Logical problems and misunderstandings in “problems with dimensionless measurement models of synchrony in biological systems”

A model for the standardized measurement of synchrony in behavioral or biological states was proposed for use in comparative analyses of synchrony's adaptive significance. A recent critique attempts to discredit dimensionless (unitless) or standardized measures of synchrony in general and the proposed model in particular. Although the critique helps define and sharpen thinking about the measurement of synchrony and makes some well‐taken points, it also arrives at questionable conclusions based upon dubious assumptions. The critique proceeds from a fictional example in which all biological states being considered always have exactly the same durations for all individuals and always start or end for all individuals at exactly the same instants in time. These unreal, biologically meaningless premises, together with other questionable assumptions, are the basis of inappropriate analyses of standardized measures. The critique's arguments reveal a failure to understand the basic underlying principle of the standardized method and are dependent upon faulty logic. A statistical discussion contains both worthy points and arguable comments based not on data or actual probabilities of real events but on irrelevant chance outcomes derived from the biologically meaningless assumptions. The critique's conclusions are not credible, and its basic question probably is not scientifically answerable. Am. J. Primatol. 47:29–42, 1999. © 1999 Wiley‐Liss, Inc.     

Rhythmic and non-rhythmic attractors in asynchronous random Boolean networks

In multi-component, discrete systems, such as Boolean networks and cellular automata, the scheme of updating of the individual elements plays a crucial role in determining their dynamic properties and their suitability as models of complex phenomena. Many interesting properties of these systems rely heavily on the use of synchronous updating of the individual elements. Considerations of parsimony have motivated the claim that, if the natural systems being modelled lack any clear evidence of synchronously driven elements, then random asynchronous updating should be used by default. The introduction of a random element precludes the possibility of strictly cyclic behaviour. In principle, this poses the question of whether asynchronously driven Boolean networks, cellular automata, etc., are inherently bad choices at the time of modelling rhythmic phenomena. This paper focuses on this subsidiary issue for the case of Asynchronous Random Boolean Networks (ARBNs). It defines measures of pseudo-periodicity between states and sufficiently relaxed statistical constraints. These measures are used to guide a genetic algorithm to find appropriate examples. Success in this search for a number of cases, and the subsequent statistical analysis lead to the conclusion that ARBNs can indeed be used as models of co-ordinated rhythmic phenomena, which may be stronger precisely because of their in-built asynchrony. The same technique is used to find non-stationary attractors that show no rhythm. Evidence suggests that the latter are more abundant than rhythmic attractor. The methodology is flexible, and allows for more demanding statistical conditions for defining pseudo-periodicity, and constraining the evolutionary search.     

Transient resetting: a novel mechanism for synchrony and its biological examples

The study of synchronization in biological systems is essential for the understanding of the rhythmic phenomena of living organisms at both molecular and cellular levels. In this paper, by using simple dynamical systems theory, we present a novel mechanism, named transient resetting, for the synchronization of uncoupled biological oscillators with stimuli. This mechanism not only can unify and extend many existing results on (deterministic and stochastic) stimulus-induced synchrony, but also may actually play an important role in biological rhythms. We argue that transient resetting is a possible mechanism for the synchronization in many biological organisms, which might also be further used in the medical therapy of rhythmic disorders. Examples of the synchronization of neural and circadian oscillators as well as a chaotic neuron model are presented to verify our hypothesis.     

A Mathematical Biologist’s Guide to Absolute and Convective Instability

Mathematical models have been highly successful at reproducing the complex spatiotemporal phenomena seen in many biological systems. However, the ability to numerically simulate such phenomena currently far outstrips detailed mathematical understanding. This paper reviews the theory of absolute and convective instability, which has the potential to redress this inbalance in some cases. In spatiotemporal systems, unstable steady states subdivide into two categories. Those that are absolutely unstable are not relevant in applications except as generators of spatial or spatiotemporal patterns, but convectively unstable steady states can occur as persistent features of solutions. The authors explain the concepts of absolute and convective instability, and also the related concepts of remnant and transient instability. They give examples of their use in explaining qualitative transitions in solution behaviour. They then describe how to distinguish different types of instability, focussing on the relatively new approach of the absolute spectrum. They also discuss the use of the theory for making quantitative predictions on how spatiotemporal solutions change with model parameters. The discussion is illustrated throughout by numerical simulations of a model for river-based predator–prey systems.     

