A method to determine quasi-steady state in constant voltage mode isotachophoresis

Identification of the steady state is very challenging in isotachophoresis (ITP); especially in complex microgeometries, such as dog-leg channels or cross-channel junctions. In this work, an elastic matching method is applied to determine the quasi-steady state in microscale ITP. In the elastic matching method, the similarity between two profiles is calculated by comparing intensity distribution of two images or profiles. To demonstrate this similarity-based analysis technique for ITP, a constant voltage mode ITP model is developed and applied to a five-component ITP system. Hydrochloric acid and caproic acid are used as the leader and terminator, respectively, while histidine is used as the counter-ion. Two sample components, acetic acid and benzoic acid, are separated under the action of an applied electric field in both straight and dog-leg microchannels. This analysis shows that conductivity profiles provide a better measure to determine the quasi-steady state in an ITP process. For a straight microchannel, the quasi-steady state is achieved in less than a minute with a total potential drop of 100?V in a 2?cm long channel. In a straight channel, a true steady state can be achieved for ITP with appropriate countercurrent flow where stationary zones are formed, but the time it takes to reach the steady state is much longer than the without counter flow case. The numerical results indicate that a steady state cannot be reached in a dog-leg microchannel because of sample dispersion and refocusing at and near the intersections and at the branch channels. However, the elastic matching method can be used to determine the quasi-steady state in a dog-leg microchannel.     

Mathematical analysis of the smallest chemical reaction system with Hopf bifurcation

Recently we presented an up to now unstudied three-dimensional dynamical system which is, according to our given definition, the smallest chemical reaction system with Hopf bifurcation. We here study the Hopf bifurcation in detail and prove that near the bifurcation point a stable limit cycle arises. In the analysis we use the methods of local bifurcation theory, especially the center manifold and the normal form theorem. In a similar way we analyse the also occurring transcritical bifurcation. Besides studying local stability, we give the proofs for global stability of the trivial steady state in the whole positive phase space and for the nontrivial steady state in a closed domain containing the steady state point.     

Exact steady state solutions in symmetrical Nernst–Planck–Poisson electrodiffusive models

We compare three one-dimensional Nernst–Planck–Poisson systems that describe ion distribution near the electrode surfaces with planar, cylindrical and spherical symmetry respectively. These three models take into account ion diffusion and migration. In particular they describe the diffusive layers formed by Li+ ions in the vicinity of the graphite electrode particles. The three types of symmetry appear due to three different ways of particle ordering inside the electrode. In this paper we construct the exact steady state solutions to these systems and approximate solutions in form of power series. Then we solve the systems numerically and compare the results. We discuss the influence of symmetry in electrode particle ordering on the steady state distribution of ions in the diffusive layer.     

EXCITED STATE pKa BEHAVIOUR OF DAPI. A RATIONALIZATION OF THE FLUORESCENCE ENHANCEMENT OF DAPI IN DAPI-NUCLEIC ACID COMPLEXES*

Abstract— The steady state and time resolved fluorescence of the drug and chromosomal staining agent, 4′,6-diamidino-2-phenylindole dihydrochloride, DAPI, was examined under different solvent conditions. In solutions between pH 3 and pH 9 the fluorescence spectral maximum of DAPI was found at 460 nm. The fluorescence decayed with double exponential kinetics, with decay times of 2.86 and 0.144 ns, at all wavelengths below 550 nm. At 550 nm single exponential decay kinetics with a lifetime of 0.153 ns was observed. The fluorescence spectrum could be resolved into two components, the 2.86 ns component having a spectral maximum near 450 nm and the 0.144 ns component having a spectral maximum near 490 nm. The results are rationalized in terms of there being two different configurations of DAPI, one of which undergoes a rapid protonation of the indole ring by proton transfer from the 6-amidinium group in the excited singlet state. The 0.144 ns component is assigned as the fluorescence from the excited state of the protonated indole ring. The results provide an explanation of the fluorescence enhancement in DAPI-nucleic acid complexes.     

