Large-scale simulations of biological cell sorting driven by differential adhesion follow diffusion-limited domain coalescence regime

Cell sorting, whereby a heterogeneous cell mixture segregates and forms distinct homogeneous tissues, is one of the main collective cell behaviors at work during development. Although differences in interfacial energies are recognized to be a possible driving source for cell sorting, no clear consensus has emerged on the kinetic law of cell sorting driven by differential adhesion. Using a modified Cellular Potts Model algorithm that allows for efficient simulations while preserving the connectivity of cells, we numerically explore cell-sorting dynamics over very large scales in space and time. For a binary mixture of cells surrounded by a medium, increase of domain size follows a power-law with exponent n = 1/4 independently of the mixture ratio, revealing that the kinetics is dominated by the diffusion and coalescence of rounded domains. We compare these results with recent numerical studies on cell sorting, and discuss the importance of algorithmic differences as well as boundary conditions on the observed scaling.     

Revising Berg-Purcell for finite receptor kinetics

From nutrient uptake to chemoreception to synaptic transmission, many systems in cell biology depend on molecules diffusing and binding to membrane receptors. Mathematical analysis of such systems often neglects the fact that receptors process molecules at finite kinetic rates. A key example is the celebrated formula of Berg and Purcell for the rate that cell surface receptors capture extracellular molecules. Indeed, this influential result is only valid if receptors transport molecules through the cell wall at a rate much faster than molecules arrive at receptors. From a mathematical perspective, ignoring receptor kinetics is convenient because it makes the diffusing molecules independent. In contrast, including receptor kinetics introduces correlations between the diffusing molecules because, for example, bound receptors may be temporarily blocked from binding additional molecules. In this work, we present a modeling framework for coupling bulk diffusion to surface receptors with finite kinetic rates. The framework uses boundary homogenization to couple the diffusion equation to nonlinear ordinary differential equations on the boundary. We use this framework to derive an explicit formula for the cellular uptake rate and show that the analysis of Berg and Purcell significantly overestimates uptake in some typical biophysical scenarios. We confirm our analysis by numerical simulations of a many-particle stochastic system.     

Stability analysis of non-autonomous reaction-diffusion systems: the effects of growing domains

By using asymptotic theory, we generalise the Turing diffusively-driven instability conditions for reaction-diffusion systems with slow, isotropic domain growth. There are two fundamental biological differences between the Turing conditions on fixed and growing domains, namely: (i) we need not enforce cross nor pure kinetic conditions and (ii) the restriction to activator-inhibitor kinetics to induce pattern formation on a growing biological system is no longer a requirement. Our theoretical findings are confirmed and reinforced by numerical simulations for the special cases of isotropic linear, exponential and logistic growth profiles. In particular we illustrate an example of a reaction-diffusion system which cannot exhibit a diffusively-driven instability on a fixed domain but is unstable in the presence of slow growth.     

Multifractality in intracellular enzymatic reactions

Enzymatic kinetics adjust well to the Michaelis-Menten paradigm in homogeneous media with dilute, perfectly mixed reactants. These conditions are quite different from the highly structured cell plasm, so applications of the classic kinetics theory to this environment are rather limited. Cytoplasmic structure produces molecular crowding and anomalous diffusion of substances, modifying the mass action kinetic laws. The reaction coefficients are no longer constant but time-variant, as stated in the fractal kinetics theory. Fractal kinetics assumes that enzymatic reactions on such heterogeneous media occur within a non-Euclidian space characterized by a certain fractal dimension, this fractal dimension gives the dependence on time of the kinetic coefficients. In this work, stochastic simulations of enzymatic reactions under molecular crowding have been completed, and kinetic coefficients for the reactions, including the Michaelis-Menten parameter KM, were calculated. The simulations results led us to confirm the time dependence of michaelian kinetic parameter for the enzymatic catalysis. Besides, other chaos related phenomena were pointed out from the obtained KM time series, such as the emergence of strange attractors and multifractality.     

