Finite element analysis of the time-dependent Smoluchowski equation for acetylcholinesterase reaction rate calculations

This article describes the numerical solution of the time-dependent Smoluchowski equation to study diffusion in biomolecular systems. Specifically, finite element methods have been developed to calculate ligand binding rate constants for large biomolecules. The resulting software has been validated and applied to the mouse acetylcholinesterase (mAChE) monomer and several tetramers. Rates for inhibitor binding to mAChE were calculated at various ionic strengths with several different time steps. Calculated rates show very good agreement with experimental and theoretical steady-state studies. Furthermore, these finite element methods require significantly fewer computational resources than existing particle-based Brownian dynamics methods and are robust for complicated geometries. The key finding of biological importance is that the rate accelerations of the monomeric and tetrameric mAChE that result from electrostatic steering are preserved under the non-steady-state conditions that are expected to occur in physiological circumstances.     

Continuum diffusion reaction rate calculations of wild-type and mutant mouse acetylcholinesterase: adaptive finite element analysis

As described previously, continuum models, such as the Smoluchowski equation, offer a scalable framework for studying diffusion in biomolecular systems. This work presents new developments in the efficient solution of the continuum diffusion equation. Specifically, we present methods for adaptively refining finite element solutions of the Smoluchowski equation based on a posteriori error estimates. We also describe new, molecular-surface-based models, for diffusional reaction boundary criteria and compare results obtained from these models with the traditional spherical criteria. The new methods are validated by comparison of the calculated reaction rates with experimental values for wild-type and mutant forms of mouse acetylcholinesterase. The results show good agreement with experiment and help to define optimal reactive boundary conditions.     

A generalized finite difference method for modeling cardiac electrical activation on arbitrary, irregular computational meshes

A generalized finite difference (GFD) method is presented that can be used to solve the bi-domain equations modeling cardiac electrical activity. Classical finite difference methods have been applied by many researchers to the bi-domain equations. However, these methods suffer from the limitation of requiring computational meshes that are structured and orthogonal. Finite element or finite volume methods enable the bi-domain equations to be solved on unstructured meshes, although implementations of such methods do not always cater for meshes with varying element topology. The GFD method solves the bi-domain equations on arbitrary and irregular computational meshes without any need to specify element basis functions. The method is useful as it can be easily applied to activation problems using existing meshes that have originally been created for use by finite element or finite difference methods. In addition, the GFD method employs an innovative approach to enforcing nodal and non-nodal boundary conditions. The GFD method performs effectively for a range of two and three-dimensional test problems and when computing bi-domain electrical activation moving through a fully anisotropic three-dimensional model of canine ventricles.     

Exploring the interaction between D-xylose isomerase and D-xylose by free energy calculation

The numerical quadrature thermodynamic integration method is used to investigate enzyme-substrate interaction of D-xylose isomerase. A screening function for the coulombic interaction is introduced into the simulation to correct the effect of finite cutoff radius for the non-bonded interaction. The binding free energy difference for D-xylose with D-xylose isomerase and its N184D mutant has been calculated, and the result 3.9 ± 1.2 kJ/mol agrees well with experimental data of 4.38 kJ/mol. In addition, the structure and dynamics of enzyme-substrate complex were simulated for mutant and wild-type enzyme, respectively. Analysis of the structures and intramolecular interactions of the complexes were found to be valuable for understanding the reaction mechanism of the enzyme D-xylose isomerase. © Wiley-Liss, Inc.     

On morphometric measurement of oxygen diffusing capacity in middle ear gas exchange.

