Electrostatic interactions in protein solution--a comparison between Poisson-Boltzmann and Monte Carlo calculations.

The accuracy of the Poisson-Boltzmann (PB) approximation and its linearized version is investigated by comparison to results obtained from Monte Carlo simulations. The dependence of the calcium binding constant of the protein calbindin as a function of salt concentration and mutation is used as a test case. The protein is modeled as a collection of charged and neutral spheres immersed in the electrolyte solution. The PB equation is solved using a finite difference technique on a grid in a spherical polar coordinate system, which is the preferred choice for a globular protein like calbindin. Both MC and PB give quantitative agreement with experimental results. The linearized PB equation is almost as accurate, but it becomes less reliable in systems with divalent ions. However, the linearized PB equation fails to describe the concentration profiles for cations and anions outside the protein even in a 1:1 salt solution.     

Energetics of charge-charge interactions in proteins

Electrostatic interactions between pairs of atoms in proteins are calculated with a model based on the linearized Poisson-Boltzmann equation. The equation is solved accurately by a method that takes into account the detailed shape of the protein. This paper presents applications to several systems. Experimental data for the interaction of ionized residues with an active site histidine in subtilisin BPN' allow the model to be tested, using various assumptions for the electrical properties of the protein and solvent. The electrostatic stabilization of the active site thiolate of rhodanese is analyzed, with attention to the influence of alpha-helices. Finally, relationships between electrostatic potential and charge-charge distance are reported for large and small globular proteins. The above results are compared with those of simpler electrostatic models, including Coulomb's law with both a distance-dependent dielectric constant (epsilon = R) and a fixed dielectric constant (epsilon = 2), and Tanford-Kirkwood theory. The primary conclusions are as follows: 1) The Poisson-Boltzmann model agrees with the subtilisin data over a range of ionic strengths; 2) two alpha-helices generate a large potential in the active site of rhodanese; 3) epsilon = R overestimates weak electrostatic interactions but yields relatively good results for strong ones; 4) Tanford-Kirkwood theory is a useful approximation to detailed solutions of the linearized Poisson-Boltzmann equation in globular proteins; and 5) the modified Tanford-Kirkwood theory over-screens the measured electrostatic interactions in subtilisin.     

Electrostatic Potential of B-DNA: Effect of Interionic Correlations

Modified Poisson-Boltzmann (MPB) equations have been numerically solved to study ionic distributions and mean electrostatic potentials around a macromolecule of arbitrarily complex shape and charge distribution. Results for DNA are compared with those obtained by classical Poisson-Boltzmann (PB) calculations. The comparisons were made for 1:1 and 2:1 electrolytes at ionic strengths up to 1 M. It is found that ion-image charge interactions and interionic correlations, which are neglected by the PB equation, have relatively weak effects on the electrostatic potential at charged groups of the DNA. The PB equation predicts errors in the long-range electrostatic part of the free energy that are only ∼1.5 kJ/mol per nucleotide even in the case of an asymmetrical electrolyte. In contrast, the spatial correlations between ions drastically affect the electrostatic potential at significant separations from the macromolecule leading to a clearly predicted effect of charge overneutralization.     

Application of polyelectrolyte theory to the study of the B-Z transition in DNA (1)

We have used the polyelectrolyte theory to study the ionic strength dependence of the B-Z equilibrium in DNA. A DNA molecule is molded as an infinitely long continuously charged cylinder of radius a with reduced linear charge density q. The parameters a and q for the B and Z forms were taken from X-ray data: aB = 1nm, qB = 4.2, aZ = 0.9 nm and qZ = 3.9. A simple theory shows that at low ionic strengths (when Debye screening length rD much greater than a) the electrostatic free energy difference FelBZ = FelZ - FelB increases with increasing ionic strength since qB greater than qZ. At high ionic strengths (when rD much less than a) the FelBZ would go on growing with increasing ionic strength if the inequality qB/aB greater than qZ/aZ were valid. In the converse case when qZ/qB greater than aZ/aB the FelBZ value decreases with increasing salt concentration at high ionic strength. Since X-ray data correspond to the latter case, theory predicts that the FelBZ value reaches a maximum at an intermediate ionic strength of about 0.1 M (where rD approximately a). We also performed rigorous calculations based on the Poisson-Boltzmann equation. These calculations have confirmed the above criterion of nonmonotonous behaviour of the FelBZ value as a function of ionic strength. Different theoretical predictions for the B-Z transition in linear and superhelical molecules are discussed. Theory predicts specifically that at a very low ionic strength the Z form may prove to be more stable than the B form.(ABSTRACT TRUNCATED AT 250 WORDS)     