Pattern sensitivity to boundary and initial conditions in reaction-diffusion models

We consider Turing-type reaction-diffusion equations and study (via computer simulations) how the relationship between initial conditions and the asymptotic steady state solutions varies as a function of the boundary conditions. The results indicate that boundary conditions which are non-homogeneous with respect to the kinetic steady state give rise to spatial patterns which are much less sensitive to variations in the initial conditions than those obtained with homogeneous boundary conditions, such as zero flux conditions. We also compare linear pattern predictions with the numerical solutions of the full nonlinear problem.This work supported in part by U.S. Army Grant DAJA 37-81-C-0220 and the Science and Engineering Research Council of Great Britain Grant GR/c/63595     

Kinetic measurements of electron transfer in coupled chromatophores from photosynthetic bacteria. A method of correction for the electrochromic effects

A quantitative study of the kinetics of electron transfer under coupled conditions in photosynthetic bacteria has so far been prevented by overlap of the electrochromic signals of carotenoids and bacteriochlorophyll with the absorbance changes of cytochromes and reaction centers. In this paper a method is presented by which the electrochromic contribution at any wavelength can be calculated from the electrochromic signal recorded at 505 nm, using a set of empirically determined polynomial functions. The electrochromic contribution to kinetic changes at any wavelength can then be subtracted to leave the true kinetics of the redox changes. The corrected redox changes of the reaction center measured at 542 and 605 nm mutually agree, thus providing an excellent test of self-consistency of the method. The corrected traces for reaction center and of cytochrome b-566 demonstrate large effects of the membrane potential on the rate and poise of electron transfer. It will be possible to study the interrelation between proton gradient and individual electron reactions under flash or steady-state illumination.     

Mechanochemical modeling of neutrophil migration based on four signaling layers,integrin dynamics,and substrate stiffness

Directional neutrophil migration during human immune responses is a highly coordinated process regulated by both biochemical and biomechanical environments. In this paper, we developed an integrative mathematical model of neutrophil migration using a lattice Boltzmann-particle method built in-house to solve the moving boundary problem with spatiotemporal regulation of biochemical components. The mechanical features of the cell cortex are modeled by a series of spring-connected nodes representing discrete cell–substrate adhesive sites. The intracellular signaling cascades responsible for cytoskeletal remodeling [e.g., small GTPases, phosphoinositide-3-kinase (PI3K), and phosphatase and tensin homolog] are built based on our previous four-layered signaling model centered on the bidirectional molecular transport mechanism and implemented as reaction–diffusion equations. Focal adhesion dynamics are determined by force-dependent integrin–ligand binding kinetics and integrin recycling and are thus integrated with cell motion. Using numerical simulations, the model reproduces the major features of cell migration in response to uniform and gradient biochemical stimuli based on the quantitative spatiotemporal regulation of signaling molecules, which agree with experimental observations. The existence of multiple types of integrins with different binding kinetics could act as an adaptation mechanism for substrate stiffness. Moreover, cells can perform reversal, U-turn, or lock-on behaviors depending on the steepness of the reversal biochemical signals received. Finally, this model is also applied to predict the responses of mutants in which PTEN is overexpressed or disrupted.     

Explicit model for searching behavior of predator

The authors present an approach for explicit modeling of spatio-temporal dynamics of predator-prey community. This approach is based on a reaction-diffusion-adjection PD (prey dependent) system. Local kinetics of population is determined by logistic reproduction function of prey, constant natural mortality of predator and Holling type 2 trophic function. Searching behavior of predator is described by the advective term in predator balance equation assuming the predator acceleration to be proportional to the prey density gradient. The model was studied with zero-flux boundary conditions. The influence of predator searching activity on the community dynamics, in particular, on the emergence of spatial heterogeneity, has been investigated by linear analysis and numerical simulations. It has been shown how searching activity may effect the persistence of species, stabilizing predator-prey interactions at very low level of pest density. It has been demonstrated that obtaining of such dynamic regimes does not require the use of complex trophic functions.     

Microsecond unfolding kinetics of sheep prion protein reveals an intermediate that correlates with susceptibility to classical scrapie

The microsecond folding and unfolding kinetics of ovine prion proteins (ovPrP) were measured under various solution conditions. A fragment comprising residues 94–233 of the full-length ovPrP was studied for four variants with differing susceptibilities to classical scrapie in sheep. The observed biexponential unfolding kinetics of ovPrP provides evidence for an intermediate species. However, in contrast to previous results for human PrP, there is no evidence for an intermediate under refolding conditions. Global analysis of the kinetic data, based on a sequential three-state mechanism, quantitatively accounts for all folding and unfolding data as a function of denaturant concentration. The simulations predict that an intermediate accumulates under both folding and unfolding conditions, but is observable only in unfolding experiments because the intermediate is optically indistinguishable from the native state. The relative population of intermediates in two ovPrP variants, both transiently and under destabilizing equilibrium conditions, correlates with their propensities for classical scrapie. The variant susceptible to classical scrapie has a larger population of the intermediate state than the resistant variant. Thus, the susceptible variant should be favored to undergo the PrP^C to PrP^Sc conversion and oligomerization.     

