The Extrapolation of Vapour–liquid Equilibrium Curves of Pure Fluids in Alternative Gibbs Ensemble Monte Carlo Implementations

Extrapolation schemes based on Taylor series expansion to determine the vapour–liquid equilibrium (VLE) curves of pure molecular fluids are presented for the NpH and μVL versions of the Gibbs ensemble Monte Carlo (GEMC) simulations. The coexistence curves of the various configurational quantities can be expressed as Taylor series around the simulated equilibrium point as a function of pressure in the NpH version and chemical potential in the μVL version. The coefficients of the Taylor series are calculated from single GEMC simulations using Clapeyron-like equations and fluctuation formulas. A Padè approximant is used to widen the range where the extrapolation is accurate. These methods are demonstrated on atomic Lennard-Jones fluid. The procedure is found to be an accurate and useful tool to calculate wide sections of the VLE curves. With this procedure the saturation heat capacity can be directly determined using the calculated derivatives.     

Quantitative estimation of magnitude and orientation of the CSA tensor from field dependence of longitudinal NMR relaxation rates

A method is presented that makes it possible to estimate both the orientation and the magnitude of the chemical shift anisotropy (CSA) tensor in molecules with a pair of spin 1/2 nuclei, typically ¹³C-¹H or ¹⁵ N-¹H. The method relies on the fact that the longitudinal cross-correlation rate as well as a linear combination of the autorelaxation rates of longitudinal heterospin magnetization, longitudinal two-spin order and longitudinal proton magnetization are proportional to the spectral density at the Larmor frequency of the heterospin. Therefore the ratio between the cross-correlation rate and the above linear combination is independent of the dynamics. From the field dependence of the ratio both the magnitude and the orientation of the CSA tensor can be estimated. The method is applicable to molecules in all motional regimes and is not limited to molecules in extreme narrowing or slow tumbling, nor is it sensitive to chemical exchange broadening. It is tested on the 22 amino acid residue peptide motilin, selectively ¹³ C labeled in the ortho positions in the ring of the single tyrosine residue. In the approximation of an axially symmetric ¹³C CSA tensor, the symmetry axis of the CSA tensor makes an angle of 23° ± 1° to the ¹³ C-¹H bond vector, and has a magnitude of 156 ± 5 ppm. This is in close agreement with solid-state NMR data on tyrosine powder [Frydman et al. (1992) Isr. J. Chem., 32, 161–164].     

Chemical Reaction Networks’ Programming for Solving Equations

The computational ability of the chemical reaction networks (CRNs) using DNA as the substrate has been verified previously. To solve more complex computational problems and perform the computational steps as expected, the practical design of the basic modules of calculation and the steps in the reactions have become the basic requirements for biomolecular computing. This paper presents a method for solving nonlinear equations in the CRNs with DNA as the substrate. We used the basic calculation module of the CRNs with a gateless structure to design discrete and analog algorithms and realized the nonlinear equations that could not be solved in the previous work, such as exponential, logarithmic, and simple triangle equations. The solution of the equation uses the transformation method, Taylor expansion, and Newton iteration method, and the simulation verified this through examples. We used and improved the basic calculation module of the CRN++ programming language, optimized the error in the basic module, and analyzed the error’s variation over time.     

Analytical uncertainty propagation in life cycle inventory and impact assessment: application to an automobile front panel

Background, aim, and scope

Uncertainty information is essential for the proper use of life cycle assessment (LCA) and environmental assessments in decision making. So far, parameter uncertainty propagation has mainly been studied using Monte Carlo techniques that are relatively computationally heavy to conduct, especially for the comparison of multiple scenarios, often limiting its use to research or to inventory only. Furthermore, Monte Carlo simulations do not automatically assess the sensitivity and contribution to overall uncertainty of individual parameters. The present paper aims to develop and apply to both inventory and impact assessment an explicit and transparent analytical approach to uncertainty. This approach applies Taylor series expansions to the uncertainty propagation of lognormally distributed parameters.