An accurate mathematical model of transmucosal gas exchange is prerequisite to understanding middle ear (ME) physiology. Current models require experimentally measured gas species time constants for all extant conditions as input parameters. However, studies on pulmonary gas exchange have shown that a morphometric model that incorporates more fundamental physiochemical and anatomic parameters accurately simulates transport from which the species time constants can be derived for all extant conditions. Here, we implemented a variant of that model for ME gas exchange that requires the measurement of diffusional length (tau) for the ME mucosa. That measure contributes to the mucosal diffusing capacity and reflects the resistance to gas flow between air space and capillary. Two methods for measuring tau have been proposed: linear distance between the air-mucosal boundary and capillary and the harmonic mean of all contributing pathway lengths. Oxygen diffusing capacity was calculated for different ME mucosal geometries by using the two tau measures, and the results were compared with those predicted by a detailed, two-dimensional finite element analysis. Predictive accuracy was improved by incorporating the harmonic tau measure, which captures important information regarding variations in capillary shape and distribution. However, compared with the oxygen diffusing capacity derived from the finite element analysis, both measures yielded nonlinear, positively biased estimates. The morphometric techniques underestimate diffusion length by failing to account for the curvilinear gas flow pathways predicted by the finite element model.     

Hemoglobin-carbon monoxide binding rate. Low temperature magneto-optical detection of spin-tunneling.

The spin-tunneling model of Hb--CO binding is used to calculate the binding rate at low temperature and high magnetic fields. The rate is calculated in second order perturbation theory assuming that spin-orbit coupling mediates the Hb iron electronic state change. The reaction which occurs at the crossing of the S = 2 and S = 0 energy vs. configuration coordinate curves is nonadiabatic, having a small electronic transition matrix element. Since detection of CO binding by polarized light in the Soret band makes it possible to observe hemes at specific orientation to the field direction, the rate is calculated for arbitrary heme orientation. Comparison with measurements at low temperature in zero field is made for spin quantization along the molecular crystal field direction.     

Comparison of the linear finite element prediction of deformation and strain of human cancellous bone to 3D digital volume correlation measurements

The mechanical properties of cancellous bone and the biological response of the tissue to mechanical loading are related to deformation and strain in the trabeculae during function. Due to the small size of trabeculae, their motion is difficult to measure. To avoid the need to measure trabecular motions during loading the finite element method has been used to estimate trabecular level mechanical deformation. This analytical approach has been empirically successful in that the analytical models are solvable and their results correlate with the macroscopically measured stiffness and strength of bones. The present work is a direct comparison of finite element predictions to measurements of the deformation and strain at near trabecular level. Using the method of digital volume correlation, we measured the deformation and calculated the strain at a resolution approaching the trabecular level for cancellous bone specimens loaded in uniaxial compression. Smoothed results from linearly elastic finite element models of the same mechanical tests were correlated to the empirical three-dimensional (3D) deformation in the direction of loading with a coefficient of determination as high as 97% and a slope of the prediction near one. However, real deformations in the directions perpendicular to the loading direction were not as well predicted by the analytical models. Our results show, that the finite element modeling of the internal deformation and strain in cancellous bone can be accurate in one direction but that this does not ensure accuracy for all deformations and strains.     

Microcalorimetric study of substrate fixation on the leucine-isoleucine-valine-binding protein from Escherichia coli.

The thermodynamical behaviour of the Leucine-Isoleucine-Valine binding protein implicated in branched chain amino acids transport has been examined. The enthalpy changes (ΔH) were measured by microcalorimetry for the binding reaction of the protein with leucine, isoleucine and valine. The binding of the amino acids resulted in a slightly endothermic reaction for leucine and in exothermic reaction for both isoleucine and valine. The free energy values (ΔG) were calculated from the respective dissociation constants measured with the three substrates by the rapid dialysis technique of Colowick and Womack. From ΔG and ΔH contributions the entropic values (ΔS) corresponding to the complex formation reactions were deduced and were found highly positive in all cases.     

