Minimum size for a nanoscale temperature discriminator based on a thermochemical system.

What are the limits of size reduction for information processing devices based on chemical reactions? In this paper, we partially answer this question. We show that a thermochemical system can be used to design a discriminator of the parameters associated with oscillations of the ambient temperature. Depending on the amplitude and frequency of the oscillations, the system exhibits sharp transitions between different types of its time evolutions. This phenomenon can be used to discriminate between different parameter values describing the oscillating environment. We investigate the reliability of the thermochemical discriminator as a function of the number of molecules involved in the reactions. A stochastic model of chemical reactions and heat exchange with the neighborhood, in which the number of molecules explicitly appears, is introduced. For the selected values of the parameters, thermochemical discriminators operating with less than 10(5) molecules appear to be unreliable.

Identification of two-step chemical mechanisms using small temperature oscillations and a single tagged species.

In order to identify two-step chemical mechanisms, we propose a method based on a small temperature modulation and on the analysis of the concentration oscillations of a single tagged species involved in the first step. The thermokinetic parameters of the first reaction step are first determined. Then, we build test functions that are constant only if the chemical system actually possesses some assumed two-step mechanism. Next, if the test functions plotted using experimental data are actually even, the mechanism is attributed and the obtained constant values provide the rate constants and enthalpy of reaction of the second step. The advantage of the protocol is to use the first step as a probe reaction to reveal the dynamics of the second step, which can hence be relieved of any tagging. The protocol is anticipated to apply to many mechanisms of biological relevance. As far as ligand binding is considered, our approach can address receptor conformational changes or dimerization as well as competition with or modulation by a second partner. The method can also be used to screen libraries of untagged compounds, relying on a tracer whose concentration can be spectroscopically monitored.

Temperature-driven coherence resonance and stochastic resonance in a thermochemical system.

We perform the stochastic analysis of a thermochemical system using a master equation which describes a chemical reaction and includes discrete and continuous temperature jumps. We study the time evolution of the system selecting the temperature of the thermostat as an easily tunable control parameter. Depending on the thermostat temperature, the system can be in an excitable, oscillatory, or stationary regime. Stochastic time series for the system temperature are generated and the distributions of interspike intervals are analyzed in the three dynamical regimes separated by a homoclinic bifurcation and a Hopf bifurcation. Different constructive roles of internal fluctuations are exhibited. A noise-induced transition is observed in the vicinity of the Hopf bifurcation. Coherence resonance and stochastic resonance are found in the oscillatory regime. In a range of thermostat temperatures, a nontrivial behavior of the highly nonlinear system is revealed by the existence of both a minimum and a maximum in the scaled standard deviation of interspike intervals as a function of particle number. This high sensitivity to system size illustrates that controlling dynamics in nanoreactors may remain a difficult task.

Determination of reaction flux from concentration fluctuations near a Hopf bifurcation.

Small open chemical systems, typically associated with far-from-equilibrium, nonlinear stochastic dynamics, offer the appropriate framework to elucidate biological phenomena at the cellular scale. Stochastic differential equations of Langevin-type are employed to establish the relation between the departure from equilibrium and the time cross-correlation functions of concentration fluctuations for chemical species susceptible to oscillate. Except in the immediate vicinity of the Hopf bifurcation, the results are in agreement with simulations of the chemical master equation but always differ from the prediction obtained for linear deterministic dynamics. In general, the magnitude of the asymmetry of time correlation functions definitely depends on the reaction flux circulating in an open system but also on the details of the nonlinearities of deterministic dynamics.

Temporal cross-correlation asymmetry and departure from equilibrium in a bistable chemical system.

