The fundamental organization of cardiac mitochondria as a network of coupled oscillators

Mitochondria can behave as individual oscillators whose dynamics may obey collective, network properties. We have shown that cardiomyocytes exhibit high-amplitude, self-sustained, and synchronous oscillations of bioenergetic parameters when the mitochondrial network is stressed to a critical state. Computational studies suggested that additional low-amplitude, high-frequency oscillations were also possible. Herein, employing power spectral analysis, we show that the temporal behavior of mitochondrial membrane potential (DeltaPsi(m)) in cardiomyocytes under physiological conditions is oscillatory and characterized by a broad frequency distribution that obeys a homogeneous power law (1/f(beta)) with a spectral exponent, beta = 1.74. Additionally, relative dispersional analysis shows that mitochondrial oscillatory dynamics exhibits long-term memory, characterized by an inverse power law that scales with a fractal dimension (D(f)) of 1.008, distinct from random behavior (D(f) = 1.5), over at least three orders of magnitude. Analysis of a computational model of the mitochondrial oscillator suggests that the mechanistic origin of the power law behavior is based on the inverse dependence of amplitude versus frequency of oscillation related to the balance between reactive oxygen species production and scavenging. The results demonstrate that cardiac mitochondria behave as a network of coupled oscillators under both physiological and pathophysiological conditions.     

The phylogeny of persistence in DNA

We continue our study, Poland [Biophysical Chemistry 110 (2004) 59-2], of the distribution of C or G (C-G for short) in the DNA of select organisms, in particular, the tendency for C-G to cluster on all scales with respect to the number of bases considered. We previously found that if we counted the number of C-G bases in consecutive, nonoverlapping boxes containing a total of m bases, then the width of the distribution function describing how many C-G bases are in a box increases with respect to m dramatically relative to the width expected for a random distribution. The relative width of the C-G composition distribution function was found to vary accurately as a power law with respect to m, the size of the box, over a very wide range of m values. We express the power law in terms of a characteristic exponent gamma, that is, the relative widths of the distributions vary as m(gamma). The enhanced relative width of the distribution functions is a direct consequence of the tendency for boxes of similar composition to follow one another. This tendency represents persistence in composition from box to box and hence we refer to gamma as the persistence exponent. The occurrence of a power law means that the tendency for C-G to cluster is present on all scales of sequence length (box size) up to the total length of the chromosome which for bacteria is the entire genome. The persistence exponent gamma that characterizes the power law is thus an important parameter describing the distribution of C-G on all scales from individual base pairs up to the total length of the DNA sample considered. In the present paper, we determine the characteristic exponent gamma and the associated fractal dimension of DNA samples for a selection of species representing all of the major types of organism, that is, we explore the phylogeny of the exponent gamma. Here we treat six prokaryotes and six eukaryotes which, together with the species we have previously treated, brings the total number of species we have examined to 15. We find the power law form for the C-G distribution for all of the species treated and hence this behavior seems to be ubiquitous. The values of the characteristic exponent gamma that we find tend to cluster around the value gamma=0.20 with no obvious pattern with respect to phylogeny. The extreme values that we obtain are gamma=0.057 (yeast) and gamma=0.386 (human). We conclude by showing that the persistence of C-G clustering on the scale of the length of a chromosome is dramatically illustrated by interpreting the C-G distribution as a random walk.     

Branching tree model with fractal vascular resistance explains fractal perfusion heterogeneity

Perfusion heterogeneities in organs such as the heart obey a power law as a function of scale, a behavior termed "fractal." An explanation of why vascular systems produce such a specific perfusion pattern is still lacking. An intuitive branching tree model is presented that reveals how this behavior can be generated as a consequence of scale-independent branching asymmetry and fractal vessel resistance. Comparison of computer simulations to experimental data from the sheep heart shows that the values of the two free model parameters are realistic. Branching asymmetry within the model is defined by the relative tissue volume being fed by each branch. Vessel ordering for fractal analysis of morphology based on fed or drained tissue volumes is preferable to the commonly used Strahler system, which is shown to depend on branching asymmetry. Recently, noninvasive imaging techniques such as PET and MRI have been used to measure perfusion heterogeneity. The model allows a physiological interpretation of the measured fractal parameters, which could in turn be used to characterize vascular morphology and function.     

