Computation of molecular electrostatics with boundary element methods.

In continuum approaches to molecular electrostatics, the boundary element method (BEM) can provide accurate solutions to the Poisson-Boltzmann equation. However, the numerical aspects of this method pose significant problems. We describe our approach, applying an alpha shape-based method to generate a high-quality mesh, which represents the shape and topology of the molecule precisely. We also describe an analytical method for mapping points from the planar mesh to their exact locations on the surface of the molecule. We demonstrate that derivative boundary integral formulation has numerical advantages over the nonderivative formulation: the well-conditioned influence matrix can be maintained without deterioration of the condition number when the number of the mesh elements scales up. Singular integrand kernels are characteristics of the BEM. Their accurate integration is an important issue. We describe variable transformations that allow accurate numerical integration. The latter is the only plausible integral evaluation method when using curve-shaped boundary elements.     

An efficient numerical method for distributed-loop models of the urine concentrating mechanism

In this study we describe an efficient numerical method, based on the semi-Lagrangian (SL) semi-implicit (SI) method and Newton's method, for obtaining steady-state (SS) solutions of equations arising in distributed-loop models of the urine concentrating mechanism. Dynamic formulations of these models contain large systems of coupled hyperbolic partial differential equations (PDEs). The SL method advances the solutions of these PDEs in time by integrating backward along flow trajectories, thus allowing large time steps while maintaining stability. The SI approach controls stiffness arising from transtubular transport terms by averaging these terms in time along flow trajectories. An approximate SS solution of a dynamic formulation obtained via the SLSI method can be used as an initial guess for a Newton-type solver, which rapidly converges to a highly accurate numerical approximation to the solution of the ordinary differential equations that arise in the corresponding SS model formulation. In general, it is difficult to specify a priori for a Newton-type solver an initial guess that falls within the radius of convergence; however, the initial guess generated by solving the dynamic formulation via the SLSI method can be made sufficiently close to the SS solution to avoid numerical instability. The combination of the SLSI method and the Newton-type solver generates stable and accurate solutions with substantially reduced computation times, when compared to previously applied dynamic methods.     

Enumeration of oligomerization states of membrane proteins in living cells by homo-FRET spectroscopy and microscopy: theory and application

Protein-protein interactions play a pivotal role in biological signaling networks. It is highly desirable to perform experiments that can directly assess the oligomerization state and degree of oligomerization of biological macromolecules in their native environment. Homo-FRET depends on the inverse sixth power of separation between interacting like fluorophores on the nanometer scale and is therefore sensitive to protein oligomerization. Homo-FRET is normally detected by steady-state or time-resolved fluorescence anisotropy measurements. Here we show by theory and simulation that an examination of the extent of homotransfer as measured by steady-state fluorescence anisotropy as a function of fluorophore labeling (or photodepletion) gives valuable information on the oligomerization state of self-associating proteins. We examine random distributions of monomers, dilute solutions of oligomers, and concentrated solutions of oligomers. The theory is applied to literature data on band 3 protein dimers in membranes, GPI-linked protein trimers in "rafts," and clustered GFP-tagged epidermal growth factor receptors in cell membranes to illustrate the general utility and applicability of our analytical approach.     

Analysis of Bone Remodeling Under Piezoelectricity Effects Using Boundary Elements

Piezoelectric materials exhibit a response to mechanical-electrical coupling,which represents an important contribution to the electrical-mechanical interaction in bone remodeling process.Therefore,the study of the piezoelectric effect on bone remodeling has high interest in applied biomechanics.The effects of mechano-regulation and electrical stimulation on bone healing are explained.The Boundary Element Method (BEM) is used to simulate piezoelectric effects on bones when sheafing forces are applied to collagen fibers to make them slip past each other.The piezoelectric fundamental solutions are obtained by using the Radon transform.The Dual Reciprocity Method (DRM) is used to simulate the particular solutions in time-dependent problems.BEM analysis showed the strong influence of electrical stimulation on bone remodeling.The examples discussed in this work showed that,as expected,the electrically loaded bone surfaces improved the bone deposition.BEM results confirmed previous findings obtained by using the Finite Element Method (FEM).This work opens very promising doors in biomechanics research,showing that mechanical loads can be replaced,in part,by electrical charges that stimulate strengthening bone density.The obtained results herein are in good agreement with those found in literature from experimental testing and/or other simulation approaches.     

