Scale invariance in biology: coincidence or footprint of a universal mechanism?

In this article, we present a self-contained review of recent work on complex biological systems which exhibit no characteristic scale. This property can manifest itself with fractals (spatial scale invariance), flicker noise or 1/f-noise where f denotes the frequency of a signal (temporal scale invariance) and power laws (scale invariance in the size and duration of events in the dynamics of the system). A hypothesis recently put forward to explain these scale-free phenomomena is criticality, a notion introduced by physicists while studying phase transitions in materials, where systems spontaneously arrange themselves in an unstable manner similar, for instance, to a row of dominoes. Here, we review in a critical manner work which investigates to what extent this idea can be generalized to biology. More precisely, we start with a brief introduction to the concepts of absence of characteristic scale (power-law distributions, fractals and 1/f-noise) and of critical phenomena. We then review typical mathematical models exhibiting such properties: edge of chaos, cellular automata and self-organized critical models. These notions are then brought together to see to what extent they can account for the scale invariance observed in ecology, evolution of species, type III epidemics and some aspects of the central nervous system. This article also discusses how the notion of scale invariance can give important insights into the workings of biological systems.     

The self-organizing fractal theory as a universal discovery method: the phenomenon of life

A universal discovery method potentially applicable to all disciplines studying organizational phenomena has been developed. This method takes advantage of a new form of global symmetry, namely, scale-invariance of self-organizational dynamics of energy/matter at all levels of organizational hierarchy, from elementary particles through cells and organisms to the Universe as a whole. The method is based on an alternative conceptualization of physical reality postulating that the energy/matter comprising the Universe is far from equilibrium, that it exists as a flow, and that it develops via self-organization in accordance with the empirical laws of nonequilibrium thermodynamics. It is postulated that the energy/matter flowing through and comprising the Universe evolves as a multiscale, self-similar structure-process, i.e., as a self-organizing fractal. This means that certain organizational structures and processes are scale-invariant and are reproduced at all levels of the organizational hierarchy. Being a form of symmetry, scale-invariance naturally lends itself to a new discovery method that allows for the deduction of missing information by comparing scale-invariant organizational patterns across different levels of the organizational hierarchy.     

The endogenous circadian pacemaker imparts a scale-invariant pattern of heart rate fluctuations across time scales spanning minutes to 24 hours

Heartbeat fluctuations in mammals display a robust temporal structure characterized by scale-invariant/fractal patterns. These scale-invariant patterns likely confer physiological advantage because they change with cardiovascular disease and these changes are associated with reduced survival. Models of physical systems imply that to produce scale-invariant patterns, factors influencing the system at different time scales must be coupled via a network of feedback interactions. A similar cardiac control network is hypothesized to be responsible for the scale-invariant pattern in heartbeat dynamics, although the essential network components have not been determined. Here is shown that scale-invariant cardiac control occurs across time scales from minutes to approximately 24 h, and that lesioning the mammalian circadian pacemaker (suprachiasmatic nucleus; SCN) completely abolishes the scale-invariant pattern at time scales>or approximately 4 h. At time scales相似文献   

A Characterization of Scale Invariant Responses in Enzymatic Networks

An ubiquitous property of biological sensory systems is adaptation: a step increase in stimulus triggers an initial change in a biochemical or physiological response, followed by a more gradual relaxation toward a basal, pre-stimulus level. Adaptation helps maintain essential variables within acceptable bounds and allows organisms to readjust themselves to an optimum and non-saturating sensitivity range when faced with a prolonged change in their environment. Recently, it was shown theoretically and experimentally that many adapting systems, both at the organism and single-cell level, enjoy a remarkable additional feature: scale invariance, meaning that the initial, transient behavior remains (approximately) the same even when the background signal level is scaled. In this work, we set out to investigate under what conditions a broadly used model of biochemical enzymatic networks will exhibit scale-invariant behavior. An exhaustive computational study led us to discover a new property of surprising simplicity and generality, uniform linearizations with fast output (ULFO), whose validity we show is both necessary and sufficient for scale invariance of three-node enzymatic networks (and sufficient for any number of nodes). Based on this study, we go on to develop a mathematical explanation of how ULFO results in scale invariance. Our work provides a surprisingly consistent, simple, and general framework for understanding this phenomenon, and results in concrete experimental predictions.     

