Dynamical properties of a gene-protein model

A major limitation of the classical random Boolean network model of gene regulatory networks is its synchronous updating, which implies that all the proteins decay at the same rate. Here a model is discussed, where the network is composed of two different sets of nodes, labelled G and P with reference to “genes” and “proteins”. Each gene corresponds to a protein (the one it codes for), while several proteins can simultaneously affect the expression of a gene. Both kinds of nodes take Boolean values. If we look at the genes only, it is like adding some memory terms, so the new state of the gene subnetwork network does no longer depend upon its previous state only. In general, these terms tend to make the dynamics of the network more ordered than that of the corresponding memoryless network. The analysis is focused here mostly on dynamical critical states. It has been shown elsewhere that the usual way of computing the Derrida parameter, starting from purely random initial conditions, can be misleading in strongly non-ergodic systems. So here the effects of perturbations on both genes’ and proteins’ levels is analysed, using both the canonical Derrida procedure and an “extended” one. The results are discussed. Moreover, the stability of attractors is also analysed, measured by counting the fraction of perturbations where the system eventually falls back onto the initial attractor

Mechanisms of gap gene expression canalization in the Drosophila blastoderm

<p>Abstract</p> <p>Background</p> <p>Extensive variation in early gap gene expression in the <it>Drosophila </it>blastoderm is reduced over time because of gap gene cross regulation. This phenomenon is a manifestation of canalization, the ability of an organism to produce a consistent phenotype despite variations in genotype or environment. The canalization of gap gene expression can be understood as arising from the actions of attractors in the gap gene dynamical system.</p> <p>Results</p> <p>In order to better understand the processes of developmental robustness and canalization in the early <it>Drosophila </it>embryo, we investigated the dynamical effects of varying spatial profiles of Bicoid protein concentration on the formation of the expression border of the gap gene <it>hunchback</it>. At several positions on the anterior-posterior axis of the embryo, we analyzed attractors and their basins of attraction in a dynamical model describing expression of four gap genes with the Bicoid concentration profile accounted as a given input in the model equations. This model was tested against a family of Bicoid gradients obtained from individual embryos. These gradients were normalized by two independent methods, which are based on distinct biological hypotheses and provide different magnitudes for Bicoid spatial variability. We showed how the border formation is dictated by the biological initial conditions (the concentration gradient of maternal Hunchback protein) being attracted to specific attracting sets in a local vicinity of the border. Different types of these attracting sets (point attractors or one dimensional attracting manifolds) define several possible mechanisms of border formation. The <it>hunchback </it>border formation is associated with intersection of the spatial gradient of the maternal Hunchback protein and a boundary between the attraction basins of two different point attractors. We demonstrated how the positional variability for <it>hunchback </it>is related to the corresponding variability of the basin boundaries. The observed reduction in variability of the <it>hunchback </it>gene expression can be accounted for by specific geometrical properties of the basin boundaries.</p> <p>Conclusion</p> <p>We clarified the mechanisms of gap gene expression canalization in early <it>Drosophila </it>embryos. These mechanisms were specified in the case of <it>hunchback </it>in well defined terms of the dynamical system theory.</p

Integrating heterogeneous knowledges for understanding biological behaviors: a probabilistic approach

Despite recent molecular technique improvements, biological knowledge remains incomplete. Reasoning on living systems hence implies to integrate heterogeneous and partial informations. Although current investigations successfully focus on qualitative behaviors of macromolecular networks, others approaches show partial quantitative informations like protein concentration variations over times. We consider that both informations, qualitative and quantitative, have to be combined into a modeling method to provide a better understanding of the biological system. We propose here such a method using a probabilistic-like approach. After its exhaustive description, we illustrate its advantages by modeling the carbon starvation response in Escherichia coli. In this purpose, we build an original qualitative model based on available observations. After the formal verification of its qualitative properties, the probabilistic model shows quantitative results corresponding to biological expectations which confirm the interest of our probabilistic approach.Comment: 10 page

Homogeneous and Scalable Gene Expression Regulatory Networks with Random Layouts of Switching Parameters

We consider a model of large regulatory gene expression networks where the thresholds activating the sigmoidal interactions between genes and the signs of these interactions are shuffled randomly. Such an approach allows for a qualitative understanding of network dynamics in a lack of empirical data concerning the large genomes of living organisms. Local dynamics of network nodes exhibits the multistationarity and oscillations and depends crucially upon the global topology of a "maximal" graph (comprising of all possible interactions between genes in the network). The long time behavior observed in the network defined on the homogeneous "maximal" graphs is featured by the fraction of positive interactions ($0\leq \eta\leq 1$) allowed between genes. There exists a critical value $\eta_c<1$ such that if $\eta<\eta_c$, the oscillations persist in the system, otherwise, when $\eta>\eta_c,$ it tends to a fixed point (which position in the phase space is determined by the initial conditions and the certain layout of switching parameters). In networks defined on the inhomogeneous directed graphs depleted in cycles, no oscillations arise in the system even if the negative interactions in between genes present therein in abundance ($\eta_c=0$). For such networks, the bidirectional edges (if occur) influence on the dynamics essentially. In particular, if a number of edges in the "maximal" graph is bidirectional, oscillations can arise and persist in the system at any low rate of negative interactions between genes ($\eta_c=1$). Local dynamics observed in the inhomogeneous scalable regulatory networks is less sensitive to the choice of initial conditions. The scale free networks demonstrate their high error tolerance.Comment: LaTeX, 30 pages, 20 picture

