Allometric scaling and maximum efficiency in physiological eigen time.

General optimization results from physics indicate that maximum efficiency of a process, in the sense of minimum overall entropy production, is achieved when the rate of entropy production is constant over time, however not in ordinary clock time but on an, in general varying, "eigen time" scale, intrinsic to the system. We identify the eigen time of a biological system with "physiological time," which generally scales with the 1/4 power of body mass, M(1/4), over a vast range of species. Since it is equally well established that metabolic rate scales as M(3/4), it follows that organisms produce entropy at the same intrinsic rate, fulfilling a necessary condition for maximum efficiency, and are all, furthermore, equally efficient on the physiological eigen time scale.

Body mass index: a measure for longevity.

Body mass index has important predictive value for mortality and morbidity both in normal subjects and in those suffering from particular pathologies. However, body mass index was introduced as a measure of body fat, which might not be expected to have such wide implications for various pathological conditions. We argue here that body mass index may actually be a measure for longevity. Our arguments are based on a well-established allometric scaling law for physiological time. The time between heart beats, the time between respirations, and longevity all scale as body weight to the 1/4 power in mammalian species ranging from shrews to blue whales. We find that body mass index also scales with body weight to the 1/4 power in humans from birth to one year of age, and again from approximately 5 to 17 years of age. On the assumption that in these two growth phases humans scale as do species, we postulate that body mass index is a measure of longevity.

Computation of action potential propagation and presynaptic bouton activation in terminal arborizations of different geometries.

Action potential propagation in axons with bifurcations involving short collaterals with synaptic boutons has been simulated using SPICE, a general purpose electrical circuit simulation program. The large electrical load of the boutons may lead to propagation failure at otherwise uncritical geometric ratios. Because the action potential gradually fails while approaching the branch point, the electrotonic spread of the failing action potential cannot depolarize the terminal boutons above an assumed threshold of 20 mV (Vrest = 0 mV) for the presynaptic calcium inflow, and therefore fails to evoke transmitter release even for boutons attached at short collaterals. For even shorter collaterals the terminal boutons can again be activated by the spread of passive current reflected at the sealed end of the bouton which increases the membrane potential above firing threshold. The action potential is then propagated in anterograde fashion into the main axon and may activate the terminal bouton on the other collateral. Differential activation of the synaptic boutons can be observed without repetitive activation of the main axon and with the assumption of uniform membrane properties. Axon enlargements above a critical size at branch points can increase the safety factor for propagation significantly and may serve a double function: they can act both as presynaptic boutons and as boosters, facilitating invasion of the action potential into the terminal arborizations. The architecture of the terminal arborizations has a profound effect on the activation pattern of synapses, suggesting that terminal arborizations not only distribute neural information to postsynaptic cells but may also be able to process neural information presynaptically.

Simulation of action potential propagation in complex terminal arborizations.

Action potential propagation in complex terminal arborizations was simulated using SPICE, a general purpose circuit simulation program. The Hodgkin-Huxley equations were used to simulate excitable membrane compartments. Conduction failure was common at branch points and regularly spaced boutons en passant. More complex arborizations had proportionally more inactive synapses than less complex arborizations. At lower temperature the safety factor for impulse propagation increased, reducing the number of silent synapses in a particular arborization. Small structural differences as well as minute changes in the discharge frequency of the action potential resulted in very different activation patterns of the arborization and terminal boutons. The results suggest that the structural diversity of terminal arborizations allows a wide range of presynaptic information processing. The results from this simulation study are discussed in the context of experimental results on the modulation of synaptic transmission.

A theoretical analysis of binding to the Ca2+-specific sites on troponin incorporated into thin filaments.

Recent data on the binding of Ca2+ to the specific sites on troponin, alone, in regulated actin, and in regulated actomyosin, as well as data on the Ca2+ activation of the actomyosin ATPase (Grabarek, Z., J. Grabarek, P.C. Leavis, and J. Gergely, 1983, J. Biol. Chem., 258:14098-14102.), are analyzed on the basis of a model used previously for qualitative theoretical studies of the Ca2+ activation of muscle contraction (Shiner and Solaro, 1982). The data allow and require an extension of the model to consider the effects of tropomyosin explicitly. Three major results of the analysis are at variance with previous investigations. A repulsive interaction between tropomyosins; and an attractive interaction between actins (or myosin heads attached to actin) are found, whereas others have found or assumed an attractive tropomyosin-tropomyosin interaction and no actin-actin interaction. The parameter values found here predict hysteresis under the conditions of the ATPase experiments; no other existing model for the interactions manifest in the Ca2+ activation of contraction can predict hysteresis. The prediction is of increased interest in light of experimental reports of hysteresis in the Ca2+ activation of isometric force (Ridgeway, E. B., A. M. Gordon, and D. A. Martyn, 1983, Science (Wash. DC), 219:1075-1077; Gordon, A. M., E. B. Ridgeway, and D. A. Martyn, 1984, Plenum Publishing Corp., New York, 553-563; Brandt, P. W., B. Gluck, M. Mini, and C. Cerri, 1985, J. Mus. Res. Cell Motil. 6:197-205.).

