The drift approximation solves the Poisson, Nernst-Planck, and continuum equations in the limit of large external voltages

Nearly linear current-voltage curves are frequently found in biological ion channels. Using the drift limit of the substantially non-linear Poisson-Nernst-Planck equations, we explain such behavior of diffusion-controlled charge transport systems. Starting from Gauss' law, drift, and continuity equations we derive a simple analytical current-voltage relation, which accounts for this deviation from linearity. As shown previously, the drift limit of the Nernst-Planck equation applies if the total electric current is dominated by the electric field, and integral contributions from concentration gradients are small. The simple analytical form of the drift current-voltage relations makes it an ideal tool to analyze experiment current-voltage curves. We also solved the complete Poisson-Nernst-Planck equations numerically, and determined current-voltage curves over a wide range of voltages, concentrations, and Debye lengths. The simulation fully supports the analytical estimate that the current-voltage curves of simple charge transport systems are dominated by the drift mechanism. Even those relations containing the most extensive approximations remained qualitatively within the correct order of magnitude. Received: 24 September 1998 / Revised version: 22 January 1999 / Accepted: 22 January 1999     

Kinetic criteria of solid state mechanisms in biology

Kinetic criteria for solid state physical mechanisms of electron and ion transport in biological systems are summarized, and the mechanisms are discussed. A reaction which is rate-limited by electron or ion transport across a particle or membrane in accord with Ohm's law will show first order kinetics, with an hyperbolic relationship between rate constant and the sum of substrate plus product. Larger initial substrate concentrations produce smaller rate constants, thus giving the appearance of substrate inhibition. Examples are cytochrome oxidase and peroxidase, and pyruvate carboxylase. Ohmic transport mechanisms may be caused by electron conduction or superconduction through protein, by electron conduction through water, or by conduction of ions through membranes. A reaction which is rate-limited by charge transport across an activation energy barrier at an interface in accord with a logarithmic voltage-current law will show reaction kinetics conforming to the Elovich equation, and will have the appearance of a pair of simultaneous first order processes. Examples include decay of photogenerated free radicals in eye melanin particles and in photosynthetic particles of bacteria, and sodium and potassium ion transport across cell surfaces. The logarithmic voltage-current law may be regarded as an empirical relationship describing behavior of interfaces, justified by extensive experimental data on many types of interfaces, or it may be derived theoretically for individual cases from statistical mechanical and/or solid state physical considerations. Dedicated to Prof. N. Rashevsky and to his enlightened editorial policy, especially to his policy of publishing that which is new, even when he disagrees with it.     

Thermodynamic principles of the behavior of biological systems

A theory of the behavior of biological systems is proposed which is an extension of the conception of biological evolution (Gladyshev, 1977, Gladyshev, 1978) based on classical (equilibrium) thermodynamics. A thermodynamic theory of homeostasis is presented, in accordance with which homeostatic mechanisms of regulation are connected with a compensative shift of a fundamental quasi-equilibrium. The principle of least compulsion is formulated on the basis of thermodynamic laws and describes behavior of biological systems. A fundamental thermodynamic equation of behavioral processes is introduced. The Weber-Fechner law is shown to be a corollary of the fundamental thermodynamic equation.     

Logarithmic and Power Law Input-Output Relations in Sensory Systems with Fold-Change Detection

Two central biophysical laws describe sensory responses to input signals. One is a logarithmic relationship between input and output, and the other is a power law relationship. These laws are sometimes called the Weber-Fechner law and the Stevens power law, respectively. The two laws are found in a wide variety of human sensory systems including hearing, vision, taste, and weight perception; they also occur in the responses of cells to stimuli. However the mechanistic origin of these laws is not fully understood. To address this, we consider a class of biological circuits exhibiting a property called fold-change detection (FCD). In these circuits the response dynamics depend only on the relative change in input signal and not its absolute level, a property which applies to many physiological and cellular sensory systems. We show analytically that by changing a single parameter in the FCD circuits, both logarithmic and power-law relationships emerge; these laws are modified versions of the Weber-Fechner and Stevens laws. The parameter that determines which law is found is the steepness (effective Hill coefficient) of the effect of the internal variable on the output. This finding applies to major circuit architectures found in biological systems, including the incoherent feed-forward loop and nonlinear integral feedback loops. Therefore, if one measures the response to different fold changes in input signal and observes a logarithmic or power law, the present theory can be used to rule out certain FCD mechanisms, and to predict their cooperativity parameter. We demonstrate this approach using data from eukaryotic chemotaxis signaling.     

