Modelling circadian rhythms of protein KaiA, KaiB and KaiC interactions in cyanobacteria

Cyanobacteria are the simplest organisms known that exhibit circadian rhythms. The mechanism of circadian rhythm generation in cyanobacteria is different from eukaryotes. Based on the recent experiments about the interaction of KaiA, KaiB, and KaiC proteins with the generation of circadian rhythms in vitro, we developed a mathematical model to describe post-translational oscillations and the possible chemical reactions involved in the circadian clock mechanism of cyanobacteria. In this model, a series of differential equations, with linear kinetics for binding of proteins, Michaelis - Menten kinetics for enzymatic processes and a term including an explicit delay for dissociation of the KaiA/KaiB/phospho-KaiC complex, are proposed describing the dynamics of the chemistry. It is demonstrated that the mathematical system can lead to circadian oscillation within a range of parameter values.     

Approximations and their consequences for dynamic modelling of signal transduction pathways

Signal transduction is the process by which the cell converts one kind of signal or stimulus into another. This involves a sequence of biochemical reactions, carried out by proteins. The dynamic response of complex cell signalling networks can be modelled and simulated in the framework of chemical kinetics. The mathematical formulation of chemical kinetics results in a system of coupled differential equations. Simplifications can arise through assumptions and approximations. The paper provides a critical discussion of frequently employed approximations in dynamic modelling of signal transduction pathways. We discuss the requirements for conservation laws, steady state approximations, and the neglect of components. We show how these approximations simplify the mathematical treatment of biochemical networks but we also demonstrate differences between the complete system and its approximations with respect to the transient and steady state behavior.     

P680(+)* reduction kinetics and redox transition probability of the water oxidizing complex as a function of pH and H/D isotope exchange in spinach thylakoids.

The rise of fluorescence as an indicator for P680(+)* reduction by YZ and the period-four oscillation of oxygen yield induced by a train of saturating flashes were measured in dark-adapted thylakoids as a function of pH in the absence of exogenous electron acceptors. The results reveal that: (i) the average amplitude of the nanosecond kinetics and the average of the maximum fluorescence attained at 100 micros after the flash in the acidic range decrease with decreasing pH; (ii) the oxygen yield exhibits a pronounced period-four oscillation at pH 6.5 and higher damping at both pH 5.0 and pH 8.0; (iii) the probability of misses in the Si-state transitions of the water oxidizing complex is affected characteristically when exchangeable protons are replaced by deuterons [at pH <6.5, the ratio alpha(D)/alpha(H) is larger than 1 whereas at pH >7.0 values of <1 are observed]. The results are discussed within the framework of a combined mechanism for P680(+)* reduction where the nanosecond kinetics reflect an electron transfer coupled with a "rocket-type" proton shift within a hydrogen bridge from YZ to a nearby basic group, X [Eckert, H.-J., and Renger, G. (1988) FEBS Lett. 236, 425-431], and subsequent relaxations within a network of hydrogen bonds. It is concluded that in the acidic region the hydrogen bond between YZ and X (most likely His 190 of polypeptide D1) is interrupted either by direct protonation of X or by conformational changes due to acid-induced Ca2+ release. This gives rise to a decreased P680(+)* reduction by nanosecond kinetics and an increase of dissipative P680(+)* recombination at low pH. A different mechanism is responsible for the almost invariant amplitude of nanosecond kinetics and increase of alpha in the alkaline region.     

Following autolysis in proteases by NMR: insights into multiple unfolding pathways and mutational plasticities

Recently there has been significant interest in deducing the form of the rate laws for chemical reactions occurring in the intracellular environment. This environment is typically characterized by low-dimensionality and a high macromolecular content; this leads to a spatial heterogeneity not typical of the well stirred in vitro environments. For this reason, the classical law of mass action has been presumed to be invalid for modeling intracellular reactions. Using lattice-gas automata models, it has recently been postulated [H. Berry, Monte Carlo simulations of enzyme reactions in two dimensions: Fractal kinetics and spatial segregation, Biophys. J. 83 (2002) 1891-1901; S. Schnell, T.E. Turner, Reaction kinetics in intracellular environments with macromolecular crowding: simulations and rate laws, Prog. Biophys. Mol. Biol. 85 (2004) 235-260] that the reaction kinetics is fractal-like. In this article we systematically investigate for the first time how the rate laws describing intracellular reactions vary as a function of: the geometry and size of the intracellular surface on which the reactions occur, the mobility of the macromolecules responsible for the crowding effects, the initial reactant concentrations and the probability of reaction between two reactant molecules. We also compare the rate laws valid in heterogeneous environments in which there is an underlying spatial lattice, for example crystalline alloys, with the rate laws valid in heterogeneous environments where there is no such natural lattice, for example in intracellular environments. Our simulations indicate that: (i) in intracellular environments both fractal kinetics and mass action can be valid, the major determinant being the probability of reaction, (ii) the geometry and size of the intracellular surface on which reactions are occurring does not significantly affect the rate law, (iii) there are considerable differences between the rate laws valid in heterogeneous non-living structures such as crystals and those valid in intracellular environments. Deviations from mass action are less pronounced in intracellular environments than in a crystalline material of similar heterogeneity.     

