The European Physical Journal Plus

The Poisson-Nernst-Planck (PNP) system is an accurate model of electrodiffusion of ionic species. It is commonly used in situations where nanoscale resolution is required, for instance close to ion channels in the membranes of biological cells. The inherent stiffness of the equations has made them challenging to solve and has limited the applicability of the system. In particular, the time step required for stable solutions has typically needed to be very short (nanoseconds), which makes simulations on the time scale of an action potential (milliseconds) difficult. Recently, it has been observed that avoiding operator splitting and instead solving the concentration equations and the electrostatic equation in a coupled manner relaxes the time-step limitation considerably. However, no theoretical explanation of this observation has been provided. Here, we aim to explain why the coupled scheme allows much larger time steps. We illustrate the mechanism by considering special cases that define necessary, but not sufficient, conditions for stability. We also show that these conditions remain relevant for the fully coupled PNP model in 3D.

This work focuses on the global and partial identification problem for fractional differential equations. We provide a general numerical procedure based on global and local optimization algorithms with two refinements for biological systems that ensure solution positivity and homogeneous parameter units. The method is applied to a new fractional model of Dengue outbreak called the Fractional Homogeneous Nishiura (FHN) model, calibrated using data of newly infected people in Cape Verde. We show that our identification method yields a better fit between data and model solutions than previous approaches and that our FHN model captures the dynamics of Dengue more closely than existing systems.

A major challenge in neuroscience is to predict how currents in nanodomains affect voltage and ionic concentrations. Cable and Rall theory provide analytic current-voltage relations by neglecting concentration gradients, and the impact of concentration gradients is usually studied numerically with the Poisson-Nernst-Planck (PNP) model. A precise quantitative understanding of the combined dynamics remains limited because analytic current-voltage-concentration relations are missing. In this work we derive such relations using a novel approach based on cross-diffusion equations. For narrow cylindrical domains, we derive time-dependent and steady-state expressions that explicitly show how currents affect voltage and ionic concentrations. We find that the influx of only one ion can significantly change the concentrations of all the other ions even if no channels for these ions are present. After a current injection we compute a biphasic voltage transient where the small-time asymptotic corresponds to the steady-state solution of the cable equation. We show that the accuracy of cable theory prediction for the voltage depends on how the current is distributed among the various ions. Finally, we develop an iterative method to accurately compute steady-state profiles for voltage and concentrations using first-order results by subdividing a cylinder into small segments.

Ultraviolet (UV) radiation induces DNA damage associated with cancer and aging, yet the sequence dependence of UV absorption remains to be investigated. Here, we present a systematic study of the UV absorption spectra of DNA based on all-electron Hartree-Fock calculations. We analyze all possible sequences up to four base pairs, as well as longer randomized sequences and genomic nullomers - motifs which are missing in a given genome. We observe a pronounced sequence dependence: cytosine- and guanine-rich motifs exhibit significantly enhanced absorption, whereas adenine-thymine-rich sequences absorb up to four times less in the mid-UV range. Notably, the human genome is biased toward adenine-thymine-rich sequences, giving it an increased susceptibility to UV-induced damage. In addition, we introduce a computational framework enabling spectral calculations of large DNA and RNA fragments, opening the door to large-scale optical analyses.

With the development of microfluidics, it has now become possible to assess the susceptibility of bacteria to antibiotics at the single-cell level instead of relying on population measurements. Such studies are particularly relevant when the growth of bacterial population in the presence of antibiotics is heterogeneous. Here, we build a model to describe such a case, and apply it to experimental measurements on a small population of E. Coli exposed to ciprofloxacin, a drug which is well known for triggering a bistable response.

