Steady-state electrodiffusion. Scaling, exact solution for ions of one charge, and the phase plane.

This is the first of two papers dealing with electrodiffusion theory (the Nernst-Planck equation coupled with Gauss's law) and its application to the current-voltage behavior of squid axon. New developments in the exact analysis of the steady-state electrodiffusion problem presented here include (a) a scale transformation that connects a given solution to an infinity of other solutions, suggesting the po-sibility of direct comparison of electrical data for membranes with different thicknesses and other properties; (b) a first-integral relation between the electric field and ion densities more general than analogous relations previously reported, and (c) an exact solution for the homovalent system, i.e., a membrane system permeated by various ion species of the same charge. The latter is a generalization of the known one-ion solution. The properties of the homovalent solution are investigated analytically and graphically. In particular we study the phase-plane curves, which reduce to the parabolas discussed by K. S. Cole in the special case in which the current-density parameter (a linear combination of the ionic current densities) is zero.     

Microscopic calculations of local lipid membrane permittivities and diffusion coefficients for application to electroporation analyses

Interaction of electric fields with biological systems has begun to receive considerable attention for applications that include field-assisted drug delivery, medical interventions, and genetic engineering. External fields induce the strongest effects at membranes with electroporation being a common feature. Membrane transport in this context of poration is often based on continuum approaches utilizing macroscopic parameters such as the permittivity, diffusion coefficients, and mobilities. In such modeling, field dependences, local inhomogeneities, and microscopic details are usually ignored. Here, a molecular dynamics (MD) scheme is used for a more rigorous and physically realistic evaluation of such parameters for potential application to electroporative transport model development. A suitable membrane structure containing a nanopore derived from MD analysis is used as the initial geometric configuration. Both static and frequency dependent diffusion coefficients have been evaluated. Permittivities are also calculated and shown to be dramatically non-uniform in the vicinity of membranes under high external fields. A positive feedback mechanism leading to enhanced membrane fields is discussed.     

Inferences about membrane properties from electrical noise measurements

Four sources of electrical noise in biological membranes, each with a different physical basis, are discussed; the analysis of each type of noise potentially yields a different sort of information about membrane properties. (a) From the thermal noise spectrum, the passive membrane impedance may be obtained, so that thermal noise measurements are essentially equivalent to the type of since wave analysis carried out by Cole and Curtis. (b) If adequately high frequency measurements could be made, the shot noise spectrum should give information about the average motion of a single ion within the membrane. (c) The number of charge carriers and single ion mobilities within the membrane can possibly be inferred from measurements of noise with a 1/f spectrum. Available data indicate, for example, that increases in axon membrane conductance are not achieved by modulations in the mobility of ions within the membrane. (d) Fluctuations arising from the mechanisms normally responsible for membrane conductance changes can produce a type of electrical noise. Analysis of such conductance fluctuations provides a way to assess the validity of various microscopic models for the behavior of individual channels. Two different probabilistic interpretations of the Hodgkin-Huxley equations are investigated here and shown to yield different predictions about the spectrum of conductance fluctuations; thus, appropriate noise measurements may serve to eliminate certain classes of microscopic models for membrane conductance changes. Further, it is shown how the analysis of conductance fluctuations can, in some circumstances, provide an estimate of the conductance of a single channel.     

Contribution of electrogenic ion transport to impedance of the algae Valonia utricularis and artificial membranes.

The cell membrane of Valonia utricularis contains an electrogenic carrier system for chloride (Wang et al., Biophys J. 59:235-248 (1991)). The electrical impedance of V. utricularis was measured in the frequency range between 1 Hz and 50 kHz. The analysis of the impedance spectra from V. utricularis and its comparison with equivalent circuit models showed that the transport system created a characteristic contribution to the impedance in the frequency range between 10 Hz and 5 kHz. The fit of the impedance spectra with the formalism derived from the theory of carrier-mediated transport allowed the determination of the kinetic parameters of chloride transport through the cell membrane of V. utricularis, and its passive electrical properties. Simultaneous measurements of the kinetic parameters with the charge pulse method demonstrated the equivalence of both experimental approaches with respect to the evaluation of the translocation rate constants of the free and the charged carriers and the total density of carriers within the membrane. Moreover, the impedance spectra of the protonophor-mediated proton transport by FCCP (carbonylcyanide p-trifluoromethoxyphenyl-hydrazone) were measured in model membranes. The carrier system made a substantial contribution to the impedance of the artificial membranes. The analysis of the spectra in terms of a simple carrier system (Benz and McLaughlin, 1983, Biophys. J. 41:381-398) allowed the evaluation of the kinetic and equilibrium parameters of the FCCP-mediated proton transport. The possible application of the measurement of impedance spectra for the study of biological transport systems is discussed.     