Cyclic kinetics and mathematical expression of the primary immune response to soluble antigen

The injected dose of antigen determines not only the duration of its persistence in the injection site but also the intensity of plasma cell response in the regional lymph node. It was found that the logarithmic sum of antigen quantity in the injection site was related to the sum of cell response values, the correlation coefficient approaching 1. The antigen-lymphoid system interrelations appear to obey Weber-Fechner’s law for afferent systems of the organism. The sum of plasma cells appeared to be in direct connection with the logarithm of the dose injected, with antigen persistence in the injection site and also with the tangent of the acute angle adjoining the ordinate. The basic components of the primary immune response of the organism to soluble antigen,viz. logarithm of the dose injected, antigen persistence in the injection place, plasma cell quantity, tangent of the acute angle, transition modulus from antigen to plasma cells, are interconnected by rather simple equations, which represent the structural elements of the mathematical model described in the text.     

Electrogenicity, pH-Dependence, and Stoichiometry of the Proton-Sucrose Symport

The electrogenicity, pH-dependence, and stoichiometry of the proton-sucrose symport were examined in plasma membrane vesicles isolated from sugar beet (Beta vulgaris L. cv Great Western) leaves. Symport mediated sucrose transport was electrogenic as demonstrated by the effect of membrane potential on ΔpH-dependent flux. In the absence of significant charge compensation, a low rate of sucrose transport was observed. When membrane potential was clamped at zero with symmetric potassium concentrations and valinomycin, the rate of sucrose flux was stimulated fourfold. In the presence of a negative membrane potential, transport increased six-fold. These results are consistent with electrogenic sucrose transport which results in a net flux of positive charge into the vesicles. The effect of membrane potential on the kinetics of sucrose transport was on V_max only with no apparent change in K_m. Sucrose transport rates driven by membrane potential only, i.e. in the absence of ΔpH, were comparable to ΔpH-driven flux. Both membrane potential and ΔpH-driven sucrose transport were used to examine proton binding to the symport and the apparent K_m for H⁺ was 0.7 micromolar. The kinetics of sucrose transport as a function of proton concentration exhibited a simple hyperbolic relationship. This observation is consistent with kinetic models of ion-cotransport systems when the stoichiometry of the system, ion:substrate, is 1:1. Quantitative measurements of proton and sucrose fluxes through the symport support a 1:1 stoichiometry. The biochemical details of protoncoupled sucrose transport reported here provide further evidence in support of the chemiosmotic hypothesis of nutrient transport across the plant cell plasma membrane.     

Evidence for Carrier-Mediated Transport of Monosaccharides in the Ehrlich Ascites Tumor Cell

Evidence for carrier-mediated transport of monosaccharides in the Ehrlich ascites tumor cells was provided through kinetic analysis of data obtained by: (a) studying sugar uptake by dilute cell suspensions with an optical densimetric apparatus, (b) studying sugar uptake by thicker cell suspensions by means of direct chemical analytical methods using packed cell plugs, (c) observing the effects of a competitive inhibitor upon sugar uptake with the chemical analytical method, and (d) measurement of tracer uptake of a high affinity sugar in thick cell suspensions in the absence of net movement. Quantitative application of the data obtained with the above experimental procedures to theoretical model systems derived for both carrier-mediated transport and simple passive diffusion indicated that the results were consonant with predictions for the carrier-mediated transport model, but could not be explained on the basis of uncomplicated diffusion.     