Materials and methods

We first apply the Taylor series expansion method to analyze the uncertainty propagation of a single scenario, in which case the squared geometric standard deviation of the final output is determined as a function of the model sensitivity to each input parameter and the squared geometric standard deviation of each parameter. We then extend this approach to the comparison of two or more LCA scenarios. Since in LCA it is crucial to account for both common inventory processes and common impact assessment characterization factors among the different scenarios, we further develop the approach to address this dependency. We provide a method to easily determine a range and a best estimate of (a) the squared geometric standard deviation on the ratio of the two scenario scores, “A/B”, and (b) the degree of confidence in the prediction that the impact of scenario A is lower than B (i.e., the probability that A/B<1). The approach is tested on an automobile case study and resulting probability distributions of climate change impacts are compared to classical Monte Carlo distributions.

Results

The probability distributions obtained with the Taylor series expansion lead to results similar to the classical Monte Carlo distributions, while being substantially simpler; the Taylor series method tends to underestimate the 2.5% confidence limit by 1-11% and the 97.5% limit by less than 5%. The analytical Taylor series expansion easily provides the explicit contributions of each parameter to the overall uncertainty. For the steel front end panel, the factor contributing most to the climate change score uncertainty is the gasoline consumption (>75%). For the aluminum panel, the electricity and aluminum primary production, as well as the light oil consumption, are the dominant contributors to the uncertainty. The developed approach for scenario comparisons, differentiating between common and independent parameters, leads to results similar to those of a Monte Carlo analysis; for all tested cases, we obtained a good concordance between the Monte Carlo and the Taylor series expansion methods regarding the probability that one scenario is better than the other.

Discussion

The Taylor series expansion method addresses the crucial need of accounting for dependencies in LCA, both for common LCI processes and common LCIA characterization factors. The developed approach in Eq. 8, which differentiates between common and independent parameters, estimates the degree of confidence in the prediction that scenario A is better than B, yielding results similar to those found with Monte Carlo simulations.

Conclusions

The probability distributions obtained with the Taylor series expansion are virtually equivalent to those from a classical Monte Carlo simulation, while being significantly easier to obtain. An automobile case study on an aluminum front end panel demonstrated the feasibility of this method and illustrated its simultaneous and consistent application to both inventory and impact assessment. The explicit and innovative analytical approach, based on Taylor series expansions of lognormal distributions, provides the contribution to the uncertainty from each parameter and strongly reduces calculation time.     

On the asymptotic trajectory of the roots of Lotka's equation.

A mathematical formula is given for the asymptotic trajectory of the complex roots of Lotka's equation. This formula is obtained by use of a Taylor expansion of the net fertility function in the neighborhood of the age beyond which fertility is zero. The approximate trajectory is compared with an exact trajectory obtained by use of a computational algorithm for finding complex roots suggested by Turner. For two examples, the agreement is surprisingly good.     

A method for the deductive and unique determination of the values of three parameters involved in fractional functions applicable to relaxation kinetics.

A novel method is proposed to determine deductively and uniquely the values of three parameters, a, b, and c in a fractional function of the form, y=a+bx/(c+x) where x and y are experimentally obtainable variables. This type of equation is frequently encountered in chemistry and biochemistry involving relaxation kinetics. The method of least squares with the Taylor expansion is employed for direct curve fitting of observed data to the fractional function. Approximate values of the parameters, which are always necessary prior to commending the above procedure, can be obtained by the method of rearrangement after canceling the denominator of fractional functions. This procedure is very simple, but very effective for estimating provisional values of the parameters. Deductive and unique determination of the parameters involved in the fractional function shown above can be accomplished for the first time by the combination of these two procedures. This method is extended to include the analysis of relaxation kinetic data such as those of temperature-jump method where the determination of equilibrium concentrations of reactants in addition to the three parameters is also necessary.     

Linear nonequilibrium thermodynamics describes the dynamics of an autocatalytic system.

