Dynamical systems under constant organization I. Topological analysis of a family of non-linear differential equations —A model for catalytic hypercycles

The paper presents a qualitative analysis of the following systems ofn differential equations: \(\dot x_i = x_i x_j - x_i \sum\nolimits_r^n { = 1} x_r x_s {\mathbf{ }}(j = i - 1 + n\delta _{i1} {\mathbf{ }}and{\mathbf{ }}s = r - 1 + n\delta _{r1} )\) , which show cyclic symmetry. These dynamical systems are of particular interest in the theory of selforganization and biological evolution as well as for application to other fields.     

Analysis of bias in the calculation and measurement of plant mineral uptake rates

Aims

Calculation of net ion uptake rate (F) from hydroponic solutions relies on balanced equations where F is equal to the initial minus the final ion content, plus fertilization. Knowledge is thus required of both volume (V), concentration (C) and of their temporal variations. The literature, however, proposes simplified equations that disregard variations in V and are thus strictly inaccurate. This paper studies the bias arising from such simplified formulae and also from deviations in V and C measurements.

Methods

We used our experimental data and simulation to analyse the impact of different bias sources on F calculation, and to compare setups where C is regulated, or left to drift in order to study F = f(C).

Results

This paper reports two major findings, the first being that simplified equations distort F diurnal dynamics and ion uptake isotherms, yielding underestimated Michaelis-Menten parameters. The second shows the advantage of using C-regulated over unregulated systems to determine F when biased V and C measurements cannot be avoided.

Conclusions

Regulated systems are able to minimize the biases on F, but the measurement of water uptake rate is compulsory. Therefore, simplified formulae should not be used.     

The problem of the eukaryotic genome size

The current state of knowledge concerning the unsolved problem of the huge interspecific eukaryotic genome size variations not correlating with the species phenotypic complexity (C-value enigma also known as C-value paradox) is reviewed. Characteristic features of eukaryotic genome structure and molecular mechanisms that are the basis of genome size changes are examined in connection with the C-value enigma. It is emphasized that endogenous mutagens, including reactive oxygen species, create a constant nuclear environment where any genome evolves. An original quantitative model and general conception are proposed to explain the C-value enigma. In accordance with the theory, the noncoding sequences of the eukaryotic genome provide genes with global and differential protection against chemical mutagens and (in addition to the anti-mutagenesis and DNA repair systems) form a new, third system that protects eukaryotic genetic information. The joint action of these systems controls the spontaneous mutation rate in coding sequences of the eukaryotic genome. It is hypothesized that the genome size is inversely proportional to functional efficiency of the anti-mutagenesis and/or DNA repair systems in a particular biological species. In this connection, a model of eukaryotic genome evolution is proposed.     

Categories of (ℓ, ℛ)-systems

We show that when we represent (ℓ, ℛ)-systems with fixed genome as automata (sequential machines), we get automata with output-dependent states. This yields a short proof that ((ℓ, ℛ)-systems from a subcategory of automata—and with more homomorphisms than previously exhibited. We show how ((ℓ, ℛ)-systems with variable genetic structure may be represented as automata and use this embedding to set up a larger subcategory of the category of automata. An analogy with dynamical systems is briefly discussed. This paper presents a formal exploration and extension of some of the ideas presented by Rosen (Bull. Math. Biophyss,26, 103–111, 1964;28, 141–148;28 149–151). We refer the reader to these papers, and references cited therein, for a discussion of the relevance of this material to relational biology.     

A re-examination of the relationship between thermal conductance and body weight in mammals

• 1.1. The following equation based on 230 conductance values for 192 species of mammals of body weights ranging from 3.5 to 150,000 g describes the relationship of conductance below thermal neutrality to body weight in mammals: C = 0.760 W^−0.426, where C has units of mlO₂/g·h·°C and W is body weight in g.
• 2.2. Bats, order Chiroptera, have conductance values higher than predicted from body weight; conductance is predicted by the equation : C = 1.54 W^−0.54.
• 3.3. Heteromyid and cricetid rodents have conductance values below predicted and the following equations predict conductance in these two families. C = 0.62 W^−0.44 and C = 1.03 W^−0.54, respectively.

