Impact of Negative Feedbacks on De Novo Pyrimidines Biosynthesis in Escherichia coli.

Earlier studies aimed at investigating the metabolism of endogenous nucleoside triphosphates in synchronous cultures of E. coli cells revealed an auto-oscillatory mode of functioning of the pyrimidine and purine nucleotide biosynthesis system, which the authors associated with the dynamics of cell division. Theoretically, this system has an intrinsic oscillatory potential, since the dynamics of its functioning are controlled through feedback mechanisms. The question of whether the nucleotide biosynthesis system has its own oscillatory circuit is still open. To address this issue, an integral mathematical model of pyrimidine biosynthesis was developed, taking into account all experimentally verified negative feedback in the regulation of enzymatic reactions, the data of which were obtained under in vitro conditions. Analysis of the dynamic modes of the model functioning has shown that in the pyrimidine biosynthesis system, both the steady-state and oscillatory functioning modes can be realized under certain sets of kinetic parameters that fit in the physiological boundaries of the investigated metabolic system. It has been demonstrated that the occurrence of the oscillatory nature of metabolite synthesis depended on the ratio of two parameters: the Hill coefficient, hUMP1-the nonlinearity of the UMP effect on the activity of carbamoyl-phosphate synthetase, and the parameter r characterizing the contribution of the noncompetitive mechanism of UTP inhibition to the regulation of the enzymatic reaction of UMP phosphorylation. Thus, it has been theoretically shown that the E. coli pyrimidine biosynthesis system possesses its own oscillatory circuit whose oscillatory potential depends to a significant degree on the mechanism of regulation of UMP kinase activity.

Evolution and extinction can occur rapidly: a modeling approach.

Fossil record of Earth describing the last 500 million years is characterized by evolution discontinuity as well as recurring global extinctions of some species and their replacement by new types, the causes of which are still debate. We developed a model of evolutionary self-development of a large ecosystem. This model of biota evolution based on the universal laws of living systems functioning: reproduction, dependence of reproduction efficiency and mortality on biota density, mutational variability in the process of reproduction and selection of the most adapted individuals. We have shown that global extinctions and phases of rapid growth and biodiversity stasis can be a reflection of the emergence of bistability in a self-organizing system, which is the Earth's biota. Bistability was found to be characteristic only for ecosystems with predominant sexual reproduction. The reason for the transition from one state to another is the selection of the most adapted individuals. That is, we explain the characteristics of the Earth's fossil record during the last 500 million years by the internal laws of Earth's ecosystem functioning, which appeared at a certain stage of evolution as a result of the emergence of life forms with an increased adaptive diversification associated with sexual dimorphism.

On the dynamical aspects of local translation at the activated synapse.

BACKGROUND: The key role in the dynamic regulation of synaptic protein turnover belongs to the Fragile X Mental Retardation Protein, which regulates the efficiency of dendritic mRNA translation in response to stimulation of metabotropic glutamate receptors at excitatory synapses of the hippocampal pyramidal cells. Its activity is regulated via positive and negative regulatory loops that function in different time ranges, which is an absolute factor for the formation of chaotic regimes that lead to disrupted proteome stability. The indicated condition may cause a number of neuropsychiatric diseases, including autism and epilepsy. The present study is devoted to a theoretical analysis of the local translation system dynamic properties and identification of parameters affecting the chaotic potential of the system. RESULTS: A mathematical model that describes the maintenance of a specific pool of active receptors on the postsynaptic membrane via two mechanisms - de novo synthesis of receptor proteins and restoration of protein function during the recycling process - has been developed. Analysis of the model revealed that an increase in the values of the parameters describing the impact of protein recycling on the maintenance of a pool of active receptors in the membrane, duration of the signal transduction via the mammalian target of rapamycin pathway, influence of receptors on the translation activation, as well as reduction of the rate of synthesis and integration of de novo synthesized proteins into the postsynaptic membrane - contribute to the reduced complexity of the local translation system dynamic state. Formation of these patterns significantly depends on the complexity and non-linearity of the mechanisms of exposure of de novo synthesized receptors to the postsynaptic membrane, the correct evaluation of which is currently problematic. CONCLUSIONS: The model predicts that an increase of "receptor recycling" and reduction of the rate of synthesis and integration of de novo synthesized proteins into the postsynaptic membrane contribute to the reduced complexity of the local translation system dynamic state. Herewith, stable stationary states occur much less frequently than cyclic states. It is possible that cyclical nature of functioning of the local translation system is its "normal" dynamic state.

Limit cycles in models of circular gene networks regulated by negative feedback loops.

