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The European Journal of Neuroscience Jul 1997Axon guidance by target-derived diffusible factors plays an important role in the development of the nervous system. This paper considers the constraints imposed on this...
Axon guidance by target-derived diffusible factors plays an important role in the development of the nervous system. This paper considers the constraints imposed on this process by the mathematics of diffusion. A point source continuously producing a factor into an infinite three-dimensional volume is considered as a model for both the in vivo and in vitro situation. Basic constraints for effective guidance are assumed to be that the concentration falls between certain maximum and minimum limits, and that the percentage change in concentration across the width of the growth cone exceeds a certain minimum value. The evolution of the shape of the gradient over time is analysed. Using biologically reasonable parameter values, it is shown that the maximum range over which growth cone guidance by a diffusible factor is possible for large times (several days) after the start of the production of the factor is 500-1000 microm. This maximum distance is independent of the diffusion constant of the diffusing molecule, applies to both chemoattractants and chemorepellents, and agrees with experimental data. At earlier times, however, the constraints may be satisfied for distances up to several millimetres. The time it takes for this maximum guidance distance to fall to the asymptotic value depends on the diffusion constant. This time is a few hours for a small molecule but as much as a few days for a large molecule. The model therefore predicts that guidance over distances larger than 1000 microm is possible if the start of production of the factor is carefully matched to the time when guidance is required.
Topics: Animals; Axons; Diffusion; Humans; Models, Neurological; Nervous System
PubMed: 9240399
DOI: 10.1111/j.1460-9568.1997.tb01496.x -
European Biophysics Journal : EBJ Dec 2022The intracellular diffusive movement of molecular substances that are buffered by diffusing chelators is often modeled as movement between compartments with constant...
The intracellular diffusive movement of molecular substances that are buffered by diffusing chelators is often modeled as movement between compartments with constant concentrations within which the buffering occurs. Here, an algorithm to solve such a system of time-dependent differential equations is presented. This Dynamic and Balanced Operator Splitting Scheme (DABOSS) combines dynamic time stepping and operator splitting techniques. The time stepping minimizes the number of time steps while bounding local errors. The balanced operator splitting separates the diffusion and reaction components (each of which is solved efficiently) in a way that preserves the correct steady-state behavior. Analysis shows that DABOSS scales almost linearly in the number of compartments and remains accurate over very long simulations. Moreover, DABOSS works efficiently for nanometer-sized compartments with sources of flux, showing that it is applicable to situations where more spatial resolution is desired.
Topics: Diffusion; Algorithms; Movement
PubMed: 36376400
DOI: 10.1007/s00249-022-01622-z -
Biophysical Journal Jun 2018Diffusion in cellular membranes is regulated by processes that occur over a range of spatial and temporal scales. These processes include membrane fluidity, interprotein...
Diffusion in cellular membranes is regulated by processes that occur over a range of spatial and temporal scales. These processes include membrane fluidity, interprotein and interlipid interactions, interactions with membrane microdomains, interactions with the underlying cytoskeleton, and cellular processes that result in net membrane movement. The complex, non-Brownian diffusion that results from these processes has been difficult to characterize, and moreover, the impact of factors such as membrane recycling on membrane diffusion remains largely unexplored. We have used a careful statistical analysis of single-particle tracking data of the single-pass plasma membrane protein CD93 to show that the diffusion of this protein is well described by a continuous-time random walk in parallel with an aging process mediated by membrane corrals. The overall result is an evolution in the diffusion of CD93: proteins initially diffuse freely on the cell surface but over time become increasingly trapped within diffusion-limiting membrane corrals. Stable populations of freely diffusing and corralled CD93 are maintained by an endocytic/exocytic process in which corralled CD93 is selectively endocytosed, whereas freely diffusing CD93 is replenished by exocytosis of newly synthesized and recycled CD93. This trafficking not only maintained CD93 diffusivity but also maintained the heterogeneous distribution of CD93 in the plasma membrane. These results provide insight into the nature of the biological and biophysical processes that can lead to significantly non-Brownian diffusion of membrane proteins and demonstrate that ongoing membrane recycling is critical to maintaining steady-state diffusion and distribution of proteins in the plasma membrane.
Topics: Animals; CHO Cells; Cell Membrane; Cricetulus; Diffusion; Endocytosis; Exocytosis; Membrane Glycoproteins
PubMed: 29925025
DOI: 10.1016/j.bpj.2018.04.024 -
Journal of Mathematical Biology Mar 2015By introducing linear cross-diffusion for a two-component reaction-diffusion system with activator-depleted reaction kinetics (Gierer and Meinhardt, Kybernetik 12:30-39,...