Analytical solutions to mathematical models of the surface and bulk erosion of solid polymers

The process by which polymeric materials hydrolyze and disappear into their environments is often called erosion. Two types of erosion have been defined according to how the hydrolysis takes place. If hydrolysis occurs throughout the entire specimen at the same time, it is called bulk erosion. If the hydrolysis is mainly confined to a region near the surface of the specimen and the surface continuously degrades by moving inward, it is termed surface erosion. In this article, a kinetic relationship for bulk erosion is developed. This relationship provides a method for estimating the hydrolysis kinetic constants for bulk‐eroding polymers. This same relationship is also applicable to surface erosion at a microscopic level. Through its combination with a diffusion–reaction equation and the provision of moving boundary conditions, an analytical solution to the steady‐state surface‐erosion problem is obtained. The erosion rate, erosion front width, and induction time can all be expressed as simple functions of the rate of polymer bond hydrolysis, water diffusivity, and solubility, plus other parameters that can be experimentally determined. The erosion front width is the product of the induction time and the erosion rate. The ratio of the erosion front width to the polymer specimen thickness is a parameter that determines whether the specimen undergoes surface or bulk erosion. Theoretical results are compared with experimental observations from the literature, and agreement is found. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 383–397, 2005     

Theory on the rate equation of Michaelis–Menten type single-substrate enzyme catalyzed reactions

Analytical solution to the Michaelis–Menten (MM) rate equations for single-substrate enzyme catalysed reaction is not known. Here we introduce an effective scaling scheme and identify the critical parameters which can completely characterize the entire dynamics of single substrate MM enzymes. Using this scaling framework, we reformulate the differential rate equations of MM enzymes over velocity-substrate, velocity-product, substrate-product and velocity-substrate-product spaces and obtain various approximations for both pre- and post-steady state dynamical regimes. Using this framework, under certain limiting conditions we successfully compute the timescales corresponding to steady state, pre- and post-steady states and also compute the approximate steady state values of velocity, substrate and product. We further define the dynamical efficiency of MM enzymes as the ratio between the reaction path length in the velocity-substrate-product space and the average reaction time required to convert the entire substrate into product. Here dynamical efficiency characterizes the phase-space dynamics and it would tell us how fast an enzyme can clear a harmful substrate from the environment. We finally perform a detailed error level analysis over various pre- and post-steady state approximations along with the already existing quasi steady state approximations and progress curve models and discuss the positive and negative points corresponding to various steady state and progress curve models.     

Analysis of graphitized cast irons by optical emission spectroscopy: matrix effects in the glow discharge and the spark excitation

Graphite has a substantially lower sputtering rate in glow discharge than have the other structural components of graphitized cast irons, which leads to a structure-related matrix effect, consisting of an increasing relative surface coverage by graphite of the sample surface during the initial stage of the GD-OES analysis, and, consequently, to an increasing carbon signal intensity. This effect exists inherently in any multicomponent system with different sputtering rates of the components and should be taken into account in GD-OES quantification. A simple theory is presented to describe quantitatively the changes in relative contributions of different phases to the flux of the sputtered material entering the discharge and a formula is presented, expressing elemental intensity changes as a function of sputtering rates and stoichiometry of the structural components.After reaching the steady state, there are no substantial differences in the GD-OES signal response of the analyzed elements between the graphitic and the white cast irons. To reach this steady state, long preburn times and high sputtering rates have to be used. In the spark atomization/excitation, there are very strong and complex structure-related matrix effects, which make the analysis of graphitic cast irons by spark excitation impossible.     

Comparative studies of SIMS and SNMS analyses during the build up of sputter equilibrium under oxygen and rare gas ion bombardment

Sputtering of solid surfaces by using a focused ion beam is the basis for secondary ion mass spectrometry (SIMS) and sputtered neutral mass spectrometry (SNMS). The ion bombardment initiates not only redistribution of sample atoms but also massive changes in the surface and near surface composition of the bombarded area due to the sputter process and implantation of the primary ions. Changes in the matrix-composition affects the secondary ion yields and therefore a steady state (sputter equilibrium) has to be reached before SIMS data can give quantifiable results. SNMS is much less affected by those yield effects and therefore a combination of SIMS and SNMS can establish a basis for interpretation of SIMS data before the steady state is reached. In order to determine the effects of primary ion incorporation, we applied different primary ion species successively to generate different equilibria. An oxygen ion beam oxidizes the sample surface and by using a rare gas primary ion (PI) this oxide can be removed and analyzed.     

On the dependence of the existence of the positive steady states on the rate coefficients for deficiency-one mass action systems: single linkage class

The Deficiency-One Theorem states that there exists a unique positive steady state in each positive stoichiometric class for weakly reversible deficiency-one mass action systems with one linkage class (regardless of the values of the rate coefficients). The non-emptiness of the set of positive steady states does not remain valid if we omit the weak reversibility. A recently published paper provided an equivalent condition to the existence of a positive steady state for deficiency-one mass action systems that are not weakly reversible, but still has only one linkage class. Based on that result, we characterise in this paper those of these mass action systems for which the non-emptiness of the set of positive steady states holds regardless of the values of the rate coefficients. Also, we provide an equivalent condition to the existence of rate coefficients such that the set of positive steady states is nonempty for the resulting mass action system.     