     

Bi-ionic flows through a single homogeneous membrane

The form of the equations for bi-ionic flows through a cation-exchanger membrane is investigated. Simple algebraic flow equations are given by a first-order expansion of an integral of the Nernst-Planck d.e.'s calculated under the assumption of local electroneutrality. Donnan equilibrium is used to find the ionic partition at the membrane-solution interfaces. Using standard techniques it is found that the cation flow equations can be put into the form $$J_i = \lambda (C_i^I C_j^{II} - C_i^{II} C_j^I ) + \frac{{\tau _i }}{F}I,i \ne j = 1,2,$$ where, however, λ and τ _i are functions of the mean concentrations across the membrane. Thus it is shown that the osmotic force Δπ _si=2 RT(C _i ^I -C _i ^II ) cannot be the driving force in the bi-ionic flow equations as might be expected in a generalization of the Kedem-Katchalsky equation for a single salt.     

Hierarchical analysis of piecewise affine models of gene regulatory networks

A key point in the analysis of dynamical models of biological systems is to handle systems of relatively high dimensions. In the present paper we propose a method to hierarchically organize a certain type of piecewise affine (PWA) differential systems. This specific class of systems has been extensively studied for the past few years, as it provides a good framework to model gene regulatory networks. The method, shown on several examples, allows a qualitative analysis of the asymptotic behavior of a PWA system, decomposing it into several smaller subsystems. This technique, based on the well-known strongly connected components decomposition, is not new. However, its adaptation to the non-smooth PWA differential equations turns out to be quite relevant because of the strong discrete structure underlying these equations. Its biological relevance is shown on a 7-dimensional PWA system modeling the gene network responsible for the carbon starvation response in Escherichia coli.

---

Determination of the drag coefficient of a toroidal plasma vortex in air

Forces acting on toroidal vortices in an unbounded medium (plasma vortices in air and vortex rings in air and water) are investigated. A solution to the equations describing such votrices is obtained. It is shown that this solution satisfactorily agrees with experiment. Based on the experimental results and the solution obtained, the drag coefficient C _x of such vortices is found. For the same Reynolds numbers, the value of C _x may be much less than the drag coefficient of a drop-shaped axisymmetric body (0.045), which is known to be the best streamlined object.     

The potential for non-adaptive origins of evolutionary innovations in central carbon metabolism

Background

Biological systems are rife with examples of pre-adaptations or exaptations. They range from the molecular scale – lens crystallins, which originated from metabolic enzymes – to the macroscopic scale, such as feathers used in flying, which originally served thermal insulation or waterproofing. An important class of exaptations are novel and useful traits with non-adaptive origins. Whether such origins could be frequent cannot be answered with individual examples, because it is a question about a biological system’s potential for exaptation.We here take a step towards answering this question by analyzing central carbon metabolism, and novel traits that allow an organism to survive on novel sources of carbon and energy. We have previously applied flux balance analysis to this system and predicted the viability of 10¹⁵ metabolic genotypes on each of ten different carbon sources.

Results

We here use this exhaustive genotype-phenotype map to ask whether a central carbon metabolism that is viable on a given, focal carbon source C – the equivalent of an adaptation in our framework – is usually or rarely viable on one or more other carbon sources C _new – a potential exaptation. We show that most metabolic genotypes harbor potential exaptations, that is, they are viable on one or more carbon sources C _new . The nature and number of these carbon sources depends on the focal carbon source C itself, and on the biochemical similarity between C and C _new . Moreover, metabolisms that show a higher biomass yield on C, and that are more complex, i.e., they harbor more metabolic reactions, are viable on a greater number of carbon sources C _new .

Conclusions

A high potential for exaptation results from correlations between the phenotypes of different genotypes, and such correlations are frequent in central carbon metabolism. If they are similarly abundant in other metabolic or biological systems, innovations may frequently have non-adaptive (“exaptive”) origins.