Derivation of the equations that describe the effects of unstirred water layers on the kinetic parameters of active transport processes in the intestine

Unidirectional flux of solutes into the intestinal mucosal cells is determined by the rate of movement of these molecules across both an unstirred water layer and the microvillus membrane of the epithelial cell. Therefore, an equation is derived in this paper that describes the velocity of active transport as a function of the characteristics of both the transport carrier in the membrane and the resistance of the overlying unstirred water layer. Using this equation a series of curves are presented that depict the effect on the kinetics of active transport of varying the thickness (d) or surface area (S_w) of the unstirred water layer, the free diffusion coefficient (D) of the solute, the distribution of active transport sites along the villus ($\text{?}{}_{\mathrm{<mtext>n</mtext>}}$), the maximal transport velocity (J^m_d) and the true Michaelis constant (K_m). These theoretical curves illustrate the serious limitations inherent in interpretation of previously published data dealing with active transport processes in the intestine.     

Predicting Chemical Environments of Bacteria from Receptor Signaling

Sensory systems have evolved to respond to input stimuli of certain statistical properties, and to reliably transmit this information through biochemical pathways. Hence, for an experimentally well-characterized sensory system, one ought to be able to extract valuable information about the statistics of the stimuli. Based on dose-response curves from in vivo fluorescence resonance energy transfer (FRET) experiments of the bacterial chemotaxis sensory system, we predict the chemical gradients chemotactic Escherichia coli cells typically encounter in their natural environment. To predict average gradients cells experience, we revaluate the phenomenological Weber''s law and its generalizations to the Weber-Fechner law and fold-change detection. To obtain full distributions of gradients we use information theory and simulations, considering limitations of information transmission from both cell-external and internal noise. We identify broad distributions of exponential gradients, which lead to log-normal stimuli and maximal drift velocity. Our results thus provide a first step towards deciphering the chemical nature of complex, experimentally inaccessible cellular microenvironments, such as the human intestine.     

Generalizations of the Roginsky-Zeldovich (or Elovich) equation for charge transport across biological surfaces

The Roginsky-Zeldovich (or Elovich) equation, which is −dx/dt=m exp (nx) (x=substrate concentration,t=time,m andn=constants), describes the kinetics of various biological electron and ion transport processes, and has been derived from the concept of charge transport across an activation energy barrier at an interface between dissimilar phases, driven by a difference in redox or ion potentials, with the simplifying assumptions that charge carrier concentration is constant, backward current across the interface is zero, and diffusion of substrate is fast. If charge carrier concentration is proportional to substrate concentration, then the kinetic equation is −dx/dt=mx exp (nx). If backward current is not zero, then −dx/dt=m ₁ exp (n ₁x) −m ₂ exp (n ₂ x), wherem ₁,m ₂,n ₁ andn ₂ are constants. Kinetic equations for interfacial charge transport in the presence of a significant substrate diffusion potential are also derived.     

A theory of ion transport across cell surfaces by a process analogous to electron transport across liquid-solid interfaces

A kinetic theory of ion transport across cell surfaces has been developed in a form analogous to the kinetic theory of electron transport across solid-liquid interfaces of biological particles. The ionic theory is based on the observation that, at least in one instance, the voltage-current behavior for ion conduction across a cell surface is describable by the Tafel equation, in analogy to the conduction of electrons across solid-liquid interfaces. The theory predicts that the kinetics of ion transport across cell surfaces should conform to the Elovich rate equation, which is shown to be true for various experimental data. The opinions and conclusions contained in this report are those of the author. They are not to be construed as necessarily reflecting the views or the endorsement of the Navy Department.     