Impact of external influences on characteristics of domains in biological membranes

Living systems continually interact with the environment. External influences evoke complex processes in the entire system. In this work, the impact of external influences on the behavior of the domains in biological membranes is considered both in general and on special examples. A general approach of nonequilibrium statistical thermodynamics proposed by L.A. Stratonovich along with the same approach adapted to stochastic storage processes are applied. Both the Stratonovich theory and the storage models allow characterizing specific features of the stationary non-equilibrium states as well as the behavior of the highly non-equilibrium processes in open systems, such as live organisms.     

Protonation free energy levels in complex molecular systems

All proteins, nucleic acids, and other biomolecules contain residues capable of exchanging protons with their environment. These proton transfer phenomena lead to pH sensitivity of many molecular processes underlying biological phenomena. In the course of biological evolution, Nature has invented some mechanisms to use pH gradients to regulate biomolecular processes inside cells or in interstitial fluids. Therefore, an ability to model protonation equilibria in molecular systems accurately would be of enormous value for our understanding of biological processes and for possible rational influence on them, like in developing pH dependent drugs to treat particular diseases. This work presents a derivation, by thermodynamic and statistical mechanical methods, of an expression for the free energy of a complex molecular system at arbitrary ionization state of its titratable residues. This constitutes one of the elements of modeling protonation equilibria. Starting from a consideration of a simple acid-base equilibrium of a model compound with a single tritratable group, we arrive at an expression which is of general validity for complex systems. The only approximation used in this derivation is the postulating that the interaction energy between any pair of titratable sites does not depend on the protonation states of all the remaining ionizable groups.     

Hyperbolic and kinetic models for self-organized biological aggregations and movement: a brief review

We briefly review hyperbolic and kinetic models for self-organized biological aggregations and traffic-like movement. We begin with the simplest models described by an advection-reaction equation in one spatial dimension. We then increase the complexity of models in steps. To this end, we begin investigating local hyperbolic systems of conservation laws with constant velocity. Next, we proceed to investigate local hyperbolic systems with density-dependent speed, systems that consider population dynamics (i.e., birth and death processes), and nonlocal hyperbolic systems. We conclude by discussing kinetic models in two spatial dimensions and their limiting hyperbolic models. This structural approach allows us to discuss the complexity of the biological problems investigated, and the necessity for deriving complex mathematical models that would explain the observed spatial and spatiotemporal group patterns.     

Studies of folding and misfolding using simplified models

Computer simulations are as vital to our studies of biological systems as experiments. They bridge and rationalize experimental observations, extend the experimental "field of view", which is often limited to a specific time or length scale, and, most importantly, provide novel insights into biological systems, offering hypotheses about yet-to-be uncovered phenomena. These hypotheses spur further experimental discoveries. Simplified molecular models have a special place in the field of computational biology. Branded as less accurate than all-atom protein models, they have offered what all-atom molecular dynamics simulations could not--the resolution of the length and time scales of biological phenomena. Not only have simplified models proven to be accurate in explaining or reproducing several biological phenomena, they have also offered a novel multiscale computational strategy for accessing a broad range of time and length scales upon integration with traditional all-atom simulations. Recent computer simulations of simplified models have shaken or advanced the established understanding of biological phenomena. It was demonstrated that simplified models can be as accurate as traditional molecular dynamics approaches in identifying native conformations of proteins. Their application to protein structure prediction yielded phenomenal accuracy in recapitulating native protein conformations. New studies that utilize the synergy of simplified protein models with all-atom models and experiments yielded novel insights into complex biological processes, such as protein folding, aggregation and the formation of large protein complexes.     