Characterization of immobiline membranes for application in a multicompartment electrolyzer for protein purification.

The efficient use of preparative protein purification in a multicompartment electrolyzer with Immobiline membranes depends on the knowledge of membrane characteristics. For that purpose, an experimental investigation of the effects of ionic charges on the membrane characteristics has been carried out through the measurements of membrane swelling and conductance. We also investigated the effects on the electrolyzer behaviour of operating parameters such as the Immobiline concentration and the presence of ion-exchange membranes. Data show that polyacrylamide gel degree of swelling is strongly dependent upon the pH and the ionic strength of the bathing solution as well as on the type and molarity of charges incorporated in the gel. The conductance of supported Immobiline gels in contact with uni-univalent chloride solutions has been measured by means of a mercury cell. The membrane conductance is also influenced by the ionic strength of the equilibrium solution and the presence of weak ionizable groups in the gel matrix. This study has demonstrated the close link between electrochemical and electromechanical properties of Immobiline membranes.     

Effect of pH on root membrane potentials

Summary The effect of pH on the root membrane potentials was studied using excised roots of Hordeum vulgare, Triticum durum, and Agropyron elongatum (barley, wheat, and tall wheatgrass respectively). The measured potentials were lower the lower the pH of the KCl in the electromotive cell used for the measurements, thus indicating the presence in the root of pH-sensitive electric fields capable of affecting the mobilities of the K and Cl ions and hence the measured potentials. The root membrane potentials from the three species were equal to zero at pH 3. This is due to the fact that at this pH the potential of diffusion is zero because of the deminishing of the electric force fields in the roots. re]19720801     

CONTRIBUTION TO THE THEORY OF PERMEABILITY OF MEMBRANES FOR ELECTROLYTES

From experiments on such membranes as apple skin, parchment paper membrane, and a membrane of completely dry collodion, results have been obtained which could be interpreted by the assumption that these membranes are less permeable for anions than for cations. In parchment paper there is only a relative diminution of the mobility of the anions, in the apple skin and in the dry collodion membrane there is practically no permeability for anions at all. The theory is developed which explains how the decrease or complete lack of mobility of anions influences the electromotive effects of the membrane and the diffusibility of electrolytes across a membrane. The results of the theory are compared with the experimental results. In membranes impermeable for anions the permeability for cations gives the same order of cations as for the mobilities in a free aqueous solution. But the differences of the mobilities are enormously magnified, e.g. the mobilities of H^• and Li^•, which are in the proportion of about 1:10 in aqueous solution, are in proportion of about 1:900 in the collodion membrane. The general cause for the retardation of ionic mobility within the membrane may be supposed to be the increased friction of the water envelope dragged along by the ion in the capillary canals of the membrane. The difference of the effect on the cations and on the anions may be attributed to the electric charge of the walls of the canals.     

Proliferation of endoplasmic reticulum with its enzyme, UDP- glucuronyltransferase, in chick embryo liver during culture. Effects of phenobarbital

UDP-Glucuronyltransferase (GT) activity increases in chick embryo liver during culture from zero to a steady-state level at or above adult values. The GT activity (o-aminophenol as acceptor) is located entirely in the membranes of the endoplasmic reticulum (ER) and the question arises whether ER increases along with GT. Earlier work showed that the synthesis and degradation rates of GT can be varied in culture over wide ranges by choosing embryo livers of different ages and both phenobarbital. In the present study we measured the GT activities and the concentrations of ER (using stereologic methods) in 5- and 11-day embryo liver during culture with and without phenobarbital. We found that GT and ER always increased in a constant ratio of 2.2 X 10(-9) U of GR activity per square micrometer of membrane, suggesting that the synthesis and degradation of GT are coupled to the synthesis and degradation of ER. A general structure for ER is proposed to explain this finding.     