Determination of metabolic fluxes in a non-steady-state system

Estimation of fluxes through metabolic networks from redistribution patterns of (13)C has become a well developed technique in recent years. However, the approach is currently limited to systems at metabolic steady-state; dynamic changes in metabolic fluxes cannot be assessed. This is a major impediment to understanding the behaviour of metabolic networks, because steady-state is not always experimentally achievable and a great deal of information about the control hierarchy of the network can be derived from the analysis of flux dynamics. To address this issue, we have developed a method for estimating non-steady-state fluxes based on the mass-balance of mass isotopomers. This approach allows multiple mass-balance equations to be written for the change in labelling of a given metabolite pool and thereby permits over-determination of fluxes. We demonstrate how linear regression methods can be used to estimate non-steady-state fluxes from these mass balance equations. The approach can be used to calculate fluxes from both mass isotopomer and positional isotopomer labelling information and thus has general applicability to data generated from common spectrometry- or NMR-based analytical platforms. The approach is applied to a GC-MS time-series dataset of (13)C-labelling of metabolites in a heterotrophic Arabidopsis cell suspension culture. Threonine biosynthesis is used to demonstrate that non-steady-state fluxes can be successfully estimated from such data while organic acid metabolism is used to highlight some common issues that can complicate flux estimation. These include multiple pools of the same metabolite that label at different rates and carbon skeleton rearrangements.     

Elimination of thermodynamically infeasible loops in steady-state metabolic models

The constraint-based reconstruction and analysis (COBRA) framework has been widely used to study steady-state flux solutions in genome-scale metabolic networks. One shortcoming of current COBRA methods is the possible violation of the loop law in the computed steady-state flux solutions. The loop law is analogous to Kirchhoff's second law for electric circuits, and states that at steady state there can be no net flux around a closed network cycle. Although the consequences of the loop law have been known for years, it has been computationally difficult to work with. Therefore, the resulting loop-law constraints have been overlooked. Here, we present a general mixed integer programming approach called loopless COBRA (ll-COBRA), which can be used to eliminate all steady-state flux solutions that are incompatible with the loop law. We apply this approach to improve flux predictions on three common COBRA methods: flux balance analysis, flux variability analysis, and Monte Carlo sampling of the flux space. Moreover, we demonstrate that the imposition of loop-law constraints with ll-COBRA improves the consistency of simulation results with experimental data. This method provides an additional constraint for many COBRA methods, enabling the acquisition of more realistic simulation results.     

Synergism analysis of biochemical systems. II. Tensor formulation and treatment of stoichiometric constraints

The previous paper outlined a conceptual and mathematical framework for synergism analysis of kinetic models. Though the formalism presented there is adequate for studying simple models, the analysis of large-scale models benefits from the more effective formulation achieved in this work. The present formulation is based on simple tensor operations and takes advantage of the analogy between the formalisms for synergism and log-synergism analysis presented before. Well-known relationships of first-order sensitivity analysis and new relationships for (log-)synergism coefficients of various steady-state properties are cast in the new formal setting. The formalism is then extended to models that are subject to constraints between variables, fluxes and/or parameters. This treatment, which generalizes Reder's concept of link matrices, is applied to networks that include moiety conservation cycles [C. Reder, Metabolic control theory: a structural approach, J. Theor. Biol. 135 (1988) 175]. It is also used to take advantage of flux conservation at steady-state to simplify synergism analysis. Issues of numerical effectiveness are briefly discussed, and the theory illustrated with the study of synergistic behaviour in the metabolism of reactive oxygen species and of a scheme of dynamic channelling.     

