Conventional Rate Equations Research Articles

Kinetics of diffusion-influenced multisite phosphorylation with enzyme reactivation.

The simplest way to account for the influence of diffusion on the kinetics of multisite phosphorylation is to modify the rate constants in the conventional rate equations of chemical kinetics. We have previously shown that this is not enough and new transitions between the reactants must also be introduced. Here we extend our results by considering enzymes that are inactive after modifying the substrate and need time to become active again. This generalization leads to a surprising result. The introduction of enzyme reactivation results in a diffusion-modified kinetic scheme with a new transition that has a negative rate constant. The reason for this is that mapping non-Markovian rate equations onto Markovian ones with time-independent rate constants is not a good approximation at short times. We then developed a non-Markovian theory that involves memory kernels instead of rate constants. This theory is now valid at short times, but is more challenging to use. As an example, the diffusion-modified kinetic scheme with new connections was used to calculate kinetics of double phosphorylation and steady-state response in a phosphorylation-dephosphorylation cycle. We have reproduced the loss of bistability in the phosphorylation-dephosphorylation cycle when the enzyme reactivation time decreases, which was obtained by particle-based computer simulations.

Neural network modeling of bismuth-doped fiber amplifier

Bismuth-doped fiber amplifiers offer an attractive solution for meeting continuously growing enormous demand on the bandwidth of modern communication systems. However, practical deployment of such amplifiers require massive development and optimization efforts with the numerical modeling being the core design tool. The numerical optimization of bismuth-doped fiber amplifiers is challenging due to a large number of unknown parameters in the conventional rate equations models. We propose here a new approach to develop a bismuth-doped fiber amplifier model based on a neural network purely trained with experimental data sets in E- and S-bands. This method allows a robust prediction of the amplifier operation that incorporates variations of fiber properties due to manufacturing process and any fluctuations of the amplifier characteristics. Using the proposed approach the spectral dependencies of gain and noise figure for given bi-directional pump currents and input signal powers have been obtained. The low mean (less than 0.19 dB) and standard deviation (less than 0.09 dB) of the maximum error are achieved for gain and noise figure predictions in the 1410–1490 nm spectral band.

Estimation of glomerular filtration rate in patients with cirrhosis: evaluation of equations currently used in clinical practice and validation of Royal Free Hospital cirrhosis glomerular filtration rate

Conventional creatinine-based glomerular filtration rate (GFR) equations have been reported to overestimate renal function in patients with cirrhosis. The Royal Free Hospital (RFH) cirrhosis GFR equation was developed to accurately estimate GFR in this population. The aim of this study was to evaluate the ability of widely available equations [Chronic Kidney Disease Epidemiology Collaboration equation (CKD-EPI), Modification of Diet in Renal Disease equations (MDRD-4, MDRD-6)] and the RFH equation to correctly estimate the GFR of patients with cirrhosis. We retrospectively analyzed data from patients with cirrhosis who underwent measurement of GFR with the use of 51Cr-EDTA (GFR-M). The CKD-EPI, MDRD-4, MDRD-6 and RFH equations were calculated, while bias, precision and accuracy were estimated for each one of them and then compared with paired t-tests. Bias was defined as the mean difference between the GFR-M and the result of each equation; precision was defined as the SD of the differences and accuracy was defined as the square root of the mean squared error (mean of the squared differences). Higher values are associated with worse bias and better precision/accuracy. One-hundred and thirty-four cirrhotic patients were included. Bias was estimated for CKD-EPI, MDRD-4, MDRD-6 and RFH at -5.91, -3.13, 0.92 and 18.24, respectively. Significant differences were observed between all equations (P < 0.001). Regarding precision, only the comparison between MDRD-4 (20.81) and RFH (16.6) yielded a statistically significant result (P = 0.037). Finally, CKD-EPI (19.32) and MDRD-6 (18.81) exhibited better accuracy than GFR-RFH (24.61) (P = 0.006 and 0.001). RFH demonstrates inferior accuracy in predicting renal function in patients with cirrhosis, in comparison to conventional equations.

