Maximum geometrical hindrance to diffusion in brain extracellular space surrounding uniformly spaced convex cells

Brain extracellular space (ECS) constitutes a porous medium in which diffusion is subject to hindrance, described by tortuosity, lambda = (D/D*)1/2, where D is the free diffusion coefficient and D* is the effective diffusion coefficient in brain. Experiments show that lambda is typically 1.6 in normal brain tissue although variations occur in specialized brain regions. In contrast, different theoretical models of cellular assemblies give ambiguous results: they either predict lambda-values similar to experimental data or indicate values of about 1.2. Here we constructed three different ECS geometries involving tens of thousands of cells and performed Monte Carlo simulation of 3-D diffusion. We conclude that the geometrical hindrance in the ECS surrounding uniformly spaced convex cells is independent of the cell shape and only depends on the volume fraction alpha (the ratio of the ECS volume to the whole tissue volume). This dependence can be described by the relation lambda = ((3-alpha)/2)1/2, indicating that the geometrical hindrance in such ECS cannot account for lambda > 1.225. Reasons for the discrepancy between the theoretical and experimental tortuosity values are discussed.     

低分子数下的生命

瞬时分子信号发生在微区,这些区域通常处于一种非均一非平衡状态。在这个过程中,参与的分了数很少,通常有很大的波动。于是,了解各种分子成分的空间组织结构对理解细胞信号通路非常必要。我们利用蒙特卡洛电脑仿真的方法,结合现实的三维几何构形,来研究亚细胞结构以及神经元和突触的生理学。     

A model of effective diffusion and tortuosity in the extracellular space of the brain

Tortuosity of the extracellular space describes hindrance posed to the diffusion process by a geometrically complex medium in comparison to an environment free of any obstacles. Calculating tortuosity in biologically relevant geometries is difficult. Yet this parameter has proved very important for many processes in the brain, ranging from ischemia and osmotic stress to delivery of nutrients and drugs. It is also significant for interpretation of the diffusion-weighted magnetic resonance data. We use a volume-averaging procedure to obtain a general expression for tortuosity in a complex environment. A simple approximation then leads to tortuosity estimates in a number of two-dimensional (2D) and three-dimensional (3D) geometries characterized by narrow pathways between the cellular elements. It also explains the counterintuitive fact of lower diffusion hindrance in a 3D environment. Comparison with Monte Carlo numerical simulations shows that the model gives reasonable tortuosity estimates for a number of regular and randomized 2D and 3D geometries. Importantly, it is shown that addition of dead-end pores increases tortuosity in proportion to the square root of enlarged total extracellular volume fraction. This conclusion is further supported by the previously described tortuosity decrease in ischemic brain slices where dead-end pores were partially occluded by large macromolecules introduced into the extracellular space.     

Monte Carlo Simulation of Release of Vesicular Content in Neuroendocrine Cells

The release of transmitter from the vesicle, its diffusion through the fusion pore, and the cleft and its interaction with the carbon electrode were simulated using the Monte Carlo method. According to the simulation the transmitter release is largely determined by geometric factors – the ratio of the fusion pore cross-sectional and vesicular areas, if the diffusion constant is as in the aqueous solution – but the speed of transmitter dissociation from the gel matrix plays an important role during the rise phase of release. Transmitter is not depleted near the entrance to the fusion pore and there is no cleft-to-vesicle feedback, but the depletion becomes evident if the diffusion constant is reduced, especially if the pore is wide. In general, the time course of amperometric currents closely resembles the time course of the simulated transmitter concentration in the cleft and the time course of release. Surprisingly, even a tenfold change of the electrode efficiency has only a marginal effect on the amplitude or the time course of amperometric currents. Greater electrode efficiency however lowers the cleft concentration, but only if the cleft is narrow. As the cleft widens the current amplitudes diminish and rise times lengthen, but the decay times are less affected. Moreover, the amplitude dependence of the rise and decay times becomes steeper as the cleft widens and/or as the release kinetics slows. Finally, lower diffusion constant of transmitter in the narrow cleft does not further prolong the amperometric currents, whose slow time course reflects slow release kinetics.     

