Analytical solution of the Poisson-Boltzmann equation for membrane potential

Analytical solution of the one-dimensional Poisson-Boltzmann equation for membrane potential is obtained in an equilibrium state of the Nemst-Planck Poisson system. Approximations, e.g. constant field or Debye-Hückel approximation, need not be used. Two types of solution, arctangenthyperbolic and arccotangenthyperbolic, exist for every value of membrane potential.A new approximate solution is obtained where V and V_m are electric potential inside the membrane and membrane potential respectively; R, T and F have their usual thermodynamic meanings, κ is Debye constant, τ the membrane thickness. This approximate solution fits the numerical solution by Runge-Kutta method (maximum error being less than 5 × 10^?4) around the potential being 50 mV. The fitness is considerably more accurate than that of Debye-Huckel approximation (error being c. 15 % around the potential 50 mV.).     

Kinetic analysis of the growth of Spirulina sp. in batch culture

Batch cultivation of Spirulina sp. was carried out under limited light at 30°C in the pH range of 9.2 to 9.7. The specific growth rate D was calculated from the tangent of the growth curve and the cell concentration at that time, and the amount of light energy absorbed per unit time per unit cell weight (E_x), namely, the specific absorption rate of light energy, was also calculated from the total amount of radiant flux of transmitted light at the surface of the culture vessel and cell concentration of the culture solution. A plot against E_x of D in the linear growth phase in batch culture and at various phases in continuous culture gave, for E_x of less than 1.0 kcal/g·h, points scattered near a straight line with slope m 0.037 g/kcal and an intercept on the ordinate, −b, of −0.0046 h⁻¹, and, for higher E_x values, points scattered near a curve of gradually decreasing slope which tended to approach a constant value.A Lineweaver-Burk plot of the reciprocal of D plus b against that of E_x yielded an equation for the growth rate which represented well the growth curve in batch culture. This equation also expressed the linear increase of D with increase of E_x at high cell concentration in the culture solution. The relation between cell growth rate and cell fluidity is discussed by use of a vector equation obtained by applying this relation to a culture solution contained in a given closed surface.     

Determination of reaction rate constants in the mammillary system

It is shown that the individual rate constants can be determined for the composite chemical system: $$A + B_i \rightleftarrows C_i ; i = 1...N$$ with only measurements of the unbound species,A(t), required. The dissociation rate constants can be determined by direct analysis of a single steady state tracer study. The association constants then follow from the analysis of stable equilibrium determinations reported earlier (Hart, 1965). An approximate solution when tracer methods are in-applicable is also given.     

Is DNA really a double helix?

Do the two chains of the DNA molecule coil round one another plectonemically ? If so, what is the approximate value of Lk (the linking number) for any closed, circular DNA molecule? Experiments using gel electrophoresis have shown that supercoiled DNA molecules usually migrate in a series of discrete bands. The only tenable explanation for this quantized behavior is that the molecules in one band all have the same value of Lk and that this value differs by unity from that of the adjacent bands. Various experiments in which circular DNA is unwound by known amounts show that (given this assumption) Lk for relaxed DNA is very roughly equal to $\text{N}\text{10}$ (where N is the number of base-pairs), as expected from the classical double helix.The original model for the double helix was right-handed. The experimental evidence for this feature is suggestive but not yet completely compelling.     

Gating Kinetics of Single Large-Conductance Ca2+-Activated K+ Channels in High Ca2+ Suggest a Two-Tiered Allosteric Gating Mechanism?

