Control analysis of metabolic networks. 2. Total differentials and general formulation of the connectivity relations

The mathematical background of the connectivity relations of metabolic control theory is analysed. The connectivity relations are shown to reflect general properties of total differentials of reaction rate vi, flux J, and metabolite concentration Xj. Connectivity relations hold for any metabolic network in which all vi are homogeneous functions of enzyme concentration Ei. This notion allows established algebraic methods to be used for the formulation of connectivity relations for metabolic systems in which numerous constraints are imposed on metabolite concentrations. A general procedure to derive connectivity relations for such metabolic systems is given. To encourage a broader audience to apply control theory to physiological systems, an easy-to-use graphical procedure is derived for formulating connectivity relations for biochemical systems in which no metabolite is involved in more than one constraint.     

Molecular set theory: II. An aspect of the biomathematical theory of sets

In an earlier paper (Molecular Set Theory: I.Bull. Math. Biophysics,22, 285–307, 1960) the author proposed a “Molecular Set Theory” as a formal mathematical meta-theoretic system for representing complex reactions not only of biological interest, but also of general chemical interest. The present paper is a refinement and extension of the earlier work along more formal algebraic lines. For example the beginnings of an algebra of molecular transformations is presented. It also emphasizes that this development, together with the genetical set theory of Woodger's and Rashevsky's set-theoretic contributions to Relational Biology, points to the existence of a biomathematical theory of sets which is not deducible from the general mathematical, abstract theory of sets.     

Algebraic properties of automata associated to Petri nets and applications to computation in biological systems

Biochemical and genetic regulatory networks are often modeled by Petri nets. We study the algebraic structure of the computations carried out by Petri nets from the viewpoint of algebraic automata theory. Petri nets comprise a formalized graphical modeling language, often used to describe computation occurring within biochemical and genetic regulatory networks, but the semantics may be interpreted in different ways in the realm of automata. Therefore, there are several different ways to turn a Petri net into a state-transition automaton. Here, we systematically investigate different conversion methods and describe cases where they may yield radically different algebraic structures. We focus on the existence of group components of the corresponding transformation semigroups, as these reflect symmetries of the computation occurring within the biological system under study. Results are illustrated by applications to the Petri net modelling of intermediary metabolism. Petri nets with inhibition are shown to be computationally rich, regardless of the particular interpretation method. Along these lines we provide a mathematical argument suggesting a reason for the apparent all-pervasiveness of inhibitory connections in living systems.     

Mechanical analysis of the development of left ventricular aneurysms

The left ventricle is modelled as a spherical shell with an infarcted wall segment. The mechanics of the circumstances causing this infarcted segment to develop into an aneurysm is presented. Both the wall stresses and deformations are worked out for aneurysms developing from infarcts of different sizes and percentages of wall damage. The governing equations consist of incompressibility relations, force-equilibrium relations and stress-strain relations. Newton Raphson technique is used to solve these nonlinear simultaneous algebraic equations, for the values of the myocardial stresses in the infarcted segment and the bulge values, in terms of the ventricular geometry and the damage extent (expressed in terms of the damage angle and percentage of wall damage). The results indicate that in general it is innermost layer which is severely stressed and that in the rupture of the ventricle the critical factor involved is the percentage of infarct thickness rather than the angle of damage.     

The transducer model for contrast detection and discrimination: formal relations, implications, and an empirical test

The transducer function mu for contrast perception describes the nonlinear mapping of stimulus contrast onto an internal response. Under a signal detection theory approach, the transducer model of contrast perception states that the internal response elicited by a stimulus of contrast c is a random variable with mean mu(c). Using this approach, we derive the formal relations between the transducer function, the threshold-versus-contrast (TvC) function, and the psychometric functions for contrast detection and discrimination in 2AFC tasks. We show that the mathematical form of the TvC function is determined only by mu, and that the psychometric functions for detection and discrimination have a common mathematical form with common parameters emanating from, and only from, the transducer function mu and the form of the distribution of the internal responses. We discuss the theoretical and practical implications of these relations, which have bearings on the tenability of certain mathematical forms for the psychometric function and on the suitability of empirical approaches to model validation. We also present the results of a comprehensive test of these relations using two alternative forms of the transducer model: a three-parameter version that renders logistic psychometric functions and a five-parameter version using Foley's variant of the Naka-Rushton equation as transducer function. Our results support the validity of the formal relations implied by the general transducer model, and the two versions that were contrasted account for our data equally well.     

