Predicting the asymmetric response of a genetic switch to noise

We present a simple analytical tool which gives an approximate insight into the stationary behavior of nonlinear systems undergoing the influence of a weak and rapid noise from one dominating source, e.g. the kinetic equations describing a genetic switch with the concentration of one substrate fluctuating around a constant mean. The proposed method allows for predicting the asymmetric response of the genetic switch to noise, arising from the noise-induced shift of stationary states. The method has been tested on an example model of the lac operon regulatory network: a reduced Yildirim-Mackey model with fluctuating extracellular lactose concentration. We calculate analytically the shift of the system's stationary states in the presence of noise. The results of the analytical calculation are in excellent agreement with the results of numerical simulation of the noisy system. The simulation results suggest that the structure of the kinetics of the underlying biochemical reactions protects the bistability of the lactose utilization mechanism from environmental fluctuations. We show that, in the consequence of the noise-induced shift of stationary states, the presence of fluctuations stabilizes the behavior of the system in a selective way: Although the extrinsic noise facilitates, to some extent, switching off the lactose metabolism, the same noise prevents it from switching on.     

From bistability to oscillations in a model for the isocitrate dehydrogenase reaction

Considered is a bienzymatic system consisting of isocitrate dehydrogenase (IDH, EC 1.1.1.42), which transforms NADP(+) into NADPH, and of diaphorase (DIA, EC 1.8.1.4), which catalyzes the reverse reaction. Experimental evidence as well as a theoretical model show the possibility of a coexistence between two stable steady states in this reaction system. The phenomenon originates from the regulatory properties of IDH. We extend the analysis of a theoretical model proposed for the IDH-DIA bienzymatic system and investigate the occurrence of different modes of bistability, with or without hysteresis, i.e. in the presence of two or only one limit point bounding the domain of multiple steady states. The analysis indicates that the two types of bistability may sometimes be observed sequentially as a given control parameter is progressively increased. We further obtain conditions in which sustained oscillations develop in the model. These results establish the isocitrate dehydrogenase reaction coupled to diaphorase as a suitable candidate for further experimental and theoretical studies of bistability and oscillations in biochemical systems.     

System-level feedbacks control cell cycle progression

Repetitive cell cycles, which are essential to the perpetuation of life, are orchestrated by an underlying biochemical reaction network centered around cyclin-dependent protein kinases (Cdks) and their regulatory subunits (cyclins). Oscillations of Cdk1/CycB activity between low and high levels during the cycle trigger DNA replication and mitosis in the correct order. Based on computational modeling, we proposed that the low and the high kinase activity states are alternative stable steady states of a bistable Cdk-control system. Bistability is a consequence of system-level feedback (positive and double-negative feedback signals) in the underlying control system. We have also argued that bistability underlies irreversible transitions between low and high Cdk activity states and thereby ensures directionality of cell cycle progression.     

Roles of protein ubiquitination and degradation kinetics in biological oscillations

Protein ubiquitination and degradation play important roles in many biological functions and are associated with many human diseases. It is well known that for biochemical oscillations to occur, proper degradation rates of the participating proteins are needed. In most mathematical models of biochemical reactions, linear degradation kinetics has been used. However, the degradation kinetics in real systems may be nonlinear, and how nonlinear degradation kinetics affects biological oscillations are not well understood. In this study, we first develop a biochemical reaction model of protein ubiquitination and degradation and calculate the degradation rate against the concentration of the free substrate. We show that the protein degradation kinetics mainly follows the Michaelis-Menten formulation with a time delay caused by ubiquitination and deubiquitination. We then study analytically how the Michaelis-Menten degradation kinetics affects the instabilities that lead to oscillations using three generic oscillation models: 1) a positive feedback mediated oscillator; 2) a positive-plus-negative feedback mediated oscillator; and 3) a negative feedback mediated oscillator. In all three cases, nonlinear degradation kinetics promotes oscillations, especially for the negative feedback mediated oscillator, resulting in much larger oscillation amplitudes and slower frequencies than those observed with linear kinetics. However, the time delay due to protein ubiquitination and deubiquitination generally suppresses oscillations, reducing the amplitude and increasing the frequency of the oscillations. These theoretical analyses provide mechanistic insights into the effects of specific proteins in the ubiquitination-proteasome system on biological oscillations.     

