The mechanical properties of biopolymers can be determined from a statistical analysis of the ensemble of shapes they exhibit when subjected to thermal forces. In practice, extracting information from fluorescence microscopy images can be challenging due to low signal/noise ratios and other artifacts. To address these issues, we develop a suite of tools for image processing and spectral data analysis that is based on a biopolymer contour representation expressed in a spectral basis of orthogonal polynomials. We determine biopolymer shape and stiffness using global fitting routines that optimize a utility function measuring the amount of fluorescence intensity overlapped by such contours. This approach allows for filtering of high-frequency noise and interpolation over sporadic gaps in fluorescence. We use benchmarking to demonstrate the validity of our methods, by analyzing an ensemble of simulated images generated using a simulated biopolymer with known stiffness and subjected to various types of image noise. We then use these methods to determine the persistence lengths of taxol-stabilized microtubules. We find that single microtubules are well described by the wormlike chain polymer model, and that ensembles of chemically identical microtubules show significant heterogeneity in bending stiffness, which cannot be attributed to sampling or fitting errors. We expect these approaches to be useful in the study of biopolymer mechanics and the effects of associated regulatory molecules.

We use numerical simulations of a bead-spring model chain to investigate the evolution of the conformations of long and flexible elastic fibers in a steady shear flow. In particular, for rather open initial configurations, and by varying a dimensionless elastic parameter, we identify two distinct conformational modes with different final size, shape, and orientation. Through further analysis we identify slipknots in the chain. Finally, we provide examples of initial configurations of an 'open' trefoil knot that the flow unknots and then knots again, sometimes repeating several times. and the influence of electric fields on knots [31]. Also, the untying of a knot in an extensional flow has been studied recently [32], and has the spirit of the present paper where we ask about the effects of flow. In contrast, for non-Brownian systems other kinds of forcing, such as vibration, can produce knotting of chains of linked spheres [13,14]. We are not aware of any similar studies, experimental or numerical simulations, highlighting conditions that lead to long non-Brownian chains forming knots in flows.Thus, while significant steps have been taken towards understanding the detailed dynamics of flexible non-Brownian fibers, to understand topology is much more difficult: it requires three-dimensional detail and very flexible, long and thin fibers as, in theory (when there is no flow), the probability of knotting in a random walk increases exponentially with the length of the walk [33]. Moreover, the initial conditions may matter and an operational definition is needed for a knot in an open fiber.In this paper, we focus on the dynamics and topology of long, flexible, non-Brownian chains of beads in a steady shear flow and use numerical simulations to investigate whether knots in the fiber occur as a result of the flow. We will find that unknotted fibers are capable of forming (open) knots, and in some cases we document a sequence of unknotting-knotting transitions. The numerical solution of this kind of problem requires tracking the motion of N beads, where hydrodynamic interactions between the beads mean that the dynamics of the topology of the object involve N 3 coupled nonlinear ordinary differential equations. Not surprisingly, such dynamics should be expected to be chaotic though we have not attempted in this paper to link our study of complex shapes and the unknotting-knotting transition to the underlying chaotic dynamics. Our results highlight new aspects of the dynamics of flexible filaments in flow.

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