Invertebrate vision relies on bistable visual pigments flipping upon photon absorption between rhodopsin and metarhodopsin states. In living butterflies, the UV-VIS absorption spectra of rhodopsin and metarhodopsin, respectively with 11-cis and all-trans isomers of 3-hydroxy-retinal (A3) chromophore, can be conveniently recorded from the eyeshine, the light reflected from the compound eye after passing twice through the light-guiding rhabdoms. * Here, a microscope coupled with a broadband LED source and a microspectrometer was used to record photorelaxations reported in eyeshine reflection spectra. Fitting temporal exponential relaxations to log-reflectance arrays yielded transient and baseline spectra that are analogous to absorbance difference and sum, respectively. Both types of spectra were subjected to singular value decomposition and to fitting of templated visual pigment absorption spectra. * The compound eye of the high brown fritillary Fabriciana adippe was exposed to a series of second-long broadband light pulses, causing photorelaxations with time constants between 40 and 120 ms that led to 80% metarhodopsin in equilibrium. The transient and baseline spectra were fitted with pigment templates, estimating the alpha peak wavelength 547-552 nm for rhodopsin and 496-501 nm for metarhodopsin. The metarhodopsin to rhodopsin alpha peak absorbance ratio 1.25-1.35 is consistent with the isosbestic wavelength at 530 nm. The second isosbestic wavelength indicates that rhodopsin beta (UV) peak absorbs more strongly than metarhodopsin below 405 nm. * Baseline spectra, which were not explicitly analysed in previous studies, enable concatenation of exposures, monitor long-term changes of pigment, and enhance the estimation of beta peak parameters. * The method can be directly used in many butterflies and could be adapted to other insects, particularly fruitflies, facilitating studies of the relation between the visual pigment spectra and the opsin sequences. Spectroscopic results can be complemented with physiologically measured photoreceptor spectral sensitivity datasets and analysed with the same global fitting procedure.

The spatial resolution of optical imaging systems is fundamentally restricted by the diffraction limit. However, in widefield live-cell microscopy, the achievable resolution is further constrained by the specimen motion, which indicates the existence of a fundamental spatio-temporal resolution trade-off between signal accumulation during the full frame integration and the resulting motion blur. To improve the fidelity with which moving objects can be imaged, a quantitative understanding of this spatio-temporal trade-off is necessary. Here, we present a systematic analysis of motion-induced resolution dynamics measured with spectral signal-to-noise ratio (SSNR). We developed a simulation framework which models the image formation of objects undergoing arbitrary motion, to evaluate the degradation of the spatial resolution under translational and rotational dynamics. Our results demonstrate that for translating objects, the spatial resolution is anisotropically reduced as a function of the orientation of the object relative to the motion vector, leading to the spectral signal-to-noise ratio degrading by up to 50% and the resolution by up to 40% for a 90{degrees} change in the motion direction. Furthermore, we show that for rotational motion, conventional radially averaged metrics such as the Fourier Ring Correlation are not able to quantify the effects of angular blur. On the other hand, the SSNR is able to accurately quantify this degradation. These findings underscore the necessity of an object-oriented imaging approach, in which acquisition parameters such as exposure time are tuned to specific biological spatio-temporal characteristics to optimize the trade-off between motion blur and spatial fidelity.

Vertex models are widely used within the field of developmental biology to study tissue morphogenesis. These models are well-suited for modeling deformation at the cellular level where movement is driven by local forces. However, understanding how these microscopic movements coordinate to yield macroscopic phenomena such as the shapes of entire tissues remains a challenge. Here we study a top-down approach using differentiable programming on a simplified vertex model of a laminar tissue, and investigate whether the attributes of individual cells can be tuned to make the mesh as a whole acquire a predefined shape. We let the mesh evolve according to simple rules defined by the input to each polygon, and evaluate the resulting shape against a target boundary. Additionally, we show how the high degeneracy of the output can be reduced by constraining the polygon distributions: first, by adding simple penalties on tissue-wide attributes; and second, by dividing the tissue into regions, within which we bias the attributes toward characteristic values. Our study shows how a simple vertex model can be combined with differentiable programming to model developing tissues, and provides insight into the way individual cells must coordinate to yield macroscopic phenomena such as pre-programmed shapes.