Random matrix theory in biological nuclear magnetic resonance spectroscopy

The statistical theory of energy levels or random matrix theory is presented in the context of the analysis of chemical shifts of nuclear magnetic resonance (NMR) spectra of large biological systems. Distribution functions for the spacing between nearest-neighbor energy levels are discussed for uncorrelated, correlated, and random superposition of correlated energy levels. Application of this approach to the NMR spectra of a vitamin, an antibiotic, and a protein demonstrates the state of correlation of an ensemble of energy levels that characterizes each system. The detection of coherent and dissipative structures in proteins becomes feasible with this statistical spectroscopic technique.     

Mechanical oscillations and dynamic organization of biomembranes

Biological membranes are considered as mechanochemical active media. A hypothesis is advanced according to which the biomembranes are dimeric distributed non-equilibrium systems. They provide conjugated functioning of energy transducers by means of mechanical oscillations. Transmembrane currents of substance and energy stimulate in the membranes modes determining space and time organization of the membranes characteristic of dissipative structures. The mechano-chemical conjugation of all the processes is suggested to be the physical basis of their cooperativity. Critical wavelengths for basic modes and their energetic parameters are evaluated.     

Solitary wave dynamics as a mechanism for explaining the internal motion during microtubule growth

Microtubules, which play many diverse and important roles in biological systems, are usually made up of 13 nearly axial protofilaments formed from individual tubulin molecules. In this paper, a nonlinear dynamic model has been developed to elucidate the mechanism of the internal motion occurring during the assembly of microtubules. The results derived from the model indicate that such internal motion is associated with a solitary wave, or kink, excited by the energy released from the hydrolysis of GTP ? GDP in microtubular solutions. As the kink moves forward, the individual tubulin molecules involved in the kink undergo motions that can be likened to the dislocation of atoms within the crystal lattice. Thus, the dynamic instability of microtubules may be characterized by a series of dislocation motions of the tubulin molecules. An energy estimate shows that a kink in the system possesses about 0.36–0.44 eV, which is quite close to but smaller than the 0.49 eV of energy released from the hydrolysis of GTP. Therefore, the relevant energy derived from our model is fully consistent with experimental observations; this finding also suggests that the hydrolysis energy may be responsible for exciting the solitary wave, or kink, leading to tubulin dislocation in microtubules. Our model, and its intrinsic properties, i.e., dynamic nonlinearity, thermodynamic irreversibility, as well as an energy input from a sustained source, implies that the growth of microtubules is a typical dissipative process and that their structure in vivo is typical of dissipative structures. © 1994 John Wiley & Sons, Inc.     

Generalized scale invariance and log-Poisson statistics for turbulence in the scrape-off-layer plasma in the T-10 tokamak

Results are presented from experimental observations of the statistical properties of scrape-off-layer plasma turbulence in the T-10 tokamak. The experimentally observed fluctuations in the fluxes and plasma density are intermittent in nature and obey a non-Gaussian statistics. The generalized property of plasma turbulence is its scale invariance. The experimental scalings for the moments of the distribution function of the difference in the amplitudes of fluctuations in the fluxes and plasma density are described by the log-Poisson model of strong turbulence. The self-similarity properties of turbulence that are associated with the topology of dissipative structures are investigated.     

Dissipative structures and amplification of enantiomeric excess (an experimental point of view)

Various mathematical models have been proposed to account for the origin of chiral molecules in biological systems. Most of these models invoke non-linear phenomena, and are based on the general concept of dissipative structures. These theoretical models define the fundamental criteria which must be obeyed by the experimental systems that we have investigated. Our initial approach to this problem was an extensive search of the literature data in order to select a few systems or experimental situations which would satisfy the criteria defined by the theoretical models. For these reasons, we carried out a study of the possibility of stereospecific autocatalysis in the asymmetric polymerisation of benzofuran. Similarly, the formation of spatial dissipative structures by coupling of a transport process with an interfacial reaction was investigated as a simple experimental example of symmetry breaking.     