Steady-state stability of substrate inhibited enzyme catalysis in open reaction systems

Substrate inhibited enzyme reactions occurring in systems open to mass transfer may display multiple steady-state behavior. For a simple one-substrate case it is shown theoretically that the unstable steady-state region always lies within the conversion range of 50–100%. A criterion for the stability of a steady-state point is given. Numerical solutions of the appropriate transient equation show how the system approaches stable steady state in instances where there are three possible steady-state points. The consequences of the existence of an unstable region on the systems response to changes in its parameters is discussed.     

Simplifying principles for chemical and enzyme reaction kinetics

Tihonov's Theorems for systems of first-order ordinary differential equations containing small parameters in the derivatives, which form the mathematical foundation of the steady-state approximation, are restated. A general procedure for simplifying chemical and enzyme reaction kinetics, based on the difference of characteristic time scales, is presented. Korzuhin's Theorem. which makes it possible to approximate any kinetic system by a closed chemical system, is also reported. The notions and theorems are illustrated with examples of Michaelis-Menten enzyme kinetics and of a simple autocatalytic system. Another example illustrates how the differences in the rate constants of different elementary reactions may be exploited to simplify reaction kinetics by using Tihonov's Theorem. All necessary mathematical notions are explained in the appendices. The most simple formulation of Tihonov's 1st Theorem ‘for beginners’ is also given.     

Size-Independent Differences between the Mean of Discrete Stochastic Systems and the Corresponding Continuous Deterministic Systems

In this paper, it is shown that for a class of reaction networks, the discrete stochastic nature of the reacting species and reactions results in qualitative and quantitative differences between the mean of exact stochastic simulations and the prediction of the corresponding deterministic system. The differences are independent of the number of molecules of each species in the system under consideration. These reaction networks are open systems of chemical reactions with no zero-order reaction rates. They are characterized by at least two stationary points, one of which is a nonzero stable point, and one unstable trivial solution (stability based on a linear stability analysis of the deterministic system). Starting from a nonzero initial condition, the deterministic system never reaches the zero stationary point due to its unstable nature. In contrast, the result presented here proves that this zero-state is a stable stationary state for the discrete stochastic system, and other finite states have zero probability of existence at large times. This result generalizes previous theoretical studies and simulations of specific systems and provides a theoretical basis for analyzing a class of systems that exhibit such inconsistent behavior. This result has implications in the simulation of infection, apoptosis, and population kinetics, as it can be shown that for certain models the stochastic simulations will always yield different predictions for the mean behavior than the deterministic simulations.     

Models of growth, self-reproduction and autoregulation based on consideration of reaction vessels with distensible, semipermeable walls]

We constructed non-equilibrium thermodynamics of the open physical-chemical irreversible processes in reactors with the strain semipermeable walls. This thermodynamics does not use the reciprocal relations of Onsager, so it may be applied when the stability stationary state is far from equilibrium. One of a general consequences of this thermodynamics is the statement: coordinated growth and self-reproduction are possible near the absolute equilibrium of the dissolvent and near the absolute stability stationary state of all chemicals with the absolute conservation of the differential equations of chemical kinetics. The supposition of ideal mixing is unnecessary; this condition is fulfilled automatically with diffusion. Growth and self-reproduction are not connected with positive eigenvalue of the differential equation of chemical kinetics. It is possible to construct a model of autoregulation and differentiation with this thermodynamics. The uniquness of such autoregulation follows from the mathematical theory [1]. The mathematical foundation of this thermodynamics is given in [1].     

Development of a new EPR spin trap, DOD-8C (N-[4-dodecyloxy-2-(7'-carboxyhept-1'-yloxy)benzylidene]-N-tert-butylamine N-oxide), for the trapping of lipid radicals at a predetermined depth within biological membranes

We report on the development of the first member of a new family of EPR spin-trapping agents designed to trap radicals at a predetermined depth within biological membranes. By analogy to the use of nitroxide spin labels to 'report' on the environment at specific depths within biological membranes, we set out to prepare similar reporter molecules, but with a nitrone in place of the nitroxide function. The prototype compounds were tested in a model system consisting of large unilamellar vesicles exposed to a copper-dependent radical generating system. This entailed the reduction of tert-butylhydroperoxide to the tert-butoxyl radical ((t)BuO(.-)) by a membrane-permeable Cu(I) complex, which was generated in situ by reduction of the Cu(II) complex by ascorbate. To assist in the identification of the radicals detected, preliminary studies were performed in methanolic solution, where the major radical trapped was shown to be (.-)CH(2)OH, resulting from H-atom abstraction from the alcohol by (t)BuO(.-). This conclusion was shown to be in agreement with predictions based on chemical kinetics, which were then used to support the proposal that the primary species trapped in the lipid vesicles were radicals derived from membrane fatty acids. This molecule represents the first of a new generation of spin traps which, through modification, can be used to position the radical-trapping nitrone moiety at chosen depths within biological membranes.     