Dynamics of cockroach ocellar neurons

The incremental responses from the second-order neurons of the ocellus of the cockroach, Periplaneta americana, have been measured. The stimulus was a white-noise-modulated light with various mean illuminances. The kernels, obtained by cross-correlating the white-noise input against the resulting response, provided a measure of incremental sensitivity as well as of response dynamics. We found that the incremental sensitivity of the second-order neurons was an exact Weber-Fechner function; white-noise-evoked responses from second-order neurons were linear; the dynamics of second-order neurons remain unchanged over a mean illuminance range of 4 log units; the small nonlinearity in the response of the second-order neuron was a simple amplitude compression; and the correlation between the white-noise input and spike discharges of the second-order neurons produced a first-order kernel similar to that of the cell's slow potential. We conclude that signal processing in the cockroach ocellus is simple but different from that in other visual systems, including vertebrate retinas and insect compound eyes, in which the system's dynamics depend on the mean illuminance.     

Deviations from the flux ratio equation for chloride ions in ouabain- and acetazolamide-treated frog skin

In order to establish whether or not chloride ions behave as freely moving particles in “passive”, i.e. ouabain-and acetazolamide-treated, frog skin, tracer fluxes of ³⁶Cl⁻ have been measured while a voltage (generally +40 mV, serosal side positive) across the skin was applied. Ussing's flux ratio equation has been used as a criterion for this type of transport. One group of skin samples exhibited significant exchange diffusion phenomena. Most samples in a second group either behaved according to the flux ratio equation or showed significant and extreme exchange diffusion. From flux ratios obtained at two different voltages across various skin samples, showing extreme exchange diffusion, it appeared that the simple form of Kedem and Essig's law derived from irreversible thermodynamics, which is valid for homogeneous systems, does not apply to the type of exchange diffusion found. The system can, however, be described by a 1 : 1 exchange mechanism working in parallel with a diffusional pathway. The ratio exchange flux/observed efflux must then have a constant value (0.83) at the voltages applied, which implies that the exchange flux is voltage dependent. By comparison with iodide flux experiments as carried out by Ussing, it is shown that iodide exhibits the same type of exchange diffusion. A carrier, possibly responsibe for the observed behaviour, is described.     

Carrier-Mediated Uptake of 1-(Malonylamino)cyclopropane-1-Carboxylic Acid in Vacuoles Isolated from Catharanthus roseus Cells

The uptake of 1-(malonylamino)cyclopropane-1-carboxylic acid (MACC), the conjugated form of the ethylene precursor, into vacuoles isolated from Catharanthus roseus cells has been studied by silicone layer floatation filtering. The transport across the tonoplast of MACC is stimulated fourfold by 5 millimolar MgATP, has a K_m of about 2 millimolar, an optimum pH around 7, and an optimum temperature at 30°C. Several effectors known to inhibit ATPase (N,N′-dicyclohexylcarbodiimide) and to collapse the transtonoplastic H⁺ electrochemical gradient (carbonylcyanide m-chlorophenylhydrazone, gramicidin, and benzylamine) all reduced MACC uptake. Abolishing the membrane potential with SCN⁻ and valinomycin also greatly inhibited MACC transport. Our data demonstrate that MACC accumulates in the vacuole against a concentration gradient by means of a proton motive force generated by a tonoplastic ATPase. The involvement of a protein carrier is suggested by the strong inhibition of uptake by compounds known to block SH—, OH—, and NH₂— groups. MACC uptake is antagonized competitively by malonyl-d-tryptophan, indicating that the carrier also accepts malonyl-d-amino acids. Neither the moities of these compounds taken separately [1-aminocyclopropane-1-carboxylic acid, malonate, d-tryptophan or d-phenylalanine] nor malate act as inhibitors of MACC transport. The absence of inhibition of malate uptake by MACC suggests that MACC and malate are taken up by two different carriers. We propose that the carrier identified here plays an important physiological role in withdrawing from the cytosol MACC and malonyl-d-amino acids generated under stress conditions.     

