Percolation Modeling the Randomly Connected Hemocoel

Imagine a hemocoel to be a porous brick wetted by hemolymph. We may ask what is the probability that the interior of the hemo-coel is wetted so that a particle can move from one place to another through the interior of our hemocoel brick. Using ideas from Broadbent and Hammersley's percolation model (Ref: Percolation), we can move between three dimensions when the brick or hemocoel is fully saturated towards a two-dimensional model when the hemocoel consists only of moistened surfaces that connect with each other two-dimensionally.

In two dimensions, we have the following: Let the surfaces of the hemocoel together be represented by a plane square lattice. Let probability, p, be a number between zero, where the edge of a square is dry, and one, where the edge is wet. Now we examine each edge of the lattice in sequence and declare the edge to be wet with a probability p and dry otherwise. Each edge is wet or dry independent of all other edges. The edges of the two-dimensional surface represent the inner connectivity of the hemocoel so that p becomes the proportion of passages that are wet and thus allows flow to cross them. Grossly imagined, when p = 0.25, the connected clusters of open edges are isolated and small, but as p increases, the size of each cluster increases, and the number of clusters increases, until at a critical value of p, called Pc, a cluster forms that now fills the entire space of the hemocoel.

Now, we may model any sized hemocoel using a large finite subsection of this two-dimensional surface. As dynamics change and flows vary, for the hemocoel to transmit hemolymph and continue to function as a hemocoel, the vertices and edges of the square lattice must contain somewhere a connected subgraph of the surface. A point within the model is wetted if and only if a path exists in two dimensions connecting a wet square of the surface to our point in question. The hemocoel functions as a circulation as long as 'a wet line of squares' exists from point to point across the hemocoel, but connectivity does not depend on the length or direction of this line that may change moment to moment. Percolation theory investigates the structure of this subgraph when we delete the closed edges particularly in regards to how the percolation structure depends on the numerical value of p. If p is large the probability of flow is increased over times when p is small.

For purposes of shrinking, it is evident that the fine structure of the 'passages' in the interior of the hemocoel is on a scale that is negligible when compared with the overall size of the hemocoel. In such situations a vertex in the center of the hemocoel is wetted with hemolymph and, hence, can give up and receive particles from the circulation, behaves rather similarly to the probability that this vertex forms the end vertex of an infinite path of open edges on the wet surface. It is for this reason also that a hemo-coel is robust and resists point blockages, as given a minimum of hemolymph to wet the surfaces, there are many alternative routes for particles to follow in case one path becomes obstructed. We may construct an analogous model for a three-dimensional volume. Our above model is called bond percolation on the square lattice and is the most studied of all percolation processes.

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