Structural relationships between parenchymal and stromal elements in the pars intermedia of the rat adenohypophysis as demonstrated by extracellular space markers

Summary The pars intermedia (PI) of the rat adenohypophysis was studied by light and transmission electron microscopy after conventional staining as well as ruthenium red staining, and after systemic injection of horseradish peroxidase. The studies disclose a complex and constant system of two channel types (Type I and Type II channels) formed by PI cells with specific relationships to a very rich nerve supply, to each other, and to a stellate cell type proposed here to represent an element of neuroglia. The channel system could perform a function in the movement of fluids and solutes within the PI which is virtually avascular in the rat as well as in other mammals.Supported by the Medical Research Council of Canada Grant # MA6445We thank Mr. J. Jones and Mr. A.R. Morris for their expert technical assistance     

Test of the Inverse Monte Carlo Method for the Calculation of Interatomic Potential Energies in Atomic Liquids

Abstract

The principle purpose of this paper is to demonstrate the use of the Inverse Monte Carlo technique for calculating pair interaction energies in monoatomic liquids from a given equilibrium property. This method is based on the mathematical relation between transition probability and pair potential given by the fundamental equation of the “importance sampling” Monte Carlo method. In order to have well defined conditions for the test of the Inverse Monte Carlo method a Metropolis Monte Carlo simulation of a Lennard Jones liquid is carried out to give the equilibrium pair correlation function determined by the assumed potential. Because an equilibrium configuration is prerequisite for an Inverse Monte Carlo simulation a model system is generated reproducing the pair correlation function, which has been calculated by the Metropolis Monte Carlo simulation and therefore representing the system in thermal equilibrium. This configuration is used to simulate virtual atom displacements. The resulting changes in atom distribution for each single simulation step are inserted in a set of non-linear equations defining the transition probability for the virtual change of configuration. The solution of the set of equations for pair interaction energies yields the Lennard Jones potential by which the equilibrium configuration has been determined.     

Poly[N-(2-hydroxypropyl)methacrylamide] polymers diffuse in brain extracellular space with same tortuosity as small molecules

Integrative optical imaging was used to show that long-chain synthetic poly[N-(2-hydroxypropyl)methacrylamide] (PHPMA) polymers in a range of molecular weights from 7.8 to 1057 kDa were able to diffuse through the extracellular space in rat neocortical slices. Tortuosity (square root of ratio of diffusion coefficient in aqueous medium to that in brain) measured with such polymers averaged 1.57, a value similar to that obtained previously with tetramethylammonium, a small cation. When PHPMA was conjugated with bovine serum albumin (BSA) to make a bulky polymer with molecular weight 176 kDa, the tortuosity rose to 2.27, a value similar to that obtained previously with BSA alone and with 70-kDa dextran. The method of image analysis was justified with diffusion models involving spherical and nonspherical initial distributions of the molecules.     

The use of 60Co-EDTA as an extracellular marker in frog skin

⁶⁰Co-EDTA was tested as an extracellular marker in isolated epithelia of frog skin.It was found to be non-toxic for frog skin and 0.1 mM EDTA proved to be enough to saturate any adsorption sites.Comparing with [¹⁴C]inulin, ⁶⁰Co-EDTA marks a slightly greater extracellular space and this volume is constant between 15 min and 2 h.Furthermore it is reproducible and its use simplifies the methodology of the determination of water and electrolyte contents in the cells.     

Bias in Markov models of disease

We examine bias in Markov models of diseases, including both chronic and infectious diseases. We consider two common types of Markov disease models: ones where disease progression changes by severity of disease, and ones where progression of disease changes in time or by age. We find sufficient conditions for bias to exist in models with aggregated transition probabilities when compared to models with state/time dependent transition probabilities. We also find that when aggregating data to compute transition probabilities, bias increases with the degree of data aggregation. We illustrate by examining bias in Markov models of Hepatitis C, Alzheimer’s disease, and lung cancer using medical data and find that the bias is significant depending on the method used to aggregate the data. A key implication is that by not incorporating state/time dependent transition probabilities, studies that use Markov models of diseases may be significantly overestimating or underestimating disease progression. This could potentially result in incorrect recommendations from cost-effectiveness studies and incorrect disease burden forecasts.     

Maximum Likelihood Estimation of Genetic Parameters in Cell Populations

In this paper we consider a cell population such as bacteria consisting of two types of cells, mutant and nonmutant. Under the mutation and homogeneous pure birth processes, this paper derives a maximum likelihood estimation procedure for estimating mutation rate and birth rate. The method is applied to Newcombe's data; further some Monte Carlo studies are generated. The numerical results indicate that the method is quite efficient for estimating genetic parameters in cell populations.     

