Differentiation between equilibrium and nonequilibrium kinetic systems by noise analysis

Methods for the differentiation between equilibrium and nonequilibrium steady-state kinetic mechanisms based on fluctuation and noise analysis are discussed. Specifically, the "sharpening" in the auto noise power spectrum is shown to be a useful indicator in identifying a nonequilibrium steady state.

Testicular receptors for luteinizing hormone after immunoneutralization of gonadotrophin releasing hormone in the male rat

The effects of immunoneutralization of endogenous gonadotrophin releasing hormone (GnRH) on the serum concentrations of testosterone and gonadotrophins and the binding of 125I-labelled human chorionic gonadotrophin (HCG) to testicular membrane fractions were studied in adult male rats. Four days after the administration of 1 ml anti-GnRH serum, the level of testosterone in the serum decreased to 44% of the concentration before the injection, whereas administration of normal rabbit serum had no effect. Multiple injections of anti-GnRH serum for 4 days dramatically suppressed the secretion of gonadotrophins in rats orchidectomized 2 months earlier. In intact male rats treated identically, immunoneutralization of GnRH decreased the level of serum testosterone to 32% of the concentration present in saline-treated controls, but did not decrease the number of testicular binding sites for HCG (LH). Administration of testosterone or oestradiol for 3 or 6 days caused a marked reduction in the concentration of serum gonadotrophins but did not decrease the number of LH receptors. This study provides further support for the concept that one releasing hormone governs secretion of both FSH and LH. In addition, these studies indicate that selective reduction of gonadotrophins for 3-6 days has no effect on the number of testicular LH receptors. This suggests that pituitary hormones other than gonadotrophins may be important in the maintenance of testicular receptors for LH.

Steroid and FSH action on LH receptors and LH-sensitive testicular responsiveness during sexual maturation of the rat

This study examines the effect of FSH, testosterone and estradiol on testicular LH receptors and in vitro testicular responsiveness to LH in immature rats under various conditions. FSH treatment of 15-day-old immature rats significantly increased the number of LH receptors but did not alter testicular responsiveness. FSH treatment of hypophysectomized immature rats increased the number of LH receptors and markedly increased testicular responsiveness. Treatment of hypophysectomized rats with testosterone proprionate for 4 days, followed by a 5-day treatment with FSH, enhanced the effect of FSH on the number of LH receptors but did not increase the effect of FSH on testicular responsiveness. In contrast, treatment with estradiol for 4 days before FSH treatment had no effect on the FSH-induced increase in LH receptors but completely inhibited the FSH-induced increase in testicular responsiveness. These observations suggest that during male sexual maturation (1) regulation of LH receptors is distinct from regulation of testicular responsiveness to LH, (2) estradiol may be a factor in the regulation of testicular responsiveness to LH, and (3) testosterone may enhance the FSH-induced increase of LH receptors.

Pituitary regulation of Leydig cell function in the adult male rat

The effects of hypophysectomy on serum testosterone, 125I-labelled hCG binding to testicular membranes and on testicular responsiveness were studied in adult rats. Serum testosterone decreased rapidly over the first 6 h after hypophysectomy. LH receptors were determined (pmol/testis) by measuring the specific binding of 125I-labelled hCG in membrane preparations of testes of rats hypophysectomized 1, 2, 3, 6, 9, or 15 days earlier. Hypophysectomy did not result in a decrease in 125I-labelled hCG binding on day 1 but this had decreased to 40% of that in intact controls by day 2. A gradual decline was found between days 2 and 6 at which time hCG binding had decreased to 15%. No further decrease occurred between days 6 and 15. Scatchard analysis indicated that the decline in hCG binding was due to a decreaffinity. FSH, testosterone, dihydrotestosterone, and oestradiol were unable to prevent the decline in hCG binding. Although serum testosterone, testicular testosterone content, and 125I-labelled hCG binding decreased rapidly after hypophysectomy, testicular responsiveness to LH was biphasic. The intraperitoneal administration of 25 microgram LH 2 h before decapitation increased testosterone in the circulation to a greater extent extent in animals hypophysectomized for 1 day than in intact controls while hCG binding affinities and capacities had not changed. Two or three days after hypophysectomy testicular responsiveness to LH was similar to that of intact controls even though hCG binding in hypophysectomized animals had decreased to 40 and 28% of intact controls respectively. It is concluded that (1) the testis is dependent on anterior pituitary hormones for maintenance of testicular LH receptors and testosterone secretion, (2) FSH, testosterone, dihydrotestosterone, or oestradiol cannot prevent the decline in testicular LH receptors resulting from hypophysectomy, and (3) steroidogenic capacity of the testis persists significantly longer than the hCG binding capacity of the testis.

Differentiation of channel models by noise analysis

Differentiation of membrane channel models based on fluctuation (or noise) analysis is discussed. The theory is particularly useful in distinguishing a single-conductance model (Hodgkin-Huxley formalism) from a multiconductance model. When applied to the frog node of Ranvier, it seems likely that the potassium channels of the membrane may have multiconductance states.

