The Dirichlet problem

Thesis (M.A.)--Boston UniversityThe problem of finding the solution to a general eliptic type partial differential equation, when the boundary values are given, is generally referred to as the Dirichlet Problem. In this paper I consider the special eliptic equation of ∇2 J=0 which is Laplace's equation, and I limit myself to the case of two dimensions. Subject to these limitations I discuss five proofs for the existence of a solution to Laplace's equation for arbitrary regions where the boundary values are given. [TRUNCATED

Facilitated diffusion and the possible role of myoglobin as a transport mechanism.

This paper deals with the facilitated diffusion of oxygen in systems containing proteins, such as ferrohemoglobin, that combine reversibly with oxygen. The phenomenon is treated in terms of translational diffusion of free and bound oxygen molecules, and of the reaction rate of the oxygen molecules with the oxygen-binding protein. These considerations lead to a differential equation (see Equation 9) for the flux; analytical or numerical solutions for this equation are not yet available. However, the assumption that chemical equilibrium exists at every point in the diffusing system leads to a simple equation already derived by other workers. This in turn permits calculation of the partial pressure of oxygen as a function of distance, in facilitated diffusion across a flat membrane, for given values of the total oxygen flux. The data given by Wittenberg in the preceding paper are analyzed in these terms, with satisfactory agreement between experiment and calculation. The possible contribution of rotary diffusion to the facilitated flux of oxygen is analyzed in detail, and it is found to be negligible compared to that of translational diffusion. Calculations of facilitated diffusion in muscle resulting from the presence of myoglobin indicate that myoglobin may be responsible for a substantial part of the transport of oxygen in muscle, especially at low partial pressures of oxygen

Facilitated diffusion. The case of carbon monoxide.

Abstract An application of a singular perturbation method to the case of carbon monoxide similar to that used previously for the case of oxygen (Murray, J. D., Proc. Roy. Soc. London B Biol. Sci., 178, 95 (1971)) shows that the absence of any observed facilitation of diffusion of this ligand by either hemoglobin or myoglobin in all of the experiments so far performed results from the very high affinity of both proteins for the gas. At every point in the solution both proteins were essentially at equilibrium with the gas but it was evidently impossible to reduce the pressure of carbon monoxide on the low pressure face of the solution sufficiently to remove a significant amount of the carbon monoxide from combination with the macromolecular carrier. The principle that a macromolecule can only function as a carrier under conditions in which its saturation with the ligand is incomplete in some region of the solution is quite general

THE EFFECT OF OXYGENATION ON THE RATE OF DIGESTION OF HUMAN HEMOGLOBINS BY CARBOXYPEPTIDASES.

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THE OXYGEN EQUILIBRIUM OF CYSTINE-TREATED HUMAN HEMOGLOBIN WITHOUT FREE SULFHYDRYL GROUPS.

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The Reactions of the Isolated α and β Chains of Human Hemoglobin with Oxygen and Carbon Monoxide

Abstract The values of the equilibrium and kinetic constants for the reactions with oxygen and carbon monoxide of the isolated α and β chains of human hemoglobin, both in the form with sulfhydryl groups blocked by p-hydroxymercuribenzoate and in the form with freely titratable sulfhydryl groups, have been determined. The data show clearly that the behavior of the isolated chains is not only unlike that of hemoglobin A, but also differs markedly from that of myoglobin. Since the isolated chains behave as simple systems without heme-heme interaction (n = 1 in the O2 equilibrium), they have been used to test the proposition that the binding of a ligand is correctly expressed as a single step reaction. In at least one case it appears that it is not

Studies on the Relations between Molecular and Functional Properties of Hemoglobin: III. THE INFLUENCE OF SALTS ON THE BOHR EFFECT IN HUMAN HEMOGLOBIN

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STUDIES ON THE OXIDATION-REDUCTION POTENTIALS OF HEME PROTEINS. V. THE OXIDATION BOHR EFFECT IN NORMAL HUMAN HEMOGLOBIN AND HUMAN HEMOGLOBIN DIGESTED WITH CARBOXYPEPTIDASE A.

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Observations on the Kinetics of the Reaction of Hemoglobin with Oxygen

Abstract The kinetics of the reaction of human deoxyhemoglobin with oxygen at high concentrations of ligand (∼10-4 m) has been studied with the use of a fast stopped flow apparatus with a dead time of approximately 300 µsec. The shape of the progress curve reveals that the apparent second order rate constant for the reaction with oxygen (k') increases with the extent of the reaction, similar to what was reported for other ligands (i.e. carbon monoxide). Within the oxygen concentration range studied, the rate of the reaction at any stage is proportional to the ligand concentration, notwithstanding the change in rate as the reaction proceeds. These results are interpreted to indicate that the ligand-linked intramolecular change in hemoglobin is a fast process which occurs in a fraction of a millisecond