Both chlorophylls a and d are essential for the photochemistry in photosystem II of the cyanobacteria, Acaryochloris marina

We have measured the flash-induced absorbance difference spectrum attributed to the formation of the secondary radical pair, P⁺Q⁻, between 270 nm and 1000 nm at 77 K in photosystem II of the chlorophyll d containing cyanobacterium, Acaryochloris marina. Despite the high level of chlorophyll d present, the flash-induced absorption difference spectrum of an approximately 2 ms decay component shows a number of features which are typical of the difference spectrum seen in oxygenic photosynthetic organisms containing no chlorophyll d. The spectral shape in the near-UV indicates that a plastoquinone is the secondary acceptor molecule (Q_A). The strong C-550 change at 543 nm confirms previous reports that pheophytin a is the primary electron acceptor. The bleach at 435 nm and increase in absorption at 820 nm indicates that the positive charge is stabilized on a chlorophyll a molecule. In addition a strong electrochromic band shift, centred at 723 nm, has been observed. It is assigned to a shift of the Qy band of the neighbouring accessory chlorophyll d, Chl_D1. It seems highly likely that it accepts excitation energy from the chlorophyll d containing antenna. We therefore propose that primary charge separation is initiated from this chlorophyll d molecule and functions as the primary electron donor. Despite its lower excited state energy (0.1 V less), as compared to chlorophyll a, this chlorophyll d molecule is capable of driving the plastoquinone oxidoreductase activity of photosystem II. However, chlorophyll a is used to stabilize the positive charge and ultimately to drive water oxidation.     

The meaning of the Weber-Fechner law and description of scenes: III. Description of the visual space

The Weber-Fechner fraction was measured in experiments where a subject was presented with a pair of lines, the length of one (reference) line being constant and the length of the other (test) one varying. The subject had to say whether the longer line was above or below the shorter one. The subject was trained to distinguish the positions of lines in areas limited in the number of reference lines located at different degrees of eccentricity. The fraction curve for each area proved to be of the same shape as the curve obtained for the entire range and reflecting the Weber-Fechner law. The fraction was originally maximum and only slightly decreased afterwards. On the basis of these data, a neural construction is suggested that serves for describing the visual space and explains the shape of the curve reflecting the Weber-Fechner law.     

Purification and characterisation of a novel DNA methyltransferase,M.AhdI

We have cloned the M and S genes of the restriction-modification (R-M) system AhdI and have purified the resulting methyltransferase to homogeneity. M.AhdI is found to form a 170 kDa tetrameric enzyme having a subunit stoichiometry M₂S₂ (where the M and S subunits are responsible for methylation and DNA sequence specificity, respectively). Sedimentation equilibrium experiments show that the tetrameric enzyme dissociates to form a heterodimer at low concentration, with K_d ≈ 2 µM. The intact (tetrameric) enzyme binds specifically to a 30 bp DNA duplex containing the AhdI recognition sequence GACN₅GTC with high affinity (K_d ≈ 50 nM), but at low enzyme concentration the DNA binding activity is governed by the dissociation of the tetramer into dimers, leading to a sigmoidal DNA binding curve. In contrast, only non-specific binding is observed if the duplex lacks the recognition sequence. Methylation activity of the purified enzyme was assessed by its ability to prevent restriction by the cognate endonuclease. The subunit structure of the M.AhdI methyltransferase resembles that of type I MTases, in contrast to the R.AhdI endonuclease which is typical of type II systems. AhdI appears to be a novel R-M system with properties intermediate between simple type II systems and more complex type I systems, and may represent an intermediate in the evolution of R-M systems.     

Mechanism of amino Acid uptake by sugarcane suspension cells

The amino acid carriers in sugarcane suspension cells were characterized for amino acid specificity and the stoichiometry of proton and potassium flux during amino acid transport.

