Probabilistic Diffusion Constrains Self-Assembly

The organizing principles that constrain and guide self-assembly include another random process, that of diffusion. We first visited the random walk in Chap. 1, where its generality was only partly explicated. Diffusion is like the Boltzmann probability distribution in that it can be biased not only by energy differences per se but also by differences in electrochemical potential, one of whose components is a concentration gradient. There are differences between solvent diffusion (self-diffusion) and the movement of a diffusible solute within a solute. Electrochemical potential gradients drive the fluxes of charged and uncharged molecules. We review the relationship between biophysical fluxes and their generalized potentials in which the shape of the object influences the diffusion coefficient through the frictional coefficient, and the properties of the solvent come into play due to its viscosity. The continuity equation is the basis for Fick’s first and second laws and the diffusion equation. The diffusion tensor is introduced along with several solutions of the diffusion equation constrained by spatial and temporal boundary and initial conditions. The probability of particle movement is developed through the diffusion propagator. The similarity between the diffusion equation and the Schrödinger equation is made manifest by a direct comparison between the diffusion propagator and the free-particle Schrödinger propagator. Measurements of diffusion coefficients can be accomplished by several means including dynamic laser light scattering, fluorescence recovery after photobleaching, fluorescence correlation spectroscopy, and both NMR spectroscopy and imaging. We present NMR imaging of diffusion of a boundary of oxygen in blood as an example. Magnetic resonance imaging is also widely used to measure the components of the diffusion tensor, especially in the human brain.