Astrocytes and extracellular matrix in extrasynaptic volume transmission

Volume transmission is a form of intercellular communication that does not require synapses; it is based on the diffusion of neuroactive substances across the brain extracellular space (ECS) and their binding to extrasynaptic high-affinity receptors on neurons or glia. Extracellular diffusion is restricted by the limited volume of the ECS, which is described by the ECS volume fraction α, and the presence of diffusion barriers, reflected by tortuosity λ, that are created, for example, by fine astrocytic processes or extracellular matrix (ECM) molecules. Organized astrocytic processes, ECM scaffolds or myelin sheets channel the extracellular diffusion so that it is facilitated in a certain direction, i.e. anisotropic. The diffusion properties of the ECS are profoundly influenced by various processes such as the swelling and morphological rebuilding of astrocytes during either transient or persisting physiological or pathological states, or the remodelling of the ECM in tumorous or epileptogenic tissue, during Alzheimer''s disease, after enzymatic treatment or in transgenic animals. The changing diffusion properties of the ECM influence neuron–glia interaction, learning abilities, the extent of neuronal damage and even cell migration. From a clinical point of view, diffusion parameter changes occurring during pathological states could be important for diagnosis, drug delivery and treatment.     

The extracellular space and blood-eye barrier in an insect retina: An ultrastructural study

Summary (1) The distribution of the extracellular space (ECS) in the outer part of the locust compound eye has been mapped with lanthanum and ruthenium red, applied to the retina. (2) In the photoreceptor zone, about 2.4% of the volume is ECS, in agreement with radiotracer and electrical estimates. Of this ECS, about 70% lies in lacunae between ommatidia, but only 1–2% adjacent to the photosensitive rhabdom. The lacunae are filled with material which binds applied tracers, and are thought to be structural spaces. (3) It has been suggested several times that such a small cation pool is insufficient to sustain more than a few large photoresponses, but this is shown to be incorrect. Enough Na⁺ lies within the rhabdomal ECS and within rapid diffusional access to it, to impose no immediate limitation. (4) The palisade vacuoles surrounding the rhabdom are intracellular, and are typical of light as well as dark-adapted eyes. (5) Tracers fail to penetrate more than about 30 m into the axon zone, in agreement with electrical, dye and radiotracer indications of a blood-eye barrier near this point. Septate and gap junctions between glial membranes proliferate at this level, the lacunae disappear, and the axonal clefts narrow, but no tight junctions were seen. Comparison is made with the barrier around the nerve cord. (6) The secondary pigment cells in the retina may function as osmotic/ionic buffers, in conjunction with the blood-eye barrier.     

Estimation of contrast agent concentration in intra- and extra-vascular spaces of brain tissue

This article presents a new method for estimating the leakage of a contrast agent out of a vessel. The proposed method is developed based on tissue homogeneity (TH) model, modified Patlak model, and Monte Carlo simulation. The analytical methods published in the literature estimate the contrast agent leakage by solving the coupled differential equations associated with the TH model under adiabatic conditions. These methods employ unrealistic simplifying assumptions and become intractable in their applications to the vessels that have a non-uniform permeability. Without making any unrealistic assumptions, our approach simply tracks the passage of the contrast agent through the capillary and its crossing of the vessel walls based on the blood flow in the vessel, the vessel's permeability, and the condition of the blood-brain barrier (BBB). These are treated as statistical processes that can be modeled reasonably well using the Monte Carlo method. In the proposed approach, the intra- and extra-vascular spaces are divided into multiple compartments, similar to the Patlak model. A real, measured arterial input function (AIF) is used as the capillary input and the concentration of the contrast agent is found as a function of time and distance, inside and outside of the capillary. This is done for normal and abnormal capillaries with uniform and non-uniform permeability. The proposed method generates concentration curves similar to those of the analytical method for simple AIF models. It also generates reasonable concentration curves for a real AIF. The proposed method does not fit a mathematical function to the measured AIF and does not make unrealistic simplifying assumptions. It is not therefore prone to the fitting errors and generates more realistic and more accurate results than the analytical methods.     

Monte Carlo Simulation of Fluids in Curved Three-dimensional Space

Abstract

This work describes the methods required to perform computer simulations of three-dimensional fluids confined to the surface of a four-dimensional hypersphere. The use of such non-Euclidian spaces, or spherical boundary conditions, is convenient in cases where spatial inhomogeneities occur over length-scales comparable to that of the entire system. The form of the pressure equation in curved space is discussed, and the results of Monte Carlo simulations of hard spheres confined to the surface of a hypersphere are presented. Comparison of the simulation results to the Carnahan-Starling equation of state in flat space provides a basis for determining when curvature effects can be neglected.     