This paper aims at determining sustained reaction fluxes in a nonlinear chemical system driven in a nonequilibrium steady state. The method relies on the computation of cross-correlation functions for the internal fluctuations of chemical species concentrations. By employing Langevin-type equations, we derive approximate analytical formulas for the cross-correlation functions associated with nonlinear dynamics. Kinetic Monte Carlo simulations of the chemical master equation are performed in order to check the validity of the Langevin equations for a bistable chemical system. The two approaches are found in excellent agreement, except for critical parameter values where the bifurcation between monostability and bistability occurs. From the theoretical point of view, the results imply that the behavior of cross-correlation functions cannot be exploited to measure sustained reaction fluxes in a specific nonlinear system without the prior knowledge of the associated chemical mechanism and the rate constants.

Reaction-diffusion approach to prevertebrae formation: effect of a local source of morphogen.

Periodic structure formation is an essential feature of embryonic development. Many models of this phenomenon, most of them based on time oscillations, have been proposed. However, temporal oscillations are not always observed during development and how a spatial periodic structure is formed still remains under question. We investigate a reaction-diffusion model, in which a Turing pattern develops without temporal oscillations, to assess its ability to account for the formation of prevertebrae. We propose a correspondence between the species of the reaction scheme and biologically relevant molecules known as morphogens. It is shown that the model satisfactorily reproduces experiments involving grafting of morphogen sources into the embryos. Using a master equation approach and the direct simulation Monte Carlo method, we examine the robustness of the results to internal fluctuations.

Identification of two-step chemical mechanisms and determination of thermokinetic parameters using frequency responses to small temperature oscillations.

Increased focus on kinetic signatures in biology, coupled with the lack of simple tools for chemical dynamics characterization, lead us to develop an efficient method for mechanism identification. A small thermal modulation is used to reveal chemical dynamics, which makes the technique compatible with in cellulo imaging. Then, the detection of concentration oscillations in an appropriate frequency range followed by a judicious analytical treatment of the data is sufficient to determine the number of chemical characteristic times, the reaction mechanism, and the full set of associated rate constants and enthalpies of reaction. To illustrate the scope of the method, dimeric protein folding is chosen as a biologically relevant example of nonlinear mechanism with one or two characteristic times.

Particle dynamics simulations of Turing patterns.

The direct simulation Monte Carlo method is used to reproduce Turing patterns at the microscopic level in reaction-diffusion systems. In order to satisfy the basic condition for the development of such a spatial structure, we propose a model involving a solvent, which allows for disparate diffusivities of individual reactive species. One-dimensional structures are simulated in systems of various lengths. Simulation results agree with the macroscopic predictions obtained by integration of the reaction-diffusion equations. Additional effects due to internal fluctuations are observed, such as temporal transitions between structures of different wavelengths in a confined system. For a structure developing behind a propagating wave front, the fluctuations suppress the induction period and accelerate the formation of the Turing pattern. These results support the ability of reaction-diffusion models to robustly reproduce axial segmentation including the formation of early vertebrae or somites in noisy biological environments.

Chemical mechanism identification from frequency response to small temperature modulation.

The description of interactions between biochemical species and the elucidation of the corresponding chemical mechanisms encounter an increasing interest both for the comprehension of biological pathways at the molecular scale and for the rationalization of drug design. Relying on powerful experimental tools such as thermal microfluidics and fluorescence detection, we propose a methodology to determine the chemical mechanism of a reaction without fitting parameters. A mechanism consistent with the accessible knowledge is assumed, and the assumption is checked through an iterative protocol. The test is based on the frequency analysis of the response of a targeted reactive species to temperature modulation. We build specific functions of the frequency that are constant for the assumed mechanism and show that the graph of these functions can be drawn from appropriate data analysis. The method is general and can be applied to any complex mechanism. It is here illustrated in detail in the case of single relaxation time mechanisms.

Master equation for a bistable chemical system with perturbed particle velocity distribution function.

We present a modified master equation for a homogeneous gaseous reactive system which includes nonequilibrium corrections due to the reaction-induced perturbation of the particle velocity distribution function. For the Schlögl model, the modified stochastic approach predicts nonequilibrium-induced transitions between different dynamical regimes, including the transformation of a monostable system into a bistable one, and vice versa. These predictions are confirmed by the comparison with microscopic simulations using the direct simulation Monte Carlo method. Compared to microscopic simulations of the particle dynamics, the modified master equation approach proves to be much more efficient.