Swarming in homogeneous environments: A social interaction based framework

A great variety of biological groups form a self-organized swarming motion at some point during their life spans, which has two prominent collective features: common velocity and constant spacings among members. In this paper, we present a general individual-based motion framework to explain such collective motion of swarms in homogeneous environments. The motion framework utilizes the concept of social interactions that has been widely accepted throughout the literature. We assume that during the motion of the swarm, each member senses and interacts with its neighbors via virtual Attraction/Alignment/Repulsion (A/A/R) forces, while perceiving and following the gradient force of the environment. During the swarm's motion, the neighborhood and the interaction relations among members may dynamically change. To explicitly consider the effect of such dynamic change on the emergence of swarm's collective behavior, we use an algebraic graph to model the topology of the interaction and the neighborhood relations among the members.By using mathematical tools of nonsmooth analysis theory and Lyapunov stability theory, we analytically prove that if the A/A/R forces have limited ranges, and the attraction/repulsion forces are balanced at a certain range, the proposed framework leads to a parallel type of collective motion of the swarm. We mathematically show that the velocities of all swarm members asymptotically converge to a common value and the spacings among neighbors remain unchanging. In addition to the mathematical analysis, a few sets of simulation results are included to demonstrate the presented framework.The contributions of this paper are twofold: First, unlike most works in the literature that mainly use computer simulations to study the swarming phenomena, this paper provides an analytical methodology to investigate how the collective group behavior is self-organized by individual motions. Second, the presented motion framework works over a general range of A/A/R interactions. In other words, we analytically prove that the commonly used A/A/R model can lead to a collective motion of the swarm. In addition, we show that the alternative model in the literature that uses only attraction/repulsion (A/R) interactions is in fact a special case of the A/A/R model.     

Lattices and lattice-valued relations in biology

Rashevsky's treatment of general binary relations between sets of biological elements is extended using the novel mathematical concept of lattice-valued relation (l.v.r.). This yields a quantitative measure of the strength of the relations between components of a biological organism, and some illustrative examples are given. Specific l.v.r.'s are used to define (more precisely than in Rashevsky's preliminary theory of binary relations) the biologically important relationships amongst hormones, metabolism and energy exchange involved in metabolic reactions. The ‘strongest link’ between the set of hormones and the set of metabolic reactions is quantified using a special l.v.r., and other specific biological realisations of lattice-valued relations in abstract-relational biology are presented. L.v.r.'s may also be regarded as a form ofG-relation in relational biology, or as a particular case of generating diagrams. Further possible developments of this approach, using more complex tools of the newly developed mathematical theory of lattice-valued relations, such as function space l.v.r., group l.v.r., l.v.r. morphisms, l.v.r. homology andn-ary l.v.r.'s are suggested.     

The matrix representation of pharmacokinetic models.

An equation is developed from the matrix of rate constants which describes the behaviour of linear pharmacokinetic models for any initial condition as a function of time. This general matrix equation is then used to derive analogous expressions for drug distribution after a period of infusion, at the steady state, or during a multiple constant-dosage regimen. Matrix expressions are also derived for areas under drug concentration curves for any compartment after single doses or during multiple dosing. General matrix equations are shown to yield loading dosage schedules to achieve plateau concentrations throughout any open system.It is suggested that matrix methods have advantages over previously used mathematical techniques in pharmacokinetics in the simplicity of the algebraic expressions, and their ease of manipulation. An algebraic example of an open two-compartment model is worked to indicate the applicability of the general expressions.     

