk. Approximately 6 percent of variation was attributable to between-community differences, while observable structural resources explained only part of this heterogeneity. Structural resilience and LCPC were weakly correlated (r = 0.155). Moderation analyses provided no clear evidence that structural resilience altered the flood-depression association, while LCPC showed a directionally consistent but uncertain buffering pattern. Predictive models incorporating community-level information improved discrimination, with the best-performing model reaching an ROC-AUC of approximately 0.71. The findings suggest that observable resource-based indices provide an incomplete account of community-level mental health vulnerability and that residual contextual measures may provide complementary information, while requiring cautious interpretation and independent validation.

Calcium oxalates are naturally occurring minerals, with the monohydrate form CaC2O4xH2O (COM) being the most stable. COM contains two crystallographically different water sites. We investigate the details of water internal dynamics in the high and low temperature phases of COM using 17O central transition solid-state NMR line shapes, as well as laboratory and rotating frame relaxation rates. The measurements were performed either under static or magic angle spinning conditions and in a wide temperature range from 343 to 180 K. The combination of all measurements allows for precise constraints on motional mechanisms, rate constants, and amplitudes of motions. The high temperature phase is dominated by large-angle fluctuations with an amplitude of about 100 degrees, identical in both sites. During the phase transition between 323 to 300 K, these large-angle jumps freeze out in one of the water sites, while remaining active in the other. In the low temperature phase from 280 to 180 K, small-angle fluctuations of 2-8 degrees in amplitude dominate the relaxation. Transverse relaxation rates also point to the existence of a very slow collective rocking motion down to about 220-200 K.

Diffusion within the cytoplasm is fundamental to numerous biological processes. Fluorescence recovery after photobleaching (FRAP) is one of the most common method for quantifying molecular diffusivity in living cells using standard laser scanning confocal microscopy (LSCM). However, accurately measuring fast cytoplasmic diffusion (typically >10 m^2/s) is challenging due to rapid recovery kinetics, weak signal-to-noise ratios, post-bleach signal artifacts, and spatial restrictions affecting normalization. While individual challenges have been addressed in specific contexts, a simple and robust framework to quantify cytoplasmic diffusivity remains elusive. Here, we present a FRAP methodology specifically designed to overcome these obstacles. By utilizing the Gaussian function -- the impulse response (ImpRes) of the diffusion equation in an infinite medium -- our approach leverages the full spatiotemporal dataset through a single-equation three-parameter fitting procedure, thus releasing restrictions to small regions of interest and arbitrary initial time-points. The methodology was validated on three datasets of increasing complexity: in silico simulated recovery profiles, in vitro data from FITC-dextran in glycerol solution, and live-cell imaging of free cytoplasmic GFP. Systematic comparison with existing models demonstrates that the ImpRes approach significantly reduces sensitivity to noise and imperfect fluorescence normalization, while remaining robust against short-term biases, such as transient probe photo-activation. Given its robustness under realistic experimental conditions and its ease of implementation, the proposed FRAP methodology provides a reliable tool for quantitative cytoplasmic analysis.

Recent experiments show that protein condensates sitting on opposite surfaces of a flat lipid membrane move together and prefer to overlap, even though they cannot touch each other. This points to an indirect, membrane-mediated interaction. Two mechanisms could be responsible: a curvature-induced interaction, which is energetic in origin, and a fluctuation-induced interaction, which is entropic. Here we study both with coarse-grained molecular dynamics simulations, using Cookes implicit-solvent lipid model together with a generic bead-spring polymer model for the condensate. We compute the potential of mean force between two condensates across the membrane. For condensates of the same size, full overlap is unfavorable, and the pair instead settles into a partially overlapping state that bends the membrane into an S-like shape. When the two condensates differ strongly in size, full overlap becomes favorable. We explain this with a simple geometric picture. The condensate wets the membrane as a thin film and imposes curvature only along its rim, while membrane tension flattens the membrane under its interior. The resulting ring of curvature can trap a smaller condensate on the opposite side. We also compare the bending undulations and the effective bending modulus of a bare membrane, a membrane with one condensate, and a membrane with condensates on both sides. A wetting condensate suppresses the undulation modes and stiffens the membrane, but whether this makes overlap entropically favorable remains inconclusive. Our results indicate that the coupling is driven mainly by curvature, and that it depends on the wetting mechanism and on the membrane tension.