Luminescence and Tm3+-to-Dy3+ energy transfer in TeO2:ZnO glass under NIR/UV excitation

A series of Tm3+/Dy3+ co-doped TeO2:ZnO glass samples were synthetized via fusion method and frequency up- and down-conversion (FUC and FDC) luminescence properties was investigated. A characterization through Raman spectrum and optical absorption of the glass host was first performed, followed by optical absorption and excitation spectra of the Tm3+/Dy3+ doped glass samples. Multi-wavelength light emission was observed in the blue (455 and 480 nm), yellow (580 nm), red (655 nm), and NIR (750, 800, 1400, and 1600 nm) regions, generated via up- or down-conversion, exploiting either NIR (1000–1240 nm) or UV (355 nm) excitation sources, respectively. The excitation routes of both FUC and FDC were proposed. The quasi-resonant overlap of Tm3+ and Dy3+ absorption, excitation, and luminescence spectra in the NIR and UV-VIS spectral regions led to efficient Tm3+-to-Dy3+ energy-transfer processes (Tm3+ [3F4] + Dy3+ [6H15/2] → Tm3+ [3H6] + Dy3+ [6H11/2], Tm3+ [1G4] + Dy3+ [6H15/2] → Tm3+ [3H6] + Dy3+ [4F9/2], and Tm3+ [3H4] + Dy3+ [6H15/2] → Tm3+ [3H6] + Dy3+ [6F5/2]). Time resolve fluorescence confirm Tm3+ ions as donors in the energy-transfer mechanism. The dependence of the luminescence and efficiency of the energy-transfer process upon concentration was evaluated using conventional rate equations.

Highly radiative nature of ultra-thin c-plane Al-rich AlGaN/AlN quantum wells for deep ultraviolet emitters

Temperature-dependent photoluminescence (PL) lifetimes were measured for a series of ultra-thin c-plane Al0.61Ga0.39N/AlN multiple quantum wells (QWs) on bulk AlN substrates with the well thickness varying from 0.6 to 2 nm. At temperatures below 75 K, estimates of the internal quantum efficiency indicate that the recombination is primarily radiative, with a lifetime of ∼160 ps for the 0.6 nm QWs, comparable to the low temperature PL lifetime observed in bulk AlGaN films of a similar Al content. This short lifetime is observed despite the presence of layer thickness fluctuations and the quantum-confined Stark effect associated with the large polarization field in the heterostructures, which tend to increase the radiative lifetime. This behavior is explained using many-body calculations of radiative recombination rates that extend beyond the conventional ABC rate equation model by accounting for both excitons and free carriers within a nonequilibrium Green's function formalism. The results indicate that the combination of the large wave function overlap integral (∼0.65) and exciton binding energy (1.82 times the 3D Rydberg) for the 0.6 nm QWs leads to an ∼20-fold increase in the radiative recombination rate relative to that obtained for the 2 nm QWs. This greater radiative recombination rate competes favorably with trapping at interface fluctuations and defect-induced nonradiative recombination that dominates recombination at higher temperatures.

Determination of internal quantum efficiency in GaInN-based light-emitting diode under electrical injection: carrier recombination dynamics analysis

We proposed a method to determine the internal quantum efficiency (IQE) of GaInN-based light-emitting diode (LED). For the accurate determination thereof, we carefully reviewed a conventional carrier rate equation and then proposed a set of advanced formulae, which can comprehensively explain the carrier dynamics in a modern GaInN-based LED. Based on our proposed formula, this convenient method to determine the IQE is presented. Next, to identify the proposed carrier rate equations and recombination dynamics, we investigated carrier lifetime. We also discuss the physical origins of IQE degradation in a modern GaInN-based LED such as efficiency droop and green gap.

Modes, stability, and small-signal response of photonic crystal Fano lasers.

Photonic crystal Fano lasers have recently been realised experimentally, showing useful properties such as pinned single-mode lasing and passive pulse generation. Here the fundamental properties of the modes of the Fano laser are analysed, showing how the laser functionality depends sensitively on the system configuration. Furthermore the laser stability is investigated and linked to the small-signal response, which shows additional dynamics that cannot be explained with a conventional rate equation model, including a damping of relaxation oscillations and a frequency modulation bandwidth that is only limited by the nanocavity response.