By introducing linear cross-diffusion for a two-component reaction-diffusion system with activator-depleted reaction kinetics (Gierer and Meinhardt, Kybernetik 12:30-39, 1972; Prigogine and Lefever, J Chem Phys 48:1695-1700, 1968; Schnakenberg, J Theor Biol 81:389-400, 1979), we derive cross-diffusion-driven instability conditions and show that they are a generalisation of the classical diffusion-driven instability conditions in the absence of cross-diffusion. Our most revealing result is that, in contrast to the classical reaction-diffusion systems without cross-diffusion, it is no longer necessary to enforce that one of the species diffuse much faster than the other. Furthermore, it is no longer necessary to have an activator-inhibitor mechanism as premises for pattern formation, activator-activator, inhibitor-inhibitor reaction kinetics as well as short-range inhibition and long-range activation all have the potential of giving rise to cross-diffusion-driven instability. To support our theoretical findings, we compute cross-diffusion induced parameter spaces and demonstrate similarities and differences to those obtained using standard reaction-diffusion theory. Finite element numerical simulations on planary square domains are presented to back-up theoretical predictions. For the numerical simulations presented, we choose parameter values from and outside the classical Turing diffusively-driven instability space; outside, these are chosen to belong to cross-diffusively-driven instability parameter spaces. Our numerical experiments validate our theoretical predictions that parameter spaces induced by cross-diffusion in both the [Formula: see text] and [Formula: see text] components of the reaction-diffusion system are substantially larger and different from those without cross-diffusion. Furthermore, the parameter spaces without cross-diffusion are sub-spaces of the cross-diffusion induced parameter spaces. Our results allow experimentalists to have a wider range of parameter spaces from which to select reaction kinetic parameter values that will give rise to spatial patterning in the presence of cross-diffusion.
Topics: Biological Transport, Active; Computer Simulation; Diffusion; Finite Element Analysis; Kinetics; Linear Models; Mathematical Concepts; Models, Biological
PubMed: 24671430
DOI: 10.1007/s00285-014-0779-6 -
Journal of Physics. Condensed Matter :... Jun 2011The problem of a particle diffusion in a fluctuating scalar field is studied. In contrast to most studies of advection diffusion in random fields, ours analyzes the case...
The problem of a particle diffusion in a fluctuating scalar field is studied. In contrast to most studies of advection diffusion in random fields, ours analyzes the case where the particle position is also coupled to the dynamics of the field. Physical realizations of this problem are numerous and range from the diffusion of proteins in fluctuating membranes to the diffusion of localized magnetic fields in spin systems. We present exact results for the diffusion constant of particles diffusing in dynamical Gaussian fields in the adiabatic limit, where the field evolution is much faster than the particle diffusion. In addition we compute the diffusion constant perturbatively, in the weak coupling limit, when the interaction of the particle with the field is small, using a Kubo-type relation. Finally we construct a simple toy model which can be solved exactly, and which extrapolates between the adiabatic limit, for fields with rapid dynamics, and the limit where the field is quenched or frozen.
Topics: Diffusion; Models, Theoretical
PubMed: 21613699
DOI: 10.1088/0953-8984/23/23/234114 -
The Journal of Chemical Physics Aug 2020Mathematical models of diffusive transport underpin our understanding of chemical, biochemical, and biological transport phenomena. Analysis of such models often focuses...
Mathematical models of diffusive transport underpin our understanding of chemical, biochemical, and biological transport phenomena. Analysis of such models often focuses on relatively simple geometries and deals with diffusion through highly idealized homogeneous media. In contrast, practical applications of diffusive transport theory inevitably involve dealing with more complicated geometries as well as dealing with heterogeneous media. One of the most fundamental properties of diffusive transport is the concept of mean particle lifetime or mean exit time, which are particular applications of the concept of first passage time and provide the mean time required for a diffusing particle to reach an absorbing boundary. Most formal analysis of mean particle lifetime applies to relatively simple geometries, often with homogeneous (spatially invariant) material properties. In this work, we present a general framework that provides exact mathematical insight into the mean particle lifetime, and higher moments of particle lifetime, for point particles diffusing in heterogeneous discs and spheres with radial symmetry. Our analysis applies to geometries with an arbitrary number and arrangement of distinct layers, where transport in each layer is characterized by a distinct diffusivity. We obtain exact closed-form expressions for the mean particle lifetime for a diffusing particle released at an arbitrary location, and we generalize these results to give exact, closed-form expressions for any higher-order moment of particle lifetime for a range of different boundary conditions. Finally, using these results, we construct new homogenization formulas that provide an accurate simplified description of diffusion through heterogeneous discs and spheres.
Topics: Diffusion; Models, Chemical
PubMed: 32828075
DOI: 10.1063/5.0010810 -
Bulletin of Mathematical Biology Nov 2017We present a multiscale approach to model diffusion in a crowded environment and its effect on the reaction rates. Diffusion in biological systems is often modeled by a...
