PubMed · 727282
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F E Yates. 1978. Complexity and the limits to knowledge.. https://doi.org/10.1152/ajpregu.1978.235.5.r201
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The Brownian Dynamics technique was used to model a diffusion-controlled intramolecular reaction of supercoiled DNA (2500 basepairs) in 0.1 M sodium chloride solution. The distance between the reactive groups along the DNA contour was 470 basepairs. The reaction radius was varied from 6 to 20 nm. The results are presented in terms of the probability distribution P(F)(t) of the first collision time. The general form of the function P(F)(t) could be correctly predicted by a simple analytical model of one-dimensional diffusion of the superhelix ends along the DNA contour. The distribution P(F)(t) is essentially non-exponential: within a large initial time interval, it scales as P(F)(t) approximately t(-1/2), which is typical for one-dimensional diffusion. However, the mean time of the first collision is inversely proportional to the reaction radius, as in three dimensions. A visual inspection of the simulated conformations showed that a considerable part of the collisions is caused by the bending of the superhelix axis in the regions of the end loops, where the axis is most flexible. This fact explains why the distribution P(F)(t) combines the features of one- and three-dimensional diffusion. The simulations were repeated for a DNA chain with a permanent bend of 100 degrees in the middle position between the reactive groups along the DNA contour. The permanent bend changes dramatically the form of the distribution P(F)(t) and reduces the mean time of the first collision by approximately one order of magnitude.
Molecular dynamics simulations of a simple, bead-spring model of semiflexible polyelectrolytes such as DNA are performed. All charges are explicitly treated. Starting from extended, noncondensed conformations, condensed structures form in the simulations with tetravalent or trivalent counterions. No condensates form or are stable for divalent counterions. The mechanism by which condensates form is described. Briefly, condensation occurs because electrostatic interactions dominate entropy, and the favored coulombic structure is a charge-ordered state. Condensation is a generic phenomenon and occurs for a variety of polyelectrolyte parameters. Toroids and rods are the condensate structures. Toroids form preferentially when the molecular stiffness is sufficiently strong.
Forces generated by microtubule polymerization and depolymerization are important for the biological functioning of cells. The mean growth velocity, V, under an opposing force, F, has been measured by; Science 278:856-860) for single microtubules growing in vitro, but their analysis of the data suggested that V decreased more rapidly with F than equilibrium (or "thermodynamic") theory predicted and entailed negative values for the dissociation rate and undefined (or unreasonable) values for the stall force, at which V vanishes. By contrast, considering the mean work done against the external load and allowing for load-distribution factors for the "on" and "off" rates, we find good agreement with a simple theory that yields a plausible stalling force. Although specific numerical results are sensitive to choice of fitting criteria, about 80% of the variation with load is carried by the "off" (or dissociation) rate, but, since that is small (in accordance with independent observations), the dominant force dependence comes from the "on" rate, which is associated with a displacement length, d(1), significantly longer than d(0) approximately 1/13(8.2 nm), the mean length increase per added tubulin dimer. Measuring the dispersion in length of the growing microtubules could provide a check. The theory implies that the stationary stall state (at V = 0) is not one of simple associative thermal equilibrium, as previously supposed; rather, it appears to be dissipative and kinetically controlled.