The hill coefficient for the Ca2+-activation of striated muscle contraction.

The following arguments are presented for the observation that curves relating free Ca2+ and force development of thin filament regulated myofilaments of skinned muscle fibers have Hill coefficient (n) greater than 4, which is the number of Ca2+ binding sites on troponin: Activation of the myofilaments is a process relaxing to a nonequilibrium steady state or stationary state. Systems operating at nonequilibrium stationary states are known to display Hill coefficients greater than the number of interacting sites and similar results have been obtained for Ca2+ activation of myofilament isometric force. The size of the basic subunit of thin filament regulated muscle may be the entire thin filament rather than seven actins, one tropomyosin, and one troponin. In this case the number of interacting sites may be on the order of hundreds. Hysteresis in the Ca2+ activation of isometric force might result from multiple stationary states and also might give rise to Hill coefficients greater than 4.

Activation of thin-filament-regulated muscle by calcium ion: considerations based on nearest-neighbor lattice statistics.

We discuss the activation of thin-filament-regulated muscles by calcium ion in terms of a qualitative model based on nearest-neighbor lattice statistics. For the most part, the model takes into account only the essential features of the phenomenon--that there must be an interaction between calcium adsorption to troponin and crossbridge reaction with actin for calcium ion to activate contraction and that the relevant stationary states are nonequilibrium ones. Even so, the model predicts the following features which are seen experimentally but have generally not been considered in previous models: (i) the relative activations of stationary-state isometric force and ATPase are not equal; (ii) in general, neither activation of force nor that of ATPase is proportional to calcium adsorption to the activating sites; and (iii) the slopes of the relations between the activations and the logarithm of the calcium ion concentration generally depend on the necessary interaction between calcium ion adsorption and crossbridge reaction with actin. Thus, these relations show cooperative effects even if these is no interaction between calcium adsorption sites.

The effects of modifiers on enzyme catalysis: a non-classical nearest neighbor approach.

We present a nearest neighbor lattice model of the effects of modifiers on two-state enzyme catalysis of the reaction S in equilibrium with p. We do not in general make the assumptions of the classical approach to cooperative catalysis that yield (1) adsorption isotherms of the same form as those for the corresponding equilibrium system and (2) a rate of the catalyzed reaction proportional to the number of occupied catalytic sites. Closed form results are obtained for two approximations, the Bragg-Williams and the quasi-chemical. The latter requires (1), but is exact for several simple cases, including the concerted model, under this condition. Under (1) it is found that an interaction between modifier and catalytic sites, whether attractive or repulsive, increases the magnitudes of the slopes of the adsorption isotherms but that interactions between identical sites (catalytic or modifier) increase these magnitudes if attractive and decrease them if repulsive. Thus, the former interaction allows for phase transitions if sufficiently attractive or repulsive, but the latter only if sufficiently attractive. Herein also lies the explanation for why the concerted model displays only "positive cooperativity". It is further seen that it is not possible to classify a modifier as an activator or inhibitor of the catalyzed reaction solely on the basis of the sign of the interaction energy between catalytic and modifier sites. For a given energy, the rate of the reaction may increase or decrease in response to the modifier, or it may respond biphasically. Similarly, the rate may respond biphasically to the activities of s or p, leading to instabilities. Thus, possibilities of multiple nonequilibrium stationary states or spatio-temporal patterns are raised.

Modulation of Ca2+ control of dog and rabbit cardiac myofibrils by Mg2+. Comparison with rabbit skeletal myofibrils.

Increases in free Mg2+ from 0.04 to 10.0 mM with constant pH 7.0 TO 0.10 M ionic strength, and 2 mM MgATP2- caused a rightward shift of the free Ca-relative ATPase relation for both cardiac skeletal myofibrils. The specific activity of cardiac myofibrillar ATPase over a wide range of free Ca2+ was, however, depressed in 0.04 vs. 1.0 mM Mg2+, whereas a similar decrease in free Mg2+ slightly enhanced skeletal myofibrillar ATPase. Lowering free Mg2+ from 1.0 to 0.04 mM caused similar increases in cardiac and skeletal myofibrillar bound calcium, which were largely attributable to increased calcium binding to myofibrillar myosin. Raising free Mg2+ from 1.0 to 10.0 mM caused only a slight decrease of skeletal myofibrillar bound calcium, and this change was attributable to myofibrillar myosin. The same increase in free Mg2+ caused cardiac myofibrils to bind increased amounts of calcium and this change was not attributable to myofibrillar myosin. By subtracting calcium bound to myofibrillar myosin, we were able to estimate calcium binding by myofibrillar troponin. The transition between basal and maximal ATPase in 1.0 and 10 mM Mg2+ was found to be assocciated with binding of an additional 2 mol/mol of either skeletal or cardiac myofibrillar troponin.