An ESR study of the effect of an electrostatic field on binding of divalent cations to the surface of serum low-density lipoproteins

The ESR technique has been used to study binding of Mn(II) ions to low-density lipoprotein (LDL) in solutions of various electrolyte ionic strengths. A model of the binding has been proposed which describes all the observations in electrolytes of ten different concentrations in terms of two types of binding sites and two corresponding sets of intrinsic binding parameters (n1 = 8, Kd1 = 1.31 X 10(-3) mol X l-1 and n2 = 170, Kd2 = 5.71 X 10(-2) mol X l-1). These parameters, together with the values of the potential (phi 0) responsible for binding of the ions to specific charged sites on the surface, reproduce the observed binding curves well in all the systems studied. The phi 0 values are obtained as an appropriate solution of the Poisson-Boltzmann equation.     

Effect of delay on auto-oscillations in cell populations]

A model of homogenous cell population is analysed, the mitotic activity of which is controled by an inhibitor produced by the cells themselves. While deducing the model it is assumed that the inhibitor concentration is proportional to the population density at some preceding time moment. The density change in time is described by a delay differential equation: dx/dt=alpha x/(1+x(v)(t--0))--x. According to this equation oscillation of cell number may be observed in the population with an increase of the critical value of delay in the mechanism of mitotic activity inhibition. The oscillation period determined analytically in linear approximation equals the quadruplicated time of delay. Numerical integration of the model has shown that the period observed is close to the calculated one.     

Protein Electrostatics: Rapid Multigrid-Based Newton Algorithm for Solution of the Full Nonlinear Poisson-Boltzmann Equation

Abstract

A new method for solving the full nonlinear Poisson-Boltzmann equation is outlined. This method is robust and efficient, and uses a combination of the multigrid and inexact Newton algorithms. The novelty of this approach lies in the appropriate combination of the two methods, neither of which by themselves are capable of solving the nonlinear problem accurately. Features of the Poisson-Boltzmann equation are fully exploited by each component of the hybrid algorithm to provide robustness and speed. The advantages inherent in this method increase with the size of the problem. The efficacy of the method is illustrated by calculations of the electrostatic potential around the enzyme Superoxide Dismutase. The CPU time required to solve the full nonlinear equation is less than half that needed for a conjugate gradient solution of the corresponding linearized Poisson-Boltzmann equation. The solutions reveal that the field around the active sites is significantly reduced as compared to that obtained by solving the corresponding linearized Poisson-Boltzmann equation. This new method for the nonlinear Poisson-Boltzmann equation will enable fast and accurate solutions of large protein electrostatics problems.     

A fast and simple method to calculate protonation states in proteins.

A simple model for electrostatic interactions in proteins, based on a distance and position dependent screening of the electrostatic potential, is presented. It is applied in conjunction with a Monte Carlo algorithm to calculate pK(alpha) values of ionizable groups in proteins. The purpose is to furnish a simple, fast, and sufficiently accurate model to be incorporated into molecular dynamic simulations. This will allow for dynamic protonation calculations and for coupling between changes in structure and protonation state during the simulation. The best method of calculating protonation states available today is based on solving the linearized Poisson-Boltzmann equation on a finite difference grid. However, this model consumes far too much computer time to be a practical alternative. Tests are reported for fixed structures on bacteriorhodopsin, lysozyme, myoglobin, and calbindin. The studies include comparisons with Poisson-Boltzmann calculations with dielectric constants 4 and 20 inside the protein, a model with uniform dielectric constant 80 and distance-dependent dielectric models. The accuracy is comparable to that of Poisson-Boltzmann calculations with dielectric constant 20, and it is considerably better than that with epsilon = 4. The time to calculate the protonation at one pH value is at least 100 times less than that of a Poisson-Boltzmann calculation. Proteins 1999;36:474-483.     