Oscillations and bistability predicted by a model for a cyclical bienzymatic system involving the regulated isocitrate dehydrogenase reaction

We analyze the dynamics of a bienzymatic system consisting of isocitrate dehydrogenase (IDH, EC. 1.1.1.42), which transforms NADP+ into NADPH, and of diaphorase (DIA, EC 1.8.1.4), which catalyzes the reverse reaction. Experimental evidence as well as a theoretical model showed the possibility of a coexistence between two stable steady states in this reaction system G.M. Guidi et al. Biophys. J. 74 (1998) 1229-1240[, owing to the regulatory properties of IDH. Here we extend this analysis by considering the behavior of the model proposed for the IDH-DIA bienzymatic system in conditions where the system is open to an influx of its substrates isocitrate and NADP+ and to an efflux of all metabolic species. The analysis indicates that in addition to different modes of bistability (including mushrooms and isolas), sustained oscillations can be observed in such conditions. These results point to the isocitrate dehydrogenase reaction coupled to diaphorase as a suitable candidate for further experimental and theoretical studies of bistability and oscillations in biochemical systems. The results obtained in this particular bienzymatic system bear on other enzymatic systems possessing a cyclical nature, which are known to play significant roles in a variety of metabolic and cellular regulatory processes.     

Split Histidine Kinases Enable Ultrasensitivity and Bistability in Two-Component Signaling Networks

Bacteria sense and respond to their environment through signaling cascades generally referred to as two-component signaling networks. These networks comprise histidine kinases and their cognate response regulators. Histidine kinases have a number of biochemical activities: ATP binding, autophosphorylation, the ability to act as a phosphodonor for their response regulators, and in many cases the ability to catalyze the hydrolytic dephosphorylation of their response regulator. Here, we explore the functional role of “split kinases” where the ATP binding and phosphotransfer activities of a conventional histidine kinase are split onto two distinct proteins that form a complex. We find that this unusual configuration can enable ultrasensitivity and bistability in the signal-response relationship of the resulting system. These dynamics are displayed under a wide parameter range but only when specific biochemical requirements are met. We experimentally show that one of these requirements, namely segregation of the phosphatase activity predominantly onto the free form of one of the proteins making up the split kinase, is met in Rhodobacter sphaeroides. These findings indicate split kinases as a bacterial alternative for enabling ultrasensitivity and bistability in signaling networks. Genomic analyses reveal that up 1.7% of all identified histidine kinases have the potential to be split and bifunctional.     

Three steady state situation in an open chemical reaction system. I

In an isothermal continuously stirred tank reactor (open chemical reaction system) fed by sulphuric acid solutions of bromate, bromide and cerium)(III) bistability (three steady state situation) is experimentally observed. This remarkable behavior, based on the instability of one steady state, has important consequences for the understanding of excitability and biochemical control mechanisms. The mass-balance equations for the reactor and the chemical mechanism of the reaction are combined into a simple mathematical model. The behavior of the resulting nonlinear differential equations is examined analytically and by a graphical integration procedure (method of isoclines). Using realistic kinetic data, the model shows the same behavior as observed in the experiment.     

Chemical networks with inflows and outflows: A positive linear differential inclusions approach

Certain mass‐action kinetics models of biochemical reaction networks, although described by nonlinear differential equations, may be partially viewed as state‐dependent linear time‐varying systems, which in turn may be modeled by convex compact valued positive linear differential inclusions. A result is provided on asymptotic stability of such inclusions, and applied to a ubiquitous biochemical reaction network with inflows and outflows, known as the futile cycle. We also provide a characterization of exponential stability of general homogeneous switched systems which is not only of interest in itself, but also plays a role in the analysis of the futile cycle. © 2009 American Institute of Chemical Engineers Biotechnol. Prog., 2009     