Theory of Self‐pulsing in Photonic Crystal Fano Lasers

AbstractLaser self‐pulsing was a phenomenon exclusive to macroscopic lasers until recently, where self‐starting laser pulsation in a microscopic photonic crystal Fano laser was reported. In this paper a theoretical model is developed to describe the Fano laser, including descriptions of the highly‐dispersive Fano mirror, the laser frequency and the threshold gain. The model is based upon a combination of conventional laser rate equations and coupled‐mode theory. The dynamical model is used to demonstrate how the laser has two regimes of operation, continuous‐wave output and self‐pulsing, and these regimes are characterised using phase diagrams, establishing the regime of self‐pulsing numerically. Furthermore, the physics behind the self‐pulsing mechanism are explained in detail and it is demonstrated how cavity absorption makes the Fano mirror function as a saturable absorber, leading to Q‐switched pulse generation. A stability analysis is used to demonstrate how the dominant mechanism of instability is relaxation oscillations becoming un‐damped. Finally the effect of varying key self‐pulsing parameters is investigated by characterisation of the change in self‐pulsing regions.

Theory for voltage modulation of transistor lasers using Franz-Keldysh absorption in the presence of optoelectronic feedback.

Compared with typical diode lasers (DLs), transistor lasers (TLs) support not only current-controlled but also voltage-controlled modulation. In this work, we theoretically investigate the small-signal voltage modulation of TLs based on the Franz-Keldysh (F-K) absorption and related optoelectronic feedback. In addition to the conventional rate equations relevant to DLs, our model physically includes various F-K effects. An optically induced current due to the F-K absorption may dramatically alter the voltage response of TLs. A model composed of the intrinsic optical response and an electrical transfer function which is fed back by this optical response is proposed to explain the true behaviors of voltage modulation in TLs.

Silicon micro-ring tunable laser for coherent optical communication.

A compact external cavity tunable laser based on a silicon hybrid micro-ring resonator is demonstrated. A theoretical model is also employed for design and analysis of the wavelength tuning performance of the device. In this model, the gain section of the device is simulated by a conventional multimode rate equation model, whereas all rest passive sections are modeled by the frequency domain method. Experimental results have shown that the output power of this device can reach 29 mW, with a linewidth less than 150 kHz. The tuning range is more than 17 nm in C-band with 60 dB side-mode-suppression-ratio (SMSR). This device shows a comparable performance with the commercial narrow linewidth laser as the source in coherent transmission systems.

Analysis of amplification properties of a photonic crystal fiber

In this work, effects of geometrical parameters and dopant concentration on amplification performance of Yb doped photonic crystal fiber amplifiers (YDPCFAs) are investigated. To this end, numerical results obtained from fully vectorial effective index method, are combined with conventional rate equations. The variation of the normalized frequency of the photonic crystal fiber with respect to air filling factor for different air hole spacing are depicted. Using obtained results, the single-mode region at signal and pump wavelengths are determined. In addition, the influence of dopant concentration on the maximum gain and optimum length are investigated. It is shown that by changing photonic crystal fiber cross-section parameters such as pitch and air filling factor, rather than increasing dopant concentration, one can obtain a higher gain at shorter optimum length. These results are useful for the design of YDPCFAs with less nonlinearity effects such as cooperative up conversion and Yb3+ ion quenching.

A simple and accurate dynamical modeling of quantum-dot semiconductor optical amplifiers

SUMMARY In this article, quantum-dot semiconductor optical amplifiers (QD-SOAs) have been modelled using state space method. To derive this model, we have manipulated the rate equation model of the QD-SOA, where the average values of the occupation probabilities along the QD-SOA cavity are considered as the state variables of the system. Using these variables, the distance dependence of the rate equations is eliminated. The derived state space model gives the optical gain and output signal of the amplifier with a high accuracy. Simulation results show that the derived model is not only much simpler and faster than conventional rate equation models, but also the optical gain and output signal of the investigated QD-SOA are calculated with a higher precision compared to the rate equation model. Copyright © 2013 John Wiley & Sons, Ltd.

