Entropy and the complexity of graphs: IV. Entropy measures and graphical structure

The structural information contentI _g (X) of a graphX was treated in detail in three previous papers (Mowshowitz 1968a, 1968b, 1968c). Those investigations ofI _g point up the desirability of defining and examining other entropy-like measures on graphs. To this end the chromatic information contentI _c (X) of a graphX is defined as the minimum entropy over all finite probability schemes constructed from chromatic decompositions having rank equal to the chromatic number ofX. Graph-theoretic results concerning chromatic number are used to establish basic properties ofI _c on arbitrary graphs. Moreover, the behavior ofI _c on certain special classes of graphs is examined. The peculiar structural characteristics of a graph on which the respective behaviors of the entropy-like measuresI _c andI _g depend are also discussed.     

Entropy and the complexity of graphs: I. An index of the relative complexity of a graph

The structural information content (Rashevsky, 1955; Trucco 1956a, b)I _g (X) of a graphX is defined as the entropy of the finite probability scheme constructed from the orbits of its automorphism groupG(X). The behavior ofI _g on various graph operations—complement, sum, join, cartesian product and composition, is examined. The principal result of the paper is the characterization of a class of graph product operations on whichI _g is semi-additive. That is to say, conditions are found for binary operations o and ∇ defined on graphs and groups, respectively, which are sufficient to insure thatI _g (X o Y)=I _g (X)+I _g (Y)−H _XY , whereH _XY is a certain conditional entropy defined relative to the orbits ofG(X o Y) andG(X) ∇G(Y).     

Entropy and the complexity of graphs: III. Graphs with prescribed information content

The connection between the adjacency matrix and the automorphisms of a digraph is used to develop a method for studying the automorphism group and, thus, the information content (Mowshowitz 1968a, b) of a digraph. An algorithm is given for constructing digraphs with zero information content, and the properties of such digraphs are examined. Moreover, an algorithm for computing the automorphism group of a digraph is presented and is used to find conditions which insure that two digraphs have the same information content. This algorithm is further used to determine the information content of digraphs whose adjacency matrices have prescribed properties.     

Soluble and bound forms of intracellular acid carboxypeptidase inAspergillus saitoi

The enzymological, physical, and immunological properties of soluble and bound forms of intracellular acid carboxypeptidase isolated from fresh mycelia ofAspergillus saitoi are reported. In the broken mycelia, about 60% of the total activity was found in the 2,000×g precipitate, with most of the remainder in the 100,000×g supernantant. The highly purified enzymes, I_a and I_b, from the 100,000×g supernatant were found to be homogeneous by such criteria as disc gel electrophoresis at pH 9.4 The bound enzyme, II, was solubilized from the 2,000×g precipitate by self-digestion at pH 6.4 and was highly purified by chromotography. The two forms of intracellular enzymes, the soluble enzymes (I_a and I_b) from the 100,00×g supernatant and the solubilized enzyme (II) from the 2,000×g precipitate, were closely related to, but not completely identical with, the extracellular acid carboxypeptidase.     

Quantal charge redistributions accompanying the structural transitions of sodium channels

Asymmetric displacement currents, I _g , associated with the gating of nerve sodium channels have been recorded in cell-attached macropatches of Xenopus laevis oocytes injected with exogenous mRNA coding for rat-brain-II sodium channels. The I _g properties were found to be similar to those of gating currents previously observed in native nerve preparations. I _g fluctuations were measured in order to ascertain the discreteness of the conformational changes which precede the channel opening. The autocorrelation of the fluctuations is consistent with a shot-like character of the elementary I _g contributions. The variance of the fluctuations indicates that most of the gating-charge movement that accompanies the activation of a single sodium channel occurs in 2 to 3 brief packets, each carrying an equivalent of about 2.3 electron charges.     