A mixed finite element formulation for a non-linear,transversely isotropic material model for the cardiac tissue

In this paper we present a mixed finite element method for modeling the passive properties of the myocardium. The passive properties are described by a non-linear, transversely isotropic, hyperelastic material model, and the myocardium is assumed to be almost incompressible. Single-field, pure displacement-based formulations are known to cause numerical difficulties when applied to incompressible or slightly compressible material cases. This paper presents an alternative approach in the form of a mixed formulation, where a separately interpolated pressure field is introduced as a primary unknown in addition to the displacement field. Moreover, a constraint term is included in the formulation to enforce (almost) incompressibility. Numerical results presented in the paper demonstrate the difficulties related to employing a pure displacement-based method, applying a set of physically relevant material parameter values for the cardiac tissue. The same problems are not experienced for the proposed mixed method. We show that the mixed formulation provides reasonable numerical results for compressible as well as nearly incompressible cases, also in situations of large fiber stretches. There is good agreement between the numerical results and the underlying analytical models.     

A mixed finite element formulation for a non-linear, transversely isotropic material model for the cardiac tissue

In this paper we present a mixed finite element method for modeling the passive properties of the myocardium. The passive properties are described by a non-linear, transversely isotropic, hyperelastic material model, and the myocardium is assumed to be almost incompressible. Single-field, pure displacement-based formulations are known to cause numerical difficulties when applied to incompressible or slightly compressible material cases. This paper presents an alternative approach in the form of a mixed formulation, where a separately interpolated pressure field is introduced as a primary unknown in addition to the displacement field. Moreover, a constraint term is included in the formulation to enforce (almost) incompressibility. Numerical results presented in the paper demonstrate the difficulties related to employing a pure displacement-based method, applying a set of physically relevant material parameter values for the cardiac tissue. The same problems are not experienced for the proposed mixed method. We show that the mixed formulation provides reasonable numerical results for compressible as well as nearly incompressible cases, also in situations of large fiber stretches. There is good agreement between the numerical results and the underlying analytical models.     

On modelling large deformations of heterogeneous biological tissues using a mixed finite element formulation

This study addresses the issue of modelling material heterogeneity of incompressible bodies. It is seen that when using a mixed (displacement–pressure) finite element formulation, the basis functions used for pressure field may not be able to capture the nonlinearity of material parameters, resulting in pseudo-residual stresses. This problem can be resolved by modifying the constitutive relation using Flory's decomposition of the deformation gradient. A two-parameter Mooney–Rivlin constitutive relation is used to demonstrate the methodology. It is shown that for incompressible materials, the modification does not alter the mechanical behaviour described by the original constitutive model. In fact, the modified constitutive equation shows a better predictability when compared against analytical solutions. Two strategies of describing the material variation (i.e. linear and step change) are explained, and their solutions are evaluated for an ideal two-material interfacing problem. When compared with the standard tied coupling approach, the step change method exhibited a much better agreement because of its ability to capture abrupt changes of the material properties. The modified equation in conjunction with integration point-based material heterogeneity is then used to simulate the deformations of heterogeneous biological structures to illustrate its applications.     

Mass-transfer limitations for immobilized enzyme-catalyzed kinetic resolution of racemate in a fixed-bed reactor

A mathematical model has been developed for immobilized enzyme-catalyzed kinetic resolution of racemate in a fixed-bed reactor in which the enzyme-catalyzed reaction (the irreversible uni-uni competitive Michaelis-Menten kinetics is chosen as an example) was coupled with intraparticle diffusion, external mass transfer, and axial dispersion. The effects of mass-transfer limitations, competitive inhibition of substrates, deactivation on the enzyme effective enantioselectivity, and the optical purity and yield of the desired product are examined quantitatively over a wide range of parameters using the orthogonal collocation method. For a first-order reaction, an analytical solution is derived from the mathematical model for slab-, cylindrical-, and spherical-enzyme supports. Based on the analytical solution for the steady-state resolution process, a new concise formulation is presented to predict quantitatively the mass-transfer limitations on enzyme effective enantioselectivity and optical purity and yield of the desired product for a continuous steady-state kinetic resolution process in a fixed-bed reactor.     