Membrane synthesis in synchronous cultures of Bacillus subtilis 168

Synthesis of bacterial membranes has been investigated in Bacillus subtilis by examining incorporation of amino acids and glycerol into the protein and lipid of membranes of synchronous cultures. A simple reproducible fractionation scheme divides cellular proteins into three classes (i) truly cytoplasmic, (ii) loosely membrane bound, released by chelating agents, and (iii) tightly membrane bound. These comprise approximately 75, 10, and 15%, respectively, of cellular proteins in this organism. Incorporation of radioactivity into these fractions, using steady-state and pulse labeling has been followed during the cell cycle. Cytoplasmic proteins and the loosely membrane-bound proteins are labeled at an exponential rate throughout the cell cycle. The membrane fraction is labeled discontinuously in the cell cycle, with periods of rapid synthesis over the latter part of the cycle and a period with no net synthesis during the early part of the cycle. Pulse labeling indicates that synthesis of membrane occurs at a linear rate that doubles at a fixed time in each cycle, which coincides with the period of zero net synthesis. Rates of membrane synthesis measured by pulse labeling during the period of rapid membrane synthesis are significantly less than indicated by steady-state labeling. These discrepancies are consistent with the hypothesis that during the cell cycle certain proteins are added to the membrane from the cytoplasm and that during the period of zero net synthesis there is an efflux of proteins from the membrane. Evidence in favor of this has been presented. The activity of succinic dehydrogenase (a representative of class c) varies in a step-wise manner with periods of rapid increase, approximately coincident with bursts of membrane protein synthesis, alternating with periods without any increase in activity. The activities of malate dehydrogenase (class a) and reduced nicotinamide adenine dinucleotide dehydrogenase (class b) increased throughout the cell cycle. Phospholipid synthesis is continuous throughout the cell cycle.     

Membrane potential and its electrode-recorded counterpart in an electrical model of an olfactory sensillum

Insect receptor neurons are surrounded with auxiliary cells and encased in a hair. Their electrical activity is usually recorded with an electrode located at the tip of the hair. Analytical expressions giving the membrane potential along the sensory dendrite and the tip-recorded potential are derived for a neuron in steady-state conditions. They formally close the gap between theoretical models and experimental measurements, when transduction mechanisms and active membrane properties are not taken into account. It is shown that the tip-recorded potential reflects correctly the relative variations of the dendritic membrane potential as a function of stimulus intensity over a large range of parameters. The geometric and electrical characteristics of the sensillum that need be known to compute the dendritic membrane potential from the tip-recorded potential are given.     

The steady-state properties of an ion exchange membrane with mobile sites

A study of the properties of the steady states of a system composed of two solutions separated by an ion exchange membrane having mobile sites is presented. It is assumed that the membrane is impermeable to coions; the solutions contain no more than two species of counterions, both of the same valence; and no flow of bulk solution occurs. Assuming that all ions are completely dissociated, behave ideally, and have constant mobilities throughout the membrane, explicit expressions are derived for the steady states of the electric current, individual fluxes, and concentration profiles as functions of the compositions of the solutions and of the difference of electric potential between them. The derived expressions are compared with those for an ion exchange membrane having fixed sites; and it is found that the expressions of certain quantities, such as the difference of electric potential between the two solutions for zero current or the ratio of the fluxes of the counterions as functions of the external parameters of the system, are the same for both types of membranes. On the other hand, differences in the behavior of the two types of membranes are found from other expressions-for example, the current-voltage relationship. In the mobile site ion exchanger the current asymptotically approaches finite limiting values for high positive and negative voltages while in the fixed site ion exchanger it is the conductance which approaches finite limiting values.     

Lipid-phase structure in epithelial cell membranes: comparison of renal brush border and basolateral membranes