Master equation for a chemical wave front with perturbation of local equilibrium.

In order to develop a stochastic description of gaseous reaction-diffusion systems, which includes a reaction-induced departure from local equilibrium, we derive a modified expression of the master equation from analytical calculations based on the Boltzmann equation. We apply the method to a chemical wave front of Fisher-Kolmogorov-Petrovsky-Piskunov type, whose propagation speed is known to be sensitive to small perturbations. The results of the modified master equation are compared successfully with microscopic simulations of the particle dynamics using the direct simulation Monte Carlo method. The modified master equation constitutes an efficient tool at the mesoscopic scale, which incorporates the nonequilibrium effect without need of determining the particle velocity distribution function.

Temperature modulation and quadrature detection for selective titration of two-state exchanging reactants.

Biological samples exhibit huge molecular diversity over large concentration ranges. Titrating a given compound in such mixtures is often difficult, and innovative strategies emphasizing selectivity are thus demanded. To overcome limitations inherent to thermodynamics, we here present a generic technique where discrimination relies on the dynamics of interaction between the target of interest and a probe introduced in excess. Considering an ensemble of two-state exchanging reactants submitted to temperature modulation, we first demonstrate that the amplitude of the out-of-phase concentration oscillations is maximum for every compound involved in a reaction whose equilibrium constant is equal to unity and whose relaxation time is equal to the inverse of the excitation angular frequency. Taking advantage of this feature, we next devise a highly specific detection protocol and validate it using a microfabricated resistive heater and an epifluorescence microscope, as well as labeled oligonucleotides to model species displaying various dynamic properties. As expected, quantification of a sought for strand is obtained even if interfering reagents are present in similar amounts. Moreover, our approach does not require any separation and is compatible with imaging. It could then benefit some of the numerous binding assays performed every day in life sciences.

Microscopic simulations of supersonic and subsonic exothermic chemical wave fronts and transition to detonation.

We perform microscopic simulations using the direct simulation Monte Carlo approach to an exothermic chemical wave front of Fisher-Kolmogorov, Petrovsky, Piskunov-type in a one-dimensional gaseous medium. The results confirm the existence of a transition from a weak detonation or deflagration to a Chapman-Jouguet detonation wave, that we already investigated at the macroscopic scale [G. Dumazer et al., Phys. Rev. E 78, 016309 (2008)]. In the domain of weak detonation or deflagration, the discrepancy between the propagation speeds deduced from the simulations and the macroscopic balance equations of hydrodynamics is explained by two microscopic effects, the discretization of the variables, known as cutoff effect, and the departure from local equilibrium. Remarkably, the propagation speed of a Chapman-Jouguet detonation wave is not sensitive to these perturbations of microscopic origin.

Determination of the six rate constants of a three-state enzymatic network and a noninvasive test of detailed balance.

The Michaelis-Menten mechanism is unanimously recognized by experimentalists and theoreticians as the reference model for the description of enzymatic catalysis. The recent explosion in the diversity of fluorescent probes solves the problem of in situ observation of proteins and the experimental investigation of enzymatic dynamics, which determines the Michaelis constant or a small number of relaxation times, is becoming more and more common. We propose a protocol for the full characterization of enzyme kinetics in the framework of the Michaelis-Menten mechanism. The method relies on the measurement of the oscillation amplitude of the enzymatic concentrations, when the biological medium is submitted to a temperature modulation of a few degrees. Analytical expressions of all the rate constants as functions of the concentration amplitudes are derived. The noninvasive character of the perturbation and the assessable uncertainty on the rate constant values make an in situ test of detailed balance possible.

Resonant response to temperature modulation for enzymatic dynamics characterization.