The Ca²⁺-dependent gating mechanism of large-conductance calcium-activated K⁺ (BK) channels from cultured rat skeletal muscle was examined from low (4 μM) to high (1,024 μM) intracellular concentrations of calcium (Ca²⁺ _i) using single-channel recording. Open probability (P _o) increased with increasing Ca²⁺ _i (K _0.5 11.2 ± 0.3 μM at +30 mV, Hill coefficient of 3.5 ± 0.3), reaching a maximum of ∼0.97 for Ca²⁺ _i ∼ 100 μM. Increasing Ca²⁺ _i further to 1,024 μM had little additional effect on either P _o or the single-channel kinetics. The channels gated among at least three to four open and four to five closed states at high levels of Ca²⁺ _i (>100 μM), compared with three to four open and five to seven closed states at lower Ca²⁺ _i. The ability of kinetic schemes to account for the single-channel kinetics was examined with simultaneous maximum likelihood fitting of two-dimensional (2-D) dwell-time distributions obtained from low to high Ca²⁺ _i. Kinetic schemes drawn from the 10-state Monod-Wyman-Changeux model could not describe the dwell-time distributions from low to high Ca²⁺ _i. Kinetic schemes drawn from Eigen''s general model for a ligand-activated tetrameric protein could approximate the dwell-time distributions but not the dependency (correlations) between adjacent intervals at high Ca²⁺ _i. However, models drawn from a general 50 state two-tiered scheme, in which there were 25 closed states on the upper tier and 25 open states on the lower tier, could approximate both the dwell-time distributions and the dependency from low to high Ca²⁺ _i. In the two-tiered model, the BK channel can open directly from each closed state, and a minimum of five open and five closed states are available for gating at any given Ca²⁺ _i. A model that assumed that the apparent Ca²⁺-binding steps can reach a maximum rate at high Ca²⁺ _i could also approximate the gating from low to high Ca²⁺ _i. The considered models can serve as working hypotheses for the gating of BK channels.     

The development and initial validation of a virtual dripping sweat rate and a clothing wetness ratio for use in predictive heat strain models

This paper applies the heat balance equation (HBE) for clothed subjects as a linear function of mean skin temperature (t _sk ) by a new sweating efficiency (η _sw ) and an approximation for the thermoregulatory sweat rate. The equation predicting t _sk in steady state conditions was derived as the solution of the HBE and used for a predictive heat strain scale. The heat loss from the wet clothing (WCL) area was identified with a new variable of ‘virtual dripping sweat rate VDSR’ (S _wdr ). This is a subject’s un-evaporated sweat rate in dry clothing from the regional sweat rate exceeding the maximum evaporative capacity, and adds the moisture to the clothing, reducing the intrinsic clothing insulation. The S _wdr allowed a mass balance analysis of the wet clothing area identified as clothing wetness (w _cl ). The w _cl was derived by combining the HBE at the WCL surface from which the evaporation rate and skin heat loss from WCL region are given. Experimental results on eight young male subjects wearing typical summer clothing, T-shirt and trousers verified the model for predicting t _sk with WCL thermal resistance (R _cl,w ) identified as 25 % of dry clothing (R _cl,d ).     

Cardiac chemical constant. Part I: Derivation of the characteristic phenomenological equation

A physiochemical parameter is derived and defined as the cardiac chemical equilibrium dissociation constant (K_D), K_D is based upon a phenomenological model in which the cardiac muscle chemical reaction kinetics describe the interconversion between long and short unils (i.e. the individual sarcomere is fully extended or fully contracted). K_D is defined as the ratio of the number of units in the long state to the number of units in the short state. The mathematical development proceeds through four stages: derivation of the governing differential equation during cardiac systole; simplification of the differential equation to describe the cardiac model; determination of the upper and lower limits and average value of Nt (the total number of units in a hypothetical mid-wall circumferential fibre); definition and calculation of the cardiac chemical constant (K_D). K_D is shown to describe a series of equilibrium points throughout cardiac systole. This requires that each mechanical equilibrium state (a series of static, steady-state intervals over time) is also associated with its own specific chemical equilibrium state.     