Equilibrium surface charge distribution in phospholipid vesicles. I. Method of calculation

This paper presents a method of calculation of the surface charge equilibrium distribution between the two surfaces of a spherically closed phospholipid bilayer suspended in aqueous electrolyte solution. The net surface charge is supposed to be provided by the ionized polar groups of the phospholipid molecules. Its equilibrium distribution is found by minimization of the free electrostatic energy. The procedure of minimization utilizes the solution of the Poisson-Boltzmann equation which describes the double electric layers of the membrane and an expression for the membrane potential derived under the assumption of absence of charges in the membrane phase. An analytical solution of the problem in the range of validity of the linearized Poisson-Boltzman equation is obtained. It is shown that in this case an equilibrium transmembrane potential exists, and the surface charge density is greater at the outer surface of the vesicle.     

Atomic force microscope measurements of long-range forces near lipid-coated surfaces in electrolytes.

The interaction of DMPC (L-alpha-dimyristoyl-1,2-diterradecanoyl-sn-glycero-3-phosphoch oli ne, C36H72NO8P) lipid-coated Si3N4 surfaces immersed in an electrolyte was investigated with an atomic force microscope. A long-range interaction was observed, even when the Si3N4 surfaces were covered with nominally neutral lipid layers. The interaction was attributed to Coulomb interactions of charges located at the lipid surface. The experimental force curves were compared with solutions for the linearized as well as with exact solutions of the Poisson-Boltzmann equation. The comparison suggested that in 0.5 mM KCl electrolyte the DMPC lipids carried about one unit of charge per 100 lipid molecules. The presence of this surface charge made it impossible to observe an effective charge density recently predicted for dipole layers near a dielectric when immersed in an electrolyte. A discrepancy between the theoretical results and the data at short separations was interpreted in terms of a decrease in the surface charge with separation distance.     

Modeling salt-mediated electrostatics of macromolecules: the discrete surface charge optimization algorithm and its application to the nucleosome

Much progress has been achieved on quantitative assessment of electrostatic interactions on the all-atom level by molecular mechanics and dynamics, as well as on the macroscopic level by models of continuum solvation. Bridging of the two representations-an area of active research-is necessary for studying integrated functions of large systems of biological importance. Following perspectives of both discrete (N-body) interaction and continuum solvation, we present a new algorithm, DiSCO (Discrete Surface Charge Optimization), for economically describing the electrostatic field predicted by Poisson-Boltzmann theory using a discrete set of Debye-Hückel charges distributed on a virtual surface enclosing the macromolecule. The procedure in DiSCO relies on the linear behavior of the Poisson-Boltzmann equation in the far zone; thus contributions from a number of molecules may be superimposed, and the electrostatic potential, or equivalently the electrostatic field, may be quickly and efficiently approximated by the summation of contributions from the set of charges. The desired accuracy of this approximation is achieved by minimizing the difference between the Poisson-Boltzmann electrostatic field and that produced by the linearized Debye-Hückel approximation using our truncated Newton optimization package. DiSCO is applied here to describe the salt-dependent electrostatic environment of the nucleosome core particle in terms of several hundred surface charges. This representation forms the basis for modeling-by dynamic simulations (or Monte Carlo)-the folding of chromatin. DiSCO can be applied more generally to many macromolecular systems whose size and complexity warrant a model resolution between the all-atom and macroscopic levels.     

Effects of the glycocalyx on the electrophoretic mobility of red cells and on streaming potentials in blood vessels: predictions of a structurally-based model

The polyelectrolyte layer coating mammalian cells, known as the glycocalyx, may be important in communicating flow information to the cell. In this paper, the layer is modelled as a semi-infinite, doubly periodic array of parallel charged cylinders. The electric potential and ion distributions surrounding such an array are found using the linearized Poisson-Boltzmann equation and an iterative domain decomposition technique. Similar methods are used to calculate Strokes flows, driven either by a shear at infinity or by an electric field, parallel or transverse to the cylinders. The resulting electric streaming currents due to flow over endothelial cells, and the electrophoretic mobilities of red blood cells are deduced as functions of polymer concentration and electrolyte molarity. It is shown that only the top portion of the layer is important in these effects.     

Dynamics of noninvasively estimated microvascular O2 extraction during ramp exercise.