The lipid-phase structures of brush border membrane vesicles (BBMV) and basolateral membrane vesicles (BLMV) isolated from rabbit renal cortex were compared by steady-state and phase-modulation measurements of diphenylhexatriene (DPH) and trans- and cis-parinaric acid (tPnA and cPnA) fluorescence. A temperature-scanning system was used which gave reproducible temperature profiles of steady-state and dynamic fluorescence parameters with a resolution of 0.1 degrees C. Steady-state anisotropy of DPH showed a triphasic dependence on temperature with slope discontinuities at 22 +/- 4 and 47 +/- 3 degrees C (BBMV) and at 23 +/- 2 and 48 +/- 1 degrees C (BLMV). At all temperatures, DPH anisotropy in BBMV was greater than that in BLMV. Ground-state heterogeneity analysis of tPnA and cPnA fluorescence lifetime data demonstrated the presence of long (greater than 12 ns) and short (less than 5 ns) lifetime components, interpreted in terms of solid-phase and fluid-phase lipid domains. The fraction of solid-phase phospholipid decreased from 0.9 to 0.1 for BBMV and from 0.7 to 0.3 in BLMV with increasing temperature (10-50 degrees C). In both membranes, tryptophan-PnA fluorescence energy-transfer measurements showed that membrane proteins were surrounded by a fluidlike phospholipid phase. These results demonstrate the inadequacy of steady-state DPH anisotropy data in defining the structural characteristics of complex biological membranes. Results obtained with the phase-sensitive parinaric acid probes demonstrate major differences in the phase structure of the two opposing cell membranes in both the bulk lipid and the lipid microenvironment around membrane proteins.     

A theory for the effects of neutral carriers such as the macrotetralide actin antibiotics on the electric properties of bilayer membranes

Summary To develop a quantitiative theoretical treatment for the effects of neutral macrocyclic antibiotics on the electrical properties of phospholipid bilayer membranes, this paper proceeds from the known ability of such molecules to form stoichiometric, lipid-soluble complexes with cations and deduces the electrical properties that a simple organic solvent phase would have if it were made into a membrane of the thinness of the phospholipid bilayer. In effect, we postulate that the essential barrier to ion movement across a bilayer membrane is its liquid-like hydrocarbon interior and that the neutral macrocyclic antibiotics bind monovalent cations and solubilize them in the membrane as mobile positively charged complexes. Using the Poisson-Boltzmann equation to describe the equilibrium profile of the electrical potential, it is shown that an excess of the positive complexes over all the other ions is expected in the membrane as a net space charge for appropriate conditions of membrane thickness and values of the partition coefficients of the various ionic species and without requiring the presence of fixed charges. Describing the fluxes of these complexes by the Nernst-Planck equation and neglecting the contribution to the electric current of uncomplexed ions, theoretical expressions are derived for the membrane potential in ionic mixtures, as well as for the limiting value of the membrane conductance at zero current when the membrane is interposed between identical solutions. The expressions are given in terms of the ionic activities and antibiotic concentrations in the aqueous solutions so as to be accessible to direct experimental test. Under suitable experimental conditions, the membrane potential is described by an equation recognizible as the Goldman-Hodgkin-Katz equation, in which the permeability ratios are combinations of parameters predicted from the present theory to be independently determinable from the ratio of membrane conductances in single salt solutions. Since this identity between permeability and conductance ratios is expected also for systems obeying the Independence Principle of Hodgkin and Huxley, the applicability of this principle to membranes exposed to antibiotics is discussed, and it is shown that this principle is compatible with the permeation mechanism proposed here.     

Extracellular and intracellular components of the impedance of neural tissue

Electric phenomena in brain tissue can be measured using extracellular potentials, such as the local field potential, or the electro-encephalogram. The interpretation of these signals depends on the electric structure and properties of extracellular media, but the measurements of these electric properties are still debated. Some measurements point to a model in which the extracellular medium is purely resistive, and thus parameters such as electric conductivity and permittivity should be independent of frequency. Other measurements point to a pronounced frequency dependence of these parameters, with scaling laws that are consistent with capacitive or diffusive effects. However, these experiments correspond to different preparations, and it is unclear how to correctly compare them. Here, we provide for the first time, impedance measurements (in the 1–10 kHz frequency range) using the same setup in various preparations, from primary cell cultures to acute brain slices, and a comparison with similar measurements performed in artificial cerebrospinal fluid with no biological material. The measurements show that when the current flows across a cell membrane, the frequency dependence of the macroscopic impedance between intracellular and extracellular electrodes is significant, and cannot be captured by a model with resistive media. Fitting a mean-field model to the data shows that this frequency dependence could be explained by the ionic diffusion mainly associated with Debye layers surrounding the membranes. We conclude that neuronal membranes and their ionic environment induce strong deviations to resistivity that should be taken into account to correctly interpret extracellular potentials generated by neurons.     

