A passivity-based stability criterion for a class of interconnected systems and applications to biochemical reaction networks

This paper presents a stability test for a class of interconnected nonlinear systems motivated by biochemical reaction networks. One of the main results determines global asymptotic stability of the network from the diagonal stability of a "dissipativity matrix" which incorporates information about the passivity properties of the subsystems, the interconnection structure of the network, and the signs of the interconnection terms. This stability test encompasses the "secant criterion" for cyclic networks presented in our previous paper, and extends it to a general interconnection structure represented by a graph. A second main result allows one to accommodate state products. This extension makes the new stability criterion applicable to a broader class of models, even in the case of cyclic systems. The new stability test is illustrated on a mitogen activated protein kinase (MAPK) cascade model, and on a branched interconnection structure motivated by metabolic networks. Finally, another result addresses the robustness of stability in the presence of diffusion terms in a compartmental system made out of identical systems.Comment: See http://www.math.rutgers.edu/~sontag/PUBDIR/index.html for related (p)reprint

Oscillations in I/O monotone systems under negative feedback

Oscillatory behavior is a key property of many biological systems. The Small-Gain Theorem (SGT) for input/output monotone systems provides a sufficient condition for global asymptotic stability of an equilibrium and hence its violation is a necessary condition for the existence of periodic solutions. One advantage of the use of the monotone SGT technique is its robustness with respect to all perturbations that preserve monotonicity and stability properties of a very low-dimensional (in many interesting examples, just one-dimensional) model reduction. This robustness makes the technique useful in the analysis of molecular biological models in which there is large uncertainty regarding the values of kinetic and other parameters. However, verifying the conditions needed in order to apply the SGT is not always easy. This paper provides an approach to the verification of the needed properties, and illustrates the approach through an application to a classical model of circadian oscillations, as a nontrivial ``case study,'' and also provides a theorem in the converse direction of predicting oscillations when the SGT conditions fail.Comment: Related work can be retrieved from second author's websit

Mathematical modeling of senescence in metabolic networks

Senescentie is een van de grootste mysteries die de mensheid probeert op te lossen. Senescentie manifesteert zich op meerdere niveaus van biologische organisatie: van cellen tot weefsels en hele organismen. Uiteindelijk zijn alle levensprocessen cruciaal afhankelijk van het metabolisme en de regulatie ervan. Daarom is het metabole netwerk het ultieme biologische netwerk dat senescentie ervaart. Het doel van dit proefschrift is om een ​​theoretische achtergrond te bieden voor het begrijpen van senescentie in metabole netwerken. Er worden twee benaderingen voor het modelleren van metabole netwerken gebruikt. In de eerste benadering modelleer ik de dynamica van metabolietenconcentraties. In metabole netwerken zonder regulatie is de dynamica van metabolietconcentraties eenvoudig, zelfs indien er sprake is van complexe topologie (inclusief vertakkingen, cycli). Namelijk, er vindt convergentie plaats naar een verzameling van stabiele toestanden. Daarentegen kan zelfs in eenvoudige metabole pathways de dynamica van metabolietconcentraties met positieve en negatieve feedback complex zijn (bijv. oscillaties, chaos). Metabole regulatie biedt enerzijds robuustheid voor verschillende verstoringen op een dagelijkse basis, maar kan anderzijdsverantwoordelijk zijn voor het ontstaan ​​van een complexe dynamica van metabolietconcentraties op latere leeftijden. In de tweede benadering modelleer ik de evolutie van de regeling van de metabolische flux. Twee strategieën om de metabolische flux te beheersen evolueren onder verschillende omstandigheden: snelheidsbeperkend (een of enkele enzymen beïnvloeden de flux) of verdeeld (meerdere enzymen beïnvloeden de flux). De aanwezigheid van senescentie in de vorm van afbraak van enzymactiviteit of metabolische regulatie beïnvloedt de evolutie van de fluxregeling. Algemene eigenschappen van metabole netwerken en hun robuustheid zijn met elkaar verbonden, en onderzoek naar hun wisselwerking is een stap in de richting van een beter begrip van het samenspel tussen metabolisme en senescentie

ENZYMES: Catalysis, Kinetics and Mechanisms

Onemarvelsattheintricate designoflivingsystems,andwecannotbutwonderhow life originated on this planet. Whether ?rst biological structures emerged as the selfreproducing genetic templates (genetics-?rst origin of life) or the metabolic universality preceded the genome and eventually integrated it (metabolism-?rst origin of life) is still a matter of hot scienti?c debate. There is growing acceptance that the RNA world came ?rst – as RNA molecules can perform both the functions of information storage and catalysis. Regardless of which view eventually gains acceptance, emergence of catalytic phenomena is at the core of biology. The last century has seen an explosive growth in our understanding of biological systems. The progression has involved successive emphasis on taxonomy ! physiology ! biochemistry ! molecular biology ! genetic engineering and ?nally the large-scale study of genomes. The ?eld of molecular biology became largely synonymous with the study of DNA – the genetic material. Molecular biology however had its beginnings in the understanding of biomolecular structure and function. Appreciationofproteins,catalyticphenomena,andthefunctionofenzymeshadalargeroleto play in the progress of modern biology