A physical (electromagnetic) model of differentiation. 1. Basic considerations

There are a number of biological phenomena and events that cannot yet be adequately described, such as cell growth and differentiation, which may be controlled by physical factors. Fr?hlich (1980) has discussed the principles of dissipative structures as applied to electromagnetic interactions in relation to basic couplings in biological systems. Recently, increasing evidence of photon storage and ultraweak photon emission from living systems, particularly from DNA, has suggested the concept of an electromagnetic model of differentiation, based on the known quantum optical properties of nucleic acids. This model has the advantage over all ideas so far published, that it is (1) simple; (2) universally applicable to events in living matter, because it is consistent with both the quantum mechanical and the thermodynamic properties on the one hand, and the known biological and biochemical data and phenomena at the other hand; (3) it not only describes the phenomena and events in terms of pure mathematical parameters, but it can also explain them; and (4) it escapes the difficulty of finding basic control mechanisms, which themselves do not need a regulator, ad infinitum.     

The fractal mind of pedologists (soil taxonomists and soil surveyors)

There has been little work in science dealing with the organizational, political and scientific layering of database structures as well as classifications and surveys of natural resources. There is disagreement among scientists whether taxonomies are invented (human-made constructs) or are discovered (“natural” structures) independent of the discipline involved. We believe it would be helpful to study the nature of taxonomies from different points of view in order to examine questions such as; are there common features in all taxonomic systems?, are the systems neutral?, and how are classifications and data collection (surveys) linked? It is generally accepted that much institutional work on soil classification systems was nationally biased, especially in terms of practical land management.Recent studies show that the USDA soil taxonomy has the same mathematical structure as some biological ones that conform to physical laws that dictate and optimize information flow in user friendly retrieval systems. In this paper we demonstrate that the multifractal nature of the USDA soil taxonomy is strongly linked with conventional soil survey practices. In fact most surveys are packed with power law distributions, such as: (i) hierarchic taxonomic level used according to the scale map; (ii) minimum polygon size fits the functions to the map scale; and (iii) boundary density–scale map relationship, among others [Beckett, P.H.T., Bie, S.W., 1978. Use of soil and land-system maps to provide soil information in Australia. CSIRO Aust. Div. Soils, Technical Pap. No. 33, pp. 1–76]. Consequently a plethora of power law examples appear in soil survey products and soil taxonomies. Because both activities are strongly linked it seems the minds of soil surveyors and soil taxonomists create similar fractal structures. Fractal objects and power laws are scale invariant mathematical constructs, and the products prepared by experts are also fractal in many aspects. This process could be the reason that maps devoid of legends and other information have a high resemblance and information content, and with independence of scales, they provide a clear fractal signature.In summary, the systems used by soil surveyors and soil taxonomists as a whole have fractal-like structures. We now believe that developing and using fractal structures are subconscious activities of the human brain reflecting both nature and our way of processing and representing information. Because the standards of many natural resource maps are similar to pedological ones, we suspect that scale-invariant information processing is intuitive to human beings and that a more rigorous formalization of survey-taxonomy architectures may help practitioners better understand their activities and constructs, and provide a way to improve them.     

Decrease in scale invariance of activity fluctuations with aging and in patients with suprasellar tumors

Motor activity in healthy young humans displays intrinsic fluctuations that are scale-invariant over a wide range of time scales (from minutes to hours). Human postmortem and animal lesion studies showed that the intact function of the suprachiasmatic nucleus (SCN) is required to maintain such scale-invariant patterns. We therefore hypothesized that scale invariance is degraded in patients treated for suprasellar tumors that compress the SCN. To test the hypothesis, we investigated 68 patients with nonfunctioning pituitary macroadenoma and 22 patients with craniopharyngioma, as well as 72 age-matched healthy controls (age range 21.0–70.6 years). Spontaneous wrist locomotor activity was measured for 7 days with actigraphy, and detrended fluctuation analysis was applied to assess correlations over a range of time scales from minutes to 24 h. For all the subjects, complex scale-invariant correlations were only present for time scales smaller than 1.5 h, and became more random at time scales 1.5–10 h. Patients with suprasellar tumors showed a larger decrease in correlations at 1.5–10 h as compared to healthy controls. Within healthy subject, gender and age >33 year were associated with attenuated scale invariance. Conversely, activity patterns at time scales between 10 and 24 h were significantly more regular than all other time scales, and this was mostly associated with age.