Quasi-steady-state approximation in the mathematical modeling of biochemical reaction networks

In the mathematical modeling of biochemical reaction networks the application of the quasi-steady-state approximation permits a reduction of the number of dynamic variables as well as of the number of parameters. It is shown that the quasi-steady-state approximation represents the zeroth approximate solution of the perturbation problem$\text{dX}\text{dt}\text{= RV(X)+}\text{1}\text{μ}\text{S}\text{W}\text{?}\text{(X)}$ with μ ? 1. The perturbation equation develops by subdivision of the flux rates of the model into the rates $\text{w}{}_{\mathrm{i}}\text{(X) = (1/μ)}\text{w}\text{?}{}_{\mathrm{i}}\text{(X)}$ of fast reactions and the rates v_j(X) of slow reactions. The matrix C=(R?S) denotes the stochiometric matrix of the reaction network. The analysis of this perturbation problem provides conditions for the applicability of the quasi-steady-state approximation in a given network. The paper presents a practical guide for the construction of the approximate solution.     

Evaluation of surface colonization kinetics in continuous culture

Two equations, describing surface colonization, were evaluated and compared using suspended glass slides in a continuous culture ofPseudomonas aeruginosa. These equations were used to determine surface growth rates from the number and distribution of cells present on the surface after incubation. One of these was the colonization equation which accounts for simultaneous attachment and growth of bacteria on surfaces: $$N = (A/\mu )e^{\mu t} - A/\mu $$ where N=number of cells on surface (cells field^?1); A=attachment rate (cells field^?1h^?1);μ=specific growth rate (h^?1); t=incubation period (h). The other was the surface growth rate equation which assumes that the number of colonies of a given size (C_i) will reach a constant value (C_max) which is equal to A divided byμ: $$\mu = \frac{{\ln \left( {\frac{N}{{C_i }} + 1} \right)}}{t}$$ Both equations gave similar results and the time required to approximate C_max may not be as long as was previously thought. In all cases both A andμ continuously decreased throughout the incubation period. These decreases may be due to various effects of microbial accumulation on the surface. Both equations accurately determined surface growth rates despite highly variable attachment rates. Growth rates were similar for both the liquid phase of the culture and the solid-liquid interface (0.4 h^?1). Use of the surface growth rate equation is favored over the use of the colonization equation since the former does not require a computer to solve forμ and the counting procedure is simplified.     

Detection of nanosecond time scale side-chain jumps in a protein dissolved in water/glycerol solvent

In solution, the correlation time of the overall protein tumbling, τ _R , plays a role of a natural dynamics cutoff—internal motions with correlation times on the order of τ _R or longer cannot be reliably identified on the basis of spin relaxation data. It has been proposed some time ago that the ‘observation window’ of solution experiments can be expanded by changing the viscosity of solvent to raise the value of τ _R . To further explore this concept, we prepared a series of samples of α-spectrin SH3 domain in solvent with increasing concentration of glycerol. In addition to the conventional ¹⁵N labeling, the protein was labeled in the Val, Leu methyl positions (¹³CHD₂ on a deuterated background). The collected relaxation data were used in asymmetric fashion: backbone ¹⁵N relaxation rates were used to determine τ _R across the series of samples, while methyl ¹³C data were used to probe local dynamics (side-chain motions). In interpreting the results, it has been initially suggested that addition of glycerol leads only to increases in τ _R , whereas local motional parameters remain unchanged. Thus the data from multiple samples can be analyzed jointly, with τ _R playing the role of experimentally controlled variable. Based on this concept, the extended model-free model was constructed with the intent to capture the effect of ns time-scale rotameric jumps in valine and leucine side chains. Using this model, we made a positive identification of nanosecond dynamics in Val-23 where ns motions were already observed earlier. In several other cases, however, only tentative identification was possible. The lack of definitive results was due to the approximate character of the model—contrary to what has been assumed, addition of glycerol led to a gradual ‘stiffening’ of the protein. This and other observations also shed light on the interaction of the protein with glycerol, which is one of the naturally occurring osmoprotectants. In particular, it has been found that the overall protein tumbling is controlled by the bulk solvent, and not by a thin solvation layer which contains a higher proportion of water.     