Amino acid transport by sugarcane cells is dependent upon three distinct transport systems. One system is specific for neutral amino acids and transports all neutral amino acids including glutamine, asparagine, and histidine. The uptake of neutral amino acids is coupled to the uptake of one proton per amino acid; one potassium ion leaves the cells for charge compensation. Histidine is only taken up in the neutral form so that deprotonation of the charged imidazole nitrogen has to occur prior to uptake. The basic amino acids are transported by another system as uniport with charge-compensating efflux of protons and potassium. The acidic amino acids are transported by a third system. Acidic amino acids bind to the transport site only if the distal carboxyl group is in the dissociated form (i.e. if the acidic amino acid is anionic). Two protons are withdrawn from the medium and one potassium leaves the cell for charge compensation during the uptake of acid amino acids. Common to all three uptake systems is a monovalent positively charged amino acidproton carrier complex at the transport site.

     

A general kinetic analysis of transport Tests of the carrier model based on predicted relations among experimental parameters

The analysis of transport kinetics has lacked both a unified treatment in which general rate equations are written entirely in terms of experimental parameters, and a convention by which these parameters may be designated in a concise yet immediately recognizable manner. Such a treatment is presented here in an easily accessible form, and a simple system of nomenclature is proposed resembling that in use in enzyme kinetics. The treatment is independent of assumptions about rate-limiting steps in transport, and applies to both active and facilitated systems, including obligatory exchange. A single substrate is characterized by twelve different parameters, only five of which are required in theory to calculate the others. If a second substrate is present on the trans side of the membrane there are six more parameters. All eighteen parameters are linked by multiple relationships which provide a complete set of rejection criteria for the generalized form of the mobile carrier. Relationships among parameters are also defined that give information on the rate-limiting steps in transport. Equations governing any individual experiment, involving only experimental parameters, are easily written out from the general expressions, for example under conditions of zero trans and infinite trans flux, equilibrium exchange, or competitive inhibition.     

DeltapH-Dependent Amino Acid Transport into Plasma Membrane Vesicles Isolated from Sugar Beet Leaves: I. Evidence for Carrier-Mediated, Electrogenic Flux through Multiple Transport Systems

Amino acid transport into plasma membrane vesicles isolated from mature sugar beet (Beta vulgaris L. cv Great Western) leaves was investigated. The transport of alanine, leucine, glutamine, glutamate, isoleucine, and arginine was driven by a trans-membrane proton concentration difference. ΔpH-Dependent alanine, leucine, glutamine, and glutamate transport exhibited simple Michaelis-Menten kinetics, and double-reciprocal plots of the data were linear with apparent K_m values of 272, 346, 258, and 1981 micromolar, respectively. These results are consistent with carrier mediated transport. ΔpH-Dependent isoleucine and arginine transport exhibited biphasic kinetics, suggesting these amino acids may be transported by at least two transport systems. Symport mediated alanine transport was electrogenic as demonstrated by the effect of membrane potential (ΔΨ) on ΔpH-dependent flux. In the absence of significant charge compensation, a low rate of alanine transport was observed. When ΔΨ was held at 0 millivolt with symmetric potassium concentrations and valinomycin, the rate of flux was stimulated fourfold. In the presence of a negative ΔΨ, alanine transport increased sixfold. These results are consistent with an electrogenic transport process which results in a net flux of positive charge into the vesicles. The effect of changing ΔΨ on the kinetics of alanine transport altered V_max with no apparent change in K_m. Amino acid transport was inhibited by the protein modifier diethyl pyrocarbonate, but was insensitive to N-ethylmaleimide, 4,4′-diisothiocyano-2,2′-stilbene disulfonic acid, p-chloromercuribenzenesulfonic acid, phenylglyoxal, and N,N′-dicyclohexylcarbodiimide. Four amino acid symport systems, two neutral, one acidic, and one basic, were resolved based on inter-amino acid competition experiments. One neutral system appears to be active for all neutral amino acids while the second exhibited a low affinity for isoleucine, threonine, valine, and proline. Although each symport was relatively specific for a given group of amino acids, each system exhibited some crossover specificity for amino acids in other groups.