Abstract
Voids exist in proteins as packing defects and are often associated with
protein functions. We study the statistical geometry of voids in
two-dimensional lattice chain polymers. We define voids as topological features
and develop a simple algorithm for their detection. For short chains, void
geometry is examined by enumerating all conformations. For long chains, the
space of void geometry is explored using sequential Monte Carlo importance
sampling and resampling techniques. We characterize the relationship of
geometric properties of voids with chain length, including probability of void
formation, expected number of voids, void size, and wall size of voids. We
formalize the concept of packing density for lattice polymers, and further
study the relationship between packing density and compactness, two parameters
frequently used to describe protein packing. We find that both fully extended
and maximally compact polymers have the highest packing density, but polymers
with intermediate compactness have low packing density. To study the
conformational entropic effects of void formation, we characterize the
conformation reduction factor of void formation and found that there are strong
end-effect. Voids are more likely to form at the chain end. The critical
exponent of end-effect is twice as large as that of self-contacting loop
formation when existence of voids is not required. We also briefly discuss the
sequential Monte Carlo sampling and resampling techniques used in this study.