Controllability of non-linear biochemical systems

Mathematical methods of biochemical pathway analysis are rapidly maturing to a point where it is possible to provide objective rationale for the natural design of metabolic systems and where it is becoming feasible to manipulate these systems based on model predictions, for instance, with the goal of optimizing the yield of a desired microbial product. So far, theory-based metabolic optimization techniques have mostly been applied to steady-state conditions or the minimization of transition time, using either linear stoichiometric models or fully kinetic models within biochemical systems theory (BST). This article addresses the related problem of controllability, where the task is to steer a non-linear biochemical system, within a given time period, from an initial state to some target state, which may or may not be a steady state. For this purpose, BST models in S-system form are transformed into affine non-linear control systems, which are subjected to an exact feedback linearization that permits controllability through independent variables. The method is exemplified with a small glycolytic-glycogenolytic pathway that had been analyzed previously by several other authors in different contexts.     

The mathematical theory of lateral impedance of visual activity and an application of Markov processes

The mathematical relationship describing recurrent lateral inhibition is expressed as a linear operator equation. Under quite general conditions, the operator equation is shown to have a unique nonnegative solution. It is also shown that the linear operator for recurrent impedance is representable as an integral operator and that, when in application to physiological models it is interpreted as recurrent inhibition, the corresponding linear equation assumes a form more general than the well known Hartline-Ratliff equation. Finally, we introduce a class of impedance operators based on the probabilistic theory of Markov processes, solve the corresponding linear integral equation, and apply the theoretical properties of the solution to the analysis of physiological and psychophysical phenomena.     

Hierarchical semantic composition of biosimulation models using bond graphs

Simulating complex biological and physiological systems and predicting their behaviours under different conditions remains challenging. Breaking systems into smaller and more manageable modules can address this challenge, assisting both model development and simulation. Nevertheless, existing computational models in biology and physiology are often not modular and therefore difficult to assemble into larger models. Even when this is possible, the resulting model may not be useful due to inconsistencies either with the laws of physics or the physiological behaviour of the system. Here, we propose a general methodology for composing models, combining the energy-based bond graph approach with semantics-based annotations. This approach improves model composition and ensures that a composite model is physically plausible. As an example, we demonstrate this approach to automated model composition using a model of human arterial circulation. The major benefit is that modellers can spend more time on understanding the behaviour of complex biological and physiological systems and less time wrangling with model composition.     

The optimality of the standard genetic code assessed by an eight-objective evolutionary algorithm

Background

The standard genetic code (SGC) is a unique set of rules which assign amino acids to codons. Similar amino acids tend to have similar codons indicating that the code evolved to minimize the costs of amino acid replacements in proteins, caused by mutations or translational errors. However, if such optimization in fact occurred, many different properties of amino acids must have been taken into account during the code evolution. Therefore, this problem can be reformulated as a multi-objective optimization task, in which the selection constraints are represented by measures based on various amino acid properties.

Results

To study the optimality of the SGC we applied a multi-objective evolutionary algorithm and we used the representatives of eight clusters, which grouped over 500 indices describing various physicochemical properties of amino acids. Thanks to that we avoided an arbitrary choice of amino acid features as optimization criteria. As a consequence, we were able to conduct a more general study on the properties of the SGC than the ones presented so far in other papers on this topic. We considered two models of the genetic code, one preserving the characteristic codon blocks structure of the SGC and the other without this restriction. The results revealed that the SGC could be significantly improved in terms of error minimization, hereby it is not fully optimized. Its structure differs significantly from the structure of the codes optimized to minimize the costs of amino acid replacements. On the other hand, using newly defined quality measures that placed the SGC in the global space of theoretical genetic codes, we showed that the SGC is definitely closer to the codes that minimize the costs of amino acids replacements than those maximizing them.

Conclusions

The standard genetic code represents most likely only partially optimized systems, which emerged under the influence of many different factors. Our findings can be useful to researchers involved in modifying the genetic code of the living organisms and designing artificial ones.