Developing parsimonious, mechanism-aware quantitative models that predict how biological effectiveness changes with different modifiers remains, in general, an unsolved problem. Advances in radiobiological research have created a large knowledge base of first-principles mechanistic models of radiation response that, in principle, could accurately predict radiosensitivity across different experimental and clinical conditions. However, in practice these mechanistic models come with an overabundance of parameters, the majority of which are practically unidentifiable and, moreover, likely unnecessary if one simply wishes to predict how radiosensitivity changes for some specific modifier of interest. Nevertheless, determining which few details in the full mechanistic model are relevant for a given purpose, as well as how to remove any other extraneous details, remains a highly non-trivial task. In this study, we demonstrate the potential of model reduction, starting from a detailed mechanistic description, as a systematic strategy for deriving parsimonious, experimentally falsifiable radiobiological descriptors. As a proof-of-concept demonstration, we apply the Manifold Boundary Approximation Method (MBAM) to a Mechanistic Model of DNA Repair and Survival (MEDRAS), for the problem of cell survival prediction following an acute exposure. Our findings reveal that the complete MEDRAS model for an arbitrary mixed-quality exposure can be structurally simplified to a reduced three-parameter model for an effective uniform-quality, named MEDRAS-LPL. Additional MBAM analysis on MEDRAS-LPL identifies two boundaries in parameter space, corresponding to sparsely ionizing and densely ionizing radiation. Mapping of MEDRAS-LPL parameter space on to effective LQ space further demonstrates that parameters close to the sparsely ionizing boundary line up with expectations from the theory of dual radiation, while parameters close to the densely ionizing boundary line up with expectations from a purely linear model based on a target-theory description. Moreover, our formalism predicts enhanced synergistic interactions between sparsely ionizing and densely ionizing radiation beyond the Zaider Rossi model (ZRM) paradigm, in line with empirical observations. The results highlight the potential for using reduced-order models not only for predictive applications but also for generating novel hypotheses that can inform future experimental designs and optimization strategies in radiobiology.

This study presents a proof-of-concept cyber-physical architecture integrating a SEIR epidemiological model (Susceptible-Exposed-Infectious-Recovered), implemented in MATLAB, with a simulated Internet of Things (IoT) acquisition and transmission stage based on the ESP32 microcontroller and the ThingSpeak platform. The system generates synthetic biomedical signals of body temperature and peripheral oxygen saturation (SpO2), structured across three levels: circadian variation, scheduled pathological episodes, and Gaussian noise. These signals feed a dual parametric coupling function that dynamically updates the SEIR transmission parameter as a combined function of body temperature and oxygen saturation deviations from their clinical reference values. The proposed architecture is organized into four functional phases: measurement, communication, computational processing, and feedback. Five simulated clinical scenarios were evaluated, ranging from normal conditions (T = 36.5 {degrees}C, SpO2 = 97%) to fever with severe hypoxia (T = 38.5 {degrees}C, SpO2 = 88%), yielding basic reproduction number (R0) values between 4.20 and 5.38, and peak infected proportions between 29.9% and 35.2% of the simulated population (N = 1,000). A sensitivity analysis on the coupling coefficients, with {+/-}50% variation from nominal values, showed that the oxygen saturation coefficient is the most influential parameter on R0 (range = 0.76) compared to the thermal coefficient (range = 0.42), with monotonic and predictable behavior across the entire evaluated parametric space. The primary contribution of this work is system integration: we propose a reproducible platform connecting biomedical simulation, IoT communication, and epidemiological modeling through parametric coupling in a controlled environment. All data used are entirely synthetic; a retrospective calibration with real Colombian data from the first epidemic wave of 2020 confirmed the epidemiological consistency of the model, with a calibrated R0 of 1.85 and a Pearson correlation of 0.930. Results should be interpreted as evidence of architectural feasibility, not as clinical or epidemiological validation. Author SummaryThe COVID-19 pandemic made it clear that epidemiological surveillance systems need tools that combine accessible technology with mathematical models capable of anticipating disease spread. In this work, we built a proof-of-concept platform connecting three elements: a low-cost electronic sensor based on the ESP32 microcontroller, a cloud communication platform (ThingSpeak), and a mathematical model that simulates how an epidemic spreads through a population. The sensor generates synthetic data on body temperature and oxygen saturation that, through a mathematical formula we designed, dynamically modify the rate of contagion in the model. We evaluated five clinical scenarios, ranging from normal conditions to fever with severe hypoxia, and analyzed how sensitive the results are to changes in the system parameters. We found that oxygen saturation has a greater influence on the estimated contagion potential than body temperature. Although all data are synthetic, this platform demonstrates that it is possible to integrate low-cost sensors with epidemiological models in real time, opening a viable pathway for early warning systems in resource-limited settings.