The quantal gating charge of sodium channel inactivation

Using a very low noise voltage clamp technique it has been possible to record from the squid giant axon a slow component of gating current (I _g ) during the inactivation phase of the macroscopic sodium current (I _Na ) which was hitherto buried in the baseline noise. In order to examine whether this slowI _g contains gating charge that originates from transitions between the open (O) and the inactivated (I) states, which would indicate a true voltage dependence of inactivation, or whether other transitions contribute charge to slowI _g , a new model independent analysis termed isochronic plot analysis has been developed. From a direct correlation ofI _g and the time derivative of the sodium conductance dg _Na/d the condition when only O-I transitions occur is detected. Then the ratio of the two signals is constant and a straight line appears in an isochronic plot ofI _g vs. dg _Na/d . Its slope does not depend on voltage or time and corresponds to the quantal gating charge of the O-I transition (q _h ) divided by the single channel ionic conductance (). This condition was found at voltages above – 10 mV up to + 40 mV and a figure of 1.21e ^– was obtained forq _h at temperatures of 5 and 15°C. At lower voltages additional charge from other transitions, e.g. closed to open, is displaced during macroscopic inactivation. This means that conventional Eyring rate analysis of the inactivation time constant _h is only valid above – 10 mV and here the figure forq _h was confirmed also from this analysis. It is further shown that most of the present controversies surrounding the voltage dependence of inactivation can be clarified. The validity of the isochronic plot analysis has been confirmed using simulated gating and ionic currents.Abbreviations I _g gating current - I _Na sodium ionic current - g _Na macroscopic sodium conductance - single channel conductance - C, O, I closed, open, inactivated state occupancy of channels - g _h quantal charge displaced in a single O-I transition of Na channel - e ^– equivalent electron charge - h index referring to inactivation process - S _l limiting slope in isochronic plot see Eq.(3) - fractional distance, see Fig. 4 and (4, 5) - TMA tetramethylammonium - TTX tetrodotoxin - Tris tris(hydroxymethyl)aminomethane - HEPES N-2-hydroxyethylpiperazine-N-2-ethanesulfonic acid     

Partition of the Hodgkin-Huxley type model parameter space into the regions of qualitatively different solutions

We have examined the problem of obtaining relationships between the type of stable solutions of the Hodgkin-Huxley type system, the values of its parameters and a constant applied current (I). As variable parameters of the system the maximal Na⁺(g_Na),K⁺(g_K) conductances and shifts (Gm, Gh, Gn) of the voltage-dependences have been chosen. To solve this problem it is sufficient to find points belonging to the boundary, partitioning the parameter space of the system into the regions of the qualitatively different types of stable solutions (steady states and stable periodic oscillations). Almost all over the physiological range of I, a type of stable solution is determined by the type of steady state (stable or unstable). Using this fact, the approximate solution of this problem could be obtained by analyzing the spectrum of eigenvalues of the Jacobian matrix for the linearized system. The families of the plane sections of the boundary have been constructed in the three-parameter spaces (I, g_Na,g_K), (I, Gm, Gh), (I, Gm, Gn).     

Modeling Hermissenda: I. Differential contributions of I A and I C to type-B cell plasticity

We developed a multicompartmental Hodgkin-Huxley model of the Hermissenda type-B photoreceptor and used it to address the relative contributions of reductions of two K⁺ currents, I _a and I _C, to changes in cellular excitability and synaptic strength that occur in these cells after associative learning. We found that reductions of g_C, the peak conductance of I _C, substantially increased the firing frequency of the type-B cell during the plateau phase of a simulated light response, whereas reductions of g_A had only a modest contribution to the plateau frequency. This can be understood at least in part by the contributions of these currents to the light-induced (nonspiking) generator potential, the plateau of which was enhanced by g_C reductions, but not by g_A reductions. In contrast, however, reductions of g_A broadened the type-B cell action potential, increased Ca²⁺ influx, and increased the size of the postsynaptic potential produced in a type-A cell, whereas similar reductions of g_C had only negligible contributions to these measures. These results suggest that reductions of I _A and I _C play important but different roles in type-B cell plasticity.     

Characterization of protein-interaction networks in tumors

Background  

Analyzing differential-gene-expression data in the context of protein-interaction networks (PINs) yields information on the functional cellular status. PINs can be formally represented as graphs, and approximating PINs as undirected graphs allows the network properties to be characterized using well-established graph measures.     