Geometrical thermodynamic field theory

A manifestly covariant, coordinate independent reformulation of the thermodynamic field theory (TFT) is presented. The TFT is a covariant field theory that describes the evolution of a thermodynamic system, extending the near‐equilibrium theory established by Prigogine in 1954. We introduce the minimum rate of dissipation principle, which applies to any system relaxing toward a steady state. We also derive the thermodynamic field equations, which in the case of α–α and β–β processes have already appeared in the literature. In more general cases, the equations are notably simpler than those previously encountered, and they extend beyond the weak‐field regime. Finally, we derive the equations that determine the steady states as well as the critical values of the control parameters beyond which a steady state becomes unstable. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Potential and flux decomposition for dynamical systems and non-equilibrium thermodynamics: curvature, gauge field, and generalized fluctuation-dissipation theorem

The driving force of the dynamical system can be decomposed into the gradient of a potential landscape and curl flux (current). The fluctuation-dissipation theorem (FDT) is often applied to near equilibrium systems with detailed balance. The response due to a small perturbation can be expressed by a spontaneous fluctuation. For non-equilibrium systems, we derived a generalized FDT that the response function is composed of two parts: (1) a spontaneous correlation representing the relaxation which is present in the near equilibrium systems with detailed balance and (2) a correlation related to the persistence of the curl flux in steady state, which is also in part linked to a internal curvature of a gauge field. The generalized FDT is also related to the fluctuation theorem. In the equal time limit, the generalized FDT naturally leads to non-equilibrium thermodynamics where the entropy production rate can be decomposed into spontaneous relaxation driven by gradient force and house keeping contribution driven by the non-zero flux that sustains the non-equilibrium environment and breaks the detailed balance. On any particular path, the medium heat dissipation due to the non-zero curl flux is analogous to the Wilson lines of an Abelian gauge theory.     

Existence of a new emitting singlet state of proflavine: femtosecond dynamics of the excited state processes and quantum chemical studies in different solvents

Proflavine (3,6-diaminoacridine) shows fluorescence emission with lifetime, 4.6 ± 0.2 ns, in all the solvents irrespective of the solvent polarity. To understand this unusual photophysical property, investigations were carried out using steady state and time-resolved fluorescence spectroscopy in the pico- and femtosecond time domain. Molecular geometries in the ground and low-lying excited states of proflavine were examined by complete structural optimization using ab initio quantum chemical computations at HF/6-311++G** and CIS/6-311++G** levels. Time dependent density functional theory (TDDFT) calculations were performed to study the excitation energies in the low-lying excited states. The steady state absorption and emission spectral details of proflavine are found to be influenced by solvents. The femtosecond fluorescence decay of the proflavine in all the solvents follows triexponential function with two ultrafast decay components (τ(1) and τ(2)) in addition to the nanosecond component. The ultrafast decay component, τ(1), is attributed to the solvation dynamics of the particular solvent used. The second ultrafast decay component, τ(2), is found to vary from 50 to 215 ps depending upon the solvent. The amplitudes of the ultrafast decay components vary with the wavelength and show time dependent spectral shift in the emission maximum. The observation is interpreted that the time dependent spectral shift is not only due to solvation dynamics but also due to the existence of more than one emitting state of proflavine in the solvent used. Time resolved area normalized emission spectral (TRANES) analysis shows an isoemissive point, indicating the presence of two emitting states in homogeneous solution. Detailed femtosecond fluorescence decay analysis allows us to isolate the two independent emitting components of the close lying singlet states. The CIS and TDDFT calculations also support the existence of the close lying emitting states. The near constant lifetime observed for proflavine in different solvents is suggested to be due to the similar dipole moments of the ground and the evolved emitting singlet state of the dye from the Franck-Condon excited state.     

Exact solutions for the entropy production rate of several irreversible processes

We investigate thermal conduction described by Newton's law of cooling and by Fourier's transport equation and chemical reactions based on mass action kinetics where we detail a simple example of a reaction mechanism with one intermediate. In these cases we derive exact expressions for the entropy production rate and its differential. We show that at a stationary state the entropy production rate is an extremum if and only if the stationary state is a state of thermodynamic equilibrium. These results are exact and independent of any expansions of the entropy production rate. In the case of thermal conduction we compare our exact approach with the conventional approach based on the expansion of the entropy production rate near equilibrium. If we expand the entropy production rate in a series and keep terms up to the third order in the deviation variables and then differentiate, we find out that the entropy production rate is not an extremum at a nonequilibrium steady state. If there is a strict proportionality between fluxes and forces, then the entropy production rate is an extremum at the stationary state even if the stationary state is far away from equilibrium.     

