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Journal of Mathematical Biology Apr 2022We consider a reaction-diffusion system of densities of two types of particles, introduced by Hannezo et al. (Cell 171(1):242-255.e27, 2017). It is a simple model for a...
We consider a reaction-diffusion system of densities of two types of particles, introduced by Hannezo et al. (Cell 171(1):242-255.e27, 2017). It is a simple model for a growth process: active, branching particles form the growing boundary layer of an otherwise static tissue, represented by inactive particles. The active particles diffuse, branch and become irreversibly inactive upon collision with a particle of arbitrary type. In absence of active particles, this system is in a steady state, without any a priori restriction on the amount of remaining inactive particles. Thus, while related to the well-studied FKPP-equation, this system features a game-changing continuum of steady state solutions, where each corresponds to a possible outcome of the growth process. However, simulations indicate that this system self-organizes: traveling fronts with fixed shape arise under a wide range of initial data. In the present work, we describe all positive and bounded traveling wave solutions, and obtain necessary and sufficient conditions for their existence. We find a surprisingly simple symmetry in the pairs of steady states which are joined via heteroclinic wave orbits. Our approach is constructive: we first prove the existence of almost constant solutions and then extend our results via a continuity argument along the continuum of limiting points.
Topics: Computer Simulation; Diffusion; Models, Biological
PubMed: 35482091
DOI: 10.1007/s00285-022-01753-z -
Biophysical Journal Nov 2015The efficient treatment of many ocular diseases depends on the rapid diffusive distribution of solutes such as drugs or drug delivery vehicles through the vitreous...
The efficient treatment of many ocular diseases depends on the rapid diffusive distribution of solutes such as drugs or drug delivery vehicles through the vitreous humor. However, this multicomponent hydrogel possesses selective permeability properties, which allow for the diffusion of certain molecules and particles, whereas others are immobilized. In this study, we perform an interspecies comparison showing that the selective permeability properties of the vitreous are conserved across several mammalian species. We identify the polyanionic glycosaminoglycans hyaluronic acid and heparan sulfate as two key macromolecules that establish this selective permeability. We show that electrostatic interactions between the polyanionic macromolecules and diffusing solutes can be weakened by charge screening or enzymatic glycosaminoglycan digestion. Furthermore, molecule penetration into the vitreous is also charge-dependent and only efficient as long as the net charge of the molecule does not exceed a certain threshold.
Topics: Animals; Cattle; Diffusion; Heparitin Sulfate; Humans; Hyaluronic Acid; Permeability; Sheep; Swine; Vitreous Body
PubMed: 26588575
DOI: 10.1016/j.bpj.2015.10.002 -
Journal of the American Chemical Society Oct 2022The kinetics of chemical reactions are determined by the law of mass action, which has been successfully applied to homogeneous, dilute mixtures. At nondilute...
The kinetics of chemical reactions are determined by the law of mass action, which has been successfully applied to homogeneous, dilute mixtures. At nondilute conditions, interactions among the components can give rise to coexisting phases, which can significantly alter the kinetics of chemical reactions. Here, we derive a theory for chemical reactions in coexisting phases at phase equilibrium. We show that phase equilibrium couples the rates of chemical reactions of components with their diffusive exchanges between the phases. Strikingly, the chemical relaxation kinetics can be represented as a flow along the phase equilibrium line in the phase diagram. A key finding of our theory is that differences in reaction rates between coexisting phases stem solely from phase-dependent reaction rate coefficients. Our theory is key to interpreting how concentration levels of reactive components in condensed phases control chemical reaction rates in synthetic and biological systems.
Topics: Kinetics; Diffusion
PubMed: 36241174
DOI: 10.1021/jacs.2c06265 -
Journal of the Royal Society, Interface Mar 2023How memory shapes animals' movement paths is a topic of growing interest in ecology, with connections to planning for conservation and climate change. Empirical studies...
How memory shapes animals' movement paths is a topic of growing interest in ecology, with connections to planning for conservation and climate change. Empirical studies suggest that memory has both temporal and spatial components, and can include both attractive and aversive elements. Here, we introduce reinforced diffusions (the continuous time counterpart of reinforced random walks) as a modelling framework for understanding the role that memory plays in determining animal movements. This framework includes reinforcement via functions of time before present and of distance away from a current location. Focusing on the interplay between memory and central place attraction (a component of home ranging behaviour), we explore patterns of space usage that result from the reinforced diffusion. Our efforts identify three qualitatively different behaviours: bounded wandering behaviour that does not collapse spatially, collapse to a very small area, and, most intriguingly, convergence to a cycle. Subsequent applications show how reinforced diffusion can create movement trajectories emulating the learning of movement routes by homing pigeons and consolidation of ant travel paths. The mathematically explicit manner with which assumptions about the structure of memory can be stated and subsequently explored provides linkages to biological concepts like an animal's 'immediate surroundings' and memory decay.