A model simulating oscillations in glycolysis was formulated in terms of nonequilibrium thermodynamics. In the kinetic rate equations every metabolite concentration was replaced with an exponential function of its chemical potential. This led to nonlinear relations between rates and chemical potentials. Each chemical potential was then expanded around its steady-state value as a Taylor series. The linear (first order) term of the Taylor series sufficed to simulate the dynamic behavior of the system, including the damped and even sustained oscillations at low substrate input or high free-energy load. The glycolytic system is autocatalytic in the first half. Because oscillations were obtained only in the presence of that autocatalytic feed-back loop we conclude that this type of kinetic nonlinearity was sufficient to account for the oscillatory behavior. The matrix of phenomenological coefficients of the system is nonsymmetric. Our results indicate that this is the symmetry property and not the linearity of the flow-force relations in the near equilibrium domain that precludes oscillations. Given autocatalytic properties, a system exhibiting liner flow-force relations and being outside the near equilibrium domain may show bifurcations, leading to self-organized behavior.     

Efficient RMSD measures for the comparison of two molecular ensembles. Root-mean-square deviation

Quantitative measures are presented for comparing the conformations of two molecular ensembles. The measures are based on Kabsch's formula for the root-mean-square deviation (RMSD) and the covariance matrix of atomic positions of isotropically distributed ensembles (IDE). By using a Taylor series expansion, it is shown that the RMSD can be expressed solely in terms of the IDE matrices. A fast approximate method is introduced for the pairwise RMSD determination whose computational cost scales linearly with the number of structures. A similarity measure for two structural ensembles that is based on the trace metric of the differences of powers of the IDE matrices is presented. The measures are illustrated for conformational ensembles generated by a molecular dynamics computer simulation of a partially folded A-state analog of ubiquitin.     

Kinematics of plant growth.

Many of the concepts and equations which have been used in the study of compressible fluids can be applied to problems of plant development. Growth field variables, i.e. functions of position in the plant and of time, can be specified in either Eulerian (spatial) or Lagrangian (material) terms. The two specifications coincide only when the spatial distribution of the variable is steady, and steady patterns are most likely to emerge when an apex is chosen as origin of the co-ordinate system. The growth field itself can be described locally by the magnitude and orientation of the principal axes of the rate of strain tensor and by the vorticity tensor. Material derivatives can be calculated if the temporal and spatial variation in both growth velocity, u (rate of displacement from a material origin), and the variable of interest are known. The equation of continuity shows the importance of including both growth velocity, u, and growth rate, ▽ ·u in estimates of local biosynthesis and transport rates in expanding tissue, although the classical continuity equation must be modified to accommodate the compartmentalized distributions characteristic of plant tissue. Relatively little information on spatial variation in plant organs can be found in the botanical literature, but the current availability of interactive computer graphics equipment suggests that analysis of the spatial distribution of growth rates at least is no longer difficult.     

Mutation in autocatalytic reaction networks

A class of kinetic equations describing catalysed and template induced replication, and mutation is introduced. This ODE in its most general form is split into two vector fields, a replication and a mutation field. The mutation field is considered as a perturbation of the replicator equation. The perturbation expansion is a Taylor series in a mutation parameter . First, second and higher order contributions are computed by means of the conventional Rayleigh-Schrödinger approach. Qualitative shifts in the positions of rest points and limit cycles on the boundary of the physically meaningful part of concentration space are predicted from flow topologies. The results of the topological analysis are summarized in two theorems which turned out to be useful in applications: the rest point migration theorem (RPM) and the limit cycle migration theorem (LCM). Quantitative expressions for the shifts of rest points are computed directly from the perturbation expansion. The concept is applied to a collection of selected examples from biophysical chemistry and biology.     

Influence of the O-phosphorylation of serine, threonine and tyrosine in proteins on the amidic 15N chemical shielding anisotropy tensors

Density functional theory was employed to study the influence of O-phosphorylation of serine, threonine, and tyrosine on the amidic ¹⁵N chemical shielding anisotropy (CSA) tensor in the context of the complex chemical environments of protein structures. Our results indicate that the amidic ¹⁵N CSA tensor has sensitive responses to the introduction of the phosphate group and the phosphorylation-promoted rearrangement of solvent molecules and hydrogen bonding networks in the vicinity of the phosphorylated site. Yet, the calculated ¹⁵N CSA tensors in phosphorylated model peptides were in range of values experimentally observed for non-phosphorylated proteins. The extent of the phosphorylation induced changes suggests that the amidic ¹⁵N CSA tensor in phosphorylated proteins could be reasonably well approximated with averaged CSA tensor values experimentally determined for non-phosphorylated amino acids in practical NMR applications, where chemical surrounding of the phosphorylated site is not known a priori in majority of cases. Our calculations provide estimates of relative errors to be associated with the averaged CSA tensor values in interpretations of NMR data from phosphorylated proteins.     