BACKGROUND: The regulatory feedback loops that present in structural and functional organization of molecular-genetic systems and the phenomenon of the regulatory signal delay, a time period between the moment of signal reception and its implementation, provide natural conditions for complicated dynamic regimes in these systems. The delay phenomenon at the intracellular level is a consequence of the matrix principle of data transmission, implemented through the rather complex processes of transcription and translation.However, the rules of the influence of system structure on system dynamics are not clearly understood. Knowledge of these rules is particularly important for construction of synthetic gene networks with predetermined properties. RESULTS: We study dynamical properties of models of simplest circular gene networks regulated by negative feedback mechanisms. We have shown existence and stability of oscillating trajectories (cycles) in these models. Two algorithms of construction and localization of these cycles have been proposed. For one of these models, we have solved an inverse problem of parameters identification. CONCLUSIONS: The modeling results demonstrate that non-stationary dynamics in the models of circular gene networks with negative feedback loops is achieved by a high degree of non-linearity of the mechanism of the autorepressor influence on its own expression, by the presence of regulatory signal delay, the value of which must exceed a certain critical value, and transcription/translation should be initiated from a sufficiently strong promoter/Shine-Dalgarno site. We believe that the identified patterns are key elements of the oscillating construction design.

Chaos and hyperchaos in simple gene network with negative feedback and time delays.

Today there are examples that prove the existence of chaotic dynamics at all levels of organization of living systems, except intracellular, although such a possibility has been theoretically predicted. The lack of experimental evidence of chaos generation at the intracellular level in vivo may indicate that during evolution the cell got rid of chaos. This work allows the hypothesis that one of the possible mechanisms for avoiding chaos in gene networks can be a negative evolutionary selection, which prevents fixation or realization of regulatory circuits, creating too mild, from the biological point of view, conditions for the emergence of chaos. It has been shown that one of such circuits may be a combination of negative autoregulation of expression of transcription factors at the level of their synthesis and degradation. The presence of such a circuit results in formation of multiple branches of chaotic solutions as well as formation of hyperchaos with equal and sufficiently low values of the delayed argument that can be implemented not only in eukaryotic, but in prokaryotic cells.

Chaos and Hyperchaos in a Model of Ribosome Autocatalytic Synthesis.

Any vital activities of the cell are based on the ribosomes, which not only provide the basic machinery for the synthesis of all proteins necessary for cell functioning during growth and division, but for biogenesis itself. From this point of view, ribosomes are self-replicating and autocatalytic structures. In current work we present an elementary model in which the autocatalytic synthesis of ribosomal RNA and proteins, as well as enzymes ensuring their degradation are described with two monotonically increasing functions. For certain parameter values, the model, consisting of one differential equation with delayed argument, demonstrates both stationary and oscillatory dynamics of the ribosomal protein synthesis, which can be chaotic and hyperchaotic dependent on the value of the delayed argument. The biological interpretation of the modeling results and parameter estimation suggest the feasibility of chaotic dynamics in molecular genetic systems of eukaryotes, which depends only on the internal characteristics of functioning of the translation system.

On the control mechanisms of the nitrite level in Escherichia coli cells: the mathematical model.

BACKGROUND: Due to a high toxicity of nitrite and its metabolites, it is of high interest to study mechanisms underlying the low NO2 level maintenance in the cell. During anaerobic growth of Escherichia coli the main nitrite-reducing enzymes are NrfA and NirB nitrite reductases. NrfA reductase is localized in the cell periplasm and uses NO2 as an electron acceptor to create a proton gradient; NirB reductase is restricted to the cytoplasm and metabolizes excessive nitrite inside the cell, the uptake of which is mediated by the transporter protein NirC. While it is known that these three systems, periplasmic, cytoplasmic and transport, determine nitrite uptake and assimilation in the cell as well as its excretion, little is known about their co-ordination. RESULTS: Using a mathematical model describing the nitrite utilization in E. coli cells cultured in a flow chemostat, the role of enzymes involved in nitrite metabolism and transport in controlling nitrite intracellular levels was investigated. It was demonstrated that the model adapted to the experimental data on expression of nrfA and nirB genes encoding NrfA and NirB nitrite reductases, can describe nitrite accumulation kinetics in the chemostat in the millimolar range of added substrate concentrations without any additional assumptions. According to the model, in this range, low intracellular nitrite level, weakly dependent on its concentration in the growth media, is maintained (mcM). It is not sufficient to consider molecular-genetic mechanisms of NrfA reductase activity regulation to describe the nitrite accumulation dynamics in the chemostat in the micromolar range (≤1 mM) of added nitrite concentrations. Analysis of different hypotheses has shown that the mechanism of local enzyme concentration change due to membrane potential-induced diffusion from the cytoplasm to the periplasm at low nitrite levels is sufficient to explain the nitrite accumulation dynamics in the chemostat. CONCLUSIONS: At nitrite concentrations in the media more than 2 mM, the model adapted to the experimental data on nitrite utilization dynamics in E. coli cells cultured in the flow chemostat demonstrates the largest contribution of genetic mechanisms involved in nrf and nir operons activity regulation to the control of nitrite intracellular levels. The model predicts a significant contribution of the membrane potential to the periplasmic NrfA nitrite reductase activity regulation and nitrite utilization dynamics at substrate concentrations ≤1 mM.

