Organismic supercategores: II. On multistable systems

The representation of biological systems in terms of organismic supercategories, introduced in previous papers (Bull. Math. Biophysics,30, 625–636;31, 59–70) is further discussed. To state more clearly this representation some new definitions are introduced. Also, some necessary changes in axiomatics are made. The conclusion is reached that any organismic supercategory has at least one superpushout, and this expresses the fact that biological systems are multistable. This way a connection between some results of Rashevsky’s theory of organismic sets and our results becomes obvious.     

Some remarks on topological biology

With reference to several recent papers by the author, it is pointed out that within the principle of biotopological mapping a choice of a primordial graph and of a particular transformation defines a system of abstract biology, similar to systems of abstract geometries. The study of such abstract systems is necessary before one can be found which is isomorphic to the actual biological world. A brief survey of the structure and properties of the system based on the choice of the primordial graph and of the transformationT defined in a previous paper (Bull. Math. Biophysics,16, 317–48, 1954) is made. Two more topological theorems are demonstrated, which, interpreted biologically, lead to the conclusion that the higher an organism, the more adaptable it is. Finally a criticism of that particular system of abstract biology is made, and its inadequacy for the representation of the actual biological phenomena pointed out, and a suggestion is made for a possible point set topological approach to biology.     

Organismic supercategories and qualitative dynamics of systems

The representation of biological systems by means of organismic supercategories, developed in previous papers (Bull. Math. Biophysics,30, 625–636;31, 59–71;32, 539–561), is further discussed. The different approaches to relational biology, developed by Rashevsky, Rosen and by Băianu and Marinescu, are compared with Qualitative Dynamics of Systems which was initiated by Henri Poincaré (1881). On the basis of this comparison some concrete result concerning dynamics of genetic system, development, fertilization, regeneration, analogies, and oncogenesis are derived.     

Organismic supercategories: I. Proposals for a general unitary theory of systems

The present paper is an attempt to outline an abstract unitary theory of systems. In the introduction some of the previous abstract representations of systems are discussed. Also a possible connection of abstract representations of systems with a general theory of measure is proposed. Then follow some necessary definitions and authors' proposals for an axiomatic theory of systems. Finally some concrete examples are analyzed in the light of the proposed theory.     

UML as a cell and biochemistry modeling language

The systems biology community is building increasingly complex models and simulations of cells and other biological entities, and are beginning to look at alternatives to traditional representations such as those provided by ordinary differential equations (ODE). The lessons learned over the years by the software development community in designing and building increasingly complex telecommunication and other commercial real-time reactive systems, can be advantageously applied to the problems of modeling in the biology domain. Making use of the object-oriented (OO) paradigm, the unified modeling language (UML) and Real-Time Object-Oriented Modeling (ROOM) visual formalisms, and the Rational Rose RealTime (RRT) visual modeling tool, we describe a multi-step process we have used to construct top–down models of cells and cell aggregates. The simple example model described in this paper includes membranes with lipid bilayers, multiple compartments including a variable number of mitochondria, substrate molecules, enzymes with reaction rules, and metabolic pathways. We demonstrate the relevance of abstraction, reuse, objects, classes, component and inheritance hierarchies, multiplicity, visual modeling, and other current software development best practices. We show how it is possible to start with a direct diagrammatic representation of a biological structure such as a cell, using terminology familiar to biologists, and by following a process of gradually adding more and more detail, arrive at a system with structure and behavior of arbitrary complexity that can run and be observed on a computer. We discuss our CellAK (Cell Assembly Kit) approach in terms of features found in SBML, CellML, E-CELL, Gepasi, Jarnac, StochSim, Virtual Cell, and membrane computing systems.     

Structural systems biology: modelling protein interactions

Much of systems biology aims to predict the behaviour of biological systems on the basis of the set of molecules involved. Understanding the interactions between these molecules is therefore crucial to such efforts. Although many thousands of interactions are known, precise molecular details are available for only a tiny fraction of them. The difficulties that are involved in experimentally determining atomic structures for interacting proteins make predictive methods essential for progress. Structural details can ultimately turn abstract system representations into models that more accurately reflect biological reality.     