Theoretical study of volume changes associated with the helix-coil transition of an alanine-rich peptide in aqueous solution

The changes in the partial molar volume (PMV) associated with the conformational transition of an alanine-rich peptide AK16 from the alpha-helix structure to various random coil structures are calculated by the three-dimensional interaction site model (3D-RISM) theory coupled with the Kirkwood-Buff theory. The volume change is analyzed by decomposing it into contributions from geometry and hydration: the changes in the van der Waals, void, thermal, and interaction volume. The total change in the PMV is positive. This is primarily due to the growth of void space within the peptide, which is canceled in part by the volume reduction resulting from the increase in the electrostatic interaction between the peptide and water molecules. The changes in the void and thermal volume of the coil structures are widely distributed and tend to compensate each other. Additionally, the relations between the hydration volume components and the surface properties are investigated. We categorize coil structures into extended coils with the PMV smaller than helix and general coils with the PMV larger than helix. The pressure therefore can both stabilize and destabilize the coil structures. The latter seems to be a more proper model of random coil structures of the peptide.     

Model reduction and a priori kinetic parameter identifiability analysis using metabolome time series for metabolic reaction networks with linlog kinetics

In this work, we present a time-scale analysis based model reduction and parameter identifiability analysis method for metabolic reaction networks. The method uses the information obtained from short term chemostat perturbation experiments. We approximate the time constant of each metabolite pool by their turn-over time and classify the pools accordingly into two groups: fast and slow pools. We performed a priori model reduction, neglecting the dynamic term of the fast pools. By making use of the linlog approximative kinetics, we obtained a general explicit solution for the fast pools in terms of the slow pools by elaborating the degenerate algebraic system resulting from model reduction. The obtained relations yielded also analytical relations between a subset of kinetic parameters. These relations also allow to realize an analytical model reduction using lumped reaction kinetics. After solving these theoretical identifiability problems and performing model reduction, we carried out a Monte Carlo approach to study the practical identifiability problems. We illustrated the methodology on model reduction and theoretical/practical identifiability analysis on an example system representing the glycolysis in Saccharomyces cerevisiae cells.     

Mathematical treatment of water movement in plant cells and tissue: a review

Abstract. Over the past three decades, many contributions have been made to the development of a mathematical basis for describing water transport in plant cells and tissue. This review paper attempts to summarize the more significant contributions and to outline the concepts upon which the various mathematical analyses are founded.
The paper itself is divided into three major sections. Section I deals with the quantitative water relations of single plant cells. Basic equations are developed which describe the water statics and water dynamics of such cells. Included is a discussion of the theory and methods for measuring the various parameters (permeabilities, cell wall elastic moduli, etc.) which enter into the development. The section closes with a presentation of circuit analog models for single plant cells.
Section II is devoted to a review and development of the water relations of plant tissues which contain numerous cells in series. Following a historical overview, various existing models are derived and physical tissue properties which enter the derivation are identified. The concept of 'local equilibrium' is discussed and circuit analog models for single cells are generalized and applied to several cells in series.
The final section contains two example applications of water transport theory as it applies to plant tissue. One application involves radial water movement in a soybean hypocotyl while the other deals with water transport in a growing root tip. A summary at the end of the section is largely devoted to a discussion of the limitations of mathematical models dial are presently available.     

Mathematical modeling of evolution. Solved and open problems

Evolution is a highly complex multilevel process and mathematical modeling of evolutionary phenomenon requires proper abstraction and radical reduction to essential features. Examples are natural selection, Mendel’s laws of inheritance, optimization by mutation and selection, and neutral evolution. An attempt is made to describe the roots of evolutionary theory in mathematical terms. Evolution can be studied in vitro outside cells with polynucleotide molecules. Replication and mutation are visualized as chemical reactions that can be resolved, analyzed, and modeled at the molecular level, and straightforward extension eventually results in a theory of evolution based upon biochemical kinetics. Error propagation in replication commonly results in an error threshold that provides an upper bound for mutation rates. Appearance and sharpness of the error threshold depend on the fitness landscape, being the distribution of fitness values in genotype or sequence space. In molecular terms, fitness landscapes are the results of two consecutive mappings from sequences into structures and from structures into the (nonnegative) real numbers. Some properties of genotype–phenotype maps are illustrated well by means of sequence–structure relations of RNA molecules. Neutrality in the sense that many RNA sequences form the same (coarse grained) structure is one of these properties, and characteristic for such mappings. Evolution cannot be fully understood without considering fluctuations—each mutant originates form a single copy, after all. The existence of neutral sets of genotypes called neutral networks, in particular, necessitates stochastic modeling, which is introduced here by simulation of molecular evolution in a kind of flowreactor.     