Topics: Animals; Ecology; Learning; Diffusion; Movement; Models, Biological
PubMed: 36987616
DOI: 10.1098/rsif.2022.0700 -
Journal of the Royal Society, Interface Mar 2021Swarming has been observed in various biological systems from collective animal movements to immune cells. In the cellular context, swarming is driven by the secretion...
Swarming has been observed in various biological systems from collective animal movements to immune cells. In the cellular context, swarming is driven by the secretion of chemotactic factors. Despite the critical role of chemotactic swarming, few methods to robustly identify and quantify this phenomenon exist. Here, we present a novel method for the analysis of time series of positional data generated from realizations of agent-based processes. We convert the positional data for each individual time point to a function measuring agent aggregation around a given area of interest, hence generating a functional time series. The functional time series, and a more easily visualized of agent aggregation derived from these functions, provide useful information regarding the evolution of the underlying process over time. We extend our method to build upon the modelling of collective motility using drift-diffusion partial differential equations (PDEs). Using a functional linear model, we are able to use the functional time series to estimate the drift and diffusivity terms associated with the underlying PDE. By producing an accurate estimate for the drift coefficient, we can infer the strength and range of attraction or repulsion exerted on agents, as in chemotaxis. Our approach relies solely on using agent positional data. The spatial distribution of diffusing chemokines is not required, nor do individual agents need to be tracked over time. We demonstrate our approach using random walk simulations of chemotaxis and experiments investigating cytotoxic T cells interacting with tumouroids.
Topics: Animals; Cell Tracking; Chemotactic Factors; Chemotaxis; Diffusion; Models, Biological; Movement
PubMed: 33715400
DOI: 10.1098/rsif.2020.0879 -
Journal of Colloid and Interface Science Jan 2022The use of isotropic potential models of simple colloids for describing complex protein-protein interactions is a topic of ongoing debate in the biophysical community....
The use of isotropic potential models of simple colloids for describing complex protein-protein interactions is a topic of ongoing debate in the biophysical community. This contention stems from the unavailability of synthetic protein-like model particles that are amenable to systematic experimental characterization. In this article, we test the utility of colloidal theory to capture the solution structure, interactions and dynamics of novel globular protein-mimicking, computationally designed peptide assemblies called bundlemers that are programmable model systems at the intersection of colloids and proteins. Small-angle neutron scattering (SANS) measurements of semi-dilute bundlemer solutions in low and high ionic strength solution indicate that bundlemers interact locally via repulsive interactions that can be described by a screened repulsive potential. We also present neutron spin echo (NSE) spectroscopy results that show high-Q freely-diffusive dynamics of bundlemers. Importantly, formation of clusters due to short-range attractive, inter-bundlemer interactions is observed in SANS even at dilute bundlemer concentrations, which is indicative of the complexity of the bundlemer charged surface. The similarities and differences between bundlemers and simple colloidal as well as complex protein-protein interactions is discussed in detail.
Topics: Colloids; Diffusion; Peptides; Proteins; Scattering, Small Angle
PubMed: 34749446
DOI: 10.1016/j.jcis.2021.09.184 -
Biophysical Journal Oct 2021Erk signaling regulates cellular decisions in many biological contexts. Recently, we have reported a series of Erk activity traveling waves that coordinate regeneration...
Erk signaling regulates cellular decisions in many biological contexts. Recently, we have reported a series of Erk activity traveling waves that coordinate regeneration of osteoblast tissue in zebrafish scales. These waves originate from a central source region, propagate as expanding rings, and impart cell growth, thus controlling tissue morphogenesis. Here, we present a minimal reaction-diffusion model for Erk activity waves. The model considers three components: Erk, a diffusible Erk activator, and an Erk inhibitor. Erk stimulates both its activator and inhibitor, forming a positive and negative feedback loop, respectively. Our model shows that this system can be excitable and propagate Erk activity waves. Waves originate from a pulsatile source that is modeled by adding a localized basal production of the activator, which turns the source region from an excitable to an oscillatory state. As Erk activity periodically rises in the source, it can trigger an excitable wave that travels across the entire tissue. Analysis of the model finds that positive feedback controls the properties of the traveling wavefront and that negative feedback controls the duration of Erk activity peak and the period of Erk activity waves. The geometrical properties of the waves facilitate constraints on the effective diffusivity of the activator, indicating that waves are an efficient mechanism to transfer growth factor signaling rapidly across a large tissue.
Topics: Animals; Diffusion; Models, Theoretical; Osteoblasts; Signal Transduction; Zebrafish
PubMed: 34022234
DOI: 10.1016/j.bpj.2021.05.004 -
Biophysical Journal Jun 2020Mesenchymal cell crawling is a critical process in normal development, in tissue function, and in many diseases. Quantitatively predictive numerical simulations of cell...
