The final stage of the cell cycle is cell division by cytokinesis, when the cell physically separates into two daughter cells. Improper timing or location of the division site results in incorrect segregation of chromosomes and thus genetically unstable aneuploid cells, which is associated with tumorigenesis. Cytokinesis in animal, fungal and amoeboid cells occurs through the assembly and constriction of an actomyosin contractile ring, a mechanism that dates back about one billion years in the common ancestor of these organisms. However, it is not well understood how the ring generates tension or how the rate of ring constriction is set. Long ago a sliding filament mechanism similar to skeletal muscle was proposed, but definitive evidence for muscle-like sarcomeric order in the ring is lacking. Here we build mathematical models of cytokinesis in the fission yeast Schizosaccharomyces pombe, where the most complete inventory of more than 150 cytokinesis genes have been documented. The models explicitly represent proteins in the contractile ring such as formin, myosin, actin, α-actinin, etc. and implements their quantities, biomechanical properties and organizations from the best available experimental information. At the same time, the models adopt coarse-grain approaches that are able to describe the collective behaviors of thousands of ring components, which include tension production, constriction, and disassembly of the ring. In the first part of this thesis, we modeled the extraordinarily rapid constriction of the partially unanchored ring in fission yeast cell ghosts. Experiments on isolated fission yeast rings showed sections of ring unanchoring from the membrane and shortening ~30-fold faster than normal (1). We demonstrated that anchoring of actin to the plasma membrane generates tension in the fission yeast cytokinetic ring by showing (1) unanchored segments in these experiments were tensionless, and (2) only a barbed-end anchoring of actin can generate tension in the normally anchored ring, and can explain the extraordinary behavior of unanchored segments. Molecularly explicit simulations accurately reproduced experimental constriction rates, and showed a novel non-contractile reeling-in mechanism by which the unanchored segment shortens, despite being tensionless. In the second part of this thesis, we built a highly coarse-grained model to study how ring tension is generated and how structural stability is maintained. Recently, a super-resolution microscopy study of the fission yeast ring revealed that myosins and formins that nucleate actin filaments colocalize in plasma membrane-anchored complexes called nodes in the constricting ring (2). The nodes move bidirectionally around the ring. Here we construct and analyze a coarse-grained mathematical model of the fission yeast ring to explore essential consequences of the recently discovered ring ultrastructure. The model reproduces experimentally measured values of ring tension, explains why nodes move bidirectionally and shows that tension is generated by myosin pulling on barbed-end-anchored actin filaments in a stochastic sliding-filament mechanism. This mechanism is not based on an ordered sarcomeric organization. We show that the ring is vulnerable to intrinsic contractile instabilities, and protection from these instabilities and organizational homeostasis require both component turnover and anchoring of components to the plasma membrane. In the third part of this thesis, we measured ring tension in fission yeast protoplasts. We found ~650 pN tension in wild type cells, ~65% the normal tension in myp2 deletion mutants and ~40% normal tension in myo2-E1 mutant cells with negligible ATPase activity and reduced actin binding. To understand the relation between organization and tension, we developed a molecularly explicit simulation of the fission yeast ring with the above organization. Our simulations revealed a clear division of labor between the 2 myosin-II isoforms, which maintains organization and maximal tension. (1) Myo2 anchors the ring to the plasma membrane, and transmits ring tension to the membrane. (2) Myo2, extending ~100 nm away from the membrane, bundles half (~25) of the actin filaments in the cross-section due to filament packing constraints, as only ~25 filaments are within reach. (3) To increase tension requires that the ring be thickened, as tensions in the ~25 membrane-proximal filaments are close to fracture. (4) Unanchored Myp2 indeed enables thickening, by bundling an additional ~25 filaments and doubling tension. Anchoring of these filaments to the membrane is indirect, via filaments shared with the anchored Myo2. (5) In simulated myo2-E1 rings ~20% of the actin filaments peeled away from the ring and formed Myp2-dressed bridges, as observed experimentally in myo2-E1 cells. (6) The organization in simulated Δmyp2 rings was highly disrupted, with ~ 50% of the actin filaments unbundled. Therefore, beyond their widely recognized job to pull actin and generate tension, myosin-II isoforms are vital crosslinking organizational elements of the ring. Two isoforms in the ring cooperate to organize the ring for maximal actomyosin interaction and tension.