Matrix method for fluctuations and noise in kinetic systems

In a series of papers we were concerned with the question of how to calculate the concentration noise power spectra of an ensemble of multi-state linear kinetic systems when the rate constants of the systems are assumed to be known. We have used a standard eigenvalue-eigenfunction method to solve the differential equations which govern the regression of the means and derived the noise power spectrum as a function of the eigenvalues and eigenfunctions of the relaxation matrix of the system. In this paper, we have obtained an equation which relates the noise spectrum matrix of the fluctuations directly to the relaxation matrix of the means. As a result, the noise power spectrum can be calculated through matrix operations without the necessity of an eigenvalue-eigenfunction calculation. The present formalism is particularly useful in the evaluation of kinetic rate constants when the noise spectrum data of concentration fluctuations are given. Possible applications to biochemical systems are briefly discussed.

Stochastics of cycle completions (fluxes) in biochemical kinetic diagrams

Over a long time period, the rate constants for cycle completions (one-way fluxes) in steady-state biochemical diagrams can be expressed explicitly in terms of the elementary rate constants for transitions between states of the diagram. These cycle rate constants determine the mean one-way fluxes in the diagram and also fluctuations about the means. These properties are confirmed by Monte Carlo computer simulations on special cases. Two other topics are considered briefly: the effect of the starting state or states on the numbers of cycle completions in computer simulation runs; and the more detailed stochastic approach required if individual cycle completions are to be followed (i.e., if the "long time" restriction is removed).

Some self-consistent two-state sliding filament models of muscle contraction

The general formalism required to treat two-state sliding filament models of muscle contraction, including free energy considerations, is first reviewed and amplified. This formalism is then used to examine, and modify as needed, three models studied previously by Podolsky and Nolan, in which cross-bridge attachment-detachment and ATP turnover are not tightly coupled. No attempt is made here to establish an optimal, self-consistent model of this type because our interest is primarily in methadology rather than in fitting experimental results. But it appears from this preliminary study that such a model, with satisfactory mechanical and thermodynamic properties, could be found. An extremely simple but unrealistic two-state model is also studied which is of interest because it demonstrates the fact that it is possible, in principle at least, for sliding filament models to work with very high thermodynamic efficiencies (50-100 percent). An appendix is included that is concerned with the form of the dependence of certain first-order rate constants on the concentrations of ATP, ADP, and P.

Further analysis of a simple prototypal muscle model near to and far from equilibrium

The same prototypal model used in a previous paper to illustrate proper construction of a muscle model is modified here with the much more realistic choice e(Deltap) = 10(8) rather than e(Deltap) = 100, where e(Deltap) is the ratio of physiological ATP activity to equilibrium ATP activity. For steady isotonic contractions, the range 1 </= e(Delta) </= 10(4) can be approximated quite well by use of linear terms only in expansions of F (force) and J (ATP flux) in powers of e(Delta) - 1 and v (velocity). This will presumably also be true in most cases of much more complicated models. However, this region is of theoretical interest only (irreversible thermodynamics, etc.) because F and J are very small. In addition, numerical calculations of F and J were made in the region 10(4) </= e(Delta) </= 10(8). The optimal efficiency eta(*) is larger under physiological conditions (about 1%) than at equilibrium by a factor of 2.1 x 10(4). The rate of entropy production is discussed in this connection.

Analysis of a simple prototypal muscle model near to and far from equilibrium

Accurate calculations of force generated and of ATP flux, in steady isotonic contractions, are made on a simple, two-state muscle model of the sliding filament type. The objective is to illustrate the proper formulation and use of a complete and self-consistent molecular model of muscle. Otherwise, the model is not meant to be realistic. Calculations were made near equilibrium, including linear and quadratic terms in a power series expansion, and arbitrarily far from equilibrium by direct numerical solution of the appropriate differential equation in the probability of cross-bridge attachment. The results obtained from the two different methods agree where they overlap near equilibrium. The efficiency of free energy conversion is emphasized, and the relation to linear irreversible thermodynamics is pointed out.

Fluctuations and noise in kinetic systems. Application to K+ channels in the squid axon

We consider the equilibrium or steady-state noise power density spectrum in the quantity N = Sigma(x) (i=0)a(i)N(i) for an ensemble of independent and equivalent systems each of which can exist in the discrete set of states i = 0, 1, ..., x. N(i) is the number of systems of the ensemble in state i and the a(i)'s are constants. There is a transition rate constant alpha(ij) for an arbitrary transition i --> j; the kinetic equations are linear. There are possible applications to enzyme and biochemical kinetics generally, to membrane transport, muscle contraction, binding on macromolecules, etc. In each case, noise measurements would provide information about the kinetic scheme. The particular application considered here is to K(+) channels or gates (one channel = one system) in the squid axon membrane: a(i)g(K) is the K(+) conductance of a channel in state i and the kinetic scheme is of the Hodgkin-Huxley type (HH). Here we allow an arbitrary set of a(i)'s. This is a generalization of our treatment of K(+) channel noise in an earlier paper. The theory is discussed and some calculations made using Fishman's recent experimental results on K(+) channel noise as a guide. Preliminary indications are that the HH choice of a(i)'s may be oversimplified and that a(0) congruent with 0, a(1) not equal a(0), a(x) not equal a(x-1). Quite possibly the a(i)'s increase from a(0) to a(x), though the early a(i)'s must be relatively small to give the observed induction behavior in g(K)(t). An increase in equal steps is unsatisfactory because this is essentially HH with x = 1 (no induction). More refined experiments may modify these tentative conclusions. In any case, it appears from Fishman's work that noise measurements will probably be very useful in distinguishing between rival models of K(+) channels.

