Mathematical description of gene regulatory units

Revealing the control mechanisms responsible for the cell's surprisingly well-organized functions should lead directly to a better understanding of how the cell adapts to extraordinarily changing environments. A general framework for describing models that can represent diverse biochemical regulatory functions systematically would help not only systematic interpretation of the various models proposed for certain systems but also further understanding of the general control mechanism and design principles underlying different biological systems. This article presents a unified mathematical framework for describing gene regulatory units. The proposed framework is fairly compatible with the classical control theoretical framework, so it should serve as a connecting bridge between engineering control theory and biological control mechanisms. It should also provide a unified view of different regulatory units and facilitate systematic comparison of different mathematical models proposed in a variety of literature.     

Cardiac Rhythm Regulation in Children during the Initial Period of School Education under Conditions of the North

Cardiac rhythm parameters recorded in healthy children during the initial weeks of schoolwork immediately after the transition from kindergarten to the first grade of the elementary school or from the elementary school to the fifth grade of the middle school showed that their regulatory systems were stressed to different degrees, reflecting different costs of adaptation to environmental conditions. The most pronounced changes in these parameters were observed in the first graders: because of more significant stress (sometimes, overstress) of their regulatory mechanisms, in 70% of these children complete adaptation to new living conditions was associated with greater physiological costs. In the fifth graders, a similar situation was observed only in 25% of the children examined. The less pronounced physiological changes observed in 30% of the first graders and in 75% of the fifth graders can be related to the more mature state of their cardiac regulatory mechanisms, as well as to their greater adaptability to environmental conditions during the period examined.     

A quantitative and dynamic model for plant stem cell regulation

Plants maintain pools of totipotent stem cells throughout their entire life. These stem cells are embedded within specialized tissues called meristems, which form the growing points of the organism. The shoot apical meristem of the reference plant Arabidopsis thaliana is subdivided into several distinct domains, which execute diverse biological functions, such as tissue organization, cell-proliferation and differentiation. The number of cells required for growth and organ formation changes over the course of a plants life, while the structure of the meristem remains remarkably constant. Thus, regulatory systems must be in place, which allow for an adaptation of cell proliferation within the shoot apical meristem, while maintaining the organization at the tissue level. To advance our understanding of this dynamic tissue behavior, we measured domain sizes as well as cell division rates of the shoot apical meristem under various environmental conditions, which cause adaptations in meristem size. Based on our results we developed a mathematical model to explain the observed changes by a cell pool size dependent regulation of cell proliferation and differentiation, which is able to correctly predict CLV3 and WUS over-expression phenotypes. While the model shows stem cell homeostasis under constant growth conditions, it predicts a variation in stem cell number under changing conditions. Consistent with our experimental data this behavior is correlated with variations in cell proliferation. Therefore, we investigate different signaling mechanisms, which could stabilize stem cell number despite variations in cell proliferation. Our results shed light onto the dynamic constraints of stem cell pool maintenance in the shoot apical meristem of Arabidopsis in different environmental conditions and developmental states.     

A description of dynamical graphs associated to elementary regulatory circuits

The biological and dynamical importance of feedback circuits in regulatory graphs has often been emphasized. The work presented here aims at completely describing the dynamics of isolated elementary regulatory circuits. Our analytical approach is based on a discrete formal framework, built upon the logical approach of R. Thomas. Given a regulatory circuit, we show that the structure of synchronous and asynchronous dynamical graphs depends only on the length of the circuit (number of genes) and on its sign (which depends on the parity of the number of negative interactions). This work constitutes a first step towards the analytical characterisation of discrete dynamical graphs for more complex regulatory networks in terms of contributions corresponding to their embedded elementary circuits.     

Mathematical Models Light Up Plant Signaling

Plants respond to changes in the environment by triggering a suite of regulatory networks that control and synchronize molecular signaling in different tissues, organs, and the whole plant. Molecular studies through genetic and environmental perturbations, particularly in the model plant Arabidopsis thaliana, have revealed many of the mechanisms by which these responses are actuated. In recent years, mathematical modeling has become a complementary tool to the experimental approach that has furthered our understanding of biological mechanisms. In this review, we present modeling examples encompassing a range of different biological processes, in particular those regulated by light. Current issues and future directions in the modeling of plant systems are discussed.     

