Sperm of marine invertebrates have to find eggs cells in the ocean. Turbulent flows mix sperm and egg cells up to the millimeter scale; below this, active swimming and chemotaxis become important. Previous work addressed either turbulent mixing or chemotaxis in still water. Here, we present a general theory of sperm chemotaxis inside the smallest eddies of turbulent flow, where signaling molecules released by egg cells are spread into thin concentration filaments. Sperm cells ‘surf’ along these filaments towards the egg. External flows make filaments longer, but also thinner. These opposing effects set an optimal flow strength. The optimum predicted by our theory matches flow measurements in shallow coastal waters. Our theory quantitatively agrees with two previous fertilization experiments in Taylor-Couette chambers and provides a mechanistic understanding of these early experiments. ‘Surfing along concentration filaments’ could be a paradigm for navigation in complex environments in the presence of turbulent flow. Many motile cells navigate in complex environments along concentration gradients of signaling molecules. This chemotaxis has been studied extensively both experimentally and theoretically, yet mostly for idealized conditions of perfect chemical gradients. But under physiological conditions, concentration fields are subject to distortions, e.g., by turbulent flows in the ocean. Pioneering experiments suggest that in species with external fertilization, chemotaxis of sperm cells towards the egg may even work better at an optimal flow strength compared to conditions of still water. Yet to date, the mechanistic cause for this optimum is not known. We present a general theory of chemotactic navigation in external flow. We characterize how external flow distorts concentration fields into long filaments, and show how chemotaxing cells can subsequently ‘surf’ along these filaments towards a chemoattractant source. Stronger flows make concentration filaments longer, but also thinner; together, these two counter-acting effects set an optimal flow strength. Beyond fertilization of marine invertebrates, we believe that ‘surfing along concentration filaments’ could be a more general paradigm, relevant also for the ecology of marine bacteria feeding on organic marine snow in the ocean, or chemotaxis inside multi-cellular organisms with internal flows.

Tumor heterogeneity and cellular plasticity are key determinants of tumor progression, metastatic spread, and therapy response driven by the cancer stem cell (CSC) population. Within the current study, we analyzed irradiation-induced plasticity within the aldehyde dehydrogenase (ALDH)-positive (ALDH+) population in prostate cancer. The radiosensitivity of xenograft tumors derived from ALDH+ and ALDH-negative (ALDH−) cells was determined with local tumor control analyses and demonstrated different dose–response profiles, time to relapse, and focal adhesion signaling. The transcriptional heterogeneity was analyzed in pools of 10 DU145 and PC3 cells with multiplex gene expression analyses and illustrated a higher degree of heterogeneity within the ALDH+ population that even increases upon irradiation in comparison with ALDH− cells. Phenotypic conversion and clonal competition were analyzed with fluorescence protein-labeled cells to distinguish cellular origins in competitive three-dimensional cultures and xenograft tumors. We found that the ALDH+ population outcompetes ALDH− cells and drives tumor growth, in particular upon irradiation. The observed dynamics of the cellular state compositions between ALDH+ and ALDH− cells in vivo before and after tumor irradiation was reproduced by a probabilistic Markov compartment model that incorporates cellular plasticity, clonal competition, and phenotype-specific radiosensitivities. Transcriptional analyses indicate that the cellular conversion from ALDH− into ALDH+ cells within xenograft tumors under therapeutic pressure was partially mediated through induction of the transcriptional repressor SNAI2. In summary, irradiation-induced cellular conversion events are present in xenograft tumors derived from prostate cancer cells and may be responsible for radiotherapy failure. Implications: The increase of ALDH+ cells with stem-like features in prostate xenograft tumors after local irradiation represents a putative cellular escape mechanism inducing tumor radioresistance.

Background: Multicellular tumor spheroids (MCTS) are advanced cell culture systems for assessing the impact of combinatorial radio(chemo)therapy as they exhibit therapeutically relevant in vivo–like characteristics from 3-dimensional cell–cell and cell–matrix interactions to radial pathophysiological gradients. State-of-the-art assays quantify long-term curative endpoints based on collected brightfield image time series from large treated spheroid populations per irradiation dose and treatment arm. This analyses require laborious spheroid segmentation of up to 100,000 images per treatment arm to extract relevant structural information from the images (e.g., diameter, area, volume, and circularity). While several image analysis algorithms are available for spheroid segmentation, they all focus on compact MCTS with a clearly distinguishable outer rim throughout growth. However, they often fail for the common case of treated MCTS, which may partly be detached and destroyed and are usually obscured by dead cell debris. Results: To address these issues, we successfully train 2 fully convolutional networks, UNet and HRNet, and optimize their hyperparameters to develop an automatic segmentation for both untreated and treated MCTS. We extensively test the automatic segmentation on larger, independent datasets and observe high accuracy for most images with Jaccard indices around 90%. For cases with lower accuracy, we demonstrate that the deviation is comparable to the interobserver variability. We also test against previously published datasets and spheroid segmentations. Conclusions: The developed automatic segmentation can not only be used directly but also integrated into existing spheroid analysis pipelines and tools. This facilitates the analysis of 3-dimensional spheroid assay experiments and contributes to the reproducibility and standardization of this preclinical in vitro model.