A model of chlorophyll a fluorescence induction kinetics with explicit description of structural constraints of individual photosystem II units

Chlorophyll a fluorescence induction (FI) kinetics, in the microseconds to the second range, reflects the overall performance of the photosynthetic apparatus. In this paper, we have developed a novel FI model, using a rule-based kinetic Monte Carlo method, which incorporates not only structural and kinetic information on PSII, but also a simplified photosystem I. This model has allowed us to successfully simulate the FI under normal or different treatment conditions, i.e., with different levels of measuring light, under 3-(3′,4′-dichlorophenyl)-1,1-dimethylurea treatment, under 2,5-dibromo-3-methyl-6-isopropyl-p-benzoquinone treatment, and under methyl viologen treatment. Further, using this model, we have systematically studied the mechanistic basis and factors influencing the FI kinetics. The results of our simulations suggest that (1) the J step is caused by the two-electron gate at the Q _B site; (2) the I step is caused by the rate limitation of the plastoquinol re-oxidation in the plastoquinone pool. This new model provides a framework for exploring impacts of modifying not only kinetic but also structural parameters on the FI kinetics.     

Michaelis-Menten kinetics under spatially constrained conditions: application to mibefradil pharmacokinetics

Two different approaches were used to study the kinetics of the enzymatic reaction under heterogeneous conditions to interpret the unusual nonlinear pharmacokinetics of mibefradil. Firstly, a detailed model based on the kinetic differential equations is proposed to study the enzymatic reaction under spatial constraints and in vivo conditions. Secondly, Monte Carlo simulations of the enzyme reaction in a two-dimensional square lattice, placing special emphasis on the input and output of the substrate were applied to mimic in vivo conditions. Both the mathematical model and the Monte Carlo simulations for the enzymatic reaction reproduced the classical Michaelis-Menten (MM) kinetics in homogeneous media and unusual kinetics in fractal media. Based on these findings, a time-dependent version of the classic MM equation was developed for the rate of change of the substrate concentration in disordered media and was successfully used to describe the experimental plasma concentration-time data of mibefradil and derive estimates for the model parameters. The unusual nonlinear pharmacokinetics of mibefradil originates from the heterogeneous conditions in the reaction space of the enzymatic reaction. The modified MM equation can describe the pharmacokinetics of mibefradil as it is able to capture the heterogeneity of the enzymatic reaction in disordered media.     

Engineering aspects of rate-related processes in food manufacturing

Many rate-related phenomena occur in food manufacturing processes. This review addresses four of them, all of which are topics that the author has studied in order to design food manufacturing processes that are favorable from the standpoint of food engineering. They include chromatographic separation through continuous separation with a simulated moving adsorber, lipid oxidation kinetics in emulsions and microencapsulated systems, kinetic analysis and extraction in subcritical water, and water migration in pasta.     

Migration by soaring or flapping: numerical atmospheric simulations reveal that turbulence kinetic energy dictates bee-eater flight mode

Aerial migrants commonly face atmospheric dynamics that may affect their movement and behaviour. Specifically, bird flight mode has been suggested to depend on convective updraught availability and tailwind assistance. However, this has not been tested thus far since both bird tracks and meteorological conditions are difficult to measure in detail throughout extended migratory flyways. Here, we applied, to our knowledge, the first comprehensive numerical atmospheric simulations by mean of the Regional Atmospheric Modeling System (RAMS) to study how meteorological processes affect the flight behaviour of migrating birds. We followed European bee-eaters (Merops apiaster) over southern Israel using radio telemetry and contrasted bird flight mode (flapping, soaring-gliding or mixed flight) against explanatory meteorological variables estimated by RAMS simulations at a spatial grid resolution of 250 × 250 m(2). We found that temperature and especially turbulence kinetic energy (TKE) determine bee-eater flight mode, whereas, unexpectedly, no effect of tailwind assistance was found. TKE during soaring-gliding was significantly higher and distinct from TKE during flapping. We propose that applying detailed atmospheric simulations over extended migratory flyways can elucidate the highly dynamic behaviour of air-borne organisms, help predict the abundance and distribution of migrating birds, and aid in mitigating hazardous implications of bird migration.     