Potassium Ion Current in the Squid Giant Axon: Dynamic Characteristic

Measurements of the potassium current in the squid axon membrane have been made, after changes of the membrane potential to the sodium potential of Hodgkin and Huxley (HH), from near the resting potential, from depolarizations of various durations and amplitudes, and from hyperpolarizations of up to 150 mv. The potassium currents I given by I = I_∞ {1 - exp [- (t + t₀)/τ]}²⁵, where t₀ is determined by the initial conditions, represent the new data and approximate the HH functions in the regions for which they are adequate. A corresponding modification for the sodium current does not appear necessary. The results support the HH assumptions of the independence of the potassium and sodium currents, the dependence of the potassium current upon a single parameter determined by the membrane potential, and the expression of this parameter by a first order differential equation, and, although the results drastically modify the analytical expressions, they very considerably extend the range of apparent validity of these assumptions. The delay in the potassium current after severe hyperpolarization is used to estimate a potassium ion mobility in the membrane as 10^-5 of its value in aqueous solutions.     

Extending the diffuse layer model of surface acidity constant behaviour: IV. Diffuse layer charge/potential relationships

Abstract

Most current electrostatic surface complexation models describing ionic binding at the particle/water interface rely on the use of Poisson–Boltzmann (PB) theory for relating diffuse layer charge densities to diffuse layer electrostatic potentials. PB theory is known to contain a number of implicit assumptions whose significance in environmental applications is largely unknown. This study seeks to better quantify the impact of these assumptions by: (1) comparing potentials obtained from planar analytical solutions to the PB with those obtained from Hypernetted Chain (HNC) theory (Attard, 2006), (2) assessing the accuracy of the Ohshima et al. (1982) spherical approximate analytical solution to the PB equation by comparison with published numerical values (Loeb et al., 1961), and (3) comparing interfacial potential estimates obtained from the spherical approximate analytical solution to the PB equation at and adjacent to the particle surface with potential estimates obtained from the Entropic Balanced Surface Potential (EB) model (Loux, 1985; Loux and Anderson, 2001) and published potential estimates obtained from the Hypernetted Chain/Mean Spherical Approximation procedure (HNC/MSA; Gonzalez-Tovar and Lozada-Cassou, 1989). EB potential estimates were obtained assuming a surface volume thickness equal to the Bjerrum length (0.357 nm in a room temperature monovalent electrolyte solution). Findings from the study included: (1) the planar, surficial HNC estimates compared favourably with planar surficial PB relationships at charge densities equal to or less than 0.05 C m^?2, (2) the Ohshima et al. (1982) approximate spherical analytical solution to the PB equation replicated the numerical charge density estimates required to obtain 72 datapoints over an e<img>/kT range of one to four with a maximum error of 3.37% and a coefficient of variation of 0.92%, (3) for a 0.1 μm radius particle in a room temperature 0.01 M (1 : 1) ionic strength solution, potential estimates over a surface charge density range of 0 to 0.3C m^?2 occurred in the following order: ψ_HNC/MSA,R >ψ_PB,R >_{ψHNC/MSA,R+0.2125nm} >ψ_PB,R+0.2nm ~ ψ_EB >ψ_{HNC/MSA,R+0.425nm} ~ _ψPB,R+0.4nm and (4) with 45 datapoints including both 1 μm and 10 nm radius particles over an ionic strength range of 1.0 to 0.001 M, the PB potential estimates 0.2 nm from the particle surface (ψ_PBR+02nm) closely tracked the corresponding EB estimates (ψ_EB) with a 5.3% coefficient of variation. If one assumes that interfacial potential values adjacent to the particle surface are most relevant for describing environmental phenomena and that a 10% coefficient of variation in potential estimates is acceptable, then presumably any of the non-surficial charge/potential relationships would be useful below an absolute charge density of 0.125 C m ^?2 (with monovalent electrolyte solutions).     