We present a multiscale approach to model diffusion in a crowded environment and its effect on the reaction rates. Diffusion in biological systems is often modeled by a discrete space jump process in order to capture the inherent noise of biological systems, which becomes important in the low copy number regime. To model diffusion in the crowded cell environment efficiently, we compute the jump rates in this mesoscopic model from local first exit times, which account for the microscopic positions of the crowding molecules, while the diffusing molecules jump on a coarser Cartesian grid. We then extract a macroscopic description from the resulting jump rates, where the excluded volume effect is modeled by a diffusion equation with space-dependent diffusion coefficient. The crowding molecules can be of arbitrary shape and size, and numerical experiments demonstrate that those factors together with the size of the diffusing molecule play a crucial role on the magnitude of the decrease in diffusive motion. When correcting the reaction rates for the altered diffusion we can show that molecular crowding either enhances or inhibits chemical reactions depending on local fluctuations of the obstacle density.
Topics: Computer Simulation; Diffusion; Kinetics; Macromolecular Substances; Mathematical Concepts; Models, Biological; Stochastic Processes
PubMed: 28924915
DOI: 10.1007/s11538-017-0346-6 -
The Journal of Physical Chemistry. B Aug 2014We have studied the effects of a high concentration of butanol and octanol on the phase behavior and on the lateral mobility of 1,2-palmitoyl-sn-glycero-3-phosphocholine...
We have studied the effects of a high concentration of butanol and octanol on the phase behavior and on the lateral mobility of 1,2-palmitoyl-sn-glycero-3-phosphocholine (DPPC) by means of differential scanning calorimetry and pulsed-gradient stimulated-echo (PGSTE) NMR spectroscopy. A lowering of the lipid transition from the gel to the liquid-crystalline state for the membrane-alcohol systems has been observed. NMR measurements reveal three distinct diffusions in the DPPC-alcohol systems, characterized by a high, intermediate, and slow diffusivity, ascribed to the water, the alcohol, and the lipid, respectively. The lipid diffusion process is promoted in the liquid phase while it is hindered in the interdigitated phase due to the presence of alcohols. Furthermore, in the interdigitated phase, lipid lateral diffusion coefficients show a slight temperature dependence. To the best of our knowledge, this is the first time that lateral diffusion coefficients on alcohol with so a long chain, and at low temperatures, are reported. By the Arrhenius plots of the temperature dependence of the diffusion coefficients, we have evaluated the apparent activation energy in both the liquid and in the interdigitated phase. The presence of alcohol increases this value in both phases. An explanation in terms of a free volume model that takes into account also for energy factors is proposed.
Topics: Butanols; Diffusion; Magnetic Resonance Spectroscopy; Octanols; Phospholipids; Temperature; Water
PubMed: 25036819
DOI: 10.1021/jp504218v -
Journal of the American Chemical Society Mar 2022Recent studies have sparked debate over whether catalytic reactions enhance the diffusion coefficients of enzymes. Through high statistics of the transient (600 μs)...
Recent studies have sparked debate over whether catalytic reactions enhance the diffusion coefficients of enzymes. Through high statistics of the transient (600 μs) displacements of unhindered single molecules freely diffusing in common buffers, we here quantify for four enzymes under catalytic turnovers. We thus formulate how ∼ ±1% precisions may be achieved for , and show no changes in diffusivity for catalase, urease, aldolase, and alkaline phosphatase under the application of wide concentration ranges of substrates. Our single-molecule approach thus overcomes potential limitations and artifacts underscored by recent studies to show no enhanced diffusion in enzymatic reactions.
Topics: Alkaline Phosphatase; Diffusion; Fructose-Bisphosphate Aldolase; Nanotechnology; Urease
PubMed: 35258969
DOI: 10.1021/jacs.1c12328 -
The Journal of Chemical Physics Jul 2011We present the detailed analysis of the diffusive transport of spatially inhomogeneous fluid mixtures and the interplay between structural and dynamical properties...
We present the detailed analysis of the diffusive transport of spatially inhomogeneous fluid mixtures and the interplay between structural and dynamical properties varying on the atomic scale. The present treatment is based on different areas of liquid state theory, namely, kinetic and density functional theory and their implementation as an effective numerical method via the lattice Boltzmann approach. By combining the first two methods, it is possible to obtain a closed set of kinetic equations for the singlet phase space distribution functions of each species. The interactions among particles are considered within a self-consistent approximation and the resulting effective molecular fields are analyzed. We focus on multispecies diffusion in systems with short-range hard-core repulsion between particles of unequal sizes and weak attractive long-range interactions. As a result, the attractive part of the potential does not contribute explicitly to viscosity but to diffusivity and the thermodynamic properties. Finally, we obtain a practical scheme to solve the kinetic equations by employing a discretization procedure derived from the lattice Boltzmann approach. Within this framework, we present numerical data concerning the mutual diffusion properties both in the case of a quiescent bulk fluid and shear flow inducing Taylor dispersion.
Topics: Diffusion; Hydrodynamics; Kinetics; Models, Chemical; Nanostructures
PubMed: 21806087
DOI: 10.1063/1.3608416