     

Time hierarchy in enzymatic reaction chains resulting from optimality principles

Several optimality principles reasonable for the evolution of enzyme reaction chains are formulated, considering the kinetic parameters of the enzymes to be variables. The solutions of these parametric programming problems are studied for the case of linear kinetics and turn out to be not unique in every case. The investigations are confined to stationary states. The net flux through the chain is considered a fixed parameter. The optimality criteria concern the osmotic pressure caused by the intermediates, various relaxation times, largest time scale, and controllability. They all yield distinct time hierarchies. The influence of a constraint concerning the sum of intermediate concentrations is studied. Various combinations of the single criteria are treated as multiobjective programming problems.     

Dissipative structures and amplification of enantiomeric excess (an experimental point of view)

Various mathematical models have been proposed to account for the origin of chiral molecules in biological systems. Most of these models invoke non-linear phenomena, and are based on the general concept of dissipative structures. These theoretical models define the fundamental criteria which must be obeyed by the experimental systems that we have investigated. Our initial approach to this problem was an extensive search of the literature data in order to select a few systems or experimental situations which would satisfy the criteria defined by the theoretical models. For these reasons, we carried out a study of the possibility of stereospecific autocatalysis in the asymmetric polymerisation of benzofuran. Similarly, the formation of spatial dissipative structures by coupling of a transport process with an interfacial reaction was investigated as a simple experimental example of symmetry breaking.     

How to help students understand physiology? Emphasize general models

Students generally approach topics in physiology as a series of unrelated phenomena that share few underlying principles. In many students' view, the Fick equation for cardiac output is fundamentally different from a renal clearance equation. If, however, students recognize that these apparently different situations can be viewed as examples of the same general conceptual model (e.g., conservation of mass), they may gain a more unified understanding of physiological systems. An understanding of as few as seven general models can provide students with an initial conceptual framework for analyzing most physiological systems. The general models deal with control systems, conservation of mass, mass and heat flow, elastic properties of tissues, transport across membranes, cell-to-cell communication, and molecular interaction.     

Mathematical properties of pump-leak models of cell volume control and electrolyte balance

Homeostatic control of cell volume and intracellular electrolyte content is a fundamental problem in physiology and is central to the functioning of epithelial systems. These physiological processes are modeled using pump-leak models, a system of differential algebraic equations that describes the balance of ions and water flowing across the cell membrane. Despite their widespread use, very little is known about their mathematical properties. Here, we establish analytical results on the existence and stability of steady states for a general class of pump-leak models. We treat two cases. When the ion channel currents have a linear current-voltage relationship, we show that there is at most one steady state, and that the steady state is globally asymptotically stable. If there are no steady states, the cell volume tends to infinity with time. When minimal assumptions are placed on the properties of ion channel currents, we show that there is an asymptotically stable steady state so long as the pump current is not too large. The key analytical tool is a free energy relation satisfied by a general class of pump-leak models, which can be used as a Lyapunov function to study stability.     

Erratum to: Mathematical properties of pump-leak models of cell volume control and electrolyte balance

Homeostatic control of cell volume and intracellular electrolyte content is a fundamental problem in physiology and is central to the functioning of epithelial systems. These physiological processes are modeled using pump-leak models, a system of differential algebraic equations that describes the balance of ions and water flowing across the cell membrane. Despite their widespread use, very little is known about their mathematical properties. Here, we establish analytical results on the existence and stability of steady states for a general class of pump-leak models. We treat two cases. When the ion channel currents have a linear current-voltage relationship, we show that there is at most one steady state, and that the steady state is globally asymptotically stable. If there are no steady states, the cell volume tends to infinity with time. When minimal assumptions are placed on the properties of ion channel currents, we show that there is an asymptotically stable steady state so long as the pump current is not too large. The key analytical tool is a free energy relation satisfied by a general class of pump-leak models, which can be used as a Lyapunov function to study stability.     

