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Prions are proteinaceous infectious particles that cause neurodegenerative diseases in humans and animals. The complex nature of prions, with respect to their conformations and aggregations, has been an important area of research for quite some time. Here, we develop a model of prion dynamics prior to the formation of oligomers and subsequent development of prion diseases within a stochastic framework, based on the analytical Master Equation and Stochastic Simulation Algorithm by Gillespie. The results that we obtain shows that solvent water molecules act as driving agents in the dynamics of prion aggregation. Further, it is found that aggregated and non-aggregated proteins tend to co-exist in an equilibrium state, depending upon the reaction rate constants. These results may provide a theoretical and qualitative contexts of possible therapeutic strategies in the treatment of prion diseases., Comment: 10 pages, 6 figures

Complexity is often exhibited in dynamical systems, where certain parameters evolve with time in a strange and chaotic nature. These systems lack predictability and are common in the physical world. Dissipative systems are one of such systems where the volume of the phase space contracts with time. On the other hand, we employ dimensionality reduction techniques to study complicated and complex data, which are tough to analyse. The Principal Component Analysis (PCA) is a dimensionality reduction technique used as a means to study complex data. Through PCA, we studied the reduced dimensional features of the numerical data generated by a nonlinear partial differential equation called the Korteweg de Vries (KdV) equation, which is a nonlinear dispersive system, where solitary waves travel along a specific direction with finite amplitude. Dissipative nature, specific to that of the Lorenz system, were observed in the dimensionally reduced data, which implies a transition from a dispersive system to a dissipative system.

Patterns in complex systems store hidden information of the system which is needed to be explored. We present a simple model of cytokine and T-cells interaction and studied the model within stochastic framework by constructing Master equation of the system and solving it. The solved probability distribution function of the model show classical Poisson pattern in the large population limit $M,Z\rightarrow large$ indicating the system has the tendency to attract a large number small-scale random processes of the cytokine population towards the basin of attraction of the system by segregating from nonrandom processes. Further, in the large $\langle Z\rangle$ limit, the pattern transform to classical Normal pattern, where, uncorrelated small-scale fluctuations are wiped out to form a regular but memoryless spatiotemporal aggregated pattern. The estimated noise using Fano factor shows clearly that the cytokine dynamics is noise induced process driving the system far away from equilibrium., Comment: 14 pages, 1 figure