On mutant sets

This paper sketches the outline of a new, general mathematical theory concerning the nature of a relative anti-closure property for subsets of general algebraic systems. It thus quite naturally fits into any exhaustive theory of relations and, in particular, into a theory of relations for abstract mathematical molecular biology. In addition, the theory possesses certain intuitive, but naive relationships to basic analytical studies of biological mating and mutation. At the purely mathematical level one can produce an abundance of theorems from the theory with interpretations in the context of the frequently appearing mathematical structures of groups and rings, among other algebraic structures.     

Modeling and simulation of molecular biology systems using petri nets: modeling goals of various approaches

Petri nets are a discrete event simulation approach developed for system representation, in particular for their concurrency and synchronization properties. Various extensions to the original theory of Petri nets have been used for modeling molecular biology systems and metabolic networks. These extensions are stochastic, colored, hybrid and functional. This paper carries out an initial review of the various modeling approaches based on Petri net found in the literature, and of the biological systems that have been successfully modeled with these approaches. Moreover, the modeling goals and possibilities of qualitative analysis and system simulation of each approach are discussed.     

Gene module level analysis: identification to networks and dynamics

Nature exhibits modular design in biological systems. Gene module level analysis is based on this module concept, aiming to understand biological network design and systems behavior in disease and development by emphasizing on modules of genes rather than individual genes. Module level analysis has been extensively applied in genome wide level analysis, exploring the organization of biological systems from identifying modules to reconstructing module networks and analyzing module dynamics. Such module level perspective provides a high level representation of the regulatory scenario and design of biological systems, promising to revolutionize our view of systems biology, genetic engineering as well as disease mechanisms and molecular medicine.     

Mapping of statistical physics to information theory with application to biological systems

The problem of achieving a mapping of formalisms in statistical physics and theoretical biology to information theory is discussed using an example for canonical ensembles. We extend the meaning of the Handscomb Monte-Carlo method to a general recipe for the transformation from a "configuration" space to a "sentence" space. The ensemble of "sentences" and its corresponding source uncertainty function are introduced. A possible mapping procedure based on a generalization of the Handscomb representation is described. For a biological illustration, we present a way to introduce a pathway representation to describe metabolic processes in living 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.     

Sparse representation approaches for the classification of high-dimensional biological data

Background

High-throughput genomic and proteomic data have important applications in medicine including prevention, diagnosis, treatment, and prognosis of diseases, and molecular biology, for example pathway identification. Many of such applications can be formulated to classification and dimension reduction problems in machine learning. There are computationally challenging issues with regards to accurately classifying such data, and which due to dimensionality, noise and redundancy, to name a few. The principle of sparse representation has been applied to analyzing high-dimensional biological data within the frameworks of clustering, classification, and dimension reduction approaches. However, the existing sparse representation methods are inefficient. The kernel extensions are not well addressed either. Moreover, the sparse representation techniques have not been comprehensively studied yet in bioinformatics.

Results

In this paper, a Bayesian treatment is presented on sparse representations. Various sparse coding and dictionary learning models are discussed. We propose fast parallel active-set optimization algorithm for each model. Kernel versions are devised based on their dimension-free property. These models are applied for classifying high-dimensional biological data.

Conclusions

In our experiment, we compared our models with other methods on both accuracy and computing time. It is shown that our models can achieve satisfactory accuracy, and their performance are very efficient.

     

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.     

Quantifying dynamic stability of genetic memory circuits

Bistability/Multistability has been found in many biological systems including genetic memory circuits. Proper characterization of system stability helps to understand biological functions and has potential applications in fields such as synthetic biology. Existing methods of analyzing bistability are either qualitative or in a static way. Assuming the circuit is in a steady state, the latter can only reveal the susceptibility of the stability to injected DC noises. However, this can be inappropriate and inadequate as dynamics are crucial for many biological networks. In this paper, we quantitatively characterize the dynamic stability of a genetic conditional memory circuit by developing new dynamic noise margin (DNM) concepts and associated algorithms based on system theory. Taking into account the duration of the noisy perturbation, the DNMs are more general cases of their static counterparts. Using our techniques, we analyze the noise immunity of the memory circuit and derive insights on dynamic hold and write operations. Considering cell-to-cell variations, our parametric analysis reveals that the dynamic stability of the memory circuit has significantly varying sensitivities to underlying biochemical reactions attributable to differences in structure, time scales, and nonlinear interactions between reactions. With proper extensions, our techniques are broadly applicable to other multistable biological systems.     