Movement of a spherical cell in capillaries using a boundary element method

This study aims to investigate the translation and rotation of a spherical particle in capillaries and overcomes limitations in previous studies by using a boundary element method. The capillary, a straight cylindrical tube, is filled with a Newtonian viscous fluid. A spherical particle is arbitrarily positioned in the capillary either co-centrically or eccentrically and is free to translate and rotate. Flow in the capillary is first assumed to be caused solely by the movement of the sphere under the gravity. When a steady state is reached, the net force and torque on the sphere are zero. The translating velocity and rotation of the particle are calculated from equilibrium equations. For a co-centric sphere, our result agrees to Bohlin's analytical solution (Bohlin, 1960) and the difference is less than 1%. For an eccentrically positioned sphere in the tube, there are no analytical solutions unless the eccentricity is infinitesimal. Results by boundary element method (BEM) give an improved estimations on the velocity and rotation of the sphere than earlier results by a boundary singularity method (BSM), particularly when the clearance between the tube and the sphere becomes small. Movement of a spherical particle in a capillary driven by a pressure gradient is further investigated, which has closer relevance to movement of blood cells in capillaries. The current study using BEM enables investigation on cell movement in close proximities of the capillary wall.     

Dynamic finite element modeling of poroviscoelastic soft tissue

Clinical evidences relative to biomechanical factors have demonstrated their important contribution to the behaviour of soft tissues. Finite element (FE) analysis is used to study the mechanical behaviour of soft tissue because it can provide numerical solutions to problems that are intractable to analytic solutions. This study focuses on the development of a FE model of a poroelastic biological tissue, which incorporates the viscoelastic material behaviour, finite deformation and inertial effect. The FE formulation is based on the weak form derived from the governing equation, and Newmark-beta method as well as Newton's method is incorporated into the implicit non-linear solutions. One-dimensional analytical solutions were used to verify the theoretical formulation and the numerical implementation of the proposed model. This study was further extended to analyze two-dimensional biomechanical models and the results clearly demonstrate the importance of including finite deformation, viscoelasticity and inertial effects.     

Fluorescence‐based sensor for selective and sensitive detection of amoxicillin (Amox) in aqueous medium: Application to pharmaceutical and biomedical analysis

We here for the first time demonstrate an analytical approach for the highly selective and sensitive detection of amoxicillin (Amox) in aqueous medium based on the fluorescence quenching of quantum dots (QDs). The change in fluorescence intensity of mercaptopropionic acid‐capped cadmium sulphide (MPA‐CdS) QDs is attributed to the increasing concentration of Amox. The results show that the fluorescence quenching of QDs by Amox takes place through both static and dynamic types of quenching mechanism. The fluorescence quenching of QDs with increase in concentration of Amox shows the linear range between 5 μg ml^?1 and 30 μg ml^?1 and the limit of detection (LOD) is 5.19 μg ml^?1. There is no interference of excipients, which are commonly present in pharmaceutical formulation and urine samples. For the practical application approach, the developed method has been successfully applied for the determination of Amox in pharmaceutical formulations and urine samples with acceptable results.     

Approximative kinetic formats used in metabolic network modeling

An overview is presented of the different approximative enzyme kinetic formats that have been proposed for use in metabolic modeling studies. It is considered that the following four general properties are relevant for approximative kinetics: the rate must be proportional to enzyme level; at high metabolite concentrations, there is downward concave behavior of rate versus concentration; the number of kinetic parameters should be as small as possible; analytical solutions of steady-state network balances are desirable. Six different approximative kinetic formats are evaluated (linear, logarithmic-linear, power law GMA, power law S-systems, thermokinetic, linear-logarithmic) and it is concluded that only the recently proposed linear-logarithmic approach combines all desired properties and therefore seems a most appropriate approximate kinetic format.     

