Biological transport processes and space dimension.

Abstract

We discuss the generic time behavior of reaction-diffusion processes capable of modeling various types of biological transport processes, such as ligand migration in proteins and gating fluctuations in ion channel proteins. The main observable in these two cases, the fraction of unbound ligands and the probability of finding the channel in the closed state, respectively, exhibits an algebraic t-F/2 decay at intermediate times, followed by an exponential cutoff. We provide a simple framework for understanding these observations and explain their ubiquity by showing that these qualitative results are independent of space dimension. We also derive an experimental criterion to dis h between a one-dimensional process and one whose effective dimension is higher.Transport processes in biological systems can often be described as reaction-diffusion processes either in real space or in a configuration space of some appropriate dimension. A prominent example of the former is the diffusive transport of biological molecules through the cell protoplasm from the site where they are produced or incorporated to the location where they are needed and absorbed. Adam and Delbruck (1) pointed out the importance of dimensionality in such processes, by observing that an effective reduction of the dimension of the diffusion space-e.g., by restricting the motion of those molecules to certain network structures that pervade the cell-can be utilized to speed up this transport. In this paper we will be concerned with a different question: How does the time behavior of such transport processes depend on the space dimension involved, and, consequently, can experimental observation give any information on it? Our attention was drawn to this problem while investigating ligand migration through protein matter. Consider, for example, the most thoroughly researched case: ligand migration through myoglobin (Mb) (2-9). Because no channel of sufficient size is available in the averaged static conformation of the protein (10), diffusion of ligand molecules (CO or 02) through the protein matrix cannot occur in the absence of conformational fluctuations (4,11,12). The question then arises whether the ligand motion through the protein matrix is effectively three-dimensional or whether the necessary protein fluctuations occur only within a lower-dimensional subspace (the single-channel hypothesis) (4, 13).Usually, the ligand diffusion process can be experimentally studied only indirectly: bound ligands are separated from the active site by flash photolysis and the fraction of unbound ligands after a time t is monitored (2). The nonexponential rebinding kinetics then observed at low temperatures (process I, below 170 K) is commonly ascribed to a distribution of reaction barriers (2). At higher temperatures the ligand wanders off into the protein matrix (processes II and higher), and this diffusion process is also reflected in the rebinding curve. The time dependence of the curve that results from this diffusion is typically seen to exhibit a t-...

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