Problems of biochemical organization]

Biological organization has been defined as a unity of structure, function and regulation. Biological organization of hierarchical multilevel biological systems is represented by a hierarchy of functioning controllable structures. The hierarchy of levels of material organization predetermines the existence of a hierarchy of regulatory mechanisms. Biochemical organization involves the levels of material organization corresponding to biomacromolecules, supramolecular complexes and cellular organelles. The levels of biomacromolecules and supramolecular structures effectuating elementary functions and controlled by basic regulatory mechanisms occupy key positions in biological systems. These levels play the role of standard functional blocks; their combination leads to hierarchically higher structural levels (cell, tissue, organ, systems of organs, organism) performing more complex functions and controlled by hierarchically more important regulatory mechanisms. The peculiarities of regulation of biological systems that are due to the existence of a hierarchy of regulatory mechanisms are discussed.     

Analysis of Homeostatic Mechanisms in Biochemical Networks

Cell metabolism is an extremely complicated dynamical system that maintains important cellular functions despite large changes in inputs. This “homeostasis” does not mean that the dynamical system is rigid and fixed. Typically, large changes in external variables cause large changes in some internal variables so that, through various regulatory mechanisms, certain other internal variables (concentrations or velocities) remain approximately constant over a finite range of inputs. Outside that range, the mechanisms cease to function and concentrations change rapidly with changes in inputs. In this paper we analyze four different common biochemical homeostatic mechanisms: feedforward excitation, feedback inhibition, kinetic homeostasis, and parallel inhibition. We show that all four mechanisms can occur in a single biological network, using folate and methionine metabolism as an example. Golubitsky and Stewart have proposed a method to find homeostatic nodes in networks. We show that their method works for two of these mechanisms but not the other two. We discuss the many interesting mathematical and biological questions that emerge from this analysis, and we explain why understanding homeostatic control is crucial for precision medicine.     

Perturbation analysis analyzed—mathematical modeling of intact and perturbed gene regulatory circuits for animal development

Gene regulatory networks for animal development are the underlying mechanisms controlling cell fate specification and differentiation. The architecture of gene regulatory circuits determines their information processing properties and their developmental function. It is a major task to derive realistic network models from exceedingly advanced high throughput experimental data. Here we use mathematical modeling to study the dynamics of gene regulatory circuits to advance the ability to infer regulatory connections and logic function from experimental data. This study is guided by experimental methodologies that are commonly used to study gene regulatory networks that control cell fate specification. We study the effect of a perturbation of an input on the level of its downstream genes and compare between the cis-regulatory execution of OR and AND logics. Circuits that initiate gene activation and circuits that lock on the expression of genes are analyzed. The model improves our ability to analyze experimental data and construct from it the network topology. The model also illuminates information processing properties of gene regulatory circuits for animal development.     

Mutation is modulated: implications for evolution

Evolution occurs through genome variation followed by selection. Because DNA sequence context affects the activity of enzymes that copy, move and repair DNA, there are intrinsic variations in the probability of genetic variation along a genome. These intrinsic variations can be affected by selective pressure. Codon changes that do not alter the encoded amino acids may still have effects on the local rate of sequence change. Large gene families could encode a successful genetic framework by which to evolve new, functional members. The speed of adaptation to environmental challenges may be improved when the distinct mechanisms of genetic change come under regulatory control. Natural selection operates on mechanisms that generate and modulate diversity as it does on all biological functions.     

The logical repertoire of ligand-binding proteins

Proteins whose conformation can be altered by the equilibrium binding of a regulatory ligand are one of the main building blocks of signal-processing networks in cells. Typically, such proteins switch between an 'inactive' and an 'active' state, as the concentration of the regulator varies. We investigate the properties of proteins that can bind two different ligands and show that these proteins can individually act as logical elements: their 'output', quantified by their average level of activity, depends on the two 'inputs', the concentrations of both regulators. In the case where the two ligands can bind simultaneously, we show that all of the elementary logical functions can be implemented by appropriate tuning of the ligand-binding energies. If the ligands bind exclusively, the logical repertoire is more limited. When such proteins cluster together, cooperative interactions can greatly enhance the sharpness of the response. Protein clusters can therefore act as digital logical elements whose activity can be abruptly switched from fully inactive to fully active, as the concentrations of the regulators pass threshold values. We discuss a particular instance in which this type of protein logic appears to be used in signal transduction-the chemotaxis receptors of E. coli.     