Mechanical significance of femoral head trabecular bone structure in Loris and Galago evaluated using micromechanical finite element models

Work on the interspecific and intraspecific variation of trabecular bone in the proximal femur of primates demonstrates important architectural variation between animals with different locomotor behaviors. This variation is thought to be related to the processes of bone adaptation whereby bone structure is optimized to the mechanical environment. Micromechanical finite element models were created for the proximal femur of the leaping Galago senegalensis and the climbing and quadrupedal Loris tardigradus by converting bone voxels from high-resolution X-ray computed tomography scans of the femoral head to eight-noded brick elements. The resulting models had approximately 1.8 million elements each. Loading conditions representing takeoff phase of a leap and more generalized load orientations were applied to the models, and the models were solved using the iterative "row-by-row" matrix-vector multiplication algorithm. The principal strain and Von Mises stress results for the leaping model were similar for both species at each load orientation. Similar hip joint reaction forces in the range of 4.9 x to 12 x body weight were calculated for both species under each loading condition, but the hip reaction values estimated for Loris were higher than predicted based on locomotor behavior. These results suggest that functional adaptation to hip joint loading may not fully explain the differences in femoral head trabecular bone structure in Galago and Loris. The finite element method represents a unique and useful tool for analyzing the functional adaptation of trabecular bone in a diversity of animals and for reconstructing locomotor behavior in extinct taxa.     

A new method to investigate in vivo knee behavior using a finite element model of the lower limb

Several finite element models have been developed for estimating the mechanical response of joint internal structures, where direct or indirect in vivo measurement is difficult or impossible. The quality of the predictions made by those models is largely dependent on the quality of the experimental data (e.g. load/displacement) used to drive them. Also numerical problems have been described in the literature when using implicit finite element techniques to simulate problems that involve contacts and large displacements. In this study, a unique strategy was developed combining high accuracy in vivo three-dimensional kinematics and a lower limb finite element model based on explicit finite element techniques. The method presents an analytical technique applied to a dynamic loading condition (impact during hopping on one leg). The validation of the lower limb model focused on the response of the whole model and the knee joint in particular to the imposed 3D femoral in vivo kinematics and ground reaction forces. The approach outlined in this study introduces a generic tool for the study of in vivo knee joint behavior.     

Characterization of nuclear factors binding to AT-rich element in the rat p53 promoter

In this study, we identified AT-rich element located at positions -504 to -516 in the rat p53 promoter by DNase I foot printing assay. This region was previously identified as a positive regulatory element in the murine p53 promoter and designated as PBF1 (p53 binding factor 1) binding site. However, the proteins binding to this AT-rich element have not been identified yet. Therefore, we characterized the binding protein by various biochemical methods. First, we confirmed that by the oligonucleotide competition assay, nuclear factors bound to the AT-rich element in a sequence-specific manner. Two binding proteins were identified in southwestern blotting analysis and the molecular masses of the proteins were 60 and 40 kDa, respectively. The proteins were stable to denaturants or ionic strength. Treatment of chelators showed that the binding proteins did not require divalent cation for DNA-binding activity. In addition, the binding proteins were labile to protease treatment. This study showed that 60 and 40 kDa proteins bound to AT-rich element and the physico-chemical properties provided new insights into the binding proteins.     

The flexibility during the juxtaposition of reacting groups and the upper limits of enzyme reactions

The combinations between enzymes and substrates occur only after their reacting groups are in juxtaposition with each other. This will greatly reduce the probability of their effective encounters. However, the results calculated with the finite element method show that the reaction limits will not decrease substantially if van der Waal's forces and a reasonable flexibility during such a juxtaposition are taken into account.     

Calculation of protein form birefringence using the finite element method.

An approach based on the finite element method (FEM) is employed to calculate the optical properties of macromolecules, specifically form birefringence. Macromolecules are treated as arbitrarily shaped particles suspended in a solvent of refraction index n1. The form birefringence of the solution is calculated as the difference in its refractive index when all the particles of refractive index n2 are either parallel to or normal to the direction of the polarization of light. Since the particles of interest are small compared to the wavelength of light, a quasi-static approximation for the refractive index is used, i.e., that it is equal to the square root of the dielectric constant of the suspension. The average dielectric constant of the mixture is calculated using the finite element method. This approach has been tested for ellipsoidal particles and a good agreement with theoretical results has been obtained. Also, numerical results for the motor domains of ncd and kinesin, small arbitrarily shaped proteins with known x-ray structures, show reasonable agreement with the experimental data obtained from transient electric birefringence experiments.     