Regulation of transpiration in coffee hedgerows: covariation of environmental variables and apparent responses of stomata to wind and humidity

Stomatal regulation of transpiration was studied in hedgerow coffee (Coffea arabica L.) at different stages of canopy development encompassing a range of leaf area indices (L) from 0·7 to 6·7. Stomatal (g_s) and crown (g_c) conductance attained maximum values early during the day and then declined as both leaf-to-bulk air water vapour mole fraction difference (V_a) and photosynthetically active photon flux density (I) continued to increase. Covariation of environmental variables during the day, particularly V, I, and wind speed (u), obscured stomatal responses to individual variables. This also caused diurnal hysteresis in the relationship between g_c and individual variables. Normalization of g_s and g_c by I removed the hysteresis and revealed a strong stomatal response to humidity. At the crown scale, transpiration (E) increased linearly with net radiation (R_n) and seemed to increase with increasing wind speed. Increasing wind speed imposed higher leaf interior to leaf surface water vapour mole fraction differences (V_s) at given levels of V_a. However, strong relationships between declining g_c and E and increasing wind speed were obtained when g_c and E were normalized by I and R_n, respectively, without invoking additional potential interactions involving temperature or CO₂ concentration at the leaf surface. Apparent stomatal responses to wind were thus at least partially a reflection of the stomatal response to humidity.     

Genetic and Demographic Structure of the Yakut Population: Reproduction Indices

Genetic and demographic information for the Yakut population living in the Republic of Sakha (Yakutia) is presented. The mean number of children per woman constituted 4.605. Crow's index (I _tot) and its components (I _m and I _f) were 0.483, 0.104, and 0.343, respectively.     

Canopy photosynthetic production in a Japanese larch stand. I. Seasonal and vertical changes of leaf characteristics along the light gradient in a canopy

In an 18 year old Japanese larch stand, leaf characteristics such as area, weight, gross photosynthetic rate and respiration rate were studied in order to obtain basic information on estimating canopy photosynthesis and respiration. The leaf growth courses in area and weight from bud opening were approximated by simple logistic curves. The growth coefficient for the area growth curve was 0.155–0.175 day⁻¹, while that for the weight growth was 0.112–0.117 day⁻¹. The larger growth coefficient in area growth caused the seasonal change in specific leaf area (SLA) that increased after bud opening to its peak early in May at almost 300 cm² g⁻¹ and then decreased until it leveled off at about 140 cm²g⁻¹. The change inSLA indicates the possibility that leaf area growth precedes leaf thickness growth. The relationship between the coefficientsa andb of the gross photosynthetic rate (p)-light flux density (1) curve (p=bI/(1+aI)) and the mean relative light flux density (I′/I ₀) at each canopy height were approximated by hyperbolic formulae:a=A/(I′/I ₀)+B andb=C/(I′/I ₀)+D. Leaf respiration rate was also increased with increasingI′/I ₀. Seasonal change of gross photosynthetic rate and leaf respiration rate were related to mean air temperature through linear regression on semilogarithmic co-ordinates.     

Demonstration and characterization of Ca2+ channel in tonoplast-free cells ofNitellopsis obtusa