Hybrid simulation of cellular behavior

MOTIVATION: To be valuable to biological or biomedical research, in silico methods must be scaled to complex pathways and large numbers of interacting molecular species. The correct method for performing such simulations, discrete event simulation by Monte Carlo generation, is computationally costly for large complex systems. Approximation of molecular behavior by continuous models fails to capture stochastic behavior that is essential to many biological phenomena. RESULTS: We present a novel approach to building hybrid simulations in which some processes are simulated discretely, while other processes are handled in a continuous simulation by differential equations. This approach preserves the stochastic behavior of cellular pathways, yet enables scaling to large populations of molecules. We present an algorithm for synchronizing data in a hybrid simulation and discuss the trade-offs in such simulation. We have implemented the hybrid simulation algorithm and have validated it by simulating the statistical behavior of the well-known lambda phage switch. Hybrid simulation provides a new method for exploring the sources and nature of stochastic behavior in cells.     

Quantifying synchrony among reproductive or other states

Synchrony among reproductive or other states is a widespread phenomenon of significance in evolutionary theory. A model is presented for use in the measurement of the degree of synchrony in any set of behavioral or biological states, such as menstrual cycling, pregnancy, or lactation. At a given time, maximum group synchrony occurs when all members of a group are in a single state, minimum synchrony occurs when equal numbers are in all states, and intermediate values of synchrony occur for other proportional representations of the states. The model yields group synchrony scores ranging from 0 to 100 by standardizing the average of the absolute deviations among frequencies of occurrence in pairs of states. In addition, individual synchrony is characterized by the proportion of others who share an individual's state at a specified set of times. Properties of the model are discussed, simplified calculation procedures are presented, and the method's usefulness for comparative purposes is illustrated. © 1995 Wiley-Liss, Inc.     

Quantifying the roles of space and stochasticity in computer simulations for cell biology and cellular biochemistry

Most of the fascinating phenomena studied in cell biology emerge from interactions among highly organized multimolecular structures embedded into complex and frequently dynamic cellular morphologies. For the exploration of such systems, computer simulation has proved to be an invaluable tool, and many researchers in this field have developed sophisticated computational models for application to specific cell biological questions. However, it is often difficult to reconcile conflicting computational results that use different approaches to describe the same phenomenon. To address this issue systematically, we have defined a series of computational test cases ranging from very simple to moderately complex, varying key features of dimensionality, reaction type, reaction speed, crowding, and cell size. We then quantified how explicit spatial and/or stochastic implementations alter outcomes, even when all methods use the same reaction network, rates, and concentrations. For simple cases, we generally find minor differences in solutions of the same problem. However, we observe increasing discordance as the effects of localization, dimensionality reduction, and irreversible enzymatic reactions are combined. We discuss the strengths and limitations of commonly used computational approaches for exploring cell biological questions and provide a framework for decision making by researchers developing new models. As computational power and speed continue to increase at a remarkable rate, the dream of a fully comprehensive computational model of a living cell may be drawing closer to reality, but our analysis demonstrates that it will be crucial to evaluate the accuracy of such models critically and systematically.     

Dichotomous noise with feedback and charge-conformational interactions

The states of the charge transfer chain fragment depending on the charge localization are modelled by interaction with the two-level system as a source of dichotomous noise whose features depend themselves on the fragment states. Basing on the exact solution of the stochastic problem, we describe the noise-induced phase transitions of the (mono-bi)stability type which are possible even in the simplest case of the fragment represented by an overdamped harmonic oscillator. The results are discussed in the context of the functional nonlinearities in biological charge transport processes.     

Modeling and simulation of biological systems with stochasticity

Mathematical modeling is a powerful approach for understanding the complexity of biological systems. Recently, several successful attempts have been made for simulating complex biological processes like metabolic pathways, gene regulatory networks and cell signaling pathways. The pathway models have not only generated experimentally verifiable hypothesis but have also provided valuable insights into the behavior of complex biological systems. Many recent studies have confirmed the phenotypic variability of organisms to an inherent stochasticity that operates at a basal level of gene expression. Due to this reason, development of novel mathematical representations and simulations algorithms are critical for successful modeling efforts in biological systems. The key is to find a biologically relevant representation for each representation. Although mathematically rigorous and physically consistent, stochastic algorithms are computationally expensive, they have been successfully used to model probabilistic events in the cell. This paper offers an overview of various mathematical and computational approaches for modeling stochastic phenomena in cellular systems.     