Mesenchymal cell crawling is a critical process in normal development, in tissue function, and in many diseases. Quantitatively predictive numerical simulations of cell crawling thus have multiple scientific, medical, and technological applications. However, we still lack a low-computational-cost approach to simulate mesenchymal three-dimensional (3D) cell crawling. Here, we develop a computationally tractable 3D model (implemented as a simulation in the CompuCell3D simulation environment) of mesenchymal cells crawling on a two-dimensional substrate. The Fürth equation, the usual characterization of mean-squared displacement (MSD) curves for migrating cells, describes a motion in which, for increasing time intervals, cell movement transitions from a ballistic to a diffusive regime. Recent experiments have shown that for very short time intervals, cells exhibit an additional fast diffusive regime. Our simulations' MSD curves reproduce the three experimentally observed temporal regimes, with fast diffusion for short time intervals, slow diffusion for long time intervals, and intermediate time -interval-ballistic motion. The resulting parameterization of the trajectories for both experiments and simulations allows the definition of time- and length scales that translate between computational and laboratory units. Rescaling by these scales allows direct quantitative comparisons among MSD curves and between velocity autocorrelation functions from experiments and simulations. Although our simulations replicate experimentally observed spontaneous symmetry breaking, short-timescale diffusive motion, and spontaneous cell-motion reorientation, their computational cost is low, allowing their use in multiscale virtual-tissue simulations. Comparisons between experimental and simulated cell motion support the hypothesis that short-time actomyosin dynamics affects longer-time cell motility. The success of the base cell-migration simulation model suggests its future application in more complex situations, including chemotaxis, migration through complex 3D matrices, and collective cell motion.
Topics: Cell Movement; Computer Simulation; Diffusion; Models, Biological; Motion
PubMed: 32407685
DOI: 10.1016/j.bpj.2020.04.024 -
Journal of Biomechanics Sep 2016The cartilage endplate (CEP) is implicated as the main pathway of nutrient supply to the healthy human intervertebral disc (IVD). In this study, the diffusivities of...
The cartilage endplate (CEP) is implicated as the main pathway of nutrient supply to the healthy human intervertebral disc (IVD). In this study, the diffusivities of nutrient/metabolite solutes in healthy CEP were assessed, and further correlated with tissue biochemical composition and structure. The CEPs from non-degenerated human IVD were divided into four regions: central, lateral, anterior, and posterior. The diffusivities of glucose and lactate were measured with a custom diffusion cell apparatus under 0%, 10%, and 20% compressive strains. Biochemical assays were conducted to quantify the water and glycosaminoglycan (GAG) contents. The Safranin-O and Ehrlich׳s hematoxylin and eosin staining and scanning electron microscopy (SEM) were performed to reveal the tissue structure of the CEP. Average diffusivities of glucose and lactate in healthy CEP were 2.68±0.93×10cm/s and 4.52±1.47×10cm/s, respectively. Solute diffusivities were region-dependent (p<0.0001) with the highest values in the central region, and mechanical strains impeded solute diffusion in the CEP (p<0.0001). The solute diffusivities were significantly correlated with the tissue porosities (glucose: p<0.0001, r=0.581; lactate: p<0.0001, r=0.534). Histological and SEM studies further revealed that the collagen fibers in healthy CEP are more compacted than those in the nucleus pulposus (NP) and annulus fibrosus (AF) and show no clear orientation. Compared to human AF and NP, much smaller solute diffusivities in human CEP suggested that it acts as a gateway for solute diffusion through the disc, maintaining the balance of nutritional environment in healthy human disc under mechanical loading and preventing the progression of disc degeneration.
Topics: Cartilage; Diffusion; Female; Glucose; Glycosaminoglycans; Humans; Intervertebral Disc; Intervertebral Disc Degeneration; Lactic Acid; Male; Middle Aged; Stress, Mechanical; Water
PubMed: 27338525
DOI: 10.1016/j.jbiomech.2016.06.008 -
Proceedings of the National Academy of... Dec 2022A noisy stabilized Kuramoto-Sivashinsky equation is analyzed by stochastic decomposition. For values of the control parameter for which periodic stationary patterns...
A noisy stabilized Kuramoto-Sivashinsky equation is analyzed by stochastic decomposition. For values of the control parameter for which periodic stationary patterns exist, the dynamics can be decomposed into diffusive and transverse parts which act on a stochastic potential. The relative positions of stationary states in the stochastic global potential landscape can be obtained from the topology spanned by the low-lying eigenmodes which interconnect them. Numerical simulations confirm the predicted landscape. The transverse component also predicts a universal class of vortex-like circulations around fixed points. These drive nonlinear drifting and limit cycle motion of the underlying periodic structure in certain regions of parameter space. Our findings might be relevant in studies of other nonlinear systems such as deep learning neural networks.
Topics: Diffusion; Motion; Neural Networks, Computer
PubMed: 36459639
DOI: 10.1073/pnas.2211359119