Summary The presence of a Ca²⁺ channel in the plasmalemma of tonoplast-freeNitellopsis obtusa cells was demonstrated and its characteristics were studied using current- and voltage-clamp techniques. A long-lasting inward membrane current (I _m ), recorded using a step voltage clamp, consisted of a single component without time-dependent inactivation. Increasing either [Ca²⁺] _o or [Cl^–] _o 1) enhanced the maximum amplitude of inwardI _m ((I _m ) _p ) and 2) shifted the peak voltage ((V _m ) _p ) at(I _m ) _p to more positive values under ramp-shaped voltage clamping and 3) depolarized the peak value of action potentials. This behavior is consistent with predictions based on the Nernst equation for Ca²⁺ but not for Cl^–. DIDS (4,4-diisothiocyano-2,2-disulfonic acid stilbene) did not suppress(I _m ) _p in tonoplast-free cells, in contrast with its effect on normal cells. La³⁺ and nifedipine blocked(I _m ) _p irreversibly. On the other hand, Ca²⁺ channel agonist, BAY K 8644 irreversibly enhanced(I _m ) _p . Both Sr²⁺ influx and K⁺ efflux increased upon excitation. The charge carried by Sr²⁺ influx was compensated for by K⁺ efflux. It is concluded that only the Ca²⁺ channel is activated during plasmalemma excitation in tonoplast-free cells. In terms of the magnitude of(I _m ) _p , Sr²⁺ could replace Ca²⁺, but Mn²⁺, Mg²⁺ and Ba²⁺ could not. External pH affected(I _m ) _p and the membrane conductance (g _m ) at(I _m ) _p ((g _m ) _p ). Increasing the external ionic strength caused increases in both(I _m ) _p and(g _m ) _p , and shifted(V _m ) _p to positive values. At the same time, Sr²⁺ influx increased. Thus Ca²⁺ channel activation seems to be enhanced by increasing external ionic strength. The possible involvement of surface potential is discussed.     

Conductive and Kinetic Properties of Connexin45 Hemichannels Expressed in Transfected HeLa Cells

Human HeLa cells transfected with mouse connexin Cx45 were used to examine the conductive and kinetic properties of Cx45 hemichannels. The experiments were carried out on single cells using a voltage-clamp method. Lowering the [Ca²⁺]_o revealed an extra current. Its sensitivity to extracellular Ca²⁺ and gap junction channel blockers (18α-glycyrrhetinic acid, palmitoleic acid, heptanol), and its absence in non-transfected HeLa cells suggested that it is carried by Cx45 hemichannels. The conductive and kinetic properties of this current, I _hc, were determined adopting a biphasic pulse protocol. I _hc activated at positive V _m and deactivated partially at negative V _m. The analysis of the instantaneous I _hc yielded a linear function g _hc,inst = f(V _m) with a hint of a negative slope (g _hc,inst: instantaneous conductance). The analysis of the steady-state I _hc revealed a sigmoidal function g _hc,ss = f(V _m) best described with the Boltzmann equation: V _m,0 = −1.08 mV, g _hc,min = 0.08 (g _hc,ss: steady-state conductance; V _{m, 0}:V _m at which g _hc,ss is half-maximally activated; g _hc,min: minimal conductance; major charge carriers: K⁺ and Cl⁻). The g _hc was minimal at negative V _m and maximal at positive V _m. This suggests that Cx45 connexons integrated in gap junction channels are gating with negative voltage. I _hc deactivated exponentially with time, giving rise to single time constants, τ_d. The function τ_d = f(V _m) was exponential and increased with positive V _m (τ_d = 7.6 s at V _m = 0 mV). The activation of I _hc followed the sum of two exponentials giving rise to the time constants, τ_a1 and τ_a2. The function τ_a1 = f(V _m) and τ_a2 = f(V _m) were bell-shaped and yielded a maximum of ≅ 0.6 s at V _m ≅ −20 mV and ≅ 4.9 s at V _m ≅ 15 mV, respectively. Neither τ_a1 = f(V _m) nor τ_a2 = f(V _m) coincided with τ_d = f(V _m). These findings conflict with the notion that activation and deactivation follow a simple reversible reaction scheme governed by first-order voltage-dependent processes.     

The potassium A-current,low firing rates and rebound excitation in Hodgkin-Huxley models

It is widely believed, following the work of Connor and Stevens (1971,J. Physiol. Lond. 214, 31–53) that the ability to fire action potentials over a wide frequency range, especially down to very low rates, is due to the transient, potassium A-current (I _A). Using a reduction of the classical Hodgkin-Huxley model, we study the effects ofI _A on steady firing rate, especially in the near-threshold regime for the onset of firing. A minimum firing rate of zero corresponds to a homoclinic bifurcation of periodic solutions at a critical level of stimulating current. It requires that the membrane's steady-state current-voltage relation be N-shaped rather than monotonic. For experimentally based genericI _A parameters, the model does not fire at arbitrarily low rates, although it can for the more atypicalI _A parameters given by Connor and Stevens for the crab axon. When theI _A inactivation rate is slow, we find that the transient potassium current can mediate more complex firing patterns, such as periodic bursting in some parameter regimes. The number of spikes per burst increases asg _A decreases and as inactivation rate decreases. We also study howI _A affects properties of transient voltage responses, such as threshold and firing latency for anodal break excitation. We provide mathematical explanations for several of these dynamic behaviors using bifurcation theory and averaging methods.     