Kinetic modeling of the photosynthetic growth of Chlamydomonas reinhardtii in a photobioreactor

The aim of this study was to establish and validate a model for the photosynthetic growth of Chlamydomonas reinhardtii in photobioreactors (PBRs). The proposed model is based on an energetic analysis of the excitation energy transfer in the photosynthesis apparatus (the Z-scheme for photosynthesis). This approach has already been validated in cyanobacteria (Arthorspira platensis) and is extended here to predict the volumetric biomass productivity for the microalga C. reinhardtii in autotrophic conditions, taking into consideration the two metabolic processes taking place in this eukaryotic microorganism, namely photosynthesis and respiration. The kinetic growth model obtained was then coupled to a radiative transfer model (the two-flux model) to determine the local kinetics, and thereby the volumetric biomass productivity, in a torus PBR. The model was found to predict PBR performances accurately for a broad set of operating conditions, including both light-limited and kinetic growth regimes, with a variance of less than 10% between experimental results and simulations.     

3D culture model and computer-assisted videomicroscopy to analyze migratory behavior of noninvasive and invasive bronchial epithelial cells

To date, most of the studies in the field of cell migration have been applied to two-dimensional (2D) models. To mimic the three-dimensional (3D) conditions similar to those observed in vivo during tumor invasion, we developed a 3D model of cell migration in which cells were embedded in a collagen I matrix placed in a double-compartment chamber. Using time-lapse videomicroscopy and interactive cell tracking in a four-dimensional data set, we determined the cell trajectories and their migration kinetics. We compared the 2D and 3D migratory behavior of a noninvasive cell line (16HBE) with the migratory behavior of an invasive cell line (BZR). Our results show that the 3D migration kinetics of the noninvasive cell line were lower than the migration kinetics of the invasive cell line. In contrast, in 2D models, no significant difference was observed between the two cell lines. To validate our 3D model, we further investigated the effect of epidermal growth factor (EGF), a promoter of tumor cell motility and invasion on the noninvasive cell line (16HBE). EGF increased significantly the migration kinetics of the noninvasive cell line. Our results show that the 3D model of cell migration allowed us to differentiate the migratory behavior of invasive and noninvasive cells and that such a model can help in the development of molecular targeted therapy as it approaches the in vivo conditions. tumor invasion; metastasis; image analysis; kinetic migration; epidermal growth factor     

Molecular dynamics insights on temperature and pressure effects on electroporation

Electroporation is a cell-level phenomenon caused by an ionic imbalance in the membrane, being of great relevance in various fields of knowledge. A dependence of the pore formation kinetics on the environmental conditions (temperature and pressure) of the cell membrane has already been reported, but further clarification regarding how these variables affect the pore formation/resealing dynamics and the transport of molecules through the membrane is still lacking. The objective of the present study was to investigate the temperature (288–348 K) and pressure (1–5000 atm) effects on the electroporation kinetics using coarse-grained molecular dynamics simulations. Results shown that the time for pore formation and resealing increased with pressure and decreased with temperature, whereas the maximum pore radius increased with temperature and decreased with pressure. This behavior influenced the ion migration through the bilayer, and the higher ionic mobility was obtained in the 288 K/1000 atm simulations, i.e., a combination of low temperature and (not excessively) high pressure. These results were used to discuss some experimental observations regarding the extraction of intracellular compounds applying this technique. This study contributes to a better understanding of electroporation under different thermodynamic conditions and to an optimal selection of processing parameters in practical applications which exploit this phenomenon.     