Alternative splicing can lead to chaos.

Alternative splicing is a widespread phenomenon in higher eukaryotes, where it serves as a mechanism to increase the functional diversity of proteins. This phenomenon has been described for different classes of proteins, including transcription regulatory proteins. We demonstrated that in the simplest genetic system model the formation of the alternatively spliced isoforms with opposite functions (activators and repressors) could be a cause of transition to chaotic dynamics. Under the simplest genetic system we understand a system consisting of a single gene encoding the structure of a transcription regulatory protein whose expression is regulated by a feedback mechanism. As demonstrated by numerical analysis of the models, if the synthesized isoforms regulate the expression of their own gene acting through different sites and independently of each other, for the generation of chaotic dynamics it is sufficient that the regulatory proteins have a dimeric structure. If regulatory proteins act through one site, the chaotic dynamics is generated in the system only when the repressor protein is either a tetrameric or a higher-dimensional multimer. In this case the activator can be a dimer. It was also demonstrated that if the transcription factor isoforms exhibit either activating or inhibiting activity and are lower-dimensional multimers (< 4), independently of the regulation type the model demonstrates either cyclic or stationary trajectories.

Mathematical modeling of bacterial cell cycle: the problem of coordinating genome replication with cell growth.

In this paper, we perform an analysis of bacterial cell-cycle models implementing different strategies to coordinately regulate genome replication and cell growth dynamics. It has been shown that the problem of coupling these processes does not depend directly on the dynamics of cell volume expansion, but does depend on the type of cell growth law. Our analysis has distinguished two types of cell growth laws, "exponential" and "linear", each of which may include both exponential and linear patterns of cell growth. If a cell grows following a law of the "exponential" type, including the exponential V(t) = V(0) exp (kt) and linear V(t) = V(0)(1 + kt) dynamic patterns, then the cell encounters the problem of coupling growth rates and replication. It has been demonstrated that to solve the problem, it is sufficient for a cell to have a repressor mechanism to regulate DNA replication initiation. For a cell expanding its volume by a law of the "linear" type, including exponential V(t) = V(0) + V(1) exp (kt) and linear V(t) = V(0) + kt dynamic patterns, the problem of coupling growth rates and replication does not exist. In other words, in the context of the coupling problem, a repressor mechanism to regulate DNA replication, and cell growth laws of the "linear" type displays the attributes of universality. The repressor-type mechanism allows a cell to follow any growth dynamic pattern, while the "linear" type growth law allows a cell to use any mechanism to regulate DNA replication.

A new stochastic model for subgenomic hepatitis C virus replication considers drug resistant mutants.

As an RNA virus, hepatitis C virus (HCV) is able to rapidly acquire drug resistance, and for this reason the design of effective anti-HCV drugs is a real challenge. The HCV subgenomic replicon-containing cells are widely used for experimental studies of the HCV genome replication mechanisms, for drug testing in vitro and in studies of HCV drug resistance. The NS3/4A protease is essential for virus replication and, therefore, it is one of the most attractive targets for developing specific antiviral agents against HCV. We have developed a stochastic model of subgenomic HCV replicon replication, in which the emergence and selection of drug resistant mutant viral RNAs in replicon cells is taken into account. Incorporation into the model of key NS3 protease mutations leading to resistance to BILN-2061 (A156T, D168V, R155Q), VX-950 (A156S, A156T, T54A) and SCH 503034 (A156T, A156S, T54A) inhibitors allows us to describe the long term dynamics of the viral RNA suppression for various inhibitor concentrations. We theoretically showed that the observable difference between the viral RNA kinetics for different inhibitor concentrations can be explained by differences in the replication rate and inhibitor sensitivity of the mutant RNAs. The pre-existing mutants of the NS3 protease contribute more significantly to appearance of new resistant mutants during treatment with inhibitors than wild-type replicon. The model can be used to interpret the results of anti-HCV drug testing on replicon systems, as well as to estimate the efficacy of potential drugs and predict optimal schemes of their usage.

