Minimal formulation of joint motion for biomechanisms

Biomechanical systems share many properties with mechanically engineered systems, and researchers have successfully employed mechanical engineering simulation software to investigate the mechanical behavior of diverse biological mechanisms, ranging from biomolecules to human joints. Unlike their man-made counterparts, however, biomechanisms rarely exhibit the simple, uncoupled, pure-axial motion that is engineered into mechanical joints such as sliders, pins, and ball-and-socket joints. Current mechanical modeling software based on internal-coordinate multibody dynamics can formulate engineered joints directly in minimal coordinates, but requires additional coordinates restricted by constraints to model more complex motions. This approach can be inefficient, inaccurate, and difficult for biomechanists to customize. Since complex motion is the rule rather than the exception in biomechanisms, the benefits of minimal coordinate modeling are not fully realized in biomedical research. Here we introduce a practical implementation for empirically-defined internal-coordinate joints, which we call "mobilizers." A mobilizer encapsulates the observations, measurement frame, and modeling requirements into a hinge specification of the permissible-motion manifold for a minimal set of internal coordinates. Mobilizers support nonlinear mappings that are mathematically equivalent to constraint manifolds but have the advantages of fewer coordinates, no constraints, and exact representation of the biomechanical motion-space-the benefits long enjoyed for internal-coordinate models of mechanical joints. Hinge matrices within the mobilizer are easily specified by user-supplied functions, and provide a direct means of mapping permissible motion derived from empirical data. We present computational results showing substantial performance and accuracy gains for mobilizers versus equivalent joints implemented with constraints. Examples of mobilizers for joints from human biomechanics and molecular dynamics are given. All methods and examples were implemented in Simbody?-an open source multibody-dynamics solver available at https://Simtk.org.     

Letters to the Editor

The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to the editor-in-chieX Chandler Davis.     

Competitive production scheduling: A two-firm,noncooperative finite dynamic game

In this paper, we study the production scheduling problem in a competitive environment. Two firms produce the same product and compete in a market. The demand is random and so is the production capacity of each firm, due to random breakdowns. We consider a finite planning horizon. The scheduling problem is formulated as a finite dynamic game. Algorithms are developed to determine the security, hazard, and Nash policies. Numerical examples are discussed. A single-firm optimization model is also analyzed and it is observed that the production control policy from the single-firm optimization model may not perform well in a competitive environment.     

The interaction of modified 7S soy protein with mono- and di-glycerides at the oil-water interface and its effect on the stability of concentrated corn oil-in-water emulsions

The influence of chemically modified 7S fraction of soybean protein (7MSPF), and its partial replacement by mono- and di-glycerides in various ratios, on the rate of drop coalescence in concentrated corn oil-in-water emulsions has been investigated. A total emulsifier concentration of 2.0 % (wt/wt) was used. The minimum drop coalescence rate was achieved when using 1.0% (wt/wt) 7MSPF in conjunction with 0.5% (wt/wt) monoglyceride and 0.5 % (wt/wt) di-glyceride at pH 5.5. At other mono-/di-glycerides and protein/glycerides ratios, and at other pHs, the rate of drop coalescence was higher than when 2.0% (wt/wt) 7MSPF was used. The reduction in drop coalescence rate under these conditions is attributed to association of 7MSPF with the glycerides at the oil-water interface. The influence of protein/glycerides ratio on the viscoelastic properties of mixed interfacial films supports this view.     

Determination of water with a modified Karl Fischer reagent Stability and the mechanism of reaction with water

It is shown that the water equivalent of the modified Karl Fischer reagent (standard Karl Fischer reagent in which dimethylformamide is substituted for methanol), depending on pKa and the concentration of the solvent used for preliminary titration, is not dependent on the water concentration being determined. Also discussed are different aspects of the stability of the Karl Fischer reagent and its modifications. On the basis of the literature data and the findings of this work, a mechanism of interaction between water and the modified Karl Fischer reagent is proposed: in the first stage of the reaction pyridine sulphodioxide is solvated with solvents containing active hydrogen (alcohols, organic acids and water). The lower the value of pK_a of the solvent, the greater the contribution of water to the pyridine sulphodioxide solvation reaction. The results of this work suggest that, especially in particular cases, the water equivalent of the Karl Fischer reagent and its modifications should be determined under the same conditions as the determination of water in the sample.     

Phonons in YBa2Cu3O7−x crystals with site-selective oxygen isotope substitution: Frequencies,normal vectors,non-linear properties

Lattice vibrations of YBa₂Cu₃O_7–x crystals with a site-selective isotope substitution¹⁸O¹⁶O are investigated theoretically. It is shown that shifts of the frequencies of A_1g oxygen vibrations due to this substitution are smaller than under complete one. However, phonon bands and normal vectors are changed much stronger, especially in the case when apical oxygens are substituted. On the basis of the presented analysis anharmonic properties and transition temperatures of the selectively substituted crystals are also briefly discussed.     

Paradoxical subsets of hyperbolic and spherical discs

Given a disc D of radius r in H ² (resp. S ²) with <r (resp. <r), we determine the pairs (m,n) for which there is an (m,n)-paradoxical subset of D but not an (m–1, n)-paradoxical subset of D or an (m, n–1)-paradoxical subset of D.     

bryA: an unusual modular polyketide synthase gene from the uncultivated bacterial symbiont of the marine bryozoan Bugula neritina

"Candidatus Endobugula sertula," the uncultivated bacterial symbiont of Bugula neritina, is the proposed source of the bryostatin family of anticancer compounds. We cloned a large modular polyketide synthase (PKS) gene complex from "Candidatus Endobugula sertula" and characterized one gene, bryA, which we propose is responsible for the initial steps of bryostatin biosynthesis. Typical PKS domains are present. However, acyltransferase domains are lacking in bryA, and beta-ketoacyl synthase domains of bryA cluster with those of PKSs with discrete, rather than integral, acyltransferases. We propose a model for biosynthesis of the bryostatin D-lactate starter unit by the bryA loading module, utilizing atypical domains homologous to FkbH, KR, and DH. The bryA gene product is proposed to synthesize a portion of the pharmacologically active part of bryostatin and may be useful in semisynthesis of clinically useful bryostatin analogs.