Monitoring conformational dynamics with solid-state R 1ρ experiments

A new application of solid-state rotating frame (R _1ρ) relaxation experiments to observe conformational dynamics is presented. Studies on a model compound, dimethyl sulfone (DMS), show that R _1ρ relaxation due to reorientation of a chemical shift anisotropy (CSA) tensor undergoing chemical exchange can be used to monitor slow-to-intermediate timescale conformational exchange processes. Control experiments used d ₆ -DMS and alanine to confirm that the technique is monitoring reorientation of the CSA tensor rather than dipolar interactions or methyl group rotation. The application of this method to proteins could represent a new site-specific probe of conformational dynamics.     

Associating the mesoscale fiber organization of the tongue with local strain rate during swallowing

The tongue is an intricately configured muscular organ that undergoes a stereotypical set of deformations during the course of normal human swallowing. In order to demonstrate quantitatively the relationship between 3D aligned lingual fiber organization and mechanics during swallowing, the tissue's myoarchitecture and strain rate were imaged before and during the propulsive phase of a 3.0ml water bolus swallow. Mesoscale fiber organization was imaged with high-resolution diffusion tensor imaging (DTI) and multi-voxel myofiber tracts generated along maximum diffusion vectors. Tissue compression/expansion was obtained via lingual pressure-gated phase-contrast (PC) MRI, a method which determines local strain rate as a function of the phase shift occurring along an applied gradient vector. The co-alignment of myofiber tract direction and the localized principal strain rate vectors was obtained by translating the strain rate tensor into the reference frame with the primary axis parallel to the maximum diffusion vector using Mohr's circle, resulting in the generation of fiber-aligned strain rate (FASR). DTI tractography displayed the complete fiber anatomy of the tongue, consisting of a core region of orthogonally aligned fibers encased within a longitudinal sheath, which merge with the externally connected styloglossus, hyoglossus, and genioglossus fibers. FASR images obtained in the mid-sagittal plane demonstrated that bolus propulsion was associated with prominent compressive strain aligned with the genioglossus muscle combined with expansive strain aligned with the verticalis and geniohyoid muscles. These data demonstrate that lingual deformation during swallowing involves complex interactions involving intrinsic and extrinsic muscles, whose contractility is directed by the alignment of mesoscale fiber tracts.     

Mass and Energy Transport Through Slit Pores: Application to Planar Poiseuille Flow

Abstract

We develop a simple, efficient and general statistical mechanical technique for calculating the pressure tensor and the heat flux vector in atomic fluids. The method is applied to the case of planar Poiseuille flow through a narrow slit pore and the results indicate that our technique is accurate and relatively efficient. A second method to calculate shear stress is derived from the momentum continuity equation. This mesoscopic method again is seen to be accurate with good computational efficiency.

We also find that the commonly used approximation to the Irving-Kirkwood expression for the heat flux and the pressure tensor (where the Irving-Kirkwood O_ij operator is set equal to unity-the so-called IK1 approximation), leads to incorrect results for highly inhomogeneous fluids. In such cases the pressure tensor and heat flux vector display spurious oscillations.

We calculate the spatially dependent viscosity across a narrow pore and find that it exhibits real but weak oscillations, a consequence of oscillations in the number density. Finally we point out that if the heat flux vector is coupled to the gradient of the square of the strain rate tensor such an effect will only affect the shape of the temperature profile. For planar Poiseuille flow, the temperature profile should deviate from the classical quartic form and include an additional quadratic component. The actual magnitude and shape of the heat flux vector remain exactly as they would if such a coupling did not exist.     