Accuracy of alternative representations for integrated biochemical systems

The Michaelis-Menten formalism often provides appropriate representations of individual enzyme-catalyzed reactions in vitro but is not well suited for the mathematical analysis of complex biochemical networks. Mathematically tractable alternatives are the linear formalism and the power-law formalism. Within the power-law formalism there are alternative ways to represent biochemical processes, depending upon the degree to which fluxes and concentrations are aggregated. Two of the most relevant variants for dealing with biochemical pathways are treated in this paper. In one variant, aggregation leads to a rate law for each enzyme-catalyzed reaction, which is then represented by a power-law function. In the other, aggregation produces a composite rate law for either net rate of increase or net rate of decrease of each system constituent; the composite rate laws are then represented by a power-law function. The first variant is the mathematical basis for a method of biochemical analysis called metabolic control, the latter for biochemical systems theory. We compare the accuracy of the linear and of the two power-law representations for networks of biochemical reactions governed by Michaelis-Menten and Hill kinetics. Michaelis-Menten kinetics are always represented more accurately by power-law than by linear functions. Hill kinetics are in most cases best modeled by power-law functions, but in some cases linear functions are best. Aggregation into composite rate laws for net increase or net decrease of each system constituent almost always improves the accuracy of the power-law representation. The improvement in accuracy is one of several factors that contribute to the wide range of validity of this power-law representation. Other contributing factors that are discussed include the nonlinear character of the power-law formalism, homeostatic regulatory mechanisms in living systems, and simplification of rate laws by regulatory mechanisms in vivo.     

A generalized numerical approach to steady-state enzyme kinetics: Applications to protein kinase inhibition

A generalized numerical treatment of steady-state enzyme kinetics is presented. This new approach relies on automatic computer derivation of the underlying mathematical model (a system of simultaneous nonlinear algebraic equations) from a symbolic representation of the reaction mechanism (a system of biochemical equations) provided by the researcher. The method allows experimental biochemists to analyze initial-rate enzyme kinetic data, under the steady-state approximation, without having to use any mathematical equations. An illustrative example is based on the inhibition kinetics of p56^lck kinase by an ATP competitive inhibitor. A computer implementation of the new method, in the modified software package DYNAFIT [Kuzmi?, P. (1996) Anal. Biochem. 237, 260–273], is freely available to all academic researchers.     

The stability of a stochastic CaMKII switch: dependence on the number of enzyme molecules and protein turnover

Molecular switches have been implicated in the storage of information in biological systems. For small structures such as synapses, these switches are composed of only a few molecules and stochastic fluctuations are therefore of importance. Such fluctuations could potentially lead to spontaneous switch reset that would limit the lifetime of information storage. We have analyzed a model of the calcium/calmodulin-dependent protein kinase II (CaMKII) switch implicated in long-term memory in the nervous system. The bistability of this switch arises from autocatalytic autophosphorylation of CaMKII, a reaction that is countered by a saturable phosphatase-1-mediated dephosphorylation. We sought to understand the factors that control switch stability and to determine the functional relationship between stability and the number of molecules involved. Using Monte Carlo simulations, we found that the lifetime of states of the switch increase exponentially with the number of CaMKII holoenzymes. Switch stability requires a balance between the kinase and phosphatase rates, and the kinase rate must remain high relative to the rate of protein turnover. Thus, a critical limit on switch stability is set by the observed turnover rate (one per 30 h on average). Our computational results show that, depending on the timescale of fluctuations in enzyme numbers, for a switch composed of about 15 CaMKII holoenzymes, the stable persistent activation can span from a few years to a human lifetime.     

Protein scaffolds can enhance the bistability of multisite phosphorylation systems

The phosphorylation of a substrate at multiple sites is a common protein modification that can give rise to important structural and electrostatic changes. Scaffold proteins can enhance protein phosphorylation by facilitating an interaction between a protein kinase enzyme and its target substrate. In this work we consider a simple mathematical model of a scaffold protein and show that under specific conditions, the presence of the scaffold can substantially raise the likelihood that the resulting system will exhibit bistable behavior. This phenomenon is especially pronounced when the enzymatic reactions have sufficiently large K(M), compared to the concentration of the target substrate. We also find for a closely related model that bistable systems tend to have a specific kinetic conformation. Using deficiency theory and other methods, we provide a number of necessary conditions for bistability, such as the presence of multiple phosphorylation sites and the dependence of the scaffold binding/unbinding rates on the number of phosphorylated sites.     