In conclusion, scale invariance is degraded in healthy subjects at the ages of >33 year as characterized by attenuation of correlations at time scales 1.5–10 h. In addition, scale invariance was more degraded in patients with suprasellar tumors as compared to healthy subjects.     

Entropy and information in evolving biological systems

Integrating concepts of maintenance and of origins is essential to explaining biological diversity. The unified theory of evolution attempts to find a common theme linking production rules inherent in biological systems, explaining the origin of biological order as a manifestation of the flow of energy and the flow of information on various spatial and temporal scales, with the recognition that natural selection is an evolutionarily relevant process. Biological systems persist in space and time by transfor ming energy from one state to another in a manner that generates structures which allows the system to continue to persist. Two classes of energetic transformations allow this; heat-generating transformations, resulting in a net loss of energy from the system, and conservative transformations, changing unusable energy into states that can be stored and used subsequently. All conservative transformations in biological systems are coupled with heat-generating transformations; hence, inherent biological production, or genealogical proesses, is positively entropic. There is a self-organizing phenomenology common to genealogical phenomena, which imparts an arrow of time to biological systems. Natural selection, which by itself is time-reversible, contributes to the organization of the self-organized genealogical trajectories. The interplay of genealogical (diversity-promoting) and selective (diversity-limiting) processes produces biological order to which the primary contribution is genealogical history. Dynamic changes occuring on times scales shorter than speciation rates are microevolutionary; those occuring on time scales longer than speciation rates are macroevolutionary. Macroevolutionary processes are neither redicible to, nor autonomous from, microevolutionary processes.Authorship alphabetical     

A canonical model of multistability and scale-invariance in biological systems

Multistability and scale-invariant fluctuations occur in a wide variety of biological organisms from bacteria to humans as well as financial, chemical and complex physical systems. Multistability refers to noise driven switches between multiple weakly stable states. Scale-invariant fluctuations arise when there is an approximately constant ratio between the mean and standard deviation of a system's fluctuations. Both are an important property of human perception, movement, decision making and computation and they occur together in the human alpha rhythm, imparting it with complex dynamical behavior. Here, we elucidate their fundamental dynamical mechanisms in a canonical model of nonlinear bifurcations under stochastic fluctuations. We find that the co-occurrence of multistability and scale-invariant fluctuations mandates two important dynamical properties: Multistability arises in the presence of a subcritical Hopf bifurcation, which generates co-existing attractors, whilst the introduction of multiplicative (state-dependent) noise ensures that as the system jumps between these attractors, fluctuations remain in constant proportion to their mean and their temporal statistics become long-tailed. The simple algebraic construction of this model affords a systematic analysis of the contribution of stochastic and nonlinear processes to cortical rhythms, complementing a recently proposed biophysical model. Similar dynamics also occur in a kinetic model of gene regulation, suggesting universality across a broad class of biological phenomena.     

耗散结构,等级系统理论与生态系统

耗散结构理论与其他热力学概念一起,可以解释生态学中的许多现象。生态系统是耗散系统,用耗散结构理论来分析和讨论生态平衡等问题更为合理、准确。等级系统理论是为理解和研究高度复杂系统而发展起来的系统理论。等级系统理论为研究生态系统的行为和特征提供了客观的、适用的概念构架和实践指南,并为生态系统科学的统一性理论的形成开辟了广阔前景。本文拟就耗散结构理论和等级系统理论的主要内容及其在生态学中的应用作一介绍和讨论。     