The art of community detection

Networks in nature possess a remarkable amount of structure. Via a series of data-driven discoveries, the cutting edge of network science has recently progressed from positing that the random graphs of mathematical graph theory might accurately describe real networks to the current viewpoint that networks in nature are highly complex and structured entities. The identification of high order structures in networks unveils insights into their functional organization. Recently, Clauset, Moore, and Newman, introduced a new algorithm that identifies such heterogeneities in complex networks by utilizing the hierarchy that necessarily organizes the many levels of structure. Here, we anchor their algorithm in a general community detection framework and discuss the future of community detection.     

Mathematical modelling of engineered tissue growth using a multiphase porous flow mixture theory

This paper outlines the framework of a porous flow mixture theory for the mathematical modelling of in vitro tissue growth, and gives an application of this theory to an aspect of tissue engineering. The problem is formulated as a set of partial differential equations governing the space and time dependence of the amounts of each component of the tissue (phase), together with the physical stresses in each component. The theory requires constitutive relations to specify the material properties of each phase, and also requires relations to specify the stresses developed due to mechanical interactions, both within each phase and between different phases. An application of the theory is given to the study of the mobility and aggregation of a population of cells seeded into an artificial polymeric scaffold. Stability analysis techniques show that the interplay of the forces between the tissue constituents results in two different regimes: either the cells form aggregates or disperse through the scaffold.     

Studies in the physicomathematical theory of organic form

In continuation of previous studies the theory is developed on the assumption that the form of any organism is determined by requirements to perform definite biological functions. A previously outlined theory of the form of plants is developed further, showing how the conditions of mechanical strength together with the specifications of the total mass and metabolism, may quantitatively determine not only the general form of the plant, but even the number, size, shape and shades of the leaves. Next the form of animals, as required by mechanical conditions and by the different types of possible locomotions is discussed. A mathematical theory of locomotion of snakes in relation to their shape is outlined. Next is discussed the form and locomotion of quadrupeds. A number of theoretical relations, which describe the shape of an animal, are derived and compared to available observations. After that the theory of flight of birds and insects is discussed, and again some form relations comparable with observations, are discussed. Finally a set of equations is outlined, which determines not only the external shape, but also the internal structure of animals. Different relations pertaining to some inner organs are derived and compared with available observations. The paper ends with a brief discussion on the shape of unicellular organisms.     

An electrostatic spatial resonance model for coaxial helical structures with applications to the filamentous bacteriophages.

A model is presented that treats the symmetry matching problem in structures made of two interacting coaxial helices of point charges. The charges are sources of a potential field that mediates a non-specific attractive interaction between the helices. The problem is represented in Fourier space, which affords the most generality. It is found that coaxial helices with optimally mated symmetries can lock into spatial resonance configurations that maximize their interaction. The resonances are represented as vectors in a discrete three-dimensional space. Two algebraic relations are given for the four symmetry parameters of two helices in resonance. One-start inner helices interacting with coaxial one-start or NR-start outer helices are considered. Applications are made to the filamentous bacteriophages Ff, Pf1, Xf, and Pf3. The interaction given by the linearized Poisson-Boltzmann equation is calculated in this formalism to allow comparison of the electrostatic free energy of interaction of different resonance structures. Experimental nucleotide/subunit ratios are accounted for, and models for the DNA-protein interfaces are presented, with particular emphasis on Pf1.     

Crosslinking in viral capsids via tiling theory

A vital part of a virus is its protein shell, called the viral capsid, that encapsulates and hence protects the viral genome. It has been shown in Twarock [2004. A tiling approach to vius capsids assembly explaining a structural puzzle in virology. J. Theor. Biol. 226, 477-482] that the surface structures of viruses with icosahedrally symmetric capsids can be modelled in terms of tilings that encode the locations of the protein subunits. This theory is extended here to multi-level tilings in order to model crosslinking structures. The new framework is demonstrated for the case of bacteriophage HK97, and it is shown, how the theory can be used in general to decide if crosslinking, and what type of crosslinking, is compatible from a mathematical point of view with the geometrical surface structure of a virus.