Hemodynamic forces play a key role in early cardiac morphogenesis, yet many computational studies assume Newtonian blood behavior. Here, we evaluate the impact of nonNewtonian shearthinning rheology on flow patterns, pressure distributions, and wall shear stress (WSS) during cardiac looping using idealized threedimensional models of the embryonic heart tube. Five geometries representing progressive looping stages, from a linear tube to an Sshaped configuration with ventricular ballooning, were analyzed under pulsatile flow using both Newtonian and powerlaw viscosity models. Across all stages, Reynolds numbers (Re {approx} 1-7) and Womersley numbers (Wo {approx} 0.3) indicated laminar, quasisteady flow consistent with embryonic conditions. Incorporating shearthinning rheology produced substantial deviations from Newtonian predictions, with peak systolic WSS differing by up to [~]40% and pressure drops by up to [~]20%. These effects were most pronounced in regions of increased curvature and geometric complexity. These findings demonstrate that nonNewtonian rheology significantly influences predicted hemodynamic environments during cardiac looping and should be incorporated into computational models aimed at understanding mechanobiological regulation of early heart development.

Mechanical properties of epithelial tissues play essential roles in morphogenesis and physiological function. In this study, we analytically derived the in-plane bulk modulus, shear modulus, and Poissons ratio of a three-dimensional cell vertex model of epithelial monolayers. We showed that the model can robustly reproduce a near-zero in-plane Poissons ratio, a mechanical feature reported in cultured epithelial tissues. Numerical simulations further confirmed that the theoretically predicted Poissons ratio accurately describes the response of the model under finite, biologically relevant strains. In addition, the model exhibits not only morphological bistability between squamous-like and columnar-like states, but also mechanical bistability characterized by distinct elastic responses. Together, these results provide a minimal three-dimensional framework that links cell-scale mechanical interactions and epithelial morphology to tissue-scale elastic properties.

Quantum kernel methods offer a potential advantage for classification tasks in high-dimensional feature spaces, yet their practical benefit critically depends on how input features are prepared. We compare five dimensionality reduction strategies--principal component analysis (PCA), Gaussian random projection (RP Gaussian), sparse random projection (RP Sparse), partial least squares (PLS), and uniform manifold approximation and projection (UMAP) -- as pre-processing steps for quantum kernel support vector machines (SVMs) applied to trabecular bone classification from synthetic micro-computed tomography (micro-CT) data. Using a custom procedural generator based on Gaussian random field zero-crossings, we produced 500 synthetic trabecular bone volumes with controlled morphometric properties such as bone volume fraction (BV/TV), trabecular thickness (Tb.Th), number (Tb.N) and spacing (Tb.Sp). Texture features extracted from grayscale slices are reduced to 8-dimensional quantum circuit inputs via each method, then classified using both classical radial basis function (RBF)-SVMs and quantum kernel SVMs with ZZ feature maps on a statevector simulator, both evaluated with 5 x 5 repeated stratified cross-validation (25 folds). Our results show that UMAP is the only reduction method where the quantum kernel remains competitive with the classical baseline. Under repeated cross-validation, UMAP showed a +0.032 accuracy gap favouring the quantum kernel (Dietterich 5 x 2 CV p = 0.177); however, validation on 10 fully independent datasets--each with independently generated samples, separate reduction fits, and separate kernel matrices -- reversed the sign to -0.030 (paired t-test p = 0.123; Wilcoxon p = 0.193; quantum wins 3/10 datasets), indicating that the apparent advantage was likely inflated by fold dependence. Nevertheless, UMAPs gap remains small and non-significant in both analyses, whereas all linear methods (PCA, RP Gaussian, PLS) show substantial quantum deficits of -0.090 to -0.116 across BV/TV classification, with PCA and PLS remaining significant under corrected tests (5 x 2 CV p = 0.004 and p = 0.007 respectively). We additionally evaluate quantum kernel ridge regression for continuous morphometric prediction, finding that ZZ quantum kernels fail uniformly at regression (negative R2 for all methods except PLS at 4 qubits), suggesting that the ZZ kernel captures decision boundaries but not smooth metric structure. These findings provide practical guidance for feature engineering in near-term quantum machine learning pipelines and demonstrate that the choice of dimensionality reduction can determine whether quantum kernels remain competitive with classical baselines.