Necessary and sufficient conditions for the existence of linear invariants in phylogenetic inference.

The necessary and sufficient conditions for the existence of linear invariants under semigroups of probability transition matrices are derived. It is found that a biologically meaningful nucleotide substitution model has linear invariants if and only if it is a submodel of one of the three most general models, which include the so-called balanced and unbalanced transversion models. Each of these three general models is a nucleotide substitution model with six parameters.     

Elemente einer systemtheoretischen Betrachtung des Stoffwechsels

The discussion of general formal properties of metabolizing systems reveals the justification of applying tentatively General Systems Theory as a most general background. Whereas set-theoretical and graph-theoretical considerations, because of their very abstract nature, are obviously not well suited, the application of the theory of compartmented systems, elaborated on the basis of network or linear systems theories, seems to be useful. In metabolic studies physical models in the sense of computers operating as simulators are indispensable tools for research. A comparison between cybernetic and kinetic models demonstrates the many links and similarities rather than contrasts existing between them. Some epistemological problems are also included in the extended discussion of the concepts of information and energy, basical for these models.     

Aggregation error in nonlinear ecological models

Conditions are presented for accurately representing the dynamics of several components of a nonlinear ecological system by a single state variable. For mass-balance ecosystem models, only two conditions exist that permit such aggregation without introducing error into the model. Under the first condition, components can be lumped if their environmental losses (e.g. respiration or migration) are characterized by identical linear functions. Under the second condition, the components must maintain constant proportionality throughout their transient behavior. If these conditions are violated, it is not possible to aggregate without error. However, it is shown that aggregation error can still be minimized. The theorems developed here extend related results derived for the general theory of systems and have the advantage of generality, being applicable to nonlinear models of considerable complexity.     

The impact of including tRNA content on the optimality of the genetic code

Statistical and biochemical studies have revealed nonrandom patterns in codon assignments. The canonical genetic code is known to be highly efficient in minimizing the effects of mistranslational errors and point mutations, since it is known that, when an amino acid is converted to another due to error, the biochemical properties of the resulted amino acid are usually very similar to those of the original one. In this study, we have taken into consideration both relative frequencies of amino acids and relative gene copy frequencies of tRNAs in genomic sequences in order to introduce a fitness function which models the mistranslational probabilities more accurately in modern organisms. The relative gene copy frequencies of tRNAs are used as estimates of the tRNA content. We also altered the rule previously used for the calculation of the probabilities of single base mutation occurrences. Our model signifies higher optimality of the genetic code towards load minimization and suggests the presence of a coevolution of tRNA frequency and the genetic code.     

From annotated genomes to metabolic flux models and kinetic parameter fitting

Significant advances in system-level modeling of cellular behavior can be achieved based on constraints derived from genomic information and on optimality hypotheses. For steady-state models of metabolic networks, mass conservation and reaction stoichiometry impose linear constraints on metabolic fluxes. Different objectives, such as maximization of growth rate or minimization of flux distance from a reference state, can be tested in different organisms and conditions. In particular, we have suggested that the metabolic properties of mutant bacterial strains are best described by an algorithm that performs a minimization of metabolic adjustment (MOMA) upon gene deletion. The increasing availability of many annotated genomes paves the way for a systematic application of these flux balance methods to a large variety of organisms. However, such a high throughput goal crucially depends on our capacity to build metabolic flux models in a fully automated fashion. Here we describe a pipeline for generating models from annotated genomes and discuss the current obstacles to full automation. In addition, we propose a framework for the integration of flux modeling results and high throughput proteomic data, which can potentially help in the inference of whole-cell kinetic parameters.     