We consider enzymes involved in a three-state Michaelis-Menten kinetics and submitted to well-chosen temperature modulations of small amplitude. From the first-order amplitudes of concentration oscillations, we design a response function that is maximum for targeted values of the chemical relaxation times. This resonant function can be used to screen a large set of enzymes and identify the one governed by the desired kinetics. The method gives access to all the dynamical parameters of the targeted enzyme without resorting to a fit. We show how to estimate the precision of this parameter determination and give some hints for experimental validation.

Steady dynamics of exothermic chemical wave fronts in van der Waals fluids.

We study the steady dynamics of an exothermic Fisher-Kolmogorov-Petrovsky-Piskunov chemical wave front traveling in a one-dimensional van der Waals fluid. The propagating wave is initiated by a nonuniformity in reactant concentration contrary to usual combustion ignition processes. The heat release and activation energy of the reaction play the role of control parameters. We recently proved that the propagation of an exothermic chemical wave front in a perfect gas displays a forbidden interval of stationary wave front speeds [G. Dumazer, M. Leda, B. Nowakowski, and A. Lemarchand, Phys. Rev. E 78, 016309 (2008)]. We examine how this result is modified for nonideal fluids and determine the effect of the van der Waals parameters and fluid density on the bifurcation between diffusion flames and Chapman-Jouguet detonation waves as heat release increases. Analytical predictions are confirmed by the numerical solution of the hydrodynamic equations including reaction kinetics.

Transition between an exothermic chemical wave front and a generic flame.

We consider the two classes of exothermic chemical wave fronts, propagating toward a stable or an unstable steady state. The hydrodynamic equations for stream velocity, temperature, and concentrations are solved numerically for increasing values of the reaction heat. For a critical value of the heat release, we find a transition between a chemical front, whose speed depends on the chemical dynamics, and a generic flame, whose speed is entirely determined by heat release. We derive an analytical expression of the flame speed from the invariants of the hydrodynamic equations. This result substantiates macroscopic approaches widely used in combustion, in which the chemical models include only simplified reaction mechanisms.

Sensitivity of an exothermic chemical wave front to a departure from local equilibrium.

We study the propagation of an exothermic chemical wave front in a reactive dilute gas and show that the particle velocity distribution departs from the Maxwellian form in the front zone. The analytical corrections to the balance equations for concentrations, temperature, and stream velocity induced by the departure from local equilibrium are derived from a perturbative solution of the Boltzmann equation. Our analytical predictions of the front properties, including its propagation speed, compare well with microscopic simulations of the particle dynamics.

Fourier analysis to measure diffusion coefficients and resolve mixtures on a continuous electrophoresis chip.

We report a method to measure diffusion coefficients of fluorescent solutes in the 10(2)-10(6) Da molecular mass range in a glass-PDMS chip. Upon applying a permanent electric field, the solute is introduced through a narrow channel into a wide analysis chamber where it migrates along the injection axis and diffuses in two dimensions. The diffusion coefficient is extracted after 1D Fourier transform of the resulting stationary concentration pattern. Analysis is straightforward, requiring no numerical integration or velocity field simulation. The diffusion coefficients measured for fluorescein, rhodamine green-labeled oligonucleotides, and YOYO-1-stained dsDNA fragments agree with the literature values and with our own fluorescence correlation spectroscopy measurements. As shown for 151 and 1257 base pair dsDNA mixtures, the present method allows us to rely on diffusion to quantitatively characterize the nature and the composition of binary mixtures. In particular, we implement a DNA hybridization assay to illustrate the efficiency of the proposed protocol for library screening.

Forbidden interval of propagation speed for exothermic chemical fronts.

We consider the propagation of an exothermic chemical front toward an unstable steady state. The hydrodynamic equations are solved numerically for increasing values of the activation energy of the reaction which controls the reaction front speed. For a large speed, the marginal stability criterion of the isothermal case is recovered. For a small speed, we observe two well-separated traveling waves: a heat front is preceding the reaction front. We find analytically a forbidden speed interval where the hydrodynamical system does not admit stationary traveling solutions.