The dynamic evolution of the power exponent in a universal growth model of tumors

We have previously reported that a universal growth law, as proposed by West and collaborators for all living organisms, appears to be able to describe also the growth of tumors in vivo after an initial exponential growth phase. In contrast to the assumption of a fixed power exponent p (assumed by West et al. to be equal to 3/4), we propose in this paper a dynamic evolution of p, using experimental data from the cancer literature. In analogy with results obtained by applying scaling laws to the study of fragmentation of solids, the dynamic behaviour of p is related to the evolution of the fractal topology of neoplastic vascular systems. Our model might be applied for diagnostic purposes to mark the emergence of an efficient neo-angiogenetic structure if the results of our in silico experiments are confirmed by clinical observations.     

The persistence exponent of DNA

Using the complete genome of Thermoplasma volcanium, as an example, we have examined the distribution functions for the amount of C or G in consecutive, non-overlapping blocks of m bases in this system. We find that these distributions are very much broader (by many factors) than those expected for a random distribution of bases. If we plot the widths of the C-G distributions relative to the widths expected for random distributions, as a function of the block size used, we obtain a power law with a characteristic exponent. The broadening of the C-G distributions follows from the empirical finding that blocks containing a given C-G content tend to be followed by blocks of similar C-G content thus indicating a statistical persistence of composition. The exponent associated with the power law thus measures the strength of persistence in a given DNA. This behavior can be understood using Mandelbrot's model of a fractional Brownian walk. In this model there is a hierarchy of persistence (correlation between blocks) between all parts of the system. The model gives us a way to scale the C-G distributions such that all these functions are collapsed onto a master curve. For a fractional Brownian walk, the fractal dimension of the C-G distribution is simply related to the persistence exponent for the power law. The persistence exponent for T. volcanium is found to be gamma = 0.29 while for a 10 million base segment of the human genome we obtain gamma = 0.39, similar to but not identical with the value found for the microbe.     

Fractal dimensions and multifractility in vascular branching

A definition for the fractal dimension of a vascular tree is proposed based on the hemodynamic function of the tree and in terms of two key branching parameters: the asymmetry ratio of arterial bifurcations and the power law exponent governing the relation between vessel diameter and flow. Data from the cardiovascular system, which generally exhibit considerable scatter in the values of these two parameters, are found to produce the same degree of scatter in the value of the fractal dimension. When this scatter is explored for a multifractal pattern, however, it is found that the required collapse onto a single curve is achieved in terms of the coarse H?lder exponent. Thus, the presence of multifractility is confirmed, and the legitimacy of the defined dimension is affirmed in the sense of the theoretical Hausdorff limit in as much as this limit can be reached with experimental data.     

Taylor's law and related allometric power laws in New Zealand mountain beech forests: the roles of space,time and environment

Taylor's law says that the variance of population density of a species is proportional to a power of mean population density. Density–mass allometry says that mean population density is proportional to a power of mean biomass per individual. These power laws predict a third, variance–mass allometry: the variance of population density of a species is proportional to a power of mean biomass per individual. We tested these laws using 10 censuses of New Zealand mountain beech trees in 250 plots over 30 years at spatial scales from 5 m to kilometers. We found that: 1) a single‐species forest not disrupted by humans obeyed all three laws; 2) random sampling explained the parameters of Taylor's law at a large spatial scale in 8 of 10 censuses, but not at a fine spatial scale; 3) larger spatial scale increased the exponent of Taylor's law and decreased the exponent of variance–mass allometry (this is the first empirical demonstration that the latter exponent depends on spatial scale), but affected the exponent of density–mass allometry slightly; 4) despite varying natural disturbance, the three laws varied relatively little over the 30 years; 5) self‐thinning and recruiting plots had significantly different intercepts and slopes of density–mass allometry and variance–mass allometry, but the parameters of Taylor's law were not usually significantly affected; and 6) higher soil calcium was associated with higher variance of population density in all censuses but not with a difference in the exponent of Taylor's law, while elevation above sea level and soil carbon‐to‐nitrogen ratios had little effect on the parameters of Taylor's law. In general, the three laws were remarkably robust. When their parameters were influenced by spatial scale and environmental factors, the parameters could not be species‐specific indicators. We suggest biological mechanisms that may explain some of these findings.     