耗散结构、等级系统理论与生态系统

耗散结构理论与其他热力学概念一起,可以解释生态学中的许多现象。生态系统是耗散系统,用耗散结构理论来分析和讨论生态平衡等问题更为合理、准确。等级系统理论是为理解和研究高度复杂系统而发展起来的系统理论。等级系统理论为研究生态系统的行为和特征提供了客观的、适用的概念构架和实践指南,并为生态系统科学的统一性理论的形成开辟了广阔前景。本文拟就耗散结构理论和等级系统理论的主要内容及其在生态学中的应用作一介绍和讨论。     

Critical periods as fundamental events in life

Development is not a continuous phenomenon. Rather, phenophases are interspaced with short critical periods. This phenomenon reflects an alternance between stabilization (during a phenophase) and dismantling (during a critical period) of a network of between-organ relationships generating the organism. Networks of relationships may be compared to dissipative systems in physics. In this context, a critical period represents a transient phase of isolation of the systems enabling its evolution towards equilibrium. As suggested here, this transition from dissipative to isolated system represents the source of newly emerging dissipative structures in which environmental or developmental perturbations are adaptively integrated. In contrast to non-living systems, an endogenous control of the transition towards critical period seems to exist during development. By extension to other scales of biological organization, it is suggested that the capacity to self-define its status (dissipative or close-to-equilibrium) represents the key property of living systems. This asks for a reconsideration of some basic notions about life, such as the role of genes in normal development, in physiological adaptation, and even in the emergence of evolutionary novelty.     

Characterization of agitation environments in 250 ml spinner vessel, 3 L,and 20 L reactor vessels used for animal cell microcarrier culture

Three dimensional particle tracking velocimetry (3-D PTV) was used to characterize the flow fields in the impeller region of three microcarrier reactor vessels. Three typical cell culture bioreactors were chosen: 250 ml small-scale spinner vessels, 3 L bench-scale reactor, and 20 L medium-scale reactor. Conditions studied correspond to the actual operating conditions in industrial setting and were determined based on the current scale-up paradigm: the Kolmogorov eddy length criterion. In this paper we present characterization of hydrodynamics on the basis of flow structures produced because of agitation. Flow structures were determined from 3-D mean velocity results obtained using 3-D PTV. Although the impellers used in 3 L and 20 L reactors were almost identical, the flow structures produced in the two reactors differed considerably. Results indicate that near geometric scale up does not necessarily amount to scale-up of flow patterns and indicates that intensity as well as distribution of energy may vary considerably during such a scale-up.     

Motivation and Benefits of Complex Systems Approaches in Ecology

Studies of complex systems in other disciplines provide models and analytical strategies for understanding ecosystems and landscapes. The emphasis is on invariant properties, particularly processes that create scaling relations over wide ranges of scale, both in time and space. Translations between levels of ecological organization may be accomplished by succinct characterizations of processes that operate at fine scales, followed by renormalization group analysis to reveal patterns at broad scales. The self-organized patterns found in simple ecosystem, landscape, and forest-fire models may be explained as feedback between the system state and control parameters. Critical phenomena and phase transitions are expected in open, dissipative systems where long-range correlations defy predictions based on average population densities, a concept that becomes irrelevant as nonstationary conditions prevail. Thus, complexity theory for open systems relates to the ecology of self-entailing ecosystems that function as their own environments and thereby create constraints through emergence. Received: 14 April 1998; accepted 26 May 1998     

The coordination of protein motors and the kinetic behavior of microtubule--a computational study

Utilizing the mechanical energy converted from chemical energy through hydrolysis of ATP, motor proteins drive cytoskeleton filaments to move in various biological systems. Recent technological advance has shown the potential of the motor proteins for powering future nano-bio-mechanical systems. In order to effectively use motor proteins as a biological motor, the interaction between the protein motors and bio-filaments needs to be well clarified, since such interaction is largely influenced by many factors, such as the coordination among the motors, their dynamic behavior, physical properties of microtubules, and the viscosity of solution involved, etc. In this study, a two-dimensional model was proposed to simulate the motion of a microtubule driven by protein motors based on a dissipative particle dynamics (DPD) method with attempt to correlate the microtubule's kinetic behavior to the coordination among protein motors.