Mathematical model of the Tat-Rev regulation of HIV-1 replication in an activated cell predicts the existence of oscillatory dynamics in the synthesis of viral components.

BACKGROUND: The life cycle of human immunodeficiency virus type-1 (HIV-1) makes possible the realization of regulatory strategies that can lead to complex dynamical behavior of the system. We analyze the strategy which is based on two feedback mechanisms, one mediating a positive regulation of the virus replication by Tat protein via the antitermination of the genomic RNAs transcription on TAR (transactivation responsive) element of the proviral DNA and the second mechanism providing a negative regulation of the splicing of the full-length (9 kb) RNAs and incompletely spliced (4 kb) RNAs via their transport from the nucleus to the cytoplasm. Although the existence of these two regulatory feedback loops has been considered in other mathematical models, none of them examined the conditions for the emergence of complex oscillatory patterns in the intracellular dynamics of viral components. RESULTS: We developed a mechanistic mathematical model for the Tat-Rev mediated regulation of HIV-1 replication, which considers the activation of proviral DNA transcription, the Tat-specific antitermination of transcription on TAR-element, resulting in the synthesis of the full-length 9 kb RNA, the splicing of the 9 kb RNA down to the 4 kb RNA and the 4 kb RNA to 2 kb RNA, the transport of 2 kb mRNAs from the nucleus to the cytoplasm by the intracellular mechanisms, the multiple binding of the Rev protein to RRE (Rev Response Element) sites on 9 kb and 4 kb RNA resulting in their export to the cytoplasm and the synthesis of Tat and Rev proteins in the cytoplasm followed by their transport into the nucleus. The degradation of all viral proteins and RNAs both in the cytoplasm and the nucleus is described. The model parameters values were derived from the published literature data. The model was used to examine the dynamics of the synthesis of the viral proteins Tat and Rev, the mRNAs under the intracellular conditions specific for activated HIV-1 infected macrophages. In addition, we analyzed alternative hypotheses for the re-cycling of the Rev proteins both in the cytoplasm and the nuclear pore complex. CONCLUSIONS: The quantitative mathematical model of the Tat-Rev regulation of HIV-1 replication predicts the existence of oscillatory dynamics which depends on the efficacy of the Tat and TAR interaction as well as on the Rev-mediated transport processes. The biological relevance of the oscillatory regimes for the HIV-1 life cycle is discussed.

Mathematical modeling of auxin transport in protoxylem and protophloem of Arabidopsis thaliana root tips.

Phytohormone auxin is the main regulator of plant growth and development. Nonuniform auxin distribution in plant tissue sets positional information, which determines morphogenesis. Auxin is transported in tissue by means of diffusion and active transport through the cell membrane. There is a number of auxin carriers performing its influx into a cell (AUX\LAX family) or efflux from a cell (PIN, PGP families). The paper presents mathematical models for auxin transport in vascular tissues of Arabidopsis thaliana L.root tip, namely protophloem and protoxylem. Tissue specificity of auxin active transport was considered in these models. There is PIN-mediated auxin efflux in both protoxylem and protophloem, but AUX1-mediated influx exists only in protophloem. Optimal values of parameters were adjusted for model solutions to fit the experimentally observed auxin distributions in the root tip. Based on simulation results we predicted that shoot-derived auxin flow to protophloem is lower than one to protoxylem, and the efficiency of PIN-mediated auxin transport in protophloem is higher than in protoxylem. In summary, our simulation showed that despite the same auxin distribution pattern, provascular tissues in the root tip differ in dynamics of auxin transport.

On the chaos in gene networks.

The methods for constructing "chaotic" nonlinear systems of differential equations modeling gene networks of arbitrary structure and dimensionality with various types of symmetry are considered. It has been shown that an increase in modality of the functions describing the control of gene expression efficiency allows for a decrease in the dimensionality of these systems with retention of their chaotic dynamics. Three-dimensional "chaotic" cyclic systems are considered. Symmetrical and asymmetrical attractors with "narrow" chaos having a Moebius-like structure have been detected in such systems. As has been demonstrated, a complete symmetry of the systems with respect to permutation of variables does not prevent the emergence of their chaotic dynamics.

How multiple auxin responsive elements may interact in plant promoters: a reverse problem solution.

