Using Row Reduced Echelon Form in Balancing Chemical Equations

In an earlier paper published in the Journal of Natural Sciences Research in 2015 on how to balance chemical equations using matrix algebra, Gabriel and Onwuka showed how to reduce the resulting matrix to echelon form using elementary row operations. However, they did not show how elementary row operations can be used in reducing the resulting echelon matrix to row reduced echelon form. We show that the solution obtained is actually the nullspace of the matrix. Hence, the solution can be infinitely many. In addition, we show that instead of manually using row operations to reduce the matrix to row reduced echelon form, software environments like octave or Matlab can be used to reduce the matrix directly. In all the examples presented in this paper, we reduced all matrices to row reduced echelon form showing all row operations, which was not clearly stated in the Gabriel and Onwuka paper. Most importantly, with the availability of Mathematical software, we show that we do not need to carry out these row operations by brute force.

From Chemical Langevin Equations to Fokker-Planck Equation: Application of Hodge Decomposition and Klein-Kramers Equation

The stochastic systems without detailed balance are common in various chemical reaction systems, such as metabolic network systems. In studies of these systems, the concept of potential landscape is useful However, what are the su^cient and necessary conditions of the existence of the potential function is still an open problem. Use Hodge decomposition theorem in differential form theory, we focus on the general chemical Langevin equations, which reitect complex chemical reaction systems. We analysis the conditions for the existence of potential landscape of the systems. By mapping the stochastic differential equations to a Hamiltonian mechanical system, we obtain the Fokker-Planck equation of the chemical reaction systems. The obtained Fokker-Planck equation can be used in further studies of other steady properties of complex chemical reaction systems, such as their steady state entropies.

Discrete formulation of mixed finite element methods for vapor deposition chemical reaction equations

The vapor deposition chemical reaction processes, which are of extremely extensive applications, can be classified as a mathematical model by the following governing nonlinear partial differential equations containing velocity vector, temperature field, pressure field, and gas mass field. The mixed finite element （MFE） method is employed to study the system of equations for the vapor deposition chemical reaction processes. The semidiscrete and fully discrete MFE formulations are derived. And the existence and convergence （error estimate） of the semidiscrete and fully discrete MFE solutions are demonstrated. By employing MFE method to treat the system of equations for the vapor deposition chemical reaction processes, the numerical solutions of the velocity vector, the temperature field, the pressure field, and the gas mass field can be found out simultaneously. Thus, these researches are not only of important theoretical means, but also of extremely extensive applied vistas.

Dyson-Schwinger Equations of Chiral Chemical Potential

We discuss the chiral phase transition of quantum chromodynamics （QCD） with a chiral chemical potential μ5 as an additional scale. Within the framework of Dyson-Schwinger equations, we focus particularly on the behavior of the widely accepted as well as interesting critical end point （CEP）, using a separable gluon propagator and a Gaussian gluon propagator. We find that there may be no CEP5 in the T-μ5 plane, and the phase transition in the T μ5 plane might be totally crossover. Our results have apparent consistency with the Lattice QCD calculation. On the other hand, our study may also provide some useful hints to some other studies related to μ5.

An implicit upwind parabolized Navier-Stokes code for chemically nonequilibrium flows

The previously developed single-sweep parabolized Navier-Stokes （SSPNS） space marching code for ideal gas flows has been extended to compute chemically nonequilibrium flows. In the code, the strongly coupled set of gas dynamics, species conservation, and turbulence equations is integrated with the implicit lower-upper symmetric GaussSeidel （LU-SGS） method in the streamwise direction in a space marching manner. The AUSMPW＋ scheme is used to calculate the inviscid fluxes in the crossflow direction, while the conventional central scheme for the viscous fluxes. The k-g two-equation turbulence model is used. The revised SSPNS code is validated by computing the Burrows-Kurkov non-premixed H2/air supersonic combustion flows, premixed H2/air hypersonic combustion flows in a three-dimensional duct with a 15° compression ramp, as well as the hypersonic laminar chemically nonequilibrium air flows around two 10° half-angle cones. The results of these calculations are in good agreement with those of experiments, NASA UPS or Prabhu＇s PNS codes. It can be concluded that the SSPNS code is highly efficient for steady supersonic/ hypersonic chemically reaction flows when there is no large streamwise separation.

Intrinsic noise analysis and stochastic simulation on transforming growth factor beta signal pathway

A typical biological cell lives in a small volmne at room temperature; the noise effect on the cell signal transduction pathway may play an important role in its dynamics. Here, using the transforming growth factor-β signal transduction pathway as an example, we report our stochastic simulations of the dynamics of the pathway and introduce a linear noise approximation method to calculate the transient intrinsic noise of pathway components. We compare the numerical solutions of the linear noise approximation with the statistic results of chemical Langevin equations, and find that they are quantitatively in agreement with the other. When transforming growth factor-β dose decreases to a low level, the time evolution of noise fluctuation of nuclear Smad2-Smad4 complex indicates the abnormal enhancement in the transient signal activation process.

Quantitative analysis of gene expression systems

Gene expression is a complex biochemical process, involving many specific processes such as transcription, translation, switching between promoter states, and regulation. All these biochemical processes inevitably lead to fluctuations in mRNA and protein abundances. This noise has been identified as an important factor underlying the observed phenotypic variability of genetically identical cells in homogeneous environments. Quantifying the contributions of different sources of noise using stochastic models of gene expression is an important step towards understanding fundamental cellular processes and cell-to-cell variability in expression levels. In this paper, we review progresses in quantitative study of simple gene expression systems, including some results that we have not published. We analytically show how specific processes associated with gene expression affect expression levels. In particular, we derive the analytical decomposition of expression noise, which is important for understanding the roles of the factorial noise in controlling phenotypic variability. We also introduce a new index （called attribute factor） to quantify expression noise, which has more advantages than the commonly-used noise indices such as noise intensity and Fano factor.

Exact distributions for stochastic models of gene expression with arbitrary regulation

Stochasticity in gene expression can result in fluctuations in gene product levels. Recent experiments indicated that feedback regulation plays an important role in controlling the noise in gene expression.A quantitative understanding of the feedback effect on gene expression requires analysis of the corresponding stochastic model. However, for stochastic models of gene expression with general regulation functions, exact analytical results for gene product distributions have not been given so far. Here, we propose a technique to solve a generalized ON-OFF model of stochastic gene expression with arbitrary(positive or negative, linear or nonlinear) feedbacks including posttranscriptional or posttranslational regulation. The obtained results, which generalize results obtained previously, provide new insights into the role of feedback in regulating gene expression. The proposed analytical framework can easily be extended to analysis of more complex models of stochastic gene expression.