A Review of the packing problem

Monte carlo simulations of protein assembly, disassembly, and linear motion on DNA

We use Monte Carlo simulations to analyze the simultaneous interactions of multiple proteins to a long DNA molecule. We study the time dependence of protein organization on DNA for different regimes that comprise (non)cooperative sequence-independent protein assembly, dissociation, and linear motion. A range of different behaviors is observed for the dynamics, final coverage, and cluster size distributions. We observe that the DNA substrate is almost never completely covered by protein when taking into account only (non)cooperative binding, because gaps remain on the substrate that are smaller than the binding site size of the protein. Due to these gaps, the apparent binding size of a protein during noncooperative binding can be overestimated by up to 30%. During dissociation of cooperatively bound proteins, the dissociation curve can be exponentially shaped even when allowing only end-dependent dissociation. We discuss the potential of our method for the analysis of a number of single-molecule experiments, for example, the binding of the DNA-repair proteins RecA and Rad51 to DNA.

Parking strategies for genome sequencing

The parking strategy is an iterative approach to DNA sequencing. Each iteration consists of sequencing a novel portion of target DNA that does not overlap any previously sequenced region. Subject to the constraint of no overlap, each new region is chosen randomly. A parking strategy is often ideal in the early stages of a project for rapidly generating unique data. As a project progresses, parking becomes progressively more expensive and eventually prohibitive. We present a mathematical model with a generalization to allow for overlaps. This model predicts multiple parameters, including progress, costs, and the distribution of gap sizes left by a parking strategy. The highly fragmented nature of the gaps left after an initial parking strategy may make it difficult to finish a project efficiently. Therefore, in addition to our parking model, we model gap closing by walking. Our gap-closing model is generalizable to many other strategies. Our discussion includes modified parking strategies and hybrids with other strategies. A hybrid parking strategy has been employed for portions of the Human Genome Project.

An application of random packing to the multiple fracture of composite materials

A one-dimensional packing approach is used to obtain limiting results for inter-crack distances after multiple fracture of a long brittle-matrix composite with continuous aligned fibres. The results may also be appropriate for applications of the Rényi car-parking model in which there is a reduced probability of cars parking bumper to bumper.

An application of random packing to the multiple fracture of composite materials

A one-dimensional packing approach is used to obtain limiting results for inter-crack distances after multiple fracture of a long brittle-matrix composite with continuous aligned fibres. The results may also be appropriate for applications of the Rényi car-parking model in which there is a reduced probability of cars parking bumper to bumper.

Packing densities of randomly constructed codes

Codes having all pairs of words separated by a Hamming distance of at least d are stochastically constructed by sequentially packing randomly generated q-ary n-tuples. Estimates of the random packing densities are obtained by repeated simulation. Using non-linear regression to fit the estimated densities, an asymptotic approximation formula is obtained for the packing densities which depends only on q, n, d, and an empirical constant.

Packing densities of randomly constructed codes

Codes having all pairs of words separated by a Hamming distance of at least d are stochastically constructed by sequentially packing randomly generated q-ary n-tuples. Estimates of the random packing densities are obtained by repeated simulation. Using non-linear regression to fit the estimated densities, an asymptotic approximation formula is obtained for the packing densities which depends only on q, n, d, and an empirical constant.