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Abstract Evolution is the main feature of all biological systems that allows populations to change their characteristics over successive generations. A powerful approach to understand evolutionary dynamics is to investigate fixation probabilities and fixation times of novel mutations on networks that mimic biological populations. It is now well established that the structure of such networks can have dramatic effects on evolutionary dynamics. In particular, there are population structures that might amplify the fixation probabilities while simultaneously delaying the fixation events. However, the microscopic origins of such complex evolutionary dynamics remain not well understood. We present here a theoretical investigation of the microscopic mechanisms of mutation fixation processes on inhomogeneous networks. It views evolutionary dynamics as a set of stochastic transitions between discrete states specified by different numbers of mutated cells. By specifically considering star networks, we obtain a comprehensive description of evolutionary dynamics. Our approach allows us to employ physics-inspired free-energy landscape arguments to explain the observed trends in fixation times and fixation probabilities, providing a better microscopic understanding of evolutionary dynamics in complex systems. 

Abstract It is widely believed that biological tissues evolved to lower the risks of cancer development. One of the specific ways to minimize the chances of tumor formation comes from proper spatial organization of tissues. However, the microscopic mechanisms of underlying processes remain not fully understood. We present a theoretical investigation on the role of spatial structures in cancer initiation dynamics. In our approach, the dynamics of single mutation fixations are analyzed using analytical calculations and computer simulations by mapping them to Moran processes on graphs with different connectivity that mimic various spatial structures. It is found that while the fixation probability is not affected by modifying the spatial structures of the tissues, the fixation times can change dramatically. The slowest dynamics is observed in ‘quasi-one-dimensional’ structures, while the fastest dynamics is observed in ‘quasi-three-dimensional’ structures. Theoretical calculations also suggest that there is a critical value of the degree of graph connectivity, which mimics the spatial dimension of the tissue structure, above which the spatial structure of the tissue has no effect on the mutation fixation dynamics. An effective discrete-state stochastic model of cancer initiation is utilized to explain our theoretical results and predictions. Our theoretical analysis clarifies some important aspects on the role of the tissue spatial structures in the cancer initiation processes. 

Abstract Motor proteins, also known as biological molecular motors, play important roles in various intracellular processes. Experimental investigations suggest that molecular motors interact with each other during the cellular transport, but the nature of such interactions remains not well understood. Stimulated by these observations, we present a theoretical study aimed to understand the effect of the range of interactions on dynamics of interacting molecular motors. For this purpose, we develop a new version of the totally asymmetric simple exclusion processes in which nearest-neighbor as well as the next nearest-neighbor interactions are taken into account in a thermodynamically consistent way. A theoretical framework based on a cluster mean-field approximation, which partially takes correlations into account, is developed to evaluate the stationary properties of the system. It is found that fundamental current–density relations in the system strongly depend on the strength and the sign of interactions, as well as on the range of interactions. For repulsive interactions stronger than some critical value, a mean-field theoretical approach predicts that increasing the range of interactions might lead to a change from unimodal to trimodal dependence in the flux-density fundamental diagram. However, it is not fully supported by extensive Monte Carlo computer simulations that test theoretical predictions. Although in most ranges of parameters a reasonable agreement between theoretical calculations and computer simulations is observed, there are situations when the cluster mean-field approach fails to describe properly the dynamics in the system. Theoretical arguments to explain these observations are presented. Our theoretical analysis clarifies the microscopic picture of how the range of interactions influences the dynamics of interacting molecular motors.