Abstract representations of biological systems in supercategories

An abstract representation of biological systems from the standpoint of the theory of supercategories is presented. The relevance of such representations forG-relational biologies is suggested. In section A the basic concepts of our representation, that is class, system, supercategory and measure are introduced. Section B is concerned with the mathematical representation starting with some axioms and principles which are natural extensions of the current abstract representations in biology. Likewise, some extensions of the principle of adequate design are introduced in section C. Two theorems which present the connection between categories and supercategories are proved. Two other theorems concerning the dynamical behavior of biological and biophysical systems are derived on the basis of the previous considerations. Section D is devoted to a general study of oscillatory behavior in enzymic systems, some general quantitative relations being derived from our representation. Finally, the relevance of these results for a quantum theoretic approach to biology is discussed.     

From interesting details to dynamical relevance: toward more effective use of empirical insights in theory construction

A perennial challenge in ecology is to develop dynamical systems models that appropriately abstract and characterize the dynamics of natural systems. Deriving an appropriate model of system dynamics can be a long and iterative process whose outcome depends critically on the quality of empirical data describing the long‐term behavior of a natural system. Most ecological time series are insufficient to offer insight into the way organizational hierarchies and spatial scales are causally linked to natural system dynamics. Moreover, the classic tradition of hypothesis testing in ecology is not likely to lead to those key insights. This because empirical research is geared almost exclusively toward testing model predictions based on underlying causal relationships assumed by theorists. So, empirical research relies heavily on theory for guidance on what is or is not dynamically relevant. I argue here that it is entirely possible to reduce much of this guesswork involved with deciding on causal structure by giving empirical research a new role in theory development. In this role, natural history and field observations are used to develop stochastic, individual‐based and spatially explicit computational models or IBMs that can explore the range of contingency and complexity inherent in real‐world systems.
IBMs can be used to run simulations allowing deductions to be made about the causal linkages between organizational hierarchies, spatial scales, and dynamics. These deductions can be tested under field conditions using experiments that manipulate the putative causal structure and evaluate the dynamical consequences. The emerging insights from this stage can then be used to inspire an analytical construct that embodies the dynamically relevant scales and mechanisms. In essence, computational modeling serves as an intermediate step in theory development in that a wide range of possibly important biological details are considered and then reduced to a subset that is dynamically relevant.     

Agential autonomy and biological individuality

What is a biological individual? How are biological individuals individuated? How can we tell how many individuals there are in a given assemblage of biological entities? The individuation and differentiation of biological individuals are central to the scientific understanding of living beings. I propose a novel criterion of biological individuality according to which biological individuals are autonomous agents. First, I articulate an ecological–dynamical account of natural agency according to which, agency is the gross dynamical capacity of a goal-directed system to bias its repertoire to respond to its conditions as affordances. Then, I argue that agents or agential dynamical systems can be agentially dependent on, or agentially autonomous from, other agents and that this agential dependence/autonomy can be symmetrical or asymmetrical, strong or weak. Biological individuals, I propose, are all and only those agential dynamical systems that are strongly agentially autonomous. So, to determine how many individuals there are in a given multiagent aggregate, such as multicellular organism, a colony, symbiosis, or a swarm, we first have to identify how many agential dynamical systems there are, and then what their relations of agential dependence/autonomy are. I argue that this criterion is adequate to the extent that it vindicates the paradigmatic cases, and explains why the paradigmatic cases are paradigmatic, and why the problematic cases are problematic. Finally, I argue for the importance of distinguishing between agential and causal dependence and show the relevance of agential autonomy for understanding the explanatory structure of evolutionary developmental biology.     

A theory for environmental systems

A theory for environmental systems is defined on the basis of two elements, termed ‘environmental unity’ and ‘behavior’. Environmental systems are regarded as non-living systems, each one related with only one biological system. We construct a material-energetic environmental diagram, which is introduced in terms of the theory of categories, thereby giving rise to a new categoryE. By means of two biological conditions, and the definition of static property of the biological system (related to its own environment), a set of theorems is obtained, exhibiting mathematical consequences for the represented theory.     

Global dynamics of biological systems from time-resolved omics experiments

The emergent properties of biological systems, organized around complex networks of irregularly connected elements, limit the applications of the direct scientific method to their study. The current lack of knowledge opens new perspectives to the inverse scientific paradigm where observations are accumulated and analysed by advanced data-mining techniques to enable a better understanding and the formulation of testable hypotheses about the structure and functioning of these systems. The current technology allows for the wide application of omics analytical methods in the determination of time-resolved molecular profiles of biological samples. Here it is proposed that the theory of dynamical systems could be the natural framework for the proper analysis and interpretation of such experiments. A new method is described, based on the techniques of non-linear time series analysis, which is providing a global view on the dynamics of biological systems probed with time-resolved omics experiments.     