Utilization of near-infrared spectroscopy (NIRS) in clinical exercise testing to detect microvascular abnormalities requires characterization of the responses in healthy individuals and theoretical foundation for data interpretation. We examined the profile of the deoxygenated hemoglobin signal from NIRS {deoxygenated hemoglobin + myoglobin [deoxy-(Hb+Mb)] approximately O(2) extraction} during ramp exercise to test the hypothesis that the increase in estimated O(2) extraction would be close to hyperbolic, reflecting a linear relationship between muscle blood flow (Q(m)) and muscle oxygen uptake (Vo(2)(m)) with a positive Q(m) intercept. Fifteen subjects (age 24 +/- 5 yr) performed incremental ramp exercise to fatigue (15-35 W/min). The deoxy-(Hb+Mb) response, measured by NIRS, was fitted by a hyperbolic function [f(x) = ax/(b + x), where a is the asymptotic value and b is the x value that yields 50% of the total amplitude] and sigmoidal function {f(x) = f(0) + A/[1 + e(-(-c+dx))], where f(0) is baseline, A is total amplitude, and c is a constant dependent on d, the slope of the sigmoid}, and the goodness of fit was determined by F test. Only one subject demonstrated a hyperbolic increase in deoxy-(Hb+Mb) (a = 170%, b = 193 W), whereas 14 subjects displayed a sigmoidal increase in deoxy-(Hb+Mb) (f(0) = -7 +/- 7%, A = 118 +/- 16%, c = 3.25 +/- 1.14, and d = 0.03 +/- 0.01). Computer simulations revealed that sigmoidal increases in deoxy-(Hb+Mb) reflect a nonlinear relationship between microvascular Q(m) and Vo(2)(m) during incremental ramp exercise. The mechanistic implications of our findings are that, in most healthy subjects, Q(m) increased at a faster rate than Vo(2)(m) early in the exercise test and slowed progressively as maximal work rate was approached.     

一类被开发的捕食系统的定性分析

讨论了一类食饵种群被开发的两种群捕食系统: dx/dt=x(a0+a1x-a2x2-a3y3)-h0, dy/dt=y(x-1)其中a0>0,a2>0,a3>0,h0>0,a1不定号.文中主要讨论了系统平衡点的行为以及系统的稳定性.用Pioncare切性曲线法及Dulac函数法讨论闭轨不存在的充分条件;用Hopf分支方法及张芷芬唯一性定理证明了极限环的存在性与唯一性.同时对相应结论的生态学意义给予了说明.     

Kinetic studies on partially liganded species of carboxyhemoglobin: (alpha 1CO-beta 1CO)alpha 2 beta 2 or (alpha 2CO beta 2CO)alpha 1 beta 1

Kinetics of CO combination with and dissociation from isomer III, (alpha 1CO beta 1CO)alpha 2 beta 2 or alpha 1 beta 1 (alpha 2CO beta 2CO), and Hb Rothschild have been studied using the double mixing and microperoxidase methods. Isomer III was prepared in a manner so that it was the only reactive species in the reaction mixture. The biphasic reaction time course in both the "on" and "off" reactions of isomer III and the CO combination reaction of Hb Rothschild are attributed to slow relaxation between the fast and slow CO-reacting species in the two proteins: isomer III: l'f = 6 x 10(6) M-1 s-1, l'dimer = 1.7 x 10(6) M-1 s-1, l's = 2.2 x 10(5) M-1 s-1, lf = 0.15 s-1, ls = 0.01 s-1; Hb Rothschild: l'f = 2.8 x 10(6) M-1 s-1; l's = 2.7 x 10(5) M-1 s-1.     

Micellar catalysis for oxidation of nitric oxide (NO) in the multi-phase systems in vivo.

The equation of the dependence of the third-order reaction acceleration due to concentrating the reagents in a small volume of the hydrophobic phase on the partition coefficients of reagents (Q) and on the lipophilic phase fraction (x), [k(app)/ k2 = H(Q(NO),Q(O2),x)] was analyzed. It was demonstrated that the numeric value of dH/dx at x-->0 could not be used in order to calculate the efficiency of catalysis from the experimental data. It was shown that, unlike in two-phase systems (with an aqueous and a hydrophobic phase), the dependence of H on Q in multi-phase systems, that include all in vivo systems, is different. The multiple phase state of the systems has a determining role for a regulation of NO-dependent processes and in the realization of conditions of 'NO catastrophes'.     