Analytical approach to the multi-state lasing phenomenon in quantum dot lasers

We introduce an analytical approach to describe the multi-state lasing phenomenon in quantum dot lasers. We show that the key parameter is the hole-to-electron capture rate ratio. If it is lower than a certain critical value, the complete quenching of ground-state lasing takes place at high injection levels. At higher values of the ratio, the model predicts saturation of the ground-state power. This explains the diversity of experimental results and their contradiction to the conventional rate equation model. Recently found enhancement of ground-state lasing in p-doped samples and temperature dependence of the ground-state power are also discussed.

Simple form of multimode laser diode rate equations incorporating the band filling effect

To derive a simple form of the multimode laser diode rate equations incorporating the band filling effect, the laser diode gain in the direct bandgap model is introduced into the conventional multimode laser diode rate equations. By numerically examining each modal gain under the gain-switching condition, it is found that both the differential gain coefficient and the carrier density at transparency show an approximately linear dependency on the oscillation frequency. As a result, it is possible to derive a simple form of the multimode laser diode rate equations with linearized gain, which can be used to simulate the behaviors of a gain-switched laser diode characterized by the band filling effect, in both the multimode and single-mode oscillation cases.

An effective rate equation approach to reaction kinetics in small volumes: Theory and application to biochemical reactions in nonequilibrium steady-state conditions

Chemical master equations provide a mathematical description of stochastic reaction kinetics in well-mixed conditions. They are a valid description over length scales that are larger than the reactive mean free path and thus describe kinetics in compartments of mesoscopic and macroscopic dimensions. The trajectories of the stochastic chemical processes described by the master equation can be ensemble-averaged to obtain the average number density of chemical species, i.e., the true concentration, at any spatial scale of interest. For macroscopic volumes, the true concentration is very well approximated by the solution of the corresponding deterministic and macroscopic rate equations, i.e., the macroscopic concentration. However, this equivalence breaks down for mesoscopic volumes. These deviations are particularly significant for open systems and cannot be calculated via the Fokker-Planck or linear-noise approximations of the master equation. We utilize the system-size expansion including terms of the order of Omega(-1/2) to derive a set of differential equations whose solution approximates the true concentration as given by the master equation. These equations are valid in any open or closed chemical reaction network and at both the mesoscopic and macroscopic scales. In the limit of large volumes, the effective mesoscopic rate equations become precisely equal to the conventional macroscopic rate equations. We compare the three formalisms of effective mesoscopic rate equations, conventional rate equations, and chemical master equations by applying them to several biochemical reaction systems (homodimeric and heterodimeric protein-protein interactions, series of sequential enzyme reactions, and positive feedback loops) in nonequilibrium steady-state conditions. In all cases, we find that the effective mesoscopic rate equations can predict very well the true concentration of a chemical species. This provides a useful method by which one can quickly determine the regions of parameter space in which there are maximum differences between the solutions of the master equation and the corresponding rate equations. We show that these differences depend sensitively on the Fano factors and on the inherent structure and topology of the chemical network. The theory of effective mesoscopic rate equations generalizes the conventional rate equations of physical chemistry to describe kinetics in systems of mesoscopic size such as biological cells.

Nonlinear Photoemission Properties of Carbon Nanotubes: Behavior of High-Density Excitons in One Dimension

This paper describes nonlinear photoemission properties of carbon nanotubes both in terms of experiment and theory. In experiments, we performed photoluminescence excitation spectroscopy where carbon nanotubes were excited with high-intensity optical pulses. We found that, as the intensity of excitation pulses increases, all the photoemission peaks from different structure carbon nanotubes showed clear saturation in intensity. Each peak exhibited a saturation value that was independent of excitation wavelength, indicating that there is an upper limit on the density of excitons. We propose that this behavior is a result of efficient exciton-exciton annihilation through which excitons decay nonradiatively. In order to explain the experimental results, we have developed a model taking into account diffusion-limited limited exciton-exciton annihilation and spontaneous decays of excitons in one dimension. The solution of the model reproduced the experimental results well, from which the exciton densities in carbon nanotubes were estimated. The validity of the model was confirmed by its comparison with Monte Carlo simulations. We also show that the conventional rate equation for exciton-exciton annihilation fails to fit the experimentally observed saturation behaviors, especially at high excitonic densities. Finally, we discuss possible influences of the existence of the upper density limit on some photoelectric applications of carbon nanotubes.