Fifth dimension of life and the 4/5 allometric scaling law for human brain

Brain cells are not spherical. The basal metabolic rate (B) of a spherical cell scales as B approximately r2, where r is the radius of the cell; that of a brain cell scales as B approximately r(d), where r is the characteristic radius of the cell and d is the fractal dimensionality of its contour. The fractal geometry of the cell leads to a 4/5 allometric scaling law for human brain, uniquely endowing humans with a 5th dimension and successfully explains why the scaling exponent varies during rest and exercise. A striking analogy between Kleiber's 3/4 law and Newton's second law is heuristically illustrated. A physical explanation is given for the 4th dimension of life for three-dimensional organisms and the 5th dimension for human brain.     

Swelling behavior and structural characteristics of wheat gluten polypeptide films

Wheat gluten films were subjected to controlled thermomechanical treatments to increase the percentage of aggregated sodium dodecyl sulfate (SDS)-insoluble gluten protein, the aggregation reaction being disulfide bonding. The rheological properties of the films were measured under immersion in water, where wheat gluten films are stable and show only slight swelling. The equilibrium swelling of the gluten films in water decreased with the increase of the percentage of SDS-insoluble protein aggregates, and the frequency the independent shear modulus increased sharply with increasing percentage of SDS-insoluble aggregates. Both findings confirm that disulfide bonding between gluten proteins is the predominant cross-linking reaction in the system. A relationship between shear modulus and aggregated protein compatible with a power law (of exponent 3) suggests the existence of a protein network at a molecular scale. However, the classical Flory-Rehner model failed to describe the relationship between the plateau modulus and the gluten volume fraction (a very drastic increase, compatible with a power law of an exponent of about 14). This result shows that gluten cannot be described as an entangled polymer network. The interpretation of both relationships is a network of mesoscale particles which in turn have a fractal inner structure (with a fractal dimension close to 3).     

A mathematical model for the distribution of hemodynamic parameters in the human retinal microvascular network

To quantitatively assess the arteriovenous distribution of hemodynamic parameters throughout the microvascular network of the human retina, we constructed a retinal microcirculatory model consisting of a dichotomous symmetric branching system. This system is characterized by a diameter exponent of 2.85, instead of 3 as dictated by Murray’s law, except for the capillary networks. The value of 2.85 was the sum of a fractal dimension (1.70) and a branch exponent (1.15) of the retinal vasculature. Following the feeding artery (central retinal artery), each bifurcation was recursively developed at a distance of an individual branch length [L(r) = 7.4r ^1.15] by a centrifugal scheme. The venular tree was formed in the same way. Using this model, we evaluated hemodynamic parameters, including blood pressure, blood flow, blood velocity, shear rate, and shear stress, within the retinal microcirculatory network as a function of vessel diameter. The arteriovenous distributions of blood pressure and velocity in the simulation were consistent with in vivo measurements in the human retina and other vascular beds of small animals. We therefore conclude that the current theoretical model was useful for quantifying hemodynamics as a function of vessel diameter within the retinal microvascular network.     

Fractal properties of perfusion heterogeneity in optimized arterial trees: a model study

Regional blood flows in the heart muscle are remarkably heterogeneous. It is very likely that the most important factor for this heterogeneity is the metabolic need of the tissue rather than flow dispersion by the branching network of the coronary vasculature. To model the contribution of tissue needs to the observed flow heterogeneities we use arterial trees generated on the computer by constrained constructive optimization. This method allows to prescribe terminal flows as independent boundary conditions, rather than obtaining these flows by the dispersive effects of the tree structure. We study two specific cases: equal terminal flows (model 1) and terminal flows set proportional to the volumes of Voronoi polyhedra used as a model for blood supply regions of terminal segments (model 2). Model 1 predicts, depending on the number Nterm of end-points, fractal dimensions D of perfusion heterogeneities in the range 1.20 to 1.40 and positively correlated nearest-neighbor regional flows, in good agreement with experimental data of the normal heart. Although model 2 yields reasonable terminal flows well approximated by a lognormal distribution, it fails to predict D and nearest-neighbor correlation coefficients r1 of regional flows under normal physiologic conditions: model 2 gives D = 1.69 +/- 0.02 and r1 = -0.18 +/- 0.03 (n = 5), independent of Nterm and consistent with experimental data observed under coronary stenosis and under the reduction of coronary perfusion pressure. In conclusion, flow heterogeneity can be modeled by terminal positions compatible with an existing tree structure without resorting to the flow-dispersive effects of a specific branching tree model to assign terminal flows.     