The Influence of Sodium-Free Solutions on the Membrane Potential of Frog Muscle Fibers

The membrane potential of frog sartorius muscle fibers in a Cl- and Na-free Ringer's solution when sucrose replaces NaCl is about the same as that in normal Ringer's solution. The K⁺ efflux is also about the same in the two solutions but muscles lose K and PO₄ in sucrose Ringer's solutions. The membrane potential in sucrose Ringer's solution is equal to that given by the Nernst equation for a K⁺ electrode, when corrections are made for the activity coefficients for K⁺ inside and outside the fiber. For a muscle in normal Ringer's solution, the measured membrane potential is within a few millivolts of E_K. This finding is incompatible with a 1:1 coupled Na-K pump. It is consistent with either no coupling of Na efflux to K influx, or a coupling ratio of 3 or greater.     

Processing of bacteriophage lambda DNA during its assembly into heads

DNA purified from bacteriophage λ added to a cell-free extract derived from induced λ lysogens can be packaged into infectious phage particles (Kaiser & Masuda, 1973). In this paper the structure of the DNA which is the substrate for in vitro packaging and head assembly is described. The active precursor is a multichromosomal polymer that contains covalently closed cohesive end sites. Neither circular or linear DNA monomers nor polymers with unsealed cohesive ends are packaged efficiently into heads. The unit length monomer is packaged when it is either contained in the interior of a polymer (both of its ends are in cos sites) or when it has a free left end and a cos site on its right. The monomer unit with a free right end is not a substrate for packaging.A procedure is given for the purification of λ DNA fragments that contain either the left or the right cohesive end. The fragments are produced by digesting λ DNA with the site-specific Escherichia coli R₁ endonuclease; the left and right ends are separated by sedimentation through a sucrose gradient. These fragments are used to construct small polymers that have a unit length λ monomer with (1) a free left end and a closed right end, (2) a free right end and a closed left end, or (3) both ends closed in cos sites.     

THE KINETICS OF OSMOTIC SWELLING IN LIVING CELLS

The rate of swelling of unfertilized sea urchin eggs in hypotonic sea water was investigated. Analysis of curves leads to the following conclusions. 1. The rate of swelling follows the equation, See PDF for Equation where V _eq., V ₀, and V_t stand for volume at equilibrium, at first instant, and at time t, respectively, the other symbols having their usual significance. This equation is found to hold over a wide range of temperatures and osmotic pressures. This relation is the one expected in a diffusion process. 2. The rate of swelling is found to have a high temperature coefficient (Q ₁₀ = 2 to 3, or µ = 13,000 to 19,000). This deviation from the usual effect of temperature on diffusion processes is thought to be associated with changes in cell permeability to water. The possible influence of changes in viscosity is discussed. 3. The lower the osmotic pressure of the solution, the longer it takes for swelling of the cell. Thus at 15° in 80 per cent sea water, the velocity constant has a value of 0.072, in 20 per cent sea water, of 0.006.     

Mineral ion contents and cell transmembrane electropotentials of pea and oat seedling tissue

The relationships of concentration gradients to electropotential gradients resulting from passive diffusion processes, after equilibration, are described by the Nernst equation. The primary criterion for the hypothesis that any given ion is actively transported is to establish that it is not diffusing passively. A test was made of how closely the Nernst equation describes the electrochemical equilibrium in seedling tissues. Segments of roots and epicotyl internodes of pea (Pisum sativum var. Alaska) and of roots and coleoptiles of oat (Avena sativa var. Victory) seedlings were immersed and shaken in defined nutrient solutions containing eight major nutrients (K⁺, Na⁺, Ca²⁺, Mg²⁺, Cl⁻, NO₃⁻, H₂PO₄⁻ and SO₄²⁻) at 1-fold and 10-fold concentrations. The tissue content of each ion was assayed at 0, 8, 24, and 48 hours. A near-equilibrium condition was approached by roots for most ions; however, the segments of shoot tissue generally continued to show a net accumulation of some ions, mainly K⁺ and NO₃⁻. Only K⁺ approached a reasonable fit to the Nernst equation and this was true for the 1-fold concentration but not the 10-fold. The data suggest that for Na⁺, Mg²⁺, and Ca²⁺ the electrochemical gradient is from the external solution to the cell interior; thus passive diffusion should be in an inward direction. Consequently, some mechanism must exist in plant tissue either to exclude these cations or to extrude them (e.g., by an active efflux pump). For each of the anions the electrochemical gradient is from the tissue to the solution; thus an active influx pump for anions seems required. Root segments approach ionic equilibrium with the solution concentration in which the seedlings were grown. Segments of shoot tissue, however, are far removed from such equilibration. Thus in the intact seedling the extracellular (wall space) fluid must be very different from that of the nutrient solution bathing the segments; it would appear that the root is the site of regulation of ion uptake in the intact plant although other correlative mechanisms may be involved.     

