Successful cellular function and organ development rely on the effective transport of proteins and other biomolecules to specific positions. There are two basic mechanisms for biological transport:
passive diffusion and motor-driven active transport. This thesis presents theoretical investigations of several biophysical problems in the context of active and passive transport.

In the matter of passive diffusion, we investigate fundamental processes of biological development that are governed by multiple signaling molecules that create non-uniform concentration profiles known as morphogen gradients. It is widely believed that the establishment of morphogen gradients is a result of complex processes that involve diffusion and degradation of locally produced signaling molecules. We have developed discrete-state stochastic and continuum mean field approaches to investigate the corresponding reaction-diffusion models.

In the case of active transport, we investigate the fundamental role of local interactions between molecular motors by analyzing a new class of totally asymmetric exclusion processes where interactions are accounted for in a thermodynamically consistent fashion. This allows us to explicitly connect microscopic features of motor proteins with their collective dynamic properties. Our theoretical analysis that combines various mean-field calculations and computer simulations suggests that the dynamic properties of molecular motors strongly depend on the interactions, and that the correlations are stronger for interacting motor proteins.

Furthermore, we investigate all times dynamics of continuous-time random walks (CTRWs). The concept of continuous-time random walks (CTRW) is a generalization of ordinary random walk models, and it is a powerful tool for investigating a broad spectrum of phenomena in natural, engineering, social, and economic sciences. Recently, several theoretical approaches have been developed that allowed to analyze explicitly dynamics of CTRW at all times, which is critically important for understanding mechanisms of underlying phenomena.
However, theoretical analysis has been done mostly for systems with a simple geometry. Here, we extend the original method based on generalized master equations to analyze all-time dynamics of CTRW models on complex networks.