植物代谢速率与个体生物量关系研究进展

植物的各项生理生态功能（例如,呼吸、生长和繁殖）都与个体生物量成异速生长关系。West, Brown及Enquist基于分形网络结构理论所提出的WBE模型认为：植物的代谢（呼吸）速率正比于个体生物量的3/4次幂。然而,恒定的“3/4异速生长指数”与实测数据、植物生理生态学等研究之间存在矛盾,引发激烈的争论。论文分析了不同回归方法对代谢指数的影响,重点对植物代谢速率与个体生物量异速生长关系研究进展进行了综述,分析并得出了植物代谢指数在小个体时接近1.0,并随着生物量的增加而系统减小,且其密切依赖于氮含量的调控的结论。据此,提出了进一步深入研究植物代谢速率个体生物量关系需要解决的一些科学问题。     

Departures from the physical optimality in the bronchial tree of rats (Rattus norvegicus)

We studied the departure from the physical optimality of the bronchial tree of rats using both i) the minimum volume and power and ii) the minimum surface and drag criteria, considering the bronchial junction as the unit study based on Zamir's model for vascular trees. Our results show deviations of the junctions of the bronchial tree from the expected optimums in the proximal airway that can be explained by both, the turbulent or transitional flow regime, and the airway's necessity to distribute its terminal branches in the alveolar surface filling the thoracic volume. The departures of the observed values at the optimum for the minimum volume and power were significantly different than the obtained departure values for the minimum surface and drag criteria. The departure from the optimum was directly related to the diameter of the smallest branch. The slopes of the regressions for the two criteria were different. The regression lines intercept at a bronchial diameter d2 = 0.129 mm. This result agreed with the idea that the tube diameter is limited at small values by the increasing flow resistance with decreasing tube diameter while at large values is limited by the increasing tube volume and dead space with increasing tube diameter.     

The Scaling of Blood Flow Resistance: From a Single Vessel to the Entire Distal Tree

Although the flow resistance of a single vessel segment is easy to compute, the equivalent resistance of a network of vessel segments or the entire vasculature of an organ is difficult to determine in an analytic form. Here, we propose what we believe is a novel resistance scaling law for a vascular tree (i.e., the resistance of a vessel segment scales with the equivalent resistance of the corresponding distal tree). The formulation can be written as (Rs/Rc)∝(Ls/Lc) (where Rs and Ls are the resistance and length of a vessel segment, respectively, and Rc and Lc are the equivalent resistance and total length of the corresponding distal tree, respectively), which was validated for the coronary vascular systems of the heart. The scaling law was also shown to apply to the vascular systems of the lung, mesentery, muscle, eye, and so on. The novel resistance scaling law, coupled with the 3/4-power scaling law for metabolic rates, can predict several structure-function relations of vascular trees, albeit with a different exponent. In particular, the self-similar nature of the scaling law may serve as a diagnostic tool with the help of noninvasive imaging modalities.     

Emergence of matched airway and vascular trees from fractal rules

The bronchial, arterial, and venous trees of the lung are complex interwoven structures. Their geometries are created during fetal development through common processes of branching morphogenesis. Insights from fractal geometry suggest that these extensively arborizing trees may be created through simple recursive rules. Mathematical models of Turing have demonstrated how only a few proteins could interact to direct this branching morphogenesis. Development of the airway and vascular trees could, therefore, be considered an example of emergent behavior as complex structures are created from the interaction of only a few processes. However, unlike inanimate emergent structures, the geometries of the airway and vascular trees are highly stereotyped. This review will integrate the concepts of emergence, fractals, and evolution to demonstrate how the complex branching geometries of the airway and vascular trees are ideally suited for gas exchange in the lung. The review will also speculate on how the heterogeneity of blood flow and ventilation created by the vascular and airway trees is overcome through their coordinated construction during fetal development.     