Irradiance influences contents of photosynthetic pigments and proteins in tropical grasses and legumes

Three tropical range grasses (Cenchrus ciliaris, Dichanthium annulatum, and Panicum antidotale) and two range legumes [Macroptilium atropurpureum (siratro) and Stylosanthes hamata (stylo)] were grown under four irradiances, i.e. 100 (I ₁₀₀, control), 75 (I ₇₅), 50 (I ₅₀), and 25 (I ₂₅) % of full sunlight. Accumulation of chlorophyll (Chl) b increased but that of Chl a decreased under low irradiances. The greater accumulation of Chl (a+b) in grasses (particularly in D. annulatum and P. antidotale) under shade predicted their shade adaptability. Among legumes Stylosanthes was more adaptive to the shade than Macroptilium due to its higher accumulation of Chl (a+b). Significant difference in the accumulation of carotenoids under I ₂₅ over I ₁₀₀ was observed in all the species, which shows the increase in quality of the fodder under limited irradiance. There was a significant decrease in soluble protein content in C. ciliaris under I ₇₅, however, no significant difference in protein content was observed under I ₅₀ and I ₂₅, which was also reflected in the SDS pattern with the reduction in content of polypeptides at I ₇₅ and following increase at I ₅₀ and I ₂₅. This was possibly due to reduction of light-induced protein at I ₇₅ and then expression of the stress-induced protein at further reduction of irradiance. Peroxidase activity in C. ciliaris increased with the decrease in irradiance and its isozyme pattern showed differences among all treatments, which indicated the role of different peroxidase isoforms at different irradiances.     

Alteration of anion channel kinetics in wild-type and abi1-1 transgenic Nicotiana benthamiana guard cells by abscisic acid

The influence of the plant water-stress hormone abscisic acid (ABA) on anion channel activity and its interaction with protein kinase and phosphatase antagonists was examined in stomatal guard cells of wild-type Nicotiana benthamiana L. and of transgenic plants expressing the dominant-negative (mutant) Arabidopsis abi1-1 protein phosphatase. Intact guard cells were impaled with double-barrelled micro-electrodes and membrane current was recorded under voltage clamp in the presence of 15 mM CsCI and 15 mM tetraethylammonium chloride (TEA-CI) to eliminate K⁺ channel currents. Under these conditions, the free-running voltage was situated close to 0 mV (+9 ± 6 mV, n = 18) and the membrane under voltage clamp was dominated by anion channel current (I_Cl) as indicated from tail current reversal near the expected chloride equilibrium potential, current sensitivity to the anion channel blockers 9-anthracene carboxylic acid and niflumic acid, and by its voltage-dependent kinetics. Pronounced activation of I_Cl was recorded on stepping from a conditioning voltage of ?250 mV to voltages between ?30 and +50 mV, and the current deactivated with a voltage-dependent halftime at more negative voltages (τ? 0.3 sec at ?150 mV). Challenge with 20 µM ABA increased the steady-state current conductance, g_Cl, near 0 mV by 1.2- to 2.8-fold and at ?150 mV by 4.5- to sixfold with a time constant of 40 ± 4 sec, and it slowed I_Cl deactivation as much as fourfold at voltages near ?50 mV, introducing two additional voltage-sensitive kinetic components to these current relaxations. Neither the steady-state and kinetic characteristics of I_Cl, nor its sensitivity to ABA were influenced by H7 or staurosporine, both broad-range protein kinase antagonists. However, the protein phosphatase 1/2A antagonist calyculin A mimicked the effects of ABA on g_Cl and current relaxations on its own and exhibited a synergistic interaction with ABA, enhancing I_Cl sensitivity to ABA three- to four-fold. Quantitatively similar current characteristics were recorded from guard cells of abi1-1 transgenic N. bentamiana, indicating that the abi1-1 protein phosphatase does not influence the ànion current or its response to ABA directly. These results demonstrate that ABA stimulates I_Cl and modulates its voltage sensitivity. Furthermore, they show that ABA promotes I_Cl, either by introducing additional long-lived states of the channel or by activating a second anion channel with similar permeation characteristics but with a very long dwell time in the open state. Overall, the data are broadly consistent with the view that ABA action engenders coordinate control of I_Cl together with guard cell K⁺ channels to effect solute loss and stomatal closure.     