Uses and misuses of progress curve analysis in enzyme kinetics

Progress curve analysis is a convenient tool for the characterization of enzyme action: a single reaction mixture provides multiple experimental measured points for continuously varying amounts of substrates and products with exactly the same enzyme and modulator concentrations. The determination of kinetic parameters from the progress curves, however, requires complex mathematical evaluation of the time-course data. Some freely available programs (e.g. FITSIM, DYNAFIT) are widely applied to fit kinetic parameters to user-defined enzymatic mechanisms, but users often overlook the stringent requirements of the analytic procedures for appropriate design of the input experiments. Flaws in the experimental setup result in unreliable parameters with consequent misinterpretation of the biological phenomenon under study. The present commentary suggests some helpful mathematical tools to improve the analytic procedure in order to diagnose major errors in concept and design of kinetic experiments.     

Size and Power of Test Statistics for Gene Correlation in 2 × 2 Contingency Tables

Monte Carlo simulation of size and power of two proposed tests for linkage disequilibrium between two genes each with two alleles were investigated. Results were compared with two commonly used statistics, the correlation coefficient r and the log-odds ratio tests. Depending on the sign of the linkage disequilibrium, the new tests were found to be more powerful than either of the correlation or log-odds ratio tests. However, on average (positive and negative linkage disequilibrium) the Chi-square test using the correlation coefficient was to a small extent more powerful than the other tests.     

Assessing the Risk of Hydroxybenzene Contamination in Fish Raised on a Chinese Aquaculture Farm

We estimated the risk posed by hydroxybenzene in farm-raised Crucian carps (Carassius carassius) from a pond on a fish farm in Beijing, China, by analyzing a time-series of observed hydroxybenzene concentrations in fish within a culturing season in year 2006. We used the basic linear regression to model the data and forecast the probability of hydroxybenzene concentration in pond fish exceeding selected health effect criteria (the risk) using Monte Carlo simulation. The risk is highly correlated with the time a fish stays in the pond. The results indicate that the risk–days in pond relationship resembles a sigmoid function with an inflection point around 150 days. The resulting model can be used to demonstrate the benefit of improving water quality in terms of increased fish production.     

Study of thermodynamic properties of symmetric and asymmetric electrolyte systems in mixture with neutral components: Monte Carlo simulation results and integral equation predictions

Monte Carlo simulation is conducted to obtain the structure, excess internal energy and Helmholtz energy for systems containing charged and neutral hard spheres of comparable concentrations. The results are compared with the thermodynamic properties predicted by solving Ornstein–Zernike equation with hypernetted chain (HNC) and mean spherical approximation (MSA) closures. The HNC approximation is found to well represent the simulation results for both structure and excess energy, while the excess energy of MSA deviates from the simulation results in the intermediate- and high-density range. A simple modification of MSA, referred to as KMSA, is proposed to accurately predict the excess internal and Helmholtz energy in the studied density range. KMSA is proved to capture the effects of neutral component, size and charge asymmetry, system temperature and dielectric constant of the background solvent, on the excess energy of electrolyte systems.     

Protein fold recognition and dynamics in the space of contact maps

We introduce an energy function for contact maps of proteins. In addition to the standard term, that takes into account pair-wise interactions between amino acids, our potential contains a new hydrophobic energy term. Parameters of the energy function were obtained from a statistical analysis of the contact maps of known structures. The quality of our energy function was tested extensively in a variety of ways. In particular, fold recognition experiments revealed that for a fixed sequence the native map is identified correctly in an overwhelming majority of the cases tested. We succeeded in identifying the structure of some proteins that are known to pose difficulties for such tests (BPTI, spectrin, and cro-protein). In addition, many known pairs of homologous structures were correctly identified, even when the two sequences had relatively low sequence homology. We also introduced a dynamic Monte Carlo procedure in the space of contact maps, taking topological and polymeric constraints into account by restrictive dynamic rules. Various aspects of protein dynamics, including high-temperature melting and refolding, were simulated. Perspectives of application of the energy function and the method for structure checking and fold prediction are discussed. Proteins 26:391–410 © 1996 Wiley-Liss, Inc.