A Network Characteristic That Correlates Environmental and Genetic Robustness

As scientific advances in perturbing biological systems and technological advances in data acquisition allow the large-scale quantitative analysis of biological function, the robustness of organisms to both transient environmental stresses and inter-generational genetic changes is a fundamental impediment to the identifiability of mathematical models of these functions. An approach to overcoming this impediment is to reduce the space of possible models to take into account both types of robustness. However, the relationship between the two is still controversial. This work uncovers a network characteristic, transient responsiveness, for a specific function that correlates environmental imperturbability and genetic robustness. We test this characteristic extensively for dynamic networks of ordinary differential equations ranging up to 30 interacting nodes and find that there is a power-law relating environmental imperturbability and genetic robustness that tends to linearity as the number of nodes increases. Using our methods, we refine the classification of known 3-node motifs in terms of their environmental and genetic robustness. We demonstrate our approach by applying it to the chemotaxis signaling network. In particular, we investigate plausible models for the role of CheV protein in biochemical adaptation via a phosphorylation pathway, testing modifications that could improve the robustness of the system to environmental and/or genetic perturbation.     

Biological and mathematical modeling of melanocyte development

We aim to evaluate environmental and genetic effects on the expansion/proliferation of committed single cells during embryonic development, using melanoblasts as a paradigm to model this phenomenon. Melanoblasts are a specific type of cell that display extensive cellular proliferation during development. However, the events controlling melanoblast expansion are still poorly understood due to insufficient knowledge concerning their number and distribution in the various skin compartments. We show that melanoblast expansion is tightly controlled both spatially and temporally, with little variation between embryos. We established a mathematical model reflecting the main cellular mechanisms involved in melanoblast expansion, including proliferation and migration from the dermis to epidermis. In association with biological information, the model allows the calculation of doubling times for melanoblasts, revealing that dermal and epidermal melanoblasts have short but different doubling times. Moreover, the number of trunk founder melanoblasts at E8.5 was estimated to be 16, a population impossible to count by classical biological approaches. We also assessed the importance of the genetic background by studying gain- and loss-of-function β-catenin mutants in the melanocyte lineage. We found that any alteration of β-catenin activity, whether positive or negative, reduced both dermal and epidermal melanoblast proliferation. Finally, we determined that the pool of dermal melanoblasts remains constant in wild-type and mutant embryos during development, implying that specific control mechanisms associated with cell division ensure half of the cells at each cell division to migrate from the dermis to the epidermis. Modeling melanoblast expansion revealed novel links between cell division, cell localization within the embryo and appropriate feedback control through β-catenin.     

Pathways database system

During the next phase of the Human Genome Project, research will focus on functional studies of attributing functions to genes, their regulatory elements, and other DNA sequences. To facilitate the use of genomic information in such studies, a new modeling perspective is needed to examine and study genome sequences in the context of many kinds of biological information. Pathways are the logical format for modeling and presenting such information in a manner that is familiar to biological researchers. In this paper, we introduce an integrated system, called "Pathways Database System," with a set of software tools for modeling, storing, analyzing, visualizing, and querying biological pathways data at different levels of genetic, molecular, biochemical and organismal detail.     

CircRNA: a rising star in plant biology

Circular RNAs (circRNAs) are covalently closed single-stranded RNA molecules, which are widespread in eukaryotic cells. As regulatory molecules, circRNAs have various functions, such as regulating gene expression, binding miRNAs or proteins, and being translated into proteins, which are important for cell proliferation and cell differentiation, individual growth and development, as well as many other biological processes. However, compared with that in animal models, studies of circRNAs in plants lags behind and, particularly, the regulatory mechanisms of biogenesis and molecular functions of plant circRNAs remain elusive. Recent studies have shown that circRNAs are wide spread in plants with tissue- or development-specific expression patterns and are responsive to a variety of environmental stresses. In this review, we summarize these advances, focusing on the regulatory mechanisms of biogenesis, molecular and biological functions of circRNAs, and the methods for investigating circRNAs. We also discuss the challenges and the prospects of plant circRNA studies.