Understanding the complex dynamics of tumor growth to develop more efficient therapeutic strategies is one of the most challenging problems in biomedicine. Three-dimensional (3D) tumor spheroids, reflecting avascular microregions within a tumor, are an advanced in vitro model system to assess the curative effect of combinatorial radio(chemo)therapy. Tumor spheroids exhibit particular crucial pathophysiological characteristics such as a radial oxygen gradient that critically affect the sensitivity of the malignant cell population to treatment. However, spheroid experiments remain laborious, and determining long-term radio(chemo)therapy outcomes is challenging. Mathematical models of spheroid dynamics have the potential to enhance the informative value of experimental data, and can support study design; however, they typically face one of two limitations: while non-spatial models are computationally cheap, they lack the spatial resolution to predict oxygen-dependent radioresponse, whereas models that describe spatial cell dynamics are computationally expensive and often heavily parameterized, impeding the required calibration to experimental data. Here, we present an effectively one-dimensional mathematical model based on the cell dynamics within and across radial spheres which fully incorporates the 3D dynamics of tumor spheroids by exploiting their approximate rotational symmetry. We demonstrate that this radial-shell (RS) model reproduces experimental spheroid growth curves of several cell lines with and without radiotherapy, showing equal or better performance than published models such as 3D agent-based models. Notably, the RS model is sufficiently efficient to enable multi-parametric optimization within previously reported and/or physiologically reasonable ranges based on experimental data. Analysis of the model reveals that the characteristic change of dynamics observed in experiments at small spheroid volume originates from the spatial scale of cell interactions. Based on the calibrated parameters, we predict the spheroid volumes at which this behavior should be observable. Finally, we demonstrate how the generic parameterization of the model allows direct parameter transfer to 3D agent-based models.

Understanding the segregation of cells is crucial to answer questions about tissue formation in embryos or tumor progression. Steinberg proposed that separation of cells can be compared to the separation of two liquids. Such a separation is well described by the Cahn-Hilliard (CH) equations and the segregation indices exhibit an algebraic decay with exponent 1/3 with respect to time. Similar exponents are also observed in cell-based models. However, the scaling behavior in these numerical models is usually only examined in the asymptotic regime and these models have not been directly applied to actual cell segregation data. In contrast, experimental data also reveals other scaling exponents and even slow logarithmic scaling laws. These discrepancies are commonly attributed to the effects of collective motion or velocity-dependent interactions. By calibrating a 2D cellular automaton (CA) model which efficiently implements a dynamic variant of the differential adhesion hypothesis to 2D experimental data from Méhes et al., we reproduce the biological cell segregation experiments with just adhesive forces. The segregation in the cellular automaton model follows a logarithmic scaling initially, which is in contrast to the proposed algebraic scaling with exponent 1/3. However, within the less than two orders of magnitudes in time which are observable in the experiments, a logarithmic scaling may appear as a pseudo-algebraic scaling. In particular, we demonstrate that the cellular automaton model can exhibit a range of exponents ≤1/3 for such a pseudo-algebraic scaling. Moreover, the time span of the experiment falls into the transitory regime of the cellular automaton rather than the asymptotic one. We additionally develop a method for the calibration of the 2D Cahn-Hilliard model and find a match with experimental data within the transitory regime of the Cahn-Hilliard model with exponent 1/4. On the one hand this demonstrates that the transitory behavior is relevant for the experiment rather than the asymptotic one. On the other hand this corroborates the ambiguity of the scaling behavior, when segregation processes can be only observed on short time spans.

The process of cell-sorting is essential for development and maintenance of tissues. Mathematical modeling can provide the means to analyze the consequences of different hypotheses about the underlying mechanisms. With the Differential Adhesion Hypothesis, Steinberg proposed that cell-sorting is determined by quantitative differences in cell-type-specific intercellular adhesion strengths. An implementation of the Differential Adhesion Hypothesis is the Differential Migration Model by Voss-Böhme and Deutsch. There, an effective adhesion parameter was derived analytically for systems with two cell types, which predicts the asymptotic sorting pattern. However, the existence and form of such a parameter for more than two cell types is unclear. Here, we generalize analytically the concept of an effective adhesion parameter to three and more cell types and demonstrate its existence numerically for three cell types based on in silico time-series data that is produced by a cellular-automaton implementation of the Differential Migration Model. Additionally, we classify the segregation behavior using statistical learning methods and show that the estimated effective adhesion parameter for three cell types matches our analytical prediction. Finally, we demonstrate that the effective adhesion parameter can resolve a recent dispute about the impact of interfacial adhesion, cortical tension and heterotypic repulsion on cell segregation.