Function of membrane domains in rho-of-plant signaling

In a crowded environment, establishing interactions between different molecular partners can take a long time. Biological membranes have solved this issue, as they simultaneously are fluid and possess compartmentalized domains. This nanoscale organization of the membrane is often based on weak, local, and multivalent interactions between lipids and proteins. However, from local interactions at the nanoscale, different functional properties emerge at the higher scale, and these are critical to regulate and integrate cellular signaling. Rho of Plant (ROP) proteins are small guanosine triphosphate hydrolase enzymes (GTPases) involved in hormonal, biotic, and abiotic signaling, as well as fundamental cell biological properties such as polarity, vesicular trafficking, and cytoskeleton dynamics. Association with the membrane is essential for ROP function, as well as their precise targeting within micrometer-sized polar domains (i.e. microdomains) and nanometer-sized clusters (i.e. nanodomains). Here, we review our current knowledge about the formation and the maintenance of the ROP domains in membranes. Furthermore, we propose a model for ROP membrane targeting and discuss how the nanoscale organization of ROPs in membranes could determine signaling parameters like signal specificity, amplification, and integration.

The nanoscale organization of Rho of Plant proteins creates emergent properties that determine cellular signaling.     

A fluorescent sterol probe study of cholesterol/phospholipid membranes

The behavior of dehydroergosterol in -α-dimyristoylphosphatidylcholine (DMPC) unsonicated multilamellar liposomes was characterized by absorption spectroscopy and fluorescence measurements. Dehydroergosterol exhibited a lowered absorption coefficient in multilamellar liposomes whiel the steady-state fluorescence anisotropy of dehydroergosterol in these membranes decreased significantly with increasing dehydroergosterol concentration, suggesting membrane sterol-sterol interactions. The comparative steady-state anisotropy of 0.9 mole percent dehydroergosterol in multilamellar liposomes was lower than in small unilamellar vesicles suggesting different sterol environments for dehydroergosterol. Dehydroergosterol fluorescence lifetime was relatively independent of membrane sterol content and yielded similar values in sonicated and unsonicated model membranes. In multilamellar liposomes containing 5 mole percent cholesterol, the gel-to-liqui crystalline phase transition of DMPC detected by 0.9 mole percent dehydroergosterol was significantly broadened when compared to the phase transition detected by dehydroergosterol in the absence of membrane cholesterol (Smutzer, G. et al. (1986) Biochim. Biophys. Acta 862, 361–371). In multilamellar liposomes containing 10 mole percent cholesterol, the major fluorescence lifetime of dehydroergosterol did not detect the gel-to-liquid crystalline phase transition of DMPC. Time-correlated fluorescence anisotropy decays of dehydroergosterol in DMPC multilamellar liposomes in the absence and presence of 5 mole percent cholesterol exhibited a single rotational correlation time near one nanosecond that was relatively independent of temperature and low concentrations of membrane cholesterol. The limiting anisotropy of 0.9 mole percent dehydroergosterol decreased above the gel-to-liquid crystalline phase transition in membranes without cholesterol and was not significantly affected by the phase transition in membranes containing 5 mole percent cholesterol. These results suggested hindered rotational diffusion of dehydroergosterol in multilamellar liposomes. Lifetime and time-correlated fluorescence measurements of 0.9 mole percent dehydroergosterol in multilamellar liposomes further suggested this fluorophore was detecting physical properties of the bulk membrane phospholipids in membranes devoid of cholesterol and was detecting sterol-rich regions in membranes of low sterol concentration.     

A Multiscale Red Blood Cell Model with Accurate Mechanics, Rheology, and Dynamics

Red blood cells (RBCs) have highly deformable viscoelastic membranes exhibiting complex rheological response and rich hydrodynamic behavior governed by special elastic and bending properties and by the external/internal fluid and membrane viscosities. We present a multiscale RBC model that is able to predict RBC mechanics, rheology, and dynamics in agreement with experiments. Based on an analytic theory, the modeled membrane properties can be uniquely related to the experimentally established RBC macroscopic properties without any adjustment of parameters. The RBC linear and nonlinear elastic deformations match those obtained in optical-tweezers experiments. The rheological properties of the membrane are compared with those obtained in optical magnetic twisting cytometry, membrane thermal fluctuations, and creep followed by cell recovery. The dynamics of RBCs in shear and Poiseuille flows is tested against experiments and theoretical predictions, and the applicability of the latter is discussed. Our findings clearly indicate that a purely elastic model for the membrane cannot accurately represent the RBC's rheological properties and its dynamics, and therefore accurate modeling of a viscoelastic membrane is necessary.     