Plant hormone auxin is a key regulator of growth and development. Auxin affects gene expression through ARF transcription factors, which bind specifically auxin responsive elements (AuxREs). Auxin responsive genes usually have more than one AuxRE, for example, a widely used auxin sensor DR5 contains seven AuxREs. Auxin responsive regions of several plant genes have been studied using sets of transgenic constructions in which the activity of one or several AuxREs were abolished. Here we present the method for analysis of the datasets on promoter activity assays having promoter sequences, namely, number and sequences of AuxREs, altogether with their measured auxin induction level. The method for a reverse problem solution considers two extreme models of AuxRE cooperation. Additive model describes auxin induction level of a gene as a sum of the individual AuxREs impacts. Multiplicative model considers pure cooperation between the AuxREs, where the combined effect is the multiplication of the individual AuxRE impacts. The reverse problem solution allows estimating the impact of an individual AuxRE into the induction level and the model for their cooperation. For promoters of three genes belonging to different plant species we showed that the multiplicative model fits better than additive. The reverse problem solution also suggests repressive state of auxin responsive promoters before auxin induction. The developed method provides possibility to investigate AuxRE structure-activity relationship and may be used as the basis for a novel approach for AuxRE recognition.

A plausible mechanism for auxin patterning along the developing root.

BACKGROUND: In plant roots, auxin is critical for patterning and morphogenesis. It regulates cell elongation and division, the development and maintenance of root apical meristems, and other processes. In Arabidopsis, auxin distribution along the central root axis has several maxima: in the root tip, in the basal meristem and at the shoot/root junction. The distal maximum in the root tip maintains the stem cell niche. Proximal maxima may trigger lateral or adventitious root initiation. RESULTS: We propose a reflected flow mechanism for the formation of the auxin maximum in the root apical meristem. The mechanism is based on auxin's known activation and inhibition of expressed PIN family auxin carriers at low and high auxin levels, respectively. Simulations showed that these regulatory interactions are sufficient for self-organization of the auxin distribution pattern along the central root axis under varying conditions. The mathematical model was extended with rules for discontinuous cell dynamics so that cell divisions were also governed by auxin, and by another morphogen Division Factor which combines the actions of cytokinin and ethylene on cell division in the root. The positional information specified by the gradients of these two morphogens is able to explain root patterning along the central root axis. CONCLUSION: We present here a plausible mechanism for auxin patterning along the developing root, that may provide for self-organization of the distal auxin maximum when the reverse fountain has not yet been formed or has been disrupted. In addition, the proximal maxima are formed under the reflected flow mechanism in response to periods of increasing auxin flow from the growing shoot. These events may predetermine lateral root initiation in a rhyzotactic pattern. Another outcome of the reflected flow mechanism - the predominance of lateral or adventitious roots in different plant species - may be based on the different efficiencies with which auxin inhibits its own transport in different species, thereby distinguishing two main types of plant root architecture: taproot vs. fibrous.

Mathematical model for suppression of subgenomic hepatitis C virus RNA replication in cell culture.

A mathematical model for suppression of the hepatitis C virus RNA replicon replication in Huh-7 cell culture in the presence of potential drugs was built. There was a good agreement between the experimental and theoretical kinetic data for the decrease in the level of viral RNA in the cell in the presence of the competitive HCV NS3 protease inhibitor. Using the model, we verified the estimates for the efficiency of the effect of potential drugs on replication of viral RNA and viral protein processing. It was demonstrated that the tested drugs are most efficient at the replication step of viral RNA. The efficiency of the combined action of real and putative inhibitors target on the host and viral proteins was also studied. It was found that the action of the inhibitor at low concentrations on the host factors considerably enhances the suppressive effect on viral RNA replication in the presence of even the low affine NS3 protease inhibitors. The developed mathematical model may serve as a tool for the evaluation of the efficiency of potential drugs on the HCV genome.

A cellular automaton to model the development of primary shoot meristems of Arabidopsis thaliana.

Development of organisms is a very complex process in which a lot of gene networks of different cell types are integrated. Development of a cellular automaton (Ermentrout and Edelshtein-Keshet, J Theor Biol 160:97-133, 1993) that models the morphodynamics of different cell types is the first step in understanding and analysis of the regulatory mechanisms underlying the functioning of developmental gene networks. A model of a cellular automaton has been developed, which simulates the embryonic development of shoot meristem in Arabidopsis thaliana. The model adequately describes the basic stages in development of this organ in wild and mutant types.

GeneNet database: description and modeling of gene networks.

Almost all cellular processes in an organism are controlled by gene networks. Here we report on the analysis of gene networks functioning using two associated methods - data accumulation in GeneNet system and generalized chemical kinetic method for mathematical simulation of gene network functional dynamics. The technology of the usage of these methods is shown on the example of the gene network of macrophage activation.