Kinetics of cold-set diffusion-limited aggregations of denatured whey protein isolate colloids

The CaCl2-induced cold-set aggregation kinetics of the denatured whey protein isolate (WPI) colloids has been investigated under dilute diffusion-limited cluster aggregation (DLCA) conditions, using small-angle light scattering. In particular, the structure factor, the scattered intensity at zero angle and the average radius of gyration have been measured for the aggregating system as a function of time. It is found that the fractal dimension of the clusters is df= 1.85, in the range typical of clusters aggregated under DLCA conditions. The aggregation kinetics in this transition region can be described by a power law relation in the initial stage of the aggregation, but the exponent of the power law is equal to 0.7, i.e., significantly larger than 1/df= 0.54, which is the typical value of the DLCA kinetics. Since it is found that the average gyration radius of the clusters has reached a value of 80 microm, leading to a cumulative volume fraction of clusters equal to 0.25, it is legitimate to expect that the process is in the region of transition from aggregation to gelation. This confirmed by the fact that, at the later stage of the aggregation, the growth of the average cluster size further accelerates with time and eventually becomes explosive, leading to gelation. The observed aggregation kinetics has been compared with that reported in the literature from DLCA Monte Carlo simulations, and a good agreement has been found with the data corresponding to the transition region from aggregation to gelation. Numerical simulations using the Smoluchowski kinetic model have also been carried out in order to support the experimental findings.     

Investigations of reaction kinetics for immobilized enzymes--identification of parameters in the presence of diffusion limitation

A method is proposed for identification of kinetic parameters when diffusion of substrates is limiting in reactions catalyzed by immobilized enzymes. This method overcomes conventional sequential procedures, which assume immobilization does not affect the conformation of the enzyme and, thus, consider intrinsic and inherent kinetics to be the same. The coupled equations describing intraparticle mass transport are solved simultaneously using numerical methods and are used for direct estimation of kinetic parameters by fitting modeling results to time-course measurements in a stirred tank reactor. While most traditional procedures were based on Michaelis-Menten kinetics, the method presented here is applicable to more complex kinetic mechanisms involving multiple state variables, such as ping-pong bi-bi. The method is applied to the kinetic resolution of (R/S)-1-methoxy-2-propanol with vinyl acetate catalyzed by Candida antarctica lipase B. A mathematical model is developed consisting of irreversible ping-pong bi-bi kinetics, including competitive inhibition of both enantiomers. The kinetic model, which fits to experimental data over a wide range of both substrates (5-95%) and temperatures (5-56 degrees C), is used for simulations to study typical behavior of immobilized enzyme systems.     

Effect of flow pulsatility on modeling the hemodynamics in the total cavopulmonary connection

Total cavopulmonary connection is the result of a series of palliative surgical repairs performed on patients with single ventricle heart defects. The resulting anatomy has complex and unsteady hemodynamics characterized by flow mixing and flow separation. Although varying degrees of flow pulsatility have been observed in vivo, non-pulsatile (time-averaged) boundary conditions have traditionally been assumed in hemodynamic modeling, and only recently have pulsatile conditions been incorporated without completely characterizing their effect or importance. In this study, 3D numerical simulations with both pulsatile and non-pulsatile boundary conditions were performed for 24 patients with different anatomies and flow boundary conditions from Georgia Tech database. Flow structures, energy dissipation rates and pressure drops were compared under rest and simulated exercise conditions. It was found that flow pulsatility is the primary factor in determining the appropriate choice of boundary conditions, whereas the anatomic configuration and cardiac output had secondary effects. Results show that the hemodynamics can be strongly influenced by the presence of pulsatile flow. However, there was a minimum pulsatility threshold, identified by defining a weighted pulsatility index (wPI), above which the influence was significant. It was shown that when wPI<30%, the relative error in hemodynamic predictions using time-averaged boundary conditions was less than 10% compared to pulsatile simulations. In addition, when wPI<50, the relative error was less than 20%. A correlation was introduced to relate wPI to the relative error in predicting the flow metrics with non-pulsatile flow conditions.     

A brief comparison of model simulations of glacial period maximum atmospheric circulation

The results of studies which have investigated the response of the simulated atmospheric circulation to inclusion of glacial period boundary conditions are discussed. Five numerical models are briefly described and the boundary conditions used in the experiments are discussed. All of the models found that the glacial period atmosphere was cooler and drier with the northern hemisphere jet stream strengthened and shifted south. Some variables, e.g. kinetic energy, surface temperature, monsoon circulation and Hadley circulation responded differently in the different simulations, some of the discrepancies are discussed.