Behavior of chemical reaction catalyzed by allosteric enzyme: threshold phenomena and discontinuous transition to oscillatory state

A theoretical study is made on a chemical reaction system catalyzed by an allosteric protein, especially on its behavior in far-from-equilibrium situations. The reaction system, which was introduced in a previous paper, consists of two chemical species, S and P, and an allosteric enzyme, E, which catalyzes the reaction of interconversion between them. This system is kept far-from-equilibrium by an interaction with its environment. This interaction is characterized by four parameters. For certain values of the parameters, the system was previously shown to have multiple steady states. In the present paper it is shown that a sustained oscillation takes place in a certain region of the control-parameter space. On one part of the boundary of this region, the system undergoes a discontinuous transition from a steady state to a state oscillating with finite amplitude, while on the other part of the boundary the amplitude of oscillation is vanishingly small right after the transition. It is also shown that this system exhibits a threshold phenomenon. A few possible mechanisms are discussed by which the assumed interaction of the system with its environment can be realized.     

Analysis of the electron transfer from Pheo to QA in PS II membrane fragments from spinach by time resolved 325 nm absorption changes in the picosecond domain

Absorption changes at 325 nm (delta A325) induced by 15 ps laser flashes (lambda = 650 nm) in PS II membrane fragments were measured with picosecond time-resolution. In samples with the reaction centers (RCs) kept in the open state (P I QA) the signals are characterized by a very fast rise (not resolvable by our equipment) followed by only small changes within our time window of 1.6 ns. In the closed state (PI QA-) of the reaction center the signal decays with an average half-life time of about 250 ps. It is shown that under our excitation conditions (E = 2 x 10(14) photons/cm2 per pulse) subtraction of the absorption changes in closed RCs (delta A closed 325) from those in open RCs (delta A open 325) leads to a difference signal which is dominated by the reduction kinetics of QA. From the rise kinetics of this signal and by comparison with data in the literature it is inferred that QA becomes reduced by direct electron transfer from Pheo- with a time constant of about 350 +/- 100 ps.     

Slow oscillations in blood pressure via a nonlinear feedback model

Blood pressure is well established to contain a potential oscillation between 0.1 and 0.4 Hz, which is proposed to reflect resonant feedback in the baroreflex loop. A linear feedback model, comprising delay and lag terms for the vasculature, and a linear proportional derivative controller have been proposed to account for the 0.4-Hz oscillation in blood pressure in rats. However, although this model can produce oscillations at the required frequency, some strict relationships between the controller and vasculature parameters must be true for the oscillations to be stable. We developed a nonlinear model, containing an amplitude-limiting nonlinearity that allows for similar oscillations under a very mild set of assumptions. Models constructed from arterial pressure and sympathetic nerve activity recordings obtained from conscious rabbits under resting conditions suggest that the nonlinearity in the feedback loop is not contained within the vasculature, but rather is confined to the central nervous system. The advantage of the model is that it provides for sustained stable oscillations under a wide variety of situations even where gain at various points along the feedback loop may be altered, a situation that is not possible with a linear feedback model. Our model shows how variations in some of the nonlinearity characteristics can account for growth or decay in the oscillations and situations where the oscillations can disappear altogether. Such variations are shown to accord well with observed experimental data. Additionally, using a nonlinear feedback model, it is straightforward to show that the variation in frequency of the oscillations in blood pressure in rats (0.4 Hz), rabbits (0.3 Hz), and humans (0.1 Hz) is primarily due to scaling effects of conduction times between species.     

Reactions chimiques couplees considerees en tant que filtres passe-bande

An investigation is made as to whether or not the existence of a band-pass filter function, analogous to that in electronics, can be proved from the fundamental laws of chemical kinetics. The problem is important for better understanding of the preference of certain biological rhythms to others. It is shown with simple examples that such behavior is possible for a number of systems of coupled chemical reactions far enough from the thermodynamic equilibrium. It is of interest to generalize this behavior since it could conceivably play a role in the transmission of “usable information” in biology.      

