We have investigated the shape, size, and motility of a minimal model of an adherent biological cell using the Monte Carlo method. The cell is modeled as a two dimensional ring polymer on the square lattice enclosing continuously polymerizing and depolymerizing actin networks. Our lattice model is an approximate representation of a real cell at a resolution of one actin molecule, 5 nm. The polymerization kinetics for the actin network are controlled by appropriate reaction probabilities which correspond to the correct experimental reaction rates. Using the simulation data we establish various scaling laws relating the size of the model cell to the concentration of polymerized and unpolymerized actin molecules and the length of the enclosing membrane. The computed drift velocities, which characterize the motility of the cell, exhibit a maximum at a certain fraction of polymerized actin which agrees with physiological fractions observed in experiments. The appearance of the maximum is related to the competition between the polymerization-induced protrusion of the membrane and the concomitant suppression of membrane fluctuations.

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Mixed entangled states are generic resource for quantum teleportation. Optimal teleportation fidelity measures the success of quantum teleportation. The relevance of rank in the teleportation process is investigated by constructing three new maximally entangled mixed states (MEMS) of different ranks. Linear entropy, concurrence, optimal teleportation fidelity and Bell function are obtained for each of these states analytically. It is found that mixed states with higher rank are better resource for teleportation. In order to achieve a fixed value of optimal teleportation fidelity, we find that low rank states must have high concurrence. Further, for each of ranks 2, 3 and 4, we numerically generate 30000 maximally entangled mixed states. The analysis of these states reveals the existence of a rank dependent upper bound on optimal teleportation fidelity for a fixed purity. In order to achieve a fixed optimal teleportation fidelity, we find MEMS exhibit a rank dependent lower bound on concurrence. MEMS are classified in terms of their degree of nonlocality. The results are found to be same with logarithmic negativity used as a measure of entanglement.

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