Fast Soft Tissue Deformation and Stump-Socket Interaction Toward a Computer-Aided Design System for Lower Limb Prostheses

Prosthetic technology is rapidly advancing but there's a catch. Regardless the technology or the material used, a minimum cost is still high. One of the problems relates to the fact that the conventional socket fabrication process is still used. This method is based on subjective estimations of the involved specialists and feedbacks of the patients. This process consumes remarkable amount of time, manpower and materials. Research works are needed to design new efficient and low-cost alternative techniques for the socket design. This technique should definitely be based on CAD-CAM methods. Therefore, the first step toward this objective is to establish an accurate numeric model for evaluating and optimizing the design process. In this present work, we developed a new approach to simulate the stump soft tissue deformation and stump-socket interaction using Mass-Spring System (MSS) approach and a point-to-surface contact formulation.A novel Mass-Spring System with corrective spring (MSS-CS) model was developed and evaluated. A node-to-surface contact formulation was also integrated and evaluated. The MSS-CS model and contact formulation were evaluated with primitive geometrical object and a stump-socket model. Moreover, a finite element model of the stump-socket interaction was also developed using Abaqus to evaluate the proposed approach.Obtained results show that the proposed contact formulation has a very good precision level and the contact pressures on the interface between the elastic and rigid bodies are very close to the analytical solutions. The comparison with Abaqus showed a qualitative concordance for the contact pressure. However, quantitative deviation remains high [25-50]% at the peak contact pressure due to different contact formulations. In particular, our MSS-CS approach is more efficient than Abaqus simulation in term of computational time and cost.A novel approach was proposed to model soft tissue deformation and stump-socket interaction in an efficient and accurate manner. As perspectives, this present approach for a real-time simulation of the stump-socket interaction could be used in a real-time CAD-CAM platform to provide a cost-effective socket manufacturing process.     

Isotope-assisted metabolic flux analysis as an equality-constrained nonlinear program for improved scalability and robustness

Stable isotope-assisted metabolic flux analysis (MFA) is a powerful method to estimate carbon flow and partitioning in metabolic networks. At its core, MFA is a parameter estimation problem wherein the fluxes and metabolite pool sizes are model parameters that are estimated, via optimization, to account for measurements of steady-state or isotopically-nonstationary isotope labeling patterns. As MFA problems advance in scale, they require efficient computational methods for fast and robust convergence. The structure of the MFA problem enables it to be cast as an equality-constrained nonlinear program (NLP), where the equality constraints are constructed from the MFA model equations, and the objective function is defined as the sum of squared residuals (SSR) between the model predictions and a set of labeling measurements. This NLP can be solved by using an algebraic modeling language (AML) that offers state-of-the-art optimization solvers for robust parameter estimation and superior scalability to large networks. When implemented in this manner, the optimization is performed with no distinction between state variables and model parameters. During each iteration of such an optimization, the system state is updated instead of being calculated explicitly from scratch, and this occurs concurrently with improvement in the model parameter estimates. This optimization approach starkly contrasts with traditional “shooting” methods where the state variables and model parameters are kept distinct and the system state is computed afresh during each iteration of a stepwise optimization. Our NLP formulation uses the MFA modeling framework of Wiechert et al. [1], which is amenable to incorporation of the model equations into an NLP. The NLP constraints consist of balances on either elementary metabolite units (EMUs) or cumomers. In this formulation, both the steady-state and isotopically-nonstationary MFA (inst-MFA) problems may be solved as an NLP. For the inst-MFA case, the ordinary differential equation (ODE) system describing the labeling dynamics is transcribed into a system of algebraic constraints for the NLP using collocation. This large-scale NLP may be solved efficiently using an NLP solver implemented on an AML. In our implementation, we used the reduced gradient solver CONOPT, implemented in the General Algebraic Modeling System (GAMS). The NLP framework is particularly advantageous for inst-MFA, scaling well to large networks with many free parameters, and having more robust convergence properties compared to the shooting methods that compute the system state and sensitivities at each iteration. Additionally, this NLP approach supports the use of tandem-MS data for both steady-state and inst-MFA when the cumomer framework is used. We assembled a software, eiFlux, written in Python and GAMS that uses the NLP approach and supports both steady-state and inst-MFA. We demonstrate the effectiveness of the NLP formulation on several examples, including a genome-scale inst-MFA model, to highlight the scalability and robustness of this approach. In addition to typical inst-MFA applications, we expect that this framework and our associated software, eiFlux, will be particularly useful for applying inst-MFA to complex MFA models, such as those developed for eukaryotes (e.g. algae) and co-cultures with multiple cell types.     