3D Confocal Laser Scanning Microscopy for Quantification of the Phase Behaviour in Agarose-MCC co-gels in Comparison to the Rheological Blending-law Analysis

The need for a rapid and direct alternative to the rheology-based blending laws in quantifying phase behaviour in biopolymer composite gels is explored in this study. In doing so, the efficacy of confocal laser scanning microscopy (CLSM) paired with image analysis software – FIJI and Imaris - in quantifying phase volume was studied. That was carried out in a model system of agarose with varying concentrations of microcrystalline cellulose (MCC) in comparison to the rheological blending laws. Structural studies performed using SEM, FTIR, differential scanning calorimetry and dynamic oscillation in-shear unveiled a continuous, weak agarose network supporting the hard, rod-shaped MCC inclusions where the composite gel strength increased with higher ‘filler’ concentration. The phase volumes of MCC, estimated with the microscopic protocol, matched the predictions obtained from computerized modelling using the Lewis-Nielsen blending laws. Results highlight the suitability of the microscopic protocol in estimating the water partition and effective phase volumes in the agarose-MCC composite gel.

     

Non-equilibrium thermodynamics of marginal- and conditional-open chemical systems

Every open chemical system treated in this paper is restricted to the case involving a sequence of monomolecular reactions. Various kinds of probability distribution governing it are introduced according to the situations in which it is placed. The chemical system subject to marginal distribution is given the term marginal-open system MO. The open chemical system ō discussed by Nicolis and Babloyantz can be regarded as the limiting system of MO. For an open chemical system, itself in contact with an external reservoir of finite volume, the probability distribution conditioned on the marginal distribution for the external reservoir in an arbitrarily fixed state is more appropriate. Such an open chemical system is called a conditional-open system CO. However, in the case of the external reservoir of infinite volume, although it is not certainly trivial, another conditional probability distribution has to be proposed; it is derived on the hypothesis that the probability distribution for an arbitrary total number of molecules in the open chemical system is known. The open chemical system so specified is called conditional-open system $\text{C}\text{O}\text{?}$. It is shown that for each system $\text{MO, CO}\text{and}\text{C}\text{O}\text{?}$ the change of entropy starting from the steady state provides a Liapunov function under some conditions and that the steady state is asymptotically stable. The relation of the entropy change to non-equilibrium fluctuations of chemical components in each system is discussed in comparison with that in the corresponding open chemical system ō, for which the steady state surely exists and is always stable. It is shown that the concept of $\text{C}\text{O}\text{?}$ is useful for investigating the phenomenon of steady-state coupling.     

Quaternary structure and geminate recombination in hemoglobin: flow-flash studies on alpha 2CO beta 2 and alpha 2 beta 2CO.

The kinetics of geminate recombination for the diliganded species alpha 2CO beta 2 and alpha 2 beta 2CO of human hemoglobin were studied using flash photolysis. The unstable diliganded species were generated just before photolysis by chemical reduction in a continuous flow reactor from the more stable valency hybrids alpha 2CO beta 2+ and alpha 2+ beta 2CO, which could be prepared by high pressure liquid chromatography. Before the flash photolysis studies, the hybrids had been characterized by double-mixing stopped-flow kinetics experiments. At pH 6.0 in the presence of inositol hexaphosphate (IHP) both of the diliganded species show second order kinetics for overall addition of a third CO that is clearly characteristic of the T state (l' = 1-2 x 10(5) M-1 s-1), whereas at higher pH and in the absence of IHP they show combination rates characteristic of an R state. The kinetics of geminate recombination following photolysis of a bound CO, however, showed little dependence on pH and IHP concentration. This surprising observation is explained on the basis that the kinetics of geminate recombination of CO primarily depends on the tertiary structure of the ligand binding site, which apparently does not differ much between the R state and the liganded T state formed on adding IHP in this system. Since this explanation requires distinguishing different tertiary structures within a particular quaternary structure, it amounts to a contradiction to the two-state allosteric model.     

Graph-theoretic methods for the analysis of chemical and biochemical networks. I. Multistability and oscillations in ordinary differential equation models

A chemical mechanism is a model of a chemical reaction network consisting of a set of elementary reactions that express how molecules react with each other. In classical mass-action kinetics, a mechanism implies a set of ordinary differential equations (ODEs) which govern the time evolution of the concentrations. In this article, ODE models of chemical kinetics that have the potential for multiple positive equilibria or oscillations are studied. We begin by considering some methods of stability analysis based on the digraph of the Jacobian matrix. We then prove two theorems originally given by A. N. Ivanova which correlate the bifurcation structure of a mass-action model to the properties of a bipartite graph with nodes representing chemical species and reactions. We provide several examples of the application of these theorems.     

Migration of enzymes on linear polymers

We present a simple model based on the kinetics of DNA-dependent ATPases where the probability for enzyme migration on the linear activating polymer depends on the rate equations at the steady state. It is shown how the chemical velocity of the reaction is correlated to the average kinematic velocity along the polymer. The implications of this result are discussed.