Statistical Geometry of Packing Defects of Lattice Chain Polymer from Enumeration and Sequential Monte Carlo Method

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.