Presteady-state kinetics and carrier-mediated transport: A theoretical analysis

Summary Kinetic studies of cotransport mechanisms have so far been limited to the conventional steady-state approach which does not allow in general to resolve either isomerization or ratelimiting steps and to determine the values of the individual rate constants for the elementary reactions involved along a given transport pathway. Such questions can only be answered using presteady-state or relaxation experiments which, for technical reasons, have not yet been introduced into the field of cotransport kinetics. However, since two recent reports seem compatible with the observation of such transient kinetics, it would appear that theoretical studies are needed to evaluate the validity of such claims and to critically evaluate the expectations from a presteady-state approach. We thus report such a study which was performed on a simple four-state mechanism of carrier-mediated transport. The time-dependent equation for zero-trans substrate uptake was thus derived and then extended to models withp intermediary steps. It is concluded that (p-1) exponential terms will describe the approach to the steady state but that such equations have low analytical value since the parameters of the flux equation cannot be expressed in terms of the individual rate constants of the elementary reactions for models withp>5. We thus propose realistic simplifications based on the time-scale separation hypothesis which allows replacement of the rate constants of the rapid steps by their equilibrium constants, thereby reducing the complexity of the kinetic system. Assuming that only one relaxation can be observed, this treatment generates approximate models for which analytical expressions can easily be derived and simulated through computer modeling. When performed on the four-state mechanism of carrier-mediated transport, the simulations demonstrate the validity of the approximate solutions derived according to this hypothesis. Moreover, our approach clearly shows that presteady-state kinetics, should they become applicable to (co)transport kinetics, could be invaluable in determining more precise transport mechanisms.     

Analytical solutions for distributions of chemotactic bacteria

This paper reports general and specialized results on analytical solutions to the governing phenomenological equations for chemotactic redistribution and population growth of motile bacteria. It is shown that the number of bacteria cells per unit volume,b, is proportional to a certain prescribed function ofs, the concentration of the critical substrate chemotactic agent, for steady-state solutions through an arbitrary spatial region with a boundary that is impermeable to bacteria cell transport. Moreover, it is demonstrated that the steady-state solution forb ands is unique for a prescribed total number of bacteria cells in the spatial region and a generic Robin boundary condition ons. The latter solution can be approximated to desired accuracy in terms of the Poisson-Green's function associated with the spatial region. Also, as shown by example, closed-form exact steady-state solutions are obtainable for certain consumption rate functions and geometrically symmetric spatial regions. A solutional procedure is formulated for the initialvalue problem in cases for which significant population growth is present and bacteria cell redistribution due to motility and chemotactic flow proceeds slowly relative to the diffusion of the chemoattractant substrate. Finally, a remarkably simple exact analytical solution is reported for a stradily propagating plane-wave which features motility, chemotactic motion and bacteria population growth regulated by substrate diffusion.     

Estimating friction coefficients of mixed globular/chain molecules, such as protein/DNA complexes.

Existing methods for predicting translational friction properties of complex molecules start by explicitly building up their three-dimensional shape with spherical subunits. This treatment has been used especially for two types of systems: rigid assemblies and flexible chain molecules. However, many protein/DNA complexes such as chromatin consist of a small number of globular, relatively rigid, bound protein interspersed by long stretches of flexible DNA chain. I present a higher level of treatment of such macromolecules that avoids explicit subunit modeling as much as possible. An existing analytical formulation of the hydrodynamics equations is shown to be accurate when used with the present treatment. Thus the approach is fast and can be applied to hydrodynamic studies of highly degenerate multiple equilibria, such as those encountered in problems of the regulation of chromatin structure. I demonstrate the approach by predicting the effect of a hypothetical unwinding process in dinucleosomes and by simulating the distribution of sedimentation coefficients for cooperative and random models for a chromatin saturation process.