     

Differential effects of fatty acid chain length on the viability of two species of cactophilic Drosophila

• 1.1. Two columnar cacti in the Sonoran Desert, agria and organpipe, contain medium chain (C₈−C₁₂) fatty acids.
• 2.2. Necrotic tissues of these cacti serve as feeding and breeding substrates for Drosophila mojavensis but not D. nigrospiracula.
• 3.3. Results show that capric and lauric acids are the predominant fatty acids of both cacti.
• 4.4. Fatty acid chain length exhibits a differential effect on larval viability with caprylic acid (Q) having the greatest and myristic acid (C₁₄) having the least effect.
• 5.5. Drosophila mojavensis is more tolerant of free fatty acids than D. nigrospiracula, and this partly explains the ability of D. mojavensis to utilize agria and organpipe cacti.

     

Evaluation of surface colonization kinetics in continuous culture

Two equations, describing surface colonization, were evaluated and compared using suspended glass slides in a continuous culture ofPseudomonas aeruginosa. These equations were used to determine surface growth rates from the number and distribution of cells present on the surface after incubation. One of these was the colonization equation which accounts for simultaneous attachment and growth of bacteria on surfaces: $$N = (A/\mu )e^{\mu t} - A/\mu $$ where N=number of cells on surface (cells field^?1); A=attachment rate (cells field^?1h^?1);μ=specific growth rate (h^?1); t=incubation period (h). The other was the surface growth rate equation which assumes that the number of colonies of a given size (C_i) will reach a constant value (C_max) which is equal to A divided byμ: $$\mu = \frac{{\ln \left( {\frac{N}{{C_i }} + 1} \right)}}{t}$$ Both equations gave similar results and the time required to approximate C_max may not be as long as was previously thought. In all cases both A andμ continuously decreased throughout the incubation period. These decreases may be due to various effects of microbial accumulation on the surface. Both equations accurately determined surface growth rates despite highly variable attachment rates. Growth rates were similar for both the liquid phase of the culture and the solid-liquid interface (0.4 h^?1). Use of the surface growth rate equation is favored over the use of the colonization equation since the former does not require a computer to solve forμ and the counting procedure is simplified.     

A Lyapunov function for piecewise-independent differential equations: stability of the ideal free distribution in two patch environments

In this article we construct Lyapunov functions for models described by piecewise-continuous and independent differential equations. Because these models are described by discontinuous differential equations, the theory of Lyapunov functions for smooth dynamical systems is not applicable. Instead, we use a geometrical approach to construct a Lyapunov function. Then we apply the general approach to analyze population dynamics describing exploitative competition of two species in a two-patch environment. We prove that for any biologically meaningful parameter combination the model has a globally stable equilibrium and we analyze this equilibrium with respect to parameters.      

Determination of the equilibrium constants of self-associating protein systems: XI. The application of C(r) in graphical analysis and the enumeration of interacting species in the ultracentrifuge

A greatly simplified procedure is proposed which employs C= f(r) as determined from sedimentation equilibrium measurements in graphical analysis of self-associating protein systems and in the enumeration of interacting species in the ultracentrifuge. Basic equations given here are applicable to any self-associating system. A procedure is outlined for enumeration of interacting components independent of non-ideal behavior, using principal component analysis.     

Mapping parameter spaces of biological switches

Since the seminal 1961 paper of Monod and Jacob, mathematical models of biomolecular circuits have guided our understanding of cell regulation. Model-based exploration of the functional capabilities of any given circuit requires systematic mapping of multidimensional spaces of model parameters. Despite significant advances in computational dynamical systems approaches, this analysis remains a nontrivial task. Here, we use a nonlinear system of ordinary differential equations to model oocyte selection in Drosophila, a robust symmetry-breaking event that relies on autoregulatory localization of oocyte-specification factors. By applying an algorithmic approach that implements symbolic computation and topological methods, we enumerate all phase portraits of stable steady states in the limit when nonlinear regulatory interactions become discrete switches. Leveraging this initial exact partitioning and further using numerical exploration, we locate parameter regions that are dense in purely asymmetric steady states when the nonlinearities are not infinitely sharp, enabling systematic identification of parameter regions that correspond to robust oocyte selection. This framework can be generalized to map the full parameter spaces in a broad class of models involving biological switches.