Executable cell biology

Computational modeling of biological systems is becoming increasingly important in efforts to better understand complex biological behaviors. In this review, we distinguish between two types of biological models--mathematical and computational--which differ in their representations of biological phenomena. We call the approach of constructing computational models of biological systems 'executable biology', as it focuses on the design of executable computer algorithms that mimic biological phenomena. We survey the main modeling efforts in this direction, emphasize the applicability and benefits of executable models in biological research and highlight some of the challenges that executable biology poses for biology and computer science. We claim that for executable biology to reach its full potential as a mainstream biological technique, formal and algorithmic approaches must be integrated into biological research. This will drive biology toward a more precise engineering discipline.     

Assessing evolutionary epistemology

There are two interrelated but distinct programs which go by the name evolutionary epistemology. One attempts to account for the characteristics of cognitive mechanisms in animals and humans by a straightforward extension of the biological theory of evolution to those aspects or traits of animals which are the biological substrates of cognitive activity, e.g., their brains, sensory systems, motor systems, etc. (EEM program). The other program attempts to account for the evaluation of ideas, scientific theories and culture in general by using models and metaphors drawn from evolutionary biology (EET program). The paper begins by distinguishing the two programs and discussing the relationship between them. The next section addresses the metaphorical and analogical relationship between evolutionary epistemology and evolutionary biology. Section IV treats the question of the locus of the epistemological problem in the light of an evolutionary analysis. The key questions here involve the relationship between evolutionary epistemology and traditional epistemology and the legitimacy of evolutionary epistemology as epistemology. Section V examines the underlying ontological presuppositions and implications of evolutionary epistemology. Finally, section VI, which is merely the sketch of a problem, addresses the parallel between evolutionary epistemology and evolutionary ethics.This research was supported, in part, by a grant from the National Science Foundation # SES—8308720. I want to thank Richard Lewontin and the Museum of Comparative Zoology, Harvard University for their hospitality and support during Spring 1984. I also want to thank David Hull, Charles Dyke and Michael Ruse for helpful comments on an earlier draft, and Andy Altman for pointing out to me the passage from Inherit the Wind.     

Generalized Maximum Range Tests for Pairwise Comparisons of Several Populations

A multiple comparison procedure (MCP) is proposed for the comparison of all pairs of several independent samples. This MCP is essentially the closed procedure with union-intersection tests based on given single tests Q_ij for the minimal hypotheses H_ij. In such cases where the α-levels of the nominal tests associated with the MCP can be exhausted, this MCP has a uniformly higher all pair power than any refined Bonferroni test using the same Q_ij. Two different general algorithms are described in section 3. A probability inequality for ranges of i.i.d. random variables which is useful for some algorithms is proved in section 4. Section 5 contains the application to independent normally distributed estimates and section 6 the comparisons of polynomial distributions by multivariate ranges. Further applications are possible. Tables of the 0.05-bounds for the tests of section 5 and 6 are enclosed.     

Notation-dependent and -independent representations of numbers in the parietal lobes

It is a commonly held view that numbers are represented in an abstract way in both parietal lobes. This view is based on failures to find differences between various notational representations. Here we show that by using relatively smaller voxels together with an adaptation paradigm and analyzing subjects on an individual basis it is possible to detect specialized numerical representations. The current results reveal a left/right asymmetry in parietal lobe function. In contrast to an abstract representation in the left parietal lobe, the numerical representation in the right parietal lobe is notation dependent and thus includes nonabstract representations. Our results challenge the commonly held belief that numbers are represented solely in an abstract way in the human brain.