An exact expression to calculate the derivatives of position-dependent observables in molecular simulations with flexible constraints

In this work, we introduce an algorithm to compute the derivatives of physical observables along the constrained subspace when flexible constraints are imposed on the system (i.e., constraints in which the constrained coordinates are fixed to configuration-dependent values). The presented scheme is exact, it does not contain any tunable parameter, and it only requires the calculation and inversion of a sub-block of the Hessian matrix of second derivatives of the function through which the constraints are defined. We also present a practical application to the case in which the sought observables are the Euclidean coordinates of complex molecular systems, and the function whose minimization defines the flexible constraints is the potential energy. Finally, and in order to validate the method, which, as far as we are aware, is the first of its kind in the literature, we compare it to the natural and straightforward finite-differences approach in a toy system and in three molecules of biological relevance: methanol, N-methyl-acetamide and a tri-glycine peptide.     

Interactions among biological systems: An analysis of asymptotic stability

An analysis of the interactions among asymptotically stable dynamical systems is formulated by making use of the dynamical system theory. Some results coming from previous mathematical analyses have been slightly modified to take into account some typical biological constraints as the boundedness properties of the solutions. In particular it has been shown that when the “coupling” among the subsystems is “loose” enough (in a sense that has to be made mathematically precise) the asymptotic behaviour of a complex system is the same of that of its individual components. The mathematical theory has been used to analyze two systems of biological significance: the coupling among chemical reactions and the stability properties of a 4-dimensional system describing the kinetics of a chemical transmitter.     

Biological pest control using cannibalistic predators and with provision of additional food: a theoretical study

Cannibalism is a conspecific lethal interaction, a typical phenomenon in many natural populations, which is used as a “life-boat strategy” to avoid circumstances leading to extinction. It is observed in many experimental studies that the cannibalistic nature of natural enemies deters the outcome of biological pest control programmes. One of the ways to deviate natural enemies from conspecific lethal interactions is to provide them with additional food. In this paper, using the theory of dynamical systems, we analyse the dynamics of a cannibalistic predator-prey system when predators are provided with additional food. A detailed mathematical analysis is carried out to study the permanence, stability and various bifurcations occurring in the system. The system analysis reveals several interesting phenomena. Depending on the choice of quality (characterised by the predator’s handling time towards additional food, and prey) and quantity of additional food, the system can exhibit multiple coexisting equilibria, leading to the emergence of a homoclinic loop. Further, it is observed that by varying the quality and quantity of additional food, one can not only limit and control the pest but also eradicate the predators. In the context of biological control programmes, the current theoretical study aids eco-managers in choosing the appropriate additional food that is to be supplied for enhancing the biocontrol efficiency of cannibalistic predators.     

An abstract cell model that describes the self-organization of cell function in living systems

The principal aim of systems biology is to search for general principles that govern living systems. We develop an abstract dynamic model of a cell, rooted in Mesarovi? and Takahara's general systems theory. In this conceptual framework the function of the cell is delineated by the dynamic processes it can realize. We abstract basic cellular processes, i.e., metabolism, signalling, gene expression, into a mapping and consider cell functions, i.e., cell differentiation, proliferation, etc. as processes that determine the basic cellular processes that realize a particular cell function. We then postulate the existence of a 'coordination principle' that determines cell function. These ideas are condensed into a theorem: If basic cellular processes for the control and regulation of cell functions are present, then the coordination of cell functions is realized autonomously from within the system. Inspired by Robert Rosen's notion of closure to efficient causation, introduced as a necessary condition for a natural system to be an organism, we show that for a mathematical model of a self-organizing cell the associated category must be cartesian closed. Although the semantics of our cell model differ from Rosen's (M,R)-systems, the proof of our theorem supports (in parts) Rosen's argument that living cells have non-simulable properties. Whereas models that form cartesian closed categories can capture self-organization (which is a, if not the, fundamental property of living systems), conventional computer simulations of these models (such as virtual cells) cannot. Simulations can mimic living systems, but they are not like living systems.     

Heterarchy in biological systems: a logic-based dynamical model of abstract biological network derived from time-state-scale re-entrant form

Heterarchical structure is important for understanding robustness and evolvability in a wide variety of levels of biological systems. Although many studies emphasize the heterarchical nature of biological systems, only a few computational representations of heterarchy have been created thus far. We propose here the time-state-scale re-entrant form to address the self-referential property derived from setting heterarchical structure. In this paper, we apply the time-state-scale re-entrant form to abstract self-referential modeling for a functional manifestation of biological network presented by [Tsuda, I., Tadaki, K., 1997. A logic-based dynamical theory for a genesis of biological threshold. BioSystems 42, 45-64]. The numerical results of this system show different intermittent phase transitions and power-law distribution of time spent in activating functional manifestation. The Hierarchically separated time-scales obtained from spectrum analysis imply that the reactions at different levels simultaneously appear in a dynamical system. The results verify the mutual inter-relationship between heterarchical structure in biological systems and the self-referential property of computational heterarchical systems.     