Arterial thrombosis is initiated when mechanical forces in flowing blood exceed the activation thresholds of platelets and von Willebrand factor (vWF). Despite extensive experimental characterization of shear-induced platelet aggregation, a unified theoretical framework that maps hemodynamic forcing onto clot nucleation is lacking. Here we present Force-Gated Thrombosis (FGT), a non-equilibrium mechanical theory that treats thrombus formation as a continuous phase transition driven by an effective mechanical forcing {Sigma} ={sigma} + |{nabla}{sigma}| + {beta}{varepsilon}, which combines local wall shear stress{sigma} , shear gradient |{nabla}{sigma}|, and extensional strain rate{varepsilon} . We introduce a dimensionless Thrombosis Number {Theta} = ({Sigma}/{Sigma}c)(P/P0)m(C/C0)n, which incorporates platelet concentration P and coagulation factor concentration C, and governs the transition between stable flow ({Theta} < 1) and active clot growth ({Theta} > 1). The thrombus density is represented by a scalar order parameter{varphi} whose dynamics follow a Ginzburg- Landau free energy functional. For a simplified stenosed artery we derive an analytic closed-form thrombosis onset criterion and a critical flow rate [Formula], where{delta} is stenosis severity. Linear stability analysis shows that perturbations grow at rate{omega} (k) = {Lambda}({Theta}) - D{varphi}k2, becoming unstable when {Theta} > 1. Near threshold the clot volume fraction scales as{varphi} [~] ({Theta} - 1)1/2, a mean-field critical exponent consistent with Ginzburg- Landau theory. Systematic comparison with fifteen published experimental and computational datasets spanning shear rates from 100 to 15,000 s-1 confirms that FGT correctly predicts the existence, location, and approximate severity of pathological thrombus formation across diverse vascular geometries. The theory provides a quantitative bridge between single-molecule mechanobiology and macroscale clinical thrombosis, and yields experimentally testable predictions distinguishing FGT from purely biochemical models.

As the duration of space flights increases, so does the need to optimize off-planet microbial growth. Microbes can both be unintentionally brought into space and cause human disease or be intentionally harnessed for on-site bioengineering functions. However, optimizing microbial growth is challenging due to an insufficient understanding of how microbial communities are affected by the extraterrestrial environment. To address this gap, we have modified a previously developed model for cell growth in microgravity. By improving the functional form used for cell growth as well as the code usability, we enable further research into how microbial communities are influenced by gravity. Applying this model to isolate individual effects of gravity on cell growth indicates that a lack of gravity-driven flow decreases cell growth in microgravity, while the absence of sedimentation increases cell growth in microgravity. These opposite effects likely contribute to the system-dependent effects of microgravity observed experimentally.

Fluorescent reporters cover a wide range of applications in both basic and applied research. Whether a study involves microscopic imaging to study (co)-localization of proteins, FRET, biosensing, or quantifying gene expression, fluorophores are attractive reporter candidates due to their relatively straightforward in vivo readout. For microbiological applications, a wide variety of fluorescent proteins with varying excitation and emission wavelengths, brightness levels, and maturation times are available. Careful consideration is required when selecting from this large suite of proteins, especially when choosing multiple fluorophores. This is further complicated in phototrophic organisms, which exhibit strong autofluorescence, especially towards the red part of the spectrum, effectively eliminating common candidates such as mCherry. In this study, the specific properties and performance of a selection of fluorescent proteins are systematically evaluated against the background of photosynthetic pigment-derived autofluorescence in the cyanobacterium Synechocystis sp. PCC 6803. Specific readouts of different combinations of fluorescent proteins are also analyzed using high-throughput methods, namely plate reader fluorescent scans and single-cell flow cytometry to quantify fluorescence. The ultimate goal is to assess each fluorescent protein with regard to: 1.) Its ability to be discerned from cyanobacterial autofluorescence. 2.) Its compatibility with other fluorophores in this context. 3.) Its overall suitability in cyanobacterial research. Several highly suitable fluorescent proteins for use in cyanobacteria are identified, including mTagBFP2, mNeonGreen and mScarlet-I and suitable combinations, covering nearly the whole spectrum of visible light. This study expands the knowledge and toolset for current and future researchers and uncovers a whole spectrum of possibilities for fluorescent protein selection in cyanobacterial cell biology.