Kinetic Mechanisms of Biological Regulation in Photosynthetic Organisms

Principles of regulation on different levels of photosynthetic apparatus are discussed. Mathematical models of isolated photosynthetic reaction centers and general system of energy transduction in chloroplast are developed. A general approach to model these complex metabolic systems is suggested. Regulatory mechanisms in plant cell are correlated with the different patterns of fluorescence induction curve at different internal physiological states of the cells and external (environmental) conditions. Light regulation inside photosynthetic reaction centers, diffusion processes in thylakoid membrane, generation of transmembrane electrochemical potential, coupling with processes of CO₂ fixation in Calvin Cycle are considered as stages of control of energy transformation in chloroplasts in their connection with kinetic patterns of fluorescence induction curves and other spectrophotometric data.     

Floquet theory: a useful tool for understanding nonequilibrium dynamics

Many ecological systems experience periodic variability. Theoretical investigation of population and community dynamics in periodic environments has been hampered by the lack of mathematical tools relative to equilibrium systems. Here, I describe one such mathematical tool that has been rarely used in the ecological literature but has widespread use: Floquet theory. Floquet theory is the study of the stability of linear periodic systems in continuous time. Floquet exponents/multipliers are analogous to the eigenvalues of Jacobian matrices of equilibrium points. In this paper, I describe the general theory, then give examples to illustrate some of its uses: it defines fitness of structured populations, it can be used for invasion criteria in models of competition, and it can test the stability of limit cycle solutions. I also provide computer code to calculate Floquet exponents and multipliers. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

A nonlinear elastic description of cell preferential orientations over a stretched substrate

The active response of cells to mechanical cues due to their interaction with the environment has been of increasing interest, since it is involved in many physiological phenomena, pathologies, and in tissue engineering. In particular, several experiments have shown that, if a substrate with overlying cells is cyclically stretched, they will reorient to reach a well-defined angle between their major axis and the main stretching direction. Recent experimental findings, also supported by a linear elastic model, indicated that the minimization of an elastic energy might drive this reorientation process. Motivated by the fact that a similar behaviour is observed even for high strains, in this paper we address the problem in the framework of finite elasticity, in order to study the presence of nonlinear effects. We find that, for a very large class of constitutive orthotropic models and with very general assumptions, there is a single linear relationship between a parameter describing the biaxial deformation and \(\cos ^2\theta _{\mathrm{eq}}\), where \(\theta _{\mathrm{eq}}\) is the orientation angle of the cell, with the slope of the line depending on a specific combination of four parameters that characterize the nonlinear constitutive equation. We also study the effect of introducing a further dependence of the energy on the anisotropic invariants related to the square of the Cauchy–Green strain tensor. This leads to departures from the linear relationship mentioned above, that are again critically compared with experimental data.

     

Evolution of clutch size in insects. I. A review of static optimality models

Abstract This paper reviews the importance of constraint assumptions to the predictions of static optimality models of insect clutch size. This allows us to identify predictions that distinguish between models embodying different constraints on female oviposition behaviour and hence to determine which resources or other factors limit clutch size evolutionarily. We conclude that while some models may be distinguished using qualitative criteria, others require the testing of quantitative predictions. In a companion paper (Wilson 1994) these models are tested using the bruchid beetle Callosobruchus maculatus.     

On our best behavior: optimality models in human behavioral ecology

This paper discusses problems associated with the use of optimality models in human behavioral ecology. Optimality models are used in both human and non-human animal behavioral ecology to test hypotheses about the conditions generating and maintaining behavioral strategies in populations via natural selection. The way optimality models are currently used in behavioral ecology faces significant problems, which are exacerbated by employing the so-called ‘phenotypic gambit’: that is, the bet that the psychological and inheritance mechanisms responsible for behavioral strategies will be straightforward. I argue that each of several different possible ways we might interpret how optimality models are being used for humans face similar and additional problems. I suggest some ways in which human behavioral ecologists might adjust how they employ optimality models; in particular, I urge the abandonment of the phenotypic gambit in the human case.