Derivation of a Floquet Formalism within a Natural Framework

Many biological systems experience a periodic environment. Floquet theory is a mathematical tool to deal with such time periodic systems. It is not often applied in biology, because linkage between the mathematics and the biology is not available. To create this linkage, we derive the Floquet theory for natural systems. We construct a framework, where the rotation of the Earth is causing the periodicity. Within this framework the angular momentum operator is introduced to describe the Earth’s rotation. The Fourier operators and the Fourier states are defined to link the rotation to the biological system. Using these operators, the biological system can be transformed into a rotating frame in which the environment becomes static. In this rotating frame the Floquet solution can be derived. Two examples demonstrate how to apply this natural framework.     

Causality, randomness, intelligibility, and the epistemology of the cell

Because the basic unit of biology is the cell, biological knowledge is rooted in the epistemology of the cell, and because life is the salient characteristic of the cell, its epistemology must be centered on its livingness, not its constituent components. The organization and regulation of these components in the pursuit of life constitute the fundamental nature of the cell. Thus, regulation sits at the heart of biological knowledge of the cell and the extraordinary complexity of this regulation conditions the kind of knowledge that can be obtained, in particular, the representation and intelligibility of that knowledge. This paper is essentially split into two parts. The first part discusses the inadequacy of everyday intelligibility and intuition in science and the consequent need for scientific theories to be expressed mathematically without appeal to commonsense categories of understanding, such as causality. Having set the backdrop, the second part addresses biological knowledge. It briefly reviews modern scientific epistemology from a general perspective and then turns to the epistemology of the cell. In analogy with a multi-faceted factory, the cell utilizes a highly parallel distributed control system to maintain its organization and regulate its dynamical operation in the face of both internal and external changes. Hence, scientific knowledge is constituted by the mathematics of stochastic dynamical systems, which model the overall relational structure of the cell and how these structures evolve over time, stochasticity being a consequence of the need to ignore a large number of factors while modeling relatively few in an extremely complex environment.     

Robustness as an evolutionary principle.

We suggest simulating evolution of complex organisms using a model constrained solely by the requirement of robustness in its expression patterns. This scenario is illustrated by evolving discrete logical networks with epigenetic properties. Evidence for dynamical features in the evolved networks is found that can be related to biological observables.     

GDSCalc: A Web-Based Application for Evaluating Discrete Graph Dynamical Systems

Discrete dynamical systems are used to model various realistic systems in network science, from social unrest in human populations to regulation in biological networks. A common approach is to model the agents of a system as vertices of a graph, and the pairwise interactions between agents as edges. Agents are in one of a finite set of states at each discrete time step and are assigned functions that describe how their states change based on neighborhood relations. Full characterization of state transitions of one system can give insights into fundamental behaviors of other dynamical systems. In this paper, we describe a discrete graph dynamical systems (GDSs) application called GDSCalc for computing and characterizing system dynamics. It is an open access system that is used through a web interface. We provide an overview of GDS theory. This theory is the basis of the web application; i.e., an understanding of GDS provides an understanding of the software features, while abstracting away implementation details. We present a set of illustrative examples to demonstrate its use in education and research. Finally, we compare GDSCalc with other discrete dynamical system software tools. Our perspective is that no single software tool will perform all computations that may be required by all users; tools typically have particular features that are more suitable for some tasks. We situate GDSCalc within this space of software tools.     

Dimensionality Reduction of Bistable Biological Systems

Time hierarchies, arising as a result of interactions between system’s components, represent a ubiquitous property of dynamical biological systems. In addition, biological systems have been attributed switch-like properties modulating the response to various stimuli across different organisms and environmental conditions. Therefore, establishing the interplay between these features of system dynamics renders itself a challenging question of practical interest in biology. Existing methods are suitable for systems with one stable steady state employed as a well-defined reference. In such systems, the characterization of the time hierarchies has already been used for determining the components that contribute to the dynamics of biological systems. However, the application of these methods to bistable nonlinear systems is impeded due to their inherent dependence on the reference state, which in this case is no longer unique. Here, we extend the applicability of the reference-state analysis by proposing, analyzing, and applying a novel method, which allows investigation of the time hierarchies in systems exhibiting bistability. The proposed method is in turn used in identifying the components, other than reactions, which determine the systemic dynamical properties. We demonstrate that in biological systems of varying levels of complexity and spanning different biological levels, the method can be effectively employed for model simplification while ensuring preservation of qualitative dynamical properties (i.e., bistability). Finally, by establishing a connection between techniques from nonlinear dynamics and multivariate statistics, the proposed approach provides the basis for extending reference-based analysis to bistable systems.