Conformation of the closed channel state of colicin A in proteoliposomes: an umbrella model

Colicin A (ColA) is a water-soluble toxin that forms a voltage-gated channel in the cytoplasmic membrane of Escherichia coli. Until now, two models were proposed for the closed channel state: the umbrella model and the penknife model. Mutants of ColA, each containing a single cysteine, were labeled with a nitroxide spin label, reconstituted into liposomes, and studied by electron paramagnetic resonance (EPR) spectroscopy to study the membrane-bound closed channel state. The spin-labeled ColA variants in solution and in liposomes of native E. coli lipid composition were analyzed in terms of the mobility of the nitroxide, its accessibility to paramagnetic reagents, and the polarity of its microenvironment. The EPR data determined for the soluble ColA pore-forming domain are in agreement with its crystal structure. Moreover, the EPR results show that ColA has a conformation in liposomes different from its water-soluble conformation. Residues that belong to helices H8 and H9 are significantly accessible for O₂ but not for nickel-ethylene diamine diacetic acid, indicating their location inside the membrane. In addition, the polarity values determined from the hyperfine tensor component A_zz of residues 176, 181, and 183 (H9) indicate the location of these residues close to the center of the lipid bilayer, supporting a transmembrane orientation of the hydrophobic hairpin. Furthermore, the accessibility and polarity data suggest that the spin-labeled side chains of the amphipathic helices (H1-H7 and H10) are located at the membrane-water interface. Evidence that the conformation of the closed channel state in artificial liposomes depends on lipid composition is given. The EPR results for ColA reconstituted into liposomes of E. coli lipids support the umbrella model for the closed channel state.     

A Method for Characterizing the Relation between Nutrient Concentration and Flux into Roots of Intact Plants

A method based on the rate of depletion of a nutrient from solution was developed to characterize nutrient flux of plant roots. Nutrient concentration of the solution was measured at a series of time intervals to describe the complete depletion curve. An integrated rate equation, based on a Michaelis-Menten model, was developed and fit to the data of the depletion curve using a least-square procedure. The equation contained values for V_max, the maximum rate of influx; Km, the Michaelis constant; and E, efflux, which were used to describe the relation between solution concentration and net influx rate. Models other than Michaelis-Menten could also be used. The method uses only one plant or group of plants to obtain data over a range of nutrient concentrations, is adapted particularly to the low concentration range, and measures the concentration below which net influx ceases. With this method the plant is in steady state absorption prior to the experiment and continues at this steady state until near the end of the experiment.     

A Brownian ratchet for protein translocation including dissociation of ratcheting sites

We study a model for the translocation of proteins across membranes through a nanopore using a ratcheting mechanism. When the protein enters the nanopore it diffuses in and out of the pore according to a Brownian motion. Moreover, it is bound by ratcheting molecules which hinder the diffusion of the protein out of the nanopore, i.e. the Brownian motion is reflected such that no ratcheting molecule exits the pore. New ratcheting molecules bind at rate γ. Extending our previous approach (Depperschmidt and Pfaffelhuber in Stoch Processes Appl 120:901–925, 2010) we allow the ratcheting molecules to dissociate (at rate δ) from the protein (Model I). We also provide an approximate model (Model II) which assumes a Poisson equilibrium of ratcheting molecules on one side of the current reflection boundary. Using analytical methods and simulations we show that the speeds of both models are approximately the same. Our analytical results on Model II give the speed of translocation by means of a solution of an ordinary differential equation. This speed gives an approximation for the time it takes to translocate a protein of given length.