Inverse Problems via the “Generalized Collage Theorem” for Vector-Valued Lax-Milgram-Based Variational Problems

We present an extended version of the Generalized Collage Theorem to deal with inverse problems for vector-valued Lax-Milgram systems. Numerical examples show how the method works in practical cases

Bacteria-instructed synthesis of polymers for self-selective microbial binding and labelling

The detection and inactivation of pathogenic strains of bacteria continues to be an important therapeutic goal. Hence, there is a need for materials that can bind selectively to specific microorganisms, for diagnostic or anti-infective applications, but which can be formed from simple and inexpensive building blocks. Here, we exploit bacterial redox systems to induce a copper-mediated radical polymerisation of synthetic monomers at cell surfaces, generating polymers in situ that bind strongly to the microorganisms which produced them. This ‘bacteria-instructed synthesis’ can be carried out with a variety of microbial strains, and we show that the polymers produced are self-selective binding agents for the ‘instructing’ cell types. We further expand on the bacterial redox chemistries to ‘click’ fluorescent reporters onto polymers directly at the surfaces of a range of clinical isolate strains, allowing rapid, facile and simultaneous binding and visualisation of pathogens

A collage-based approach to solving inverse problems for second-order nonlinear parabolic PDEs

The essence of collage-based methods for solving inverse problems is to bound the approximation error above by a more readily minimizable distance. The original collage method applies to ordinary differential equations (ODEs) and makes use of Banach's fixed point theorem, and the collage theorem (named for its usefulness in fractal imaging) to build an upper bound on the approximation error. A similar technique was established for solving inverse problems for second-order linear elliptic partial differential equations (PDEs). In this setting the Lax-Milgram representation theorem is the driving force, and a generalized collage theorem has been established and can be used to control the approximation error. These ideas have been further extended to include nonlinear elliptic PDEs using the nonlinear Lax-Milgram representation theorem, a more general version of its linear counterpart. In this paper we develop a collage-based method for solving inverse problems for a general class of second-order nonlinear parabolic PDEs. We develop necessary background theory, discuss the complications introduced by the presence of time-dependence, establish sufficient conditions for using this method, and present examples of the method in practice

Solving inverse problems for delay integral equations using the "collage method" for fixed points

Many inverse problems in applied mathematics can be formulated as the approximation of a target element $u$ in a complete metric space $(X,d)$ by the fixed point $\bar x$ of an appropriate contraction mapping $T : X \to X$. The method of {\em collage coding} seeks to solve this problem by finding a contraction mapping $T$ that minimizes the so-called {\em collage distance} $d(x,Tx)$. In this paper, we develop a collage coding framework for inverse problems involving two classes of integral equations -- those with delay and Hammerstein-type equations. We illustrate the method with some practical example

Identification of heme oxygenase and cytochrome P-450 in the rabbit heart

The regulation of cardiac heme oxygenase and cytochrome P-450 mixed function oxidase was studied in the rabbit heart. Heme oxygenase activity is found in ventricular and atrial microsomal fractions. This activity is NADPH dependent, and is inhibited by tin and zinc protoporphyrin, but not by either SKF 525A or 7,8-benzoflavone. Immunologic studies of cardiac heme oxygenase demonstrate that antibodies prepared against human purified hepatic heme oxygenase recognize rabbit atrial heme oxygenase and inhibit the enzyme activity by 92%. In contrast, control immunoglobulin does not inhibit heme oxygenase activity. Further, the western blotting technique demonstrates that a similar band of protein with a molecular weight of 32,000 exists in cardiac microsomes and that no protein cross-reacts with purified hepatocyte heme oxygenase. Marked induction of atrial heme oxygenase is observed in microsomal fractions prepared from rabbits treated with cobalt chloride. Atrial microsomes possess 0.24 nmol of cytochrome P-450 as compared to 0.68 nmol/mg protein in microsomes from the liver. The levels of aryl hydrocarbon hydroxylase (AHH) activity, a cytochrome P-450-dependent enzyme, in ventricle and atrium are stimulated by a NADPH-generating system and are sensitive to 7,8-benzoflavone, and SKF 525A, known inhibitors of cytochrome P-450 mixed function oxidase. AHH activity in ventricular and atrial microsomes is 2-3% of that seen in liver microsomes whereas the P-450 content/mg protein is about 20% of that observed in the liver. AHH activity is mediated by a form of cytochrome P-450 that is inducible by 3-methylcholanthrene/β-naphthoflavone. A possible new role of the heart cytochrome P-450 system in cardiac function is proposed. It is also proposed that physiological regulation of cardiac cytochrome P-450 by heavy metals is mediated through the induction of cardiac heme oxygenase