On the theory of ion transport across the nerve membrane, VII. Cooperativity between channels of a large square lattice

The exact kinetics of a 10 x 10 Ising system (periodic boundary conditions) with two-state channels arranged in a square lattice was studied by computer simulation. With all three values of the cooperativity parameter used, no induction in the K(+)-current curve was obtained. This confirms one of our previous conclusions concerning K(+) channels in the squid axon membrane: models with interacting channels arranged in a twodimensional lattice (Adam's model) are apparently excluded. Other topics included: equilibrium properties; short-range pair correlation functions; phase transition.

On the theory of ion transport across the nerve membrane. V. Two models for the Cole-Moore K + hyperpolarization delay

Two illustrative molecular models, designed to explain the Cole-Moore K(+) hyperpolarization delay, are proposed and analyzed. Both introduce a process supplementary to the usual Hodgkin-Huxley (HH) one for a K(+) channel. In both cases the new process becomes involved as a consequence of the conditioning hyperpolarization of the membrane and would account for the observed delay time in the K(+) current after depolarization to near ENa. The first model uses adsorption or desorption of phospholipid molecules on the surface of the assumed protein K(+) channel or gate. The second model involves the translocation of the charged subunits of the channel in the hyperpolarizing electric field.

On the theory of ion transport across the nerve membrane. IV. Noise from the open-close kinetics of K + channels

The theoretical power density spectrum G(f) of fluctuations in the steady-state squid axon K(+) current in the 10(3) Hz region has been derived assuming that these are fluctuations in the number of open K(+) channels in the Hodgkin-Huxley (HH) model. Various modifications of the HH model were also studied. The results were negative in all cases when compared with experiment. This confirms the generally held view that the observed G(f) approximately 1/f is associated primarily with K(+) current through open K(+) channels and not with the open-close kinetics of these channels.

On the theory of ion transport across the nerve membrane. 3. Potassium ion kinetics and cooperativity (with x=4,6,9)

The calculations of Part II of this series have been extended to square (x = 4) and octahedral (x = 6) arrangements of subunits in a potassium channel (or gate). The conclusion is the same as before: experimental induction and superposition properties of gK(t), on depolarization, seem to rule out any significant degree of interaction or cooperativity between the (protein?) subunits of K(+) channel. Calculations for x = 4, 6, and 9 have also been made for a square lattice of interacting channels (periodic boundary conditions). Because of apparent rapid convergence with x, it seems fairly safe to conclude that this model is unsatisfactory. There is some difficulty with superposition but the principal shortcoming is a failure to produce induction behavior. Aggregation models for the K(+) channel are also discussed briefly here. They, too, appear rather unpromising (for the same reasons as seem to exclude conformational cooperativity within a channel).

Cooperative effects in models of steady-state transport across membranes. IV. One-site, two-site, and multisite models

Several different one-site, two-site, and multisite models of steady-state ion transport across a membrane are investigated. The basic features, including cooperative interactions between channels, are the same as in earlier papers in this series. In particular, the present paper represents a considerable elaboration of part III. The models might apply to artificial or possibly to biological membranes, but particular applications must await further elucidation of the molecular structure and operation of these membranes.

On the theory of ion transport across the nerve membrane. II. Potassium ion kinetics and cooperativity (with x = 4)

We use a tetrahedral model of four interacting protein subunits to represent the K(+) channel or gate in the squid nerve membrane. The kinetic predictions, with varying degrees of cooperativity, are compared with experimental observations, especially those of Hodgkin and Huxley (J. Physiol. 117, 500, 1952) and of Cole and Moore (Biophys. J. 1, 1, 1960). The tentative conclusion reached is that if there is any cooperativity present it must be rather weak. There is no indication here that cooperativity improves the Hodgkin-Huxley assumption of independent "subunits". Other related models will be discussed in Part III. We also find evidence against the suggestion that there is cooperativity between K(+) channels arranged in patches of a two-dimensional lattice.

Cooperative effects in models of steady-state transport across membranes. II. Oscillating phase transition

A simple model for steady-state transport of a neutral molecule across a membrane is investigated in a preliminary way. In this model, there is a possible conformation change in each membrane unit which alters the access of the binding site for the transported molecule from one bath to the other. Thus, transport cannot be accomplished without a conformation change. Furthermore, we assume a cooperative interaction between nearest-neighbor membrane units in the same conformation. Then, with suitable rate constants and bath concentrations, and if the interaction energy is large enough, the membrane will oscillate back and forth between the two conformational phases, producing a surge of flux in each cycle. The period of the cycle depends on the times necessary to nucleate the two phase transitions.