Abstract
In Chap. 11, we considered various analytically tractable models of intracellular pattern formation and waves based on deterministic RD equations. A complementary approach is to develop more realistic multi-scale computational models, which include details of the structure of individual macromolecules, the biochemical network of signaling pathways, the aqueous environment of the cytoplasm, the mechanical properties of the cytoskeleton, and the geometry of the cell [74, 278, 317, 845].
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Notes
- 1.
A normal real matrix \(\mathbf{A}\) is one for which \(\mathbf{A}{} \mathbf{A}^{\top }=\mathbf{A}^{\top }{} \mathbf{A}\). Examples include symmetric (\(\mathbf{A}^{\top }=\mathbf{A}\)) and skew-symmetric (\(\mathbf{A}^{\top }=-\mathbf{A}\)) matrices. A normal matrix has an orthogonal set of eigenvectors and is diagonalizable by a unitary matrix (\(\mathbf{UAU}^{\dagger } =\mathbf{A}_\mathrm{diag}\), \(\mathbf{U}^{\dagger }=\mathbf{U}^{-1}\)). However, the eigenvalues of a normal but non-symmetric matrix may be complex. Finally, the real part of the principal eigenvalue of a normal matrix \(\mathbf{A}\) is equal to the principal eigenvalue of the Hermitian part \(H(\mathbf{A})=(\mathbf{A}+\mathbf{A}^{\top })/2\).
- 2.
The so-called Doi-shift \(a^{\dagger } \rightarrow a^{\dagger }+1\) was not needed in the weak noise limit of Sect. 8.2, since we rescaled the occupation numbers by the system size N, so that the shift 1/N becomes negligible.
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Bressloff, P.C. (2021). Stochastic reaction–diffusion processes. In: Stochastic Processes in Cell Biology. Interdisciplinary Applied Mathematics, vol 41 . Springer, Cham. https://doi.org/10.1007/978-3-030-72519-8_16
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DOI: https://doi.org/10.1007/978-3-030-72519-8_16
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