Application of a new method of nonlinear dynamical system identification to biochemical problems

The system identification method for a variety of nonlinear dynamic models is elaborated. The problem of identification of an original nonlinear model presented as a system of ordinary differential equations in the Cauchy explicit form with a polynomial right part reduces to the solution of the system of linear equations for the constants of the dynamical model. In other words, to construct an integral model of the complex system (phenomenon), it is enough to collect some data array characterizing the time-course of dynamical parameters of the system. Collection of such a data array has always been a problem. However difficulties emerging are, as a rule, not principal and may be overcome almost without exception. The potentialities of the method under discussion are demonstrated by the example of the test problem of multiparametric nonlinear oscillator identification. The identification method proposed may be applied to the study of different biological systems and in particular the enzyme kinetics of complex biochemical reactions.     

Memory switches in chemical reaction space

Just as complex electronic circuits are built from simple Boolean gates, diverse biological functions, including signal transduction, differentiation, and stress response, frequently use biochemical switches as a functional module. A relatively small number of such switches have been described in the literature, and these exhibit considerable diversity in chemical topology. We asked if biochemical switches are indeed rare and if there are common chemical motifs and family relationships among such switches. We performed a systematic exploration of chemical reaction space by generating all possible stoichiometrically valid chemical configurations up to 3 molecules and 6 reactions and up to 4 molecules and 3 reactions. We used Monte Carlo sampling of parameter space for each such configuration to generate specific models and checked each model for switching properties. We found nearly 4,500 reaction topologies, or about 10% of our tested configurations, that demonstrate switching behavior. Commonly accepted topological features such as feedback were poor predictors of bistability, and we identified new reaction motifs that were likely to be found in switches. Furthermore, the discovered switches were related in that most of the larger configurations were derived from smaller ones by addition of one or more reactions. To explore even larger configurations, we developed two tools: the “bistabilizer,” which converts almost-bistable systems into bistable ones, and frequent motif mining, which helps rank untested configurations. Both of these tools increased the coverage of our library of bistable systems. Thus, our systematic exploration of chemical reaction space has produced a valuable resource for investigating the key signaling motif of bistability.     

Dimensionality Reduction of Bistable Biological Systems

Time hierarchies, arising as a result of interactions between system’s components, represent a ubiquitous property of dynamical biological systems. In addition, biological systems have been attributed switch-like properties modulating the response to various stimuli across different organisms and environmental conditions. Therefore, establishing the interplay between these features of system dynamics renders itself a challenging question of practical interest in biology. Existing methods are suitable for systems with one stable steady state employed as a well-defined reference. In such systems, the characterization of the time hierarchies has already been used for determining the components that contribute to the dynamics of biological systems. However, the application of these methods to bistable nonlinear systems is impeded due to their inherent dependence on the reference state, which in this case is no longer unique. Here, we extend the applicability of the reference-state analysis by proposing, analyzing, and applying a novel method, which allows investigation of the time hierarchies in systems exhibiting bistability. The proposed method is in turn used in identifying the components, other than reactions, which determine the systemic dynamical properties. We demonstrate that in biological systems of varying levels of complexity and spanning different biological levels, the method can be effectively employed for model simplification while ensuring preservation of qualitative dynamical properties (i.e., bistability). Finally, by establishing a connection between techniques from nonlinear dynamics and multivariate statistics, the proposed approach provides the basis for extending reference-based analysis to bistable systems.     

Antagonism and bistability in protein interaction networks

A protein interaction network (PIN) is a set of proteins that modulate one another's activities by regulated synthesis and degradation, by reversible binding to form complexes, and by catalytic reactions (e.g., phosphorylation and dephosphorylation). Most PINs are so complex that their dynamical characteristics cannot be deduced accurately by intuitive reasoning alone. To predict the properties of such networks, many research groups have turned to mathematical models (differential equations based on standard biochemical rate laws, e.g., mass-action, Michaelis-Menten, Hill). When using Michaelis-Menten rate expressions to model PINs, care must be exercised to avoid making inconsistent assumptions about enzyme-substrate complexes. We show that an appealingly simple model of a PIN that functions as a bistable switch is compromised by neglecting enzyme-substrate intermediates. When the neglected intermediates are put back into the model, bistability of the switch is lost. The theory of chemical reaction networks predicts that bistability can be recovered by adding specific reaction channels to the molecular mechanism. We explore two very different routes to recover bistability. In both cases, we show how to convert the original 'phenomenological' model into a consistent set of mass-action rate laws that retains the desired bistability properties. Once an equivalent model is formulated in terms of elementary chemical reactions, it can be simulated accurately either by deterministic differential equations or by Gillespie's stochastic simulation algorithm.