Inositol 1,4,5-trisphosphate (IP3) is a key second messenger that regulates diverse physiological processes. Visualization of IP3 dynamics in living cells is therefore important for understanding its signaling processes. In this study, we developed genetically encoded green fluorescent IP3 biosensors named Green iPenguins with distinct half-maximal effective concentrations (EC50) for IP3, enabling detection of IP3 signals over a range of concentrations. The biosensors displayed more than a 4-fold increase in fluorescence intensity upon IP3 and showed high specificity for IP3 over structurally related molecules. When expressed in HEK293T cells, the biosensors enabled visualization of IP3 dynamics involved in different signaling pathways. They were also compatible with dual-color imaging, allowing simultaneous monitoring of IP3 together with cAMP or Ca2+ signals. In addition, the hierarchical relationship between IP3 and Ca2+ signaling was visualized, providing insight into the temporal relationship between these two second messengers. The biosensors are expected to facilitate future studies of physiological processes involving IP3 signaling networks.

Bacterial communities grow as dynamic populations that respond to their environments. A clinically relevant example is the inactivation of beta-lactam antibiotics by intracellular beta-lactamase in E. faecalis resistant strains. In these populations, resistant bacteria act as antibiotic sinks, detoxifying the environment and allowing sensitive bacteria to survive treatment through a cooperative interaction. In this work, we study strongly coupled planktonic and biofilm populations of mixed sensitive-resistant E. faecalis bacteria under antibiotic stress using fluorescent microscopy. The presence of resistant bacteria in the system benefits both resistant and sensitive cells, leading to mixed planktonic and biofilm populations at super-inhibitory drug concentrations. We show that a beta-lactam antibiotic with or without the addition of a beta-lactam inhibitor can lead to a population inversion effect, characterized by a non-monotonic relation between initial and final fractions of resistant bacteria. The effect is observed in both the planktonic and biofilm populations and is modulated by the total initial cell density. A well-mixed model with competition mediated by resource sharing and cooperation from global degradation of toxins predicts the experimentally observed behavior. These observations suggest underlying population-level mechanisms that are largely independent of biofilm spatial structure.

Cell migration plays a central role in numerous physiological and pathological processes and emerges from the coordinated interplay between intracellular force generation, adhesion dynamics, and mechanical interactions with the environment. A minimal, mechanistically grounded understanding of these processes is required to disentangle the respective contributions of cell-intrinsic and environmental cues. Here, a two-dimensional in silico cell motility model is introduced to describe mesenchymal migration driven by intracellular traction forces generated within actin-rich protrusions anchored to a substrate. The model explicitly accounts for adhesion nucleation, maturation, force buildup and rupture, and relies on a small set of physically interpretable parameters. A systematic mechanical analysis identifies parameter regimes that permit effective cell translocation and delineates conditions leading to stalled or mobile cells. Within motile regimes, the model reproduces a broad spectrum of cell morphologies and migratory behaviours. In particular, cell trajectories exhibit the statistical features of a persistent random walk, with a crossover from ballistic to diffusive motion that arises solely from adhesion dynamics and force balance, without imposing polarization or directional bias. Cell morphology is shown to strongly regulate migration speed, persistence, and pausing behaviour. Altogether, this model provides a minimal reference framework for cell migration on non-deformable substrates and establishes a baseline for future studies of mechanically driven guidance. By construction, it is well suited for extension to deformable fibrous substrates, where cell-induced matrix remodeling and stiffness feedback are expected to bias migration and regulate cell encounters relevant to tissue morphogenesis and anastomosis.

During vertebrate development, the flat surface of the gut epithelium undergoes a dramatic transformation into densely packed arrays of finger-like projections called intestinal villi. Recent studies show that the villus formation relies on a tissue dewetting process, in which mesenchymal tissues buckle the overlying epithelial layer into periodic folds. However, the mechanisms driving subsequent elongation of these folds into finger-like villi remain largely unexplored. Here, we propose a simple mechanism for villus elongation that couples tissue dewetting to cell differentiation, which emerged as a repeated outcome of multiple independent simulations of an evolutionary-developmental Cellular Potts Model. In this mechanism, a liquid-like mesenchymal tissue continuously differentiates into a solid-like mesenchymal tissue at the interface between them. This differentiation drives the liquid-like tissue to continuously retract from the solid-like tissue in the opposite direction of the interface through dewetting, ultimately creating a finger-like projection. A merit of our proposed mechanism is that it only requires two tissues with different viscosities, high surface tension, and cell differentiation. We develop a simplified phase-field model to determine exactly how villus morphology depends on these three requirements. Since these requirements are satisfied not only in intestinal villi but also in many other developing tissues, we propose that the same mechanism could also drive the elongation of other tissues.