A Mathematical Model Applying the Random-Walk Method to the Environment of a Neuron
A mathematical model describing the supply and demand relationships existing in the environment of a brain cell (neuron of the cerebral cortex) was developed. The stochastic random-walk technique was applied to the representation and solution of the system which consisted of a neuron being supplied with nutrients by an adjoining capillary. The random-walk method incorporated a uniformly generated random number which was weighted by the normal distribution curve to determine the random walk of a molecule. The resultant weighted value was designated as defining the motion of any particular species in space. The distribution curve was a function of diffusivity and time. The method allowed the tracking of individual molecules as they proceeded through the metabolic reactions in the cell. Oxygen, glucose, carbon dioxide and lactate were selected as the primary components of study, since they represent the major input and output parameters of metabolism inside the cell. The consumption and/or production of these components were dependent on probability values assigned to each metabolic reaction into which they entered. The solution of the model was based on the number of molecules existing in the tissue as a fuction of PO2, (partial pressure of oxygen), glucose level, etc. The model was very sensitive to perturbations of metabolic scheme parameters and to PO2 levels in the capillary. The model predicted an excess of 02, (oxygen) in the tissue. The effects of edema on intercapillary distances as well as changes in the size and number of mitochondria within the neuron were examined using the model. Apart from the results which are presented later in the text, the primary limitation of the modeling method was the constraint put on by the large amount of computer time necessary for a simulation. This work is intended to be a methodology for the theoretical analysis of biochemical processes at the cellular and subcellular level. As this is the primary emphasis, the results have not been experimentally verified.
KeywordsOxygen Transport Metabolic Reaction Gaussian Distribution Function Random Step Brownian Motion Process
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