Prediction of strength and strain of the proximal femur by a CT-based finite element method

Hip fractures are the most serious complication of osteoporosis and have been recognized as a major public health problem. In elderly persons, hip fractures occur as a result of increased fragility of the proximal femur due to osteoporosis. It is essential to precisely quantify the strength of the proximal femur in order to estimate the fracture risk and plan preventive interventions. CT-based finite element analysis could possibly achieve precise assessment of the strength of the proximal femur. The purpose of this study was to create a simulation model that could accurately predict the strength and surface strains of the proximal femur using a CT-based finite element method and to verify the accuracy of our model by load testing using fresh frozen cadaver specimens. Eleven right femora were collected. The axial CT scans of the proximal femora were obtained with a calibration phantom, from which the 3D finite element models were constructed. Materially nonlinear finite element analyses were performed. The yield and fracture loads were calculated, while the sites where elements failed and the distributions of the principal strains were determined. The strain gauges were attached to the proximal femoral surfaces. A quasi-static compression test of each femur was conducted. The yield loads, fracture loads and principal strains of the prediction significantly correlated with those measured (r=0.941, 0.979, 0.963). Finite element analysis showed that the solid elements and shell elements in undergoing compressive failure were at the same subcapital region as the experimental fracture site.     

Continuum simulations of acetylcholine diffusion with reaction-determined boundaries in neuromuscular junction models

The reaction-diffusion system of the neuromuscular junction has been modeled in 3D using the finite element package FEtk. The numerical solution of the dynamics of acetylcholine with the detailed reaction processes of acetylcholinesterases and nicotinic acetylcholine receptors has been discussed with the reaction-determined boundary conditions. The simulation results describe the detailed acetylcholine hydrolysis process, and reveal the time-dependent interconversion of the closed and open states of the acetylcholine receptors as well as the percentages of unliganded/monoliganded/diliganded states during the neuro-transmission. The finite element method has demonstrated its flexibility and robustness in modeling large biological systems.     

Electrostatic effects in proteins: comparison of dielectric and charge models.

Two approaches for calculating electrostatic effects in proteins are compared and ana analysis is presented of the dependence of calculated properties on the model used to define the charge distribution. Changes in electrostatic free energy have been calculated using a screened Coulomb potential (SCP) with a distance-dependent effective dielectric permittivity to model bulk solvent effects and a finite difference approach to solve the Poisson-Boltzmann (FDPB) equation. The properties calculated include shifts in dissociation constants of ionizable groups, the effect of annihilating surface charges on the binding of metals, and shifts in redox potentials due to changes in the charge of ionizable groups. In the proteins considered the charged sites are separated by 3.5-12 A. It is shown that for the systems studied in this distance range the SCP yields calculated values which are at least as accurate as those obtained from solution of the FDPB equation. In addition, in the distance range 3-5 A the SCP gives substantially better results than the FDPB equation. Possible sources of this difference between the two methods are discussed. Shifts in binding constants and redox potentials were calculated with several standard charge sets, and the resulting values show a variation of 20-40% between the 'best' and 'worst' cases. From this study it is concluded that in most applications, changes in electrostatic free energies can be calculated economically and reliably using an SCP approach with a single functional form of the screening function.     

Analysis of pulsatile blood flow: A carotid siphon model

Numerical results for axial and secondary flow velocity and pressure in a three-dimensional model of the human carotid siphon have been calculated; the investigations were carried out under physiologically relevant pulsatile flow conditoins. Time-dependent, three-dimensional Navier-Stokes equations were solved numerically by using a special finite element method. The results of the computer simulation presented here concentrate on the secondary motion effect during the pulsatile flow cycle in multiple three-dimensional curvatures.     