Coexistence for systems governed by difference equations of Lotka-Volterra type

The question of the long term survival of species in models governed by Lotka-Volterra difference equations is considered. The criterion used is the biologically realistic one of permanence, that is populations with all initial values positive must eventually all become greater than some fixed positive number. We show that in spite of the complex dynamics associated even with the simplest of such systems, it is possible to obtain readily applicable criteria for permanence in a wide range of cases.     

Dynamical network biomarkers for identifying critical transitions and their driving networks of biologic processes

Non-smooth or even abrupt state changes exist during many biological processes, e.g., cell differentiation processes, proliferation processes, or even disease deterioration processes. Such dynamics generally signals the emergence of critical transition phenomena, which result in drastic changes of system states or eventually qualitative changes of phenotypes. Hence, it is of great importance to detect such transitions and further reveal their molecular mechanisms at network level. Here, we review the recent advances on dynamical network biomarkers (DNBs) as well as the related theoretical foundation, which can identify not only early signals of the critical transitions but also their leading networks, which drive the whole system to initiate such transitions. In order to demonstrate the effectiveness of this novel approach, examples of complex diseases are also provided to detect pre-disease stage, for which traditional methods or biomarkers failed.     

Nonlinear systems in medicine

Many achievements in medicine have come from applying linear theory to problems. Most current methods of data analysis use linear models, which are based on proportionality between two variables and/or relationships described by linear differential equations. However, nonlinear behavior commonly occurs within human systems due to their complex dynamic nature; this cannot be described adequately by linear models. Nonlinear thinking has grown among physiologists and physicians over the past century, and non-linear system theories are beginning to be applied to assist in interpreting, explaining, and predicting biological phenomena. Chaos theory describes elements manifesting behavior that is extremely sensitive to initial conditions, does not repeat itself and yet is deterministic. Complexity theory goes one step beyond chaos and is attempting to explain complex behavior that emerges within dynamic nonlinear systems. Nonlinear modeling still has not been able to explain all of the complexity present in human systems, and further models still need to be refined and developed. However, nonlinear modeling is helping to explain some system behaviors that linear systems cannot and thus will augment our understanding of the nature of complex dynamic systems within the human body in health and in disease states.     

A model for non-resolvable ambiguities

The dynamical behavior of the perception of ambiguous figures arising from the essentially nonresolvable ambiguity built in the figures themselves is analysed. Two main features of it are explored, the initial transient and the rhythmic alternation of inversions of perspective following it. The initial transient is envisaged in terms of a symmetry breaking disorder-order transformation induced by the attention of the observer. The rhythmic inversions of the ordered structures are classified by a model based on an ideally non linear decision equation for binary systems originally proposed to schematize the observed behavior of neurons. Finally it is suggested that the phenomena relative to the perception of ambiguous figures could represent the simplest ones among more complex situations arising in perception, in concept formation, in the sensing of emotion and in communication in general.     

A Design Principle of Group-level Decision Making in Cell Populations

Populations of cells often switch states as a group to cope with environmental changes such as nutrient availability and cell density. Although the gene circuits that underlie the switches are well understood at the level of single cells, the ways in which such circuits work in concert among many cells to support group-level switches are not fully explored. Experimental studies of microbial quorum sensing show that group-level changes in cellular states occur in either a graded or an all-or-none fashion. Here, we show through numerical simulations and mathematical analysis that these behaviors generally originate from two distinct forms of bistability. The choice of bistability is uniquely determined by a dimensionless parameter that compares the synthesis and the transport of the inducing molecules. The role of the parameter is universal, such that it not only applies to the autoinducing circuits typically found in bacteria but also to the more complex gene circuits involved in transmembrane receptor signaling. Furthermore, in gene circuits with negative feedback, the same dimensionless parameter determines the coherence of group-level transitions from quiescence to a rhythmic state. The set of biochemical parameters in bacterial quorum-sensing circuits appear to be tuned so that the cells can use either type of transition. The design principle identified here serves as the basis for the analysis and control of cellular collective decision making.