Contributions of <Emphasis Type="Italic">I</Emphasis><Subscript>h</Subscript> to feature selectivity in layer II stellate cells of the entorhinal cortex

Stellate cells (SCs) of the entorhinal cortex generate prominent subthreshold oscillations that are believed to be important contributors to the hippocampal theta rhythm. The slow inward rectifier I _h is expressed prominently in SCs and has been suggested to be a dominant factor in their integrative properties. We studied the input-output relationships in stellate cells (SCs) of the entorhinal cortex, both in control conditions and in the presence of the I _h antagonist ZD7288. Our results show that I _h is responsible for SCs’ subthreshold resonance, and contributes to enhanced spiking reliability to theta-rich stimuli. However, SCs still exhibit other traits of rhythmicity, such as subthreshold oscillations, under I _h blockade. To clarify the effects of I _h on SC spiking, we used a generalized form of principal component analysis to show that SCs select particular features with relevant temporal signatures from stimuli. The spike-selected mix of those features varies with the frequency content of the stimulus, emphasizing the inherent nonlinearity of SC responses. A number of controls confirmed that this selectivity represents a stimulus-induced change in the cellular input-output relationship rather than an artifact of the analysis technique. Sensitivity to slow features remained statistically significant in ZD7288. However, with I _h blocked, slow stimulus features were less predictive of spikes and spikes conveyed less information about the stimulus over long time scales. Together, these results suggest that I _h is an important contributor to the input-output relationships expressed by SCs, but that other factors in SCs also contribute to subthreshold oscillations and nonlinear selectivity to slow features. Action Editor: Xiao-Jing Wang     

Voltage dependence of cellular current and conductances in frog skin

Summary Knowledge of the voltage dependencies of apical and basolateral conductances is important in determining the factors that regulate transcellular transport. To gain this knowledge it is necessary to distinguish between cellular and paracellular currents and conductances. This is generally done by sequentially measuring transepithelial current/voltage (I _t /V _t ) and conductance/voltage (g _t /V _t ) relationships before and after the abolition of cellular sodium transport with amiloride. Often, however, there are variable time-dependent and voltage-dependent responses to voltage perturbation both in the absence and presence of amiloride, pointing to effects on the paracellular pathway. We have here investigated these phenomena systematically and found that the difficulties were significantly lessened by the use of an intermittent technique, measuringI _t andg _t before and after brief (<10 sec) exposure to amiloride at each setting ofV _t .I/V relationships were characterized by these means in frog skins (Rana pipiens, Northern variety, andRana temporaria). Cellular current,I _c , decreased with hyperpolarization (larger serosa positive clamps) ofV _t . DerivedI _c /V _t relationships betweenV _t =0 and 175 mV (serosa positive) were slightly concave upwards. Because values of cell conductance,g _c , remained finite, it was possible to demonstrate reversal ofI _c . Values of the reversal potentialV' averaged 156±14 (sd,n=18) mV. Simultaneous microelectrode measurements permitted also the calculation of apical and basolateral conductances,g _a andg _b . The apical conductance decreased monotonically with increasing positivity ofV _t (andV _a ). In contrast, in the range in which the basolateral conductance could be evaluated adequately (V _t <125 mV),g _b increased with more positive values ofV _t (andV _b ). That is, there was an inverse relation betweeng _b and cellular current at the quasi-steady state, 10–30 sec after the transepithelial voltage step.