Use of the photoelastic method and finite element analysis in the assessment of wall strain in abdominal aortic aneurysm models

Abdominal aortic aneurysm (AAA) is a significant health problem. Current clinical rupture-risk relies primarily on the maximum diameter of the AAA and also growth rate. However, AAAs are a patient-specific problem and recently, numerical tools have been employed to estimate rupture-potential. Alternatively, experimental assessment of AAA biomechanics receives less attention, yet, rigorous validation of numerical tools is required prior to clinical acceptance. This paper examines the use of the photoelastic method to assess wall strain and its validation using finite element analysis (FEA) in a small number of patient-specific AAA models. Experimental models were manufactured in-house using the injection-moulding procedure together with a commercially available photoelastic material. The material was mechanically characterised prior to testing, with models examined under three loading regimes (80, 120 and 160mmHg). Each experimental model was imaged using computed tomography (CT) and reconstructed in three dimensions (3D) for numerical analyses. Experimental wall strain was measured and numerical wall strain calculated with finite element analysis (FEA). Results were qualitatively and quantitatively compared. There was good qualitative agreement between the experimental and numerical methods, with similar trends apparent throughout all models at all pressures. Overall, acceptable percentage errors between the techniques were observed for all models. Median errors of -6.5%, -0.4% and 3.9% for the models at 80, 120 and 160mmHg pressures, respectively, were determined. The photoelastic method is a very useful experimental tool that provides instant, easy to interpret, information regarding wall strain. The technique is useful for validation of numerical AAA studies.     

Finite element modelling of contracting skeletal muscle

To describe the mechanical behaviour of biological tissues and transport processes in biological tissues, conservation laws such as conservation of mass, momentum and energy play a central role. Mathematically these are cast into the form of partial differential equations. Because of nonlinear material behaviour, inhomogeneous properties and usually a complex geometry, it is impossible to find closed-form analytical solutions for these sets of equations. The objective of the finite element method is to find approximate solutions for these problems. The concepts of the finite element method are explained on a finite element continuum model of skeletal muscle. In this case, the momentum equations have to be solved with an extra constraint, because the material behaves as nearly incompressible. The material behaviour consists of a highly nonlinear passive part and an active part. The latter is described with a two-state Huxley model. This means that an extra nonlinear partial differential equation has to be solved. The problems and solutions involved with this procedure are explained. The model is used to describe the mechanical behaviour of a tibialis anterior of a rat. The results have been compared with experimentally determined strains at the surface of the muscle. Qualitatively there is good agreement between measured and calculated strains, but the measured strains were higher.     

Quantitative analysis of exogenous IGF-1 administration of intervertebral disc through intradiscal injection

Exogenous administration of IGF-1 has been proposed as a therapy for disc degeneration. The objectives of this study were to develop a numerical model for quantitatively analysing exogenous administration of IGF-1 into the intervertebral disc (IVD) via intradiscal injection and to investigate the effects of IGF-1 administration on distribution of glucose and oxygen in the IVD. In this study, the reversible binding reaction between IGF-1 and IGF binding proteins was incorporated into the mechano-electrochemical mixture model. The model was used to numerically analyse transport of IGF-1, glucose, oxygen and lactate in the IVD after IGF-1 administration. The enhancement of IGF-1 on lactate production was also taken into account in the theoretical model. The numerical analyses using finite element method demonstrated that the binding reactions significantly affect the time-dependent distribution of IGF-1 in the IVD. It was found that the region affected by IGF-1 was smaller and the duration of the therapeutic IGF-1 level was longer in the degenerated disc with a higher concentration of IGF binding proteins. It was also found that the IGF-1 injection can reduce glucose concentration and increase lactate accumulation (i.e., lower pH) in the IVD and these influences were regulated by the IGF-1 binding reactions. This study indicated the complexity of intradiscal administration of growth factors, which needs to be fully analysed in order to achieve a successful outcome. The new theoretical model developed in this study can serve as a powerful tool in analysing and designing the optimal treatments of growth factors for disc degeneration.