Greater glycogen utilization during 1- than 2-adrenergic receptor stimulation in the isolated perfused rat heart

Differences in energy metabolism during beta(1)- and beta(2)-adrenergic receptor (AR) stimulation have been shown to translate to differences in the elicited functional responses. It has been suggested that differential access to glycogen during beta(1)- compared with beta(2)-AR stimulation may influence the peak functional response and modulation of the response during sustained adrenergic stimulation. Interleaved (13)C- and (31)P-NMR spectroscopy was used during beta(1)- and beta(2)-AR stimulation at matched peak workload (2.5 times baseline) in the isolated perfused rat heart to monitor glycogen levels, phosphorylation potential, and intracellular pH. Simultaneous measurements of left ventricular (LV) function [LV developed pressure (LVDP)], heart rate (HR), and rate-pressure product (RPP = LVDP x HR) were also performed. The heart was perfused under both substrate-free (SF) conditions and with exogenous glucose (G). The greater glycogenolysis was observed during beta(1)- than beta(2)-AR stimulation with G (54% vs. 38% reduction, P = 0.006) and SF (92% vs. 79% reduction, P = 0.04) perfusions. The greater beta(1)-AR-mediated glycogenolysis was correlated with greater ability to sustain the initial contractile response. However, with SF perfusion, the duration of this ability was limited: excessive early glycogen depletion caused an earlier decline in LVDP and phosphorylation potential during beta(1)- than beta(2)-AR stimulation. Therefore, endogenous glycogen stores are depleted earlier and to a greater extent, despite a slightly weaker overall inotropic response, during beta(1)- than beta(2)-AR stimulation. These findings are consistent with beta(1)-AR-specific PKA-dependent glycogen phosphorylase kinase signaling.     

Macromolecule transfer through mesothelium and connective tissue.

Diffusional permeability (P) to inulin (P(in)), albumin (P(alb)), and dextrans [70 (P(dx 70)), 150 (P(dx 150)), 550 (P(dx 550)), and 2, 000 (P(dx 2,000))] was determined in specimens of parietal pericardium of rabbits, which may be obtained with less damage than pleura. P(in), P(alb), P(dx 70), P(dx 150), P(dx 550), and P(dx 2, 000) were 0.51 +/- 0.06 (SE), 0.18 +/- 0.03, 0.097 +/- 0.021, 0. 047 +/- 0.011, 0.025 +/- 0.004, and 0.021 +/- 0.005 x 10(-5) cm/s, respectively. P(in), P(alb), and P(dx 70) of connective tissue, obtained after removal of mesothelium from specimens, were 10.3 +/- 1.42, 2.97 +/- 0.38, and 2.31 +/- 0.16 x 10(-5) cm/s, respectively. Hence, P(in), P(alb), and P(dx 70) of mesothelium were 0.54, 0.20, and 0.10 x 10(-5) cm/s, respectively. Inulin (like small solutes) fitted the relationship P-solute radius for restricted diffusion with a 6-nm "pore" radius, whereas macromolecules were much above it. Hence, macromolecule transfer mainly occurs through "large pores" and/or transcytosis. In line with this, the addition of phospholipids on the luminal side (which decreases pore radius to approximately 1.5 nm) halved P(in) but did not change P(alb) and P(dx 70). P(in) is roughly similar in mesothelium and capillary endothelium, whereas P to macromolecules is greater in mesothelium. The albumin diffusion coefficient through connective tissue was 17% of that in water. Mesothelium provides 92% of resistance to albumin diffusion through the pericardium.     

A piezoelectric quartz crystal impedance study on Cu(2+)-induced precipitation of bovine serum albumin in aqueous solution

Piezoelectric quartz crystal impedance (QCI) technique was used for monitoring the Cu(2+)-induced precipitation of bovine serum albumin onto the gold electrode. The critical precipitate concentration of Cu(2+) reflected by the significant decrease in the resonant frequency was estimated to be 9.98 x 10(-5) mol x l(-1), and the saturated adherence of the precipitate on the electrode occurred when the Cu(2+) concentration was greater than 9.79x10(-3) mol x l(-1). The frequency shift in air was about 85.5% of that in liquid, and the Deltaf(0)/DeltaR(1) ratio found in solution was 82.67 Hz Omega(-1), suggesting that the frequency response was predominated by the mass change due to precipitate adherence to the electrode surface. The response of the resonant frequency was analyzed using an equation Deltaf=a(0) + a(1) e(-t/tau(1)) + a(2) e(-t/tau(2)). The relationship between the total a(0) values and the Cu(2+) concentration was discussed.