Theory of chemically activated unimolecular reactions. Weak collisions and steady states

The prediction of decomposition/stabilization ratios for chemically activated unimolecular reactions is usually based on the assumption of steady state and irreversible deactivation. The levels below the activation energy E_o are treated as a sink. In the present work we assume that the reaction can be described by a weak-collision master equation. An exact analysis shows that one can expect three timescales: an initial transient, an intermediate steady state, and an asymptotic steady state. The intermediate steady state is well defined only if the eigenvalues of the corresponding thermal rate matrix can be separated into two groups one of which contains eigenvalues of substantially smaller magnitude. An efficient and accurate method for the estimation of intermediate steady-state populations and branching ratios is described. It avoids the irreversible deactivation assumption which is shown to lead to errors particularly for weak collisions. The new analysis is illustrated in a series of calculations for a model of the sec-butyl radical system. The numerical solutions are readily obtained by use of a recently developed equilibrium finite-basis-set method.     

Response analysis in biochemical chain reactions with negative feedforward and feedbackward loops

Finite difference equations can be used to study the responses of biochemical chain reactions at any step of the chain to an external stimulus. In this study, we developed mathematical models for two hypothetical chain reactions involving loops to study the responses in the chain as the length of the chain gets longer, so called transient and steady state responses. The first model is for a chain with a negative feedforward loop, and the second one is for a chain that has a negative feedback loop. Although both of the models have the same steady state equations and values, we showed that the chain with negative feedforward and negative feedback loops can produce significantly different behaviors. The former can bring the chain into oscillations with various periods and eventually chaos when the feedback is strong enough as the length of the reaction chain increases, whereas the latter is not capable of producing oscillations and more complicated dynamics.     

A multigrid method for N-component nucleation

A multigrid algorithm has been developed enabling more efficient solution of the cluster size distribution for N-component nucleation from the Becker-D?ring equations. The theoretical derivation is valid for an arbitrary number of condensing components, making the simulation of many-component nucleating systems feasible. A steady state ternary nucleation problem is defined to demonstrate its efficiency. The results are used as a validation for existing nucleation theories. The non-steady state ternary problem provides useful insight into the initial stages of the nucleation process. We observe that for the ideal mixture the main nucleation flux bypasses the saddle point.     

Near Activation and Differential Activation in Enzymatic Reactions

A framework that introduces the concept of near‐activation complexation and differential free energy of activation is presented to extend the capabilities of the classical transition‐state theory in enzymatic reactions. In our approach, reaching a near‐equilibrium energy level is assumed to be necessary for complexation near the activation point, whereas an additional differential energy level is required for the near‐equilibrium complex to activate and release reaction products. Integration of these energy levels within the transition‐state theory explains the thermodynamic nature of the Michaelis–Menten (affinity) constant and its relationship with the rate constant under the quasi‐steady‐state assumption. The concepts of near‐activation complexation and differential free energy of activation were tested on 57 independent experiments of and uptake by various microalgae and bacteria at temperatures ranging between 1 and 45°C. Results showed that near‐activation complexation was always favored, whereas the differential energy of activation led to an apparent energy barrier consistent with earlier observations. Temperature affected all energy levels within this framework but did not alter substantially their thermodynamic features. The approach (1) mutually links the thermodynamics and kinetics of Michaelis‐Menten and rate constants with a mathematical expression; (2) describes the likelihood of formation of sub‐, super‐, and activated complexes; and (3) shows direction and thermodynamic likelihood of each reaction branch within the transition state.     

Experimental study of embedding of attractors in concentration space

The embedding of attractors and their stable and unstable manifolds can be studied experimentally by controlled addition of chemical species to bring about a particular response. For stable small amplitude oscillations near a Hopf bifurcation from a steady state the embedding can be completely determined even in systems where two of the species are not observable. A quenching of the oscillations by dilution candetermine the steady state concentrations. A species that cannot quench the oscillations almost certainly cannot be an essential component of the oscillation. The method can be extended to a study of attractor associated with subharmonic and quasiperiodic bifurcations and of attractors corresponding to nonperiodic motion. We present preliminary results for a subharmonic bifurcation.     

Novel Methods for Controlling the Current During Isoelectric Focusing

While studying the effect of electrochemical reactions occurring at the electrodes on achievement of steady state in isoelectric focusing (IEF) we observed an abnormal increase of the current. Because the magnitude of the current determines the progress of IEF, knowledge gained from studies of its nature and generation may enable this effect to be controlled. We observed that addition of gelatin to the electrode solutions suppresses the magnitude of the current flowing through the system; this enables IEF to be performed under conditions closer to steady state, and steady state is achieved more quickly.