The authors developed a time dependent method to study the single molecule dynamics of a simple gene regulatory network: a repressilator with three genes mutually repressing each other. They quantitatively characterize the time evolution dynamics of the repressilator. Furthermore, they study purely dynamical issues such as statistical fluctuations and noise evolution. They illustrated some important features of the biological network such as monostability, spirals, and limit cycle oscillation. Explicit time dependent Fano factors which describe noise evolution and show statistical fluctuations out of equilibrium can be significant and far from the Poisson distribution. They explore the phase space and the interrelationships among fluctuations, order, amplitude, and period of oscillations of the repressilators. The authors found that repressilators follow ordered limit cycle orbits and are more likely to appear in the lower fluctuating regions. The amplitude of the repressilators increases as the suppressing of the genes decreases and production of proteins increases. The oscillation period of the repressilators decreases as the suppressing of the genes decreases and production of proteins increases.

We investigated the spatial-dependent stochastic effects originating from the finite number of bicoid proteins in Drosophila melanogaster, which are crucial to cell development. We obtained an exact solution to the spatial-dependent stochastic chemical master equation and recovered the usual reaction-diffusion solution for the average of the bicoid concentration, valid in the bulk. We also used the steady state probability to get the spatial potential landscape. The stochastic effects are captured by the Poisson distribution; so, as the average of the bicoid concentration decreases from the anterior (A) to the posterior (P) of the embryo, the statistical fluctuations also decrease. An alternative way of interpreting this is that the shape of the spatial potential landscape shrinks from A to P. While the mathematical result is known, we offer a simple approach to understanding why it is what it is and give associated physical intuitions. The approach can be generalized and applied to any problem with a particle that diffuses, decays, and has a stochastic source.

Integrins are transmembrane proteins that allow cells to bind to their external environment. They are the primary regulators of cell-matrix interactions, with direct roles in cell motility and signaling, which in turn regulate numerous physiological processes. Under common experimental conditions, integrins tend to cluster for sturdy and effective binding to extra-cellular matrix molecules. These clusters often evolve into focal adhesions, which regulate downstream signaling. However, integrin clusters are more pronounced and have longer lifetimes in two-dimensional assays than in more realistic three-dimensional environments. While a number of models and theoretical approaches have focused on integrin binding and diffusion, the reasons for the differences between two- and three-dimensional clustering have remained elusive. In this study, we model an individual cluster attached to a two-dimensional collagen film and attached to collagen fibers of various sizes in three-dimensional matrices. We then discuss how our results explain differences in size and lifetime, and how they hint at reasons for other differences between the two environments. Further, we make predictions regarding the stability of clusters based on different overall intracellular conditions. Our results show good agreement with experiments and provide a quantitative basis for understanding how matrix dimensionality and structure regulate integrin behavior in environments that mimic in vivo conditions.