Arterial branching within the confines of fractal L-system formalism

Parametric Lindenmayer systems (L-systems) are formulated to generate branching tree structures that can incorporate the physiological laws of arterial branching. By construction, the generated trees are de facto fractal structures, and with appropriate choice of parameters, they can be made to exhibit some of the branching patterns of arterial trees, particularly those with a preponderant value of the asymmetry ratio. The question of whether arterial trees in general have these fractal characteristics is examined by comparison of pattern with vasculature from the cardiovascular system. The results suggest that parametric L-systems can be used to produce fractal tree structures but not with the variability in branching parameters observed in arterial trees. These parameters include the asymmetry ratio, the area ratio, branch diameters, and branching angles. The key issue is that the source of variability in these parameters is not known and, hence, it cannot be accurately reproduced in a model. L-systems with a random choice of parameters can be made to mimic some of the observed variability, but the legitimacy of that choice is not clear.     

Dimensional analysis revisited

The applicability of dimensional analysis (DA) is discussed in relation to the metabolic scaling laws. The evolution of different theories of biological similarity has shown that the calculated reduced exponents (b) of Huxley's allometric equation are closely correlated with the numerical values obtained from the statistical analysis of empirical data. Body mass and body weight are not equivalent as biological reference systems, since in accordance to Newton's second law, the former has a dimension of a mass, while the latter should be dimensionally considered as a force (W = MLT-2). This distinction affects the coefficients of the mass exponent (alpha). This difference is of paramount importance in microgravity conditions (spaceflight) and of buoyancy during the fetal life in mammals. Furthermore, the coefficients (beta) of the length dimension, and (gamma) of the time dimension do not vary when mass or weight are utilized as reference systems. Consequently, the "specific metabolic time," that results from the ratio of basal oxygen consumption and body mass or body weight yields the "biological meaning" of the time dimension, which is of fractal nature.     

Evolution of reduced dispersal mortality and 'fat-tailed' dispersal kernels in autocorrelated landscapes

Models describing the evolution of dispersal strategies have mostly focused on the evolution of dispersal rates. Taking trees as a model for organisms with undirected, passive dispersal, we have developed an individual-based, spatially explicit simulation tool to investigate the evolution of the dispersal kernel, P(r), and its resulting cumulative seed-density distribution, D(r). Simulations were run on a variety of fractal landscapes differing in the fraction of suitable habitat and the spatial autocorrelation. Starting from a uniform D(r), evolution led to an increase in the fraction of seeds staying in the home cell, a reduction of the dispersal mortality (arrival in unsuitable habitat), and the evolution of 'fat-tailed' D(r) in autocorrelated landscapes and approximately uniform D(r) in random landscapes. The evolutionary process was characterized by long periods of stasis with a few bouts of rapid change in the dispersal rate.     

Computer Simulation of Vapor-Liquid Phase Separation

Abstract

A late-time growth law of domains undergoing vapor-liquid phase separation is studied for two- and three-dimensional Lennard-Jones fluids by molecular dynamics simulations. The characteristic domain size shows a power law growth in a late stage with the growth exponent of ½ for both two- and three-dimensional fluids. This study concerns also the relationship between statistical properties of domain patterns and temperatures. The asymptotic form factor of each system is obtained using scaling and the asymptotic tail of the form factor is analyzed. This tail is related to the domain-wall structure. At low system temperatures, the form factor satisfies Porod's law; the asymptotic tail decreases as S(k) ～ k ^{?(D+ 1)} where D is the system dimensionality. However, it is found that the decay of the asymptotic tail becomes slower than that of the Porod tail at higher temperatures in both two- and three-dimensional systems. This indicates that the dimension of the domain wall is fractal and increases with increasing system temperature.     

Theoretical relationship between the optimal models of the vascular tree

Two models of optimal branching structure of the vascular tree are compared. Murray’s minimum work model derived from minimum energy loss due to flow and volume in the duct system is proved to be included as a mathematical group in the authors’ model defined by the minimum volume under determinant pressure, flow and position at the terminals. The problem about heterotypical trees which are identical at the terminal conditions but different in the topological order of branch combinations are discussed, applying the results of analyses on the equivalent duct of uniform terminal pressure trees. It is proved that the minimum work tree has the least energy loss compared with its heterotypical minimum volume trees and is a better model of branching structure of the vascular tree.