Current Developments in Mathematical Biology: Proceedings of by Kazem Mahdavi, Rebecca Culshaw, John Boucher

By Kazem Mahdavi, Rebecca Culshaw, John Boucher

This quantity is a set of papers on quite a few components of present curiosity in mathematical biology, akin to epidemic affliction modeling, together with the consequences of vaccination and pressure alternative; immunology, comparable to T-Cell dynamics and the mechanism of phagocytosis; knot conception; DNA computation; and Boolean networks.

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Extra resources for Current Developments in Mathematical Biology: Proceedings of the Conference on Mathematical Biology and Dynamical Systems, the University of Texas at Tyler, ... 2005 (Series on Knots and Everything)

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Ch03 FA December 8, 2006 50 14:14 B-455 Current Developments in Mathematical Biology Trim Size: 9in x 6in L. H. Kauffman & S. Lambropoulou We shall abbreviate the expression [[a1 ], [a2 ], . . , [an ]] by writing [a1 , a2 , . . , an ], and later will use the latter expression for a numerical continued fraction as well. There should be no ambiguity between the tangle and numerical interpretations, as these will be clear from context. Figure 4 illustrates the rational tangle [2, −2, 3]. From the above discussion it makes sense to assign to a rational tangle in standard form, T = [[a1 ], [a2 ], .

We now use another branching process to model the propagation of gates that affect a given gate. The theorems about forced gates are similar to those about ineffective gates, but the proofs are simpler. Therefore we will only sketch them. This time we define the branching process by: qk = aυ(x) (1 − a)mi −υ(a) pi i:mi ≥k mi x∈{0,1} γ(φi ,x)=k for k = 0, 1, . . Then µ = λ and by our conditions that σ 2 < ∞. Again, E(Zt ) = λt , and  2 t t  σ λ (λ − 1) var(Zt ) = λ2 − λ  2 tσ ∞ i=1 pi m2i < ∞, if µ = 1, if µ = 1.

For simplicity of notation, we suppress the functional notation. Next, the in-gates for each gate are selected. If the gate has been assigned an m-argument function, then its in-gates are chosen from the n(n − 1) · · · (n − m + 1) equally likely possibilities. Finally, random initial states are independently chosen for each gate. We make several restrictions on the distribution p1 , p2 , . . , still consistent with the random networks in the literature. We assume that the average and variance of the number of arguments of a randomly selected Boolean function, or equivalently, the average and variance of the indegree of a gate, ∞ are finite.

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