Boolean network model predicts cell cycle sequence of fission yeast

A Boolean network model of the cell-cycle regulatory network of fission yeast (Schizosaccharomyces Pombe) is constructed solely on the basis of the known biochemical interaction topology. Simulating the model in the computer, faithfully reproduces the known sequence of regulatory activity patterns along the cell cycle of the living cell. Contrary to existing differential equation models, no parameters enter the model except the structure of the regulatory circuitry. The dynamical properties of the model indicate that the biological dynamical sequence is robustly implemented in the regulatory network, with the biological stationary state G1 corresponding to the dominant attractor in state space, and with the biological regulatory sequence being a strongly attractive trajectory. Comparing the fission yeast cell-cycle model to a similar model of the corresponding network in S. cerevisiae, a remarkable difference in circuitry, as well as dynamics is observed. While the latter operates in a strongly damped mode, driven by external excitation, the S. pombe network represents an auto-excited system with external damping.Comment: 10 pages, 3 figure

Growth-rate-dependent dynamics of a bacterial genetic oscillator

Gene networks exhibiting oscillatory dynamics are widespread in biology. The minimal regulatory designs giving rise to oscillations have been implemented synthetically and studied by mathematical modeling. However, most of the available analyses generally neglect the coupling of regulatory circuits with the cellular "chassis" in which the circuits are embedded. For example, the intracellular macromolecular composition of fast-growing bacteria changes with growth rate. As a consequence, important parameters of gene expression, such as ribosome concentration or cell volume, are growth-rate dependent, ultimately coupling the dynamics of genetic circuits with cell physiology. This work addresses the effects of growth rate on the dynamics of a paradigmatic example of genetic oscillator, the repressilator. Making use of empirical growth-rate dependences of parameters in bacteria, we show that the repressilator dynamics can switch between oscillations and convergence to a fixed point depending on the cellular state of growth, and thus on the nutrients it is fed. The physical support of the circuit (type of plasmid or gene positions on the chromosome) also plays an important role in determining the oscillation stability and the growth-rate dependence of period and amplitude. This analysis has potential application in the field of synthetic biology, and suggests that the coupling between endogenous genetic oscillators and cell physiology can have substantial consequences for their functionality.Comment: 14 pages, 9 figures (revised version, accepted for publication

COMPUTER SIMULATION AND COMPUTABILITY OF BIOLOGICAL SYSTEMS

The ability to simulate a biological organism by employing a computer is related to the ability of the computer to calculate the behavior of such a dynamical system, or the "computability" of the system.* However, the two questions of computability and simulation are not equivalent. Since the question of computability can be given a precise answer in terms of recursive functions, automata theory and dynamical systems, it will be appropriate to consider it first. The more elusive question of adequate simulation of biological systems by a computer will be then addressed and a possible connection between the two answers given will be considered. A conjecture is formulated that suggests the possibility of employing an algebraic-topological, "quantum" computer (Baianu, 1971b) for analogous and symbolic simulations of biological systems that may include chaotic processes that are not, in genral, either recursively or digitally computable. Depending on the biological network being modelled, such as the Human Genome/Cell Interactome or a trillion-cell Cognitive Neural Network system, the appropriate logical structure for such simulations might be either the Quantum MV-Logic (QMV) discussed in recent publications (Chiara, 2004, and references cited therein)or Lukasiewicz Logic Algebras that were shown to be isomorphic to MV-logic algebras (Georgescu et al, 2001)

Small RNAs Establish Delays and Temporal Thresholds in Gene Expression

Non-coding RNAs are crucial regulators of gene expression in prokaryotes and eukaryotes, but it remains poorly understood how they affect the dynamics of transcriptional networks. We analyzed the temporal characteristics of the cyanobacterial iron stress response by mathematical modeling and quantitative experimental analyses, and focused on the role of a recently discovered small non-coding RNA, IsrR. We found that IsrR is responsible for a pronounced delay in the accumulation of isiA mRNA encoding the late-phase stress protein, IsiA, and that it ensures a rapid decline in isiA levels once external stress triggers are removed. These kinetic properties allow the system to selectively respond to sustained (as opposed to transient) stimuli, and thus establish a temporal threshold, which prevents energetically costly IsiA accumulation under short-term stress conditions. Biological information is frequently encoded in the quantitative aspects of intracellular signals (e.g., amplitude and duration). Our simulations reveal that competitive inhibition and regulated degradation allow intracellular regulatory networks to efficiently discriminate between transient and sustained inputs