Effect of hydrodynamic interactions on the diffusion of integral membrane proteins: diffusion in plasma membranes.

Tracer diffusion coefficients of integral membrane proteins (IMPs) in intact plasma membranes are often much lower than those found in blebbed, organelle, and reconstituted membranes. We calculate the contribution of hydrodynamic interactions to the tracer, gradient, and rotational diffusion of IMPs in plasma membranes. Because of the presence of immobile IMPs, Brinkman's equation governs the hydrodynamics in plasma membranes. Solutions of Brinkman's equation enable the calculation of short-time diffusion coefficients of IMPs. There is a large reduction in particle mobilities when a fraction of them is immobile, and as the fraction increases, the mobilities of the mobile particles continue to decrease. Combination of the hydrodynamic mobilities with Monte Carlo simulation results, which incorporate excluded area effects, enable the calculation of long-time diffusion coefficients. We use our calculations to analyze results for tracer diffusivities in several different systems. In erythrocytes, we find that the hydrodynamic theory, when combined with excluded area effects, closes the gap between existing theory and experiment for the mobility of band 3, with the remaining discrepancy likely due to direct obstruction of band 3 lateral mobility by the spectrin network. In lymphocytes, the combined hydrodynamic-excluded area theory provides a plausible explanation for the reduced mobility of sIg molecules induced by binding concanavalin A-coated platelets. However, the theory does not explain all reported cases of "anchorage modulation" in all cell types in which receptor mobilities are reduced after binding by concanavalin A-coated platelets. The hydrodynamic theory provides an explanation of why protein lateral mobilities are restricted in plasma membranes and why, in many systems, deletion of the cytoplasmic tail of a receptor has little effect on diffusion rates. However, much more data are needed to test the theory definitively. We also predict that gradient and tracer diffusivities are the same to leading order. Finally, we have calculated rotational diffusion coefficients in plasma membranes. They decrease less rapidly than translational diffusion coefficients with increasing protein immobilization, and the results agree qualitatively with the limited experimental data available.     

Digital simulation of associated and nonassociated liquid membrane electrochemical properties.

The method and results of a digital simulation of electrochemical properties for associated and nonassociated liquid ion-exchange membranes are presented. It is assumed that the membranes is ideally permselective, sites are completely trapped, electroneutrality holds everywhere in the membrane, and the bathing solutions contain no more than two counterions, of which one is completely dissociated in the membrane. Electrochemical properties are simulated for the single counterion case and in the interference region. Concentration profiles, potentiometric responses, transient potential responses to activity steps, and current-voltage curves are given and the effects of ion-pairing and species mobilities are studied. It is found that ion-pairing increases the potentiometric selectivity toward the complexing ion over the noncomplexing ion. Transient responses to an ion activity step are shown to depend in a complex way on the ion-pair formation constant and the various mobilities. Current-voltage curves are simulated for varying degrees of ion-pairing and qualitative agreement is found with previous theoretical treatments, as well as quantitative agreement in those cases where closed-form expressions are known.     

Diffusion signal in magnetic resonance imaging: origin and interpretation in neurosciences

Diffusion Magnetic Resonance Imaging provides images of unquestionable diagnostic value. It is commonly used in the assessment of stroke and in white matter fiber tracking, among other applications. The diffusion coefficient has been shown to depend on cell concentration, membrane permeability, and cell orientation in the case of white matter or muscle fiber tracking; yet a clear relation between diffusion measurements and known physiological parameters is not established. The aim of this paper is to review hypotheses and actual knowledge on diffusion signal origin to provide assistance in the interpretation of diffusion MR images. Focus will be set on brain images, as most common applications of diffusion MRI are found in neuroradiology. Diffusion signal does not come from two intra- or extracellular compartments, as was first assumed. Restriction of water displacement due to membranes, hindrance in the extracellular space, and tissue heterogeneity are important factors. Unanswered questions remain on how to deal with tissue heterogeneity, and how to retrieve parameters less troublesome to work with from biological and clinical points of view. Diffusion quantification should be done with care, as many variables can lead to variation in measurements.