Existence of an upper limit on the density of excitons in carbon nanotubes by diffusion-limited exciton-exciton annihilation: Experiment and theory

Through an investigation of photoemission properties of highly photoexcited single-walled carbon nanotubes, we demonstrate that there is an upper limit on the achievable excitonic density. As the intensity of optical excitation increases, all photoluminescence emission peaks arising from different chirality single-walled carbon nanotubes showed clear saturation in intensity. Each peak exhibited a saturation value that was independent of the excitation wavelength, indicating that there is an upper limit on the excitonic density for each nanotube species. We propose that this saturation behavior is a result of efficient exciton-exciton annihilation through which excitons decay nonradiatively. In order to explain the experimental results and obtain excitonic densities in the saturation regime, we have developed a model, taking into account the generation, diffusion-limited exciton-exciton annihilation, and spontaneous decays of one-dimensional excitons. Using the model, we were able to reproduce the experimentally obtained saturation curves under certain approximations, from which the excitonic densities were estimated. The validity of the model was confirmed through comparison with Monte Carlo simulations. Finally, we show that the conventional rate equation for exciton-exciton annihilation fails to fit the experimentally observed saturation behavior, especially at high excitonic densities.

Modeling the Injection-Locked Behavior of a Quantum Dash Semiconductor Laser

Using the conventional rate equations describing an injection-locked system, a novel modulation response function is derived, which implicitly incorporates nonlinear gain through the free-running relaxation oscillation frequency and damping rate of the slave laser. In this paper, it is shown that the model presented can be used to extract the characteristic parameters of the coupled system from experimental data. The number of fitting parameters in the model is reduced by determining the fundamental slave parameters through the conventional free-running response function; these parameters are considered to be constant during the curve-fitting of the injection-locked system. Furthermore, in order to reduce the number of possible solutions generated during the least-squares-fitting process, the remaining fitting parameters are tightly constrained based on the physical limits of the coupled system. By reducing the number of unknown fitting parameters and constraining the remaining terms, a stronger confidence in the extracted parameters is achieved. Using a series of response curves measured from an injection-locked quantum dash laser, characteristic parameters of the system are extracted and validity of the model is examined. The verified model is used to analyze the impact of the linewidth enhancement factor on the characteristics of the response function in the microwave domain.

An Analytical Model of the Intracavity Optical Second Harmonic Generation in a Vertical-External-Cavity Surface-Emitting Laser

An analysis model of an intracavity optical second-harmonic generation in a vertical-external-cavity surface-emitting laser (VECSEL) is presented. The model is based on the conventional semiconductor rate equations for the carrier and the photon combined with the second harmonic generation process. The parameters used in the model are obtainable from experimental data such as power transfer curves, output spectrum, and cavity configuration. The accuracy of our model is confirmed by comparing experimental and theoretical results. The influences of physically adjustable parameters in the VECSEL on both the lasing performance and the second harmonic generation were thoroughly investigated by utilizing the proposed analysis model.

Effect of linear temperature dependence of thermoelectric properties on energy conversion efficiency

New thermal rate equations were developed by taking the temperature dependences of the electrical resistivity ρ and thermal conductivity κ of the thermoelectric (TE) materials into the thermal rate equations on the assumption that they vary linearly with temperature T. The relative energy conversion efficiency η/ η 0 for a single TE element was formulated by approximate analysis, where η and η 0 are the energy conversion efficiencies derived from the new and conventional thermal rate equations, respectively. Applying it to Si–Ge alloys, the temperature dependence of ρ is stronger than that of κ, so the former has a more significant effect on η/ η 0 than the latter. However, the degree of contribution from both of them to η/ η 0 was a little lower than 1% at the temperature difference Δ T of 600 K. When the temperature dependence of κ was increased to become equal to that of ρ, however, it was found that η/ η 0 is increased by about 10% at Δ T = 600 K. It is clarified here that the temperature dependences of ρ and κ are also important factors for an improvement in η.