The effects of group size, leaf size, and density on the performance of a leaf-mining moth

1. The effects of two factors, leaf size and group size, on the performance of the Tupelo leafminer, Antispila nysaefoliella (Lepidoptera: Heliozelidae), were examined by fitting growth models to mine expansion data using nonlinear mixed-effects models. 2. The rate of mine expansion served as a proxy for larval performance because of its correlation with both feeding activity and growth rate and is also the means by which a larva achieves its final mine size (or total consumption). 3. Leaf size was used as a measure of resource availability, and was expected to reduce the impact of resource competition and enhance larval performance. 4. In contrast to the unidirectional effects expected for leaf size (i.e. more resources should enhance performance), the direction for the effects of group size was expected to depend on the mechanism(s) driving the effect. For example, if there is resource competition among larvae in a group, then this could increase the feeding rates of some larvae or reduce the total consumption of others. However, if leaf mining induces host plant chemical defences, then larger groups might elicit a greater defensive response by the host plant (at the leaf), and hence, be characterized by reduced feeding and growth rates. 5. To investigate these interactions, two growth models, the Gompertz model and a modified version of the von Bertalanffy growth equation, were fitted to time series of the sizes of individual leaf mines using nonlinear mixed-effects models. Linear and nonlinear associations of each factor (group size or leaf size) with model parameters were then evaluated using a hierarchical testing procedure by determining: (i) whether inclusion of the factor produced a better-fit model, and (ii) if it did, the form of that relationship (i.e. linear or nonlinear). 6. Three patterns were detected with these analyses. (i) Leaf size had a significant positive, linear relationship with mine expansion rate. (ii) Group size had a significant quadratic relationship with mine expansion rate. (iii) The effects of leaf and group size on the maximum mine size were opposite to those found with growth rate.     

Stress relaxation property of the cell wall and auxin-induced cell elongation

A stress-relaxation method has been developed to measure the mechanical property of the plant cell wall, as a physically defined terms. In the method, the stress relaxation property of the cell wall is simulated with a Maxwell viscoelastic model whose character is represented by four parameters; the minimum relaxation time, To, the relaxation rate, b, the maximum relaxation time, Tm and the residual stress, c. Thus, the mechanical property of the cell wall is represented by the four parameters. Physical and physiological meanings of the parameters are discussed. Auxin effects on the parameters were also studied. The cell elongation is simply thought to be extension of the cell wall under a force. The extension of the cell wall can be simulated by the mechanical property of the cell wall. However, the calculated extension was found to be incomparable to the real cell growth, indicating that there has to be other factors limiting the rate of cell growth. Major factors governing cell growth are discussed to be the cell wall mechanical property, the osmotic potential and water movement in the apoplast. A possibility to predict cell expansion with the three factors was discussed and a novel equation representing cell growth was obtained: $$1/R = 1/R_w + 1/R_p $$ whereR is the rate of cell elongation,R _w is the rate of cell wall extension due to the osmotic pressure andR _p is the rate of cell elongation determined by water conductivity.     

Evaluation of methanogenic kinetics in an anaerobic fluidized bed reactor (AFBR)

A kinetic study of the methanogenic phase was carried out on a pilot lab scale anaerobic fluidized bed reactor (AFBR) in batch mode. An examination of the effect of initial acetate concentration, bed expansion and bed segregation is presented.Experimental data observed for the acetate removal against time were adjusted to a zero-order kinetic equation, over the chemical oxygen demand (COD) range studied (1430–5340 mg litre⁻¹), independently of the bed expansion (11–37%). The kinetic constant was calculated using robust regression analysis. The zero-order kinetic constant, K₀ was between 1180–1380 mg COD litre⁻¹ h⁻¹ on the fixed bed volume basis, and the maximum specific substrate utilization rate, k, was between 145–198 mg COD g VS⁻¹ h⁻¹.The kinetic behaviour was found to be different throughout the reactor, on the fixed bed volume basis and the activity at the bottom of the bed was lower than the activity in the upper region. However, on an attached volatile solids basis, the activity at the bottom level was the greatest.