The statistical-mechanical treatment of closed polymer chains based on algebraic topology is proposed. Using the Monte-Carlo method numerical results were obtained for the probability to knot formation during random closing of polymer chains of different length. For very rigid chains such as DNA double helix the probability of knot formation is rather great. Topological restrictions in a system of two polymer chains are shown to lead to a specific topological interaction between them.

A major challenge of current neuroscience is to elucidate the brain mechanisms that underlie cognitive function. There is no doubt that cognitive processing in the brain engages large populations of cells. This article explores the logic of investigating these problems by combining psychological studies in human subjects and neurophysiological studies of neuronal populations in the motor cortex of behaving monkeys. The results obtained show that time-varying psychological processes can be visualized in the time-varying activity of neuronal populations. Moreover, the functional interactions between cells in the motor cortex are very similar to those observed in a massively interconnected artificial network performing the same computation.

The effect of superhelicity on the base-pair opening probability and on the probability of occurrence of cruciform states in palindromic regions is theoretically treated. The calculations show that below the superhelix density value of -sigma=0.05 superhelicity does not appreciably affect the characteristics of DNA secondary structure fluctuations. In the range of physiological superhelix densities sigma (-sigma=0.05-0.09) the base-pair opening probability markedly increases. However, within this range of sigma the base-pairs are opened only transiently and permanently open regions are not formed. Permanently opened regions appear at higher negative superhelix densities (-sigma greater than 0.10). At the values of -sigma higher than 0.06 a cruciform structure in the palindromic region centred in position 3965 proves to be the most probable fluctuational disturbance in the 0x174 duplex DNA. Different experimental approaches used for probing the fluctuations in superhelical DNA have been analysed. The results suggest that most direct quantitative information can be derived from data on the nicking of closed DNA by single strand-specific endonucleases. Such data (Wang, 1974) accord with the results of theoretical calculations. Calculations show that, due to base-pair opening, the total free energy of superhelical DNA should depend parabolically on sigma only up to some critical value of sigma=sigmac. If negative superhelicity exceeds this critical value, which under physiological conditions proves to be -sigma=0.085, the free energy should increase linearly with -sigma. The biological role of supercoiling is discussed in the light of obtained results.