The Interplay between Chemistry and Mechanics in the Transduction of a Mechanical Signal into a Biochemical Function

There are many processes in biology in which mechanical forces are generated. Force-bearing networks can transduce locally developed mechanical signals very extensively over different parts of the cell or tissues. In this article we conduct an overview of this kind of mechanical transduction, focusing in particular on the multiple layers of complexity displayed by the mechanisms that control and trigger the conversion of a mechanical signal into a biochemical function. Single molecule methodologies, through their capability to introduce the force in studies of biological processes in which mechanical stresses are developed, are unveiling subtle intertwining mechanisms between chemistry and mechanics and in particular are revealing how chemistry can control mechanics. The possibility that chemistry interplays with mechanics should be always considered in biochemical studies.Comment: 50 pages, 18 figure

Systems biology approaches to the dynamics of gene expression and chemical reactions

Systems biology is an emergent interdisciplinary field of study whose main goal is to understand the global properties and functions of a biological system by investigating its structure and dynamics [74]. This high-level knowledge can be reached only with a coordinated approach involving researchers with different backgrounds in molecular biology, the various omics (like genomics, proteomics, metabolomics), computer science and dynamical systems theory. The history of systems biology as a distinct discipline began in the 1960s, and saw an impressive growth since year 2000, originated by the increased accumulation of biological information, the development of high-throughput experimental techniques, the use of powerful computer systems for calculations and database hosting, and the spread of Internet as the standard medium for information diffusion [77]. In the last few years, our research group tried to tackle a set of systems biology problems which look quite diverse, but share some topics like biological networks and system dynamics, which are of our interest and clearly fundamental for this field. In fact, the first issue we studied (covered in Part I) was the reverse engineering of large-scale gene regulatory networks. Inferring a gene network is the process of identifying interactions among genes from experimental data (tipically microarray expression profiles) using computational methods [6]. Our aim was to compare some of the most popular association network algorithms (the only ones applicable at a genome-wide level) in different conditions. In particular we verified the predictive power of similarity measures both of direct type (like correlations and mutual information) and of conditional type (partial correlations and conditional mutual information) applied on different kinds of experiments (like data taken at equilibrium or time courses) and on both synthetic and real microarray data (for E. coli and S. cerevisiae). In our simulations we saw that all network inference algorithms obtain better performances from data produced with \u201cstructural\u201d perturbations (like gene knockouts at steady state) than with just dynamical perturbations (like time course measurements or changes of the initial expression levels). Moreover, our analysis showed differences in the performances of the algorithms: direct methods are more robust in detecting stable relationships (like belonging to the same protein complex), while conditional methods are better at causal interactions (e.g. transcription factor\u2013binding site interactions), especially in presence of combinatorial transcriptional regulation. Even if time course microarray experiments are not particularly useful for inferring gene networks, they can instead give a great amount of information about the dynamical evolution of a biological process, provided that the measurements have a good time resolution. Recently, such a dataset has been published [119] for the yeast metabolic cycle, a well-known process where yeast cells synchronize with respect to oxidative and reductive functions. In that paper, the long-period respiratory oscillations were shown to be reflected in genome-wide periodic patterns in gene expression. As explained in Part II, we analyzed these time series in order to elucidate the dynamical role of post-transcriptional regulation (in particular mRNA stability) in the coordination of the cycle. We found that for periodic genes, arranged in classes according either to expression profile or to function, the pulses of mRNA abundance have phase and width which are directly proportional to the corresponding turnover rates. Moreover, the cascade of events which occurs during the yeast metabolic cycle (and their correlation with mRNA turnover) reflects to a large extent the gene expression program observable in other dynamical contexts such as the response to stresses or stimuli. The concepts of network and of systems dynamics return also as major arguments of Part III. In fact, there we present a study of some dynamical properties of the so-called chemical reaction networks, which are sets of chemical species among which a certain number of reactions can occur. These networks can be modeled as systems of ordinary differential equations for the species concentrations, and the dynamical evolution of these systems has been theoretically studied since the 1970s [47, 65]. Over time, several independent conditions have been proved concerning the capacity of a reaction network, regardless of the (often poorly known) reaction parameters, to exhibit multiple equilibria. This is a particularly interesting characteristic for biological systems, since it is required for the switch-like behavior observed during processes like intracellular signaling and cell differentiation. Inspired by those works, we developed a new open source software package for MATLAB, called ERNEST, which, by checking these various criteria on the structure of a chemical reaction network, can exclude the multistationarity of the corresponding reaction system. The results of this analysis can be used, for example, for model discrimination: if for a multistable biological process there are multiple candidate reaction models, it is possible to eliminate some of them by proving that they are always monostationary. Finally, we considered the related property of monotonicity for a reaction network. Monotone dynamical systems have the tendency to converge to an equilibrium and do not present chaotic behaviors. Most biological systems have the same features, and are therefore considered to be monotone or near-monotone [85, 116]. Using the notion of fundamental cycles from graph theory, we proved some theoretical results in order to determine how distant is a given biological network from being monotone. In particular, we showed that the distance to monotonicity of a network is equal to the minimal number of negative fundamental cycles of the corresponding J-graph, a signed multigraph which can be univocally associated to a dynamical system