Instytut Podstawowych Problemów Techniki

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DNA in cells is organized in negatively supercoiled loops. The resulting torsional and bending strain allows DNA to adopt a surprisingly wide variety of 3-D shapes. This interplay between negative supercoiling, looping, and shape influences how DNA is stored, replicated, transcribed, repaired, and likely every other aspect of DNA activity. To understand the consequences of negative supercoiling and curvature on the hydrodynamic properties of DNA, we submitted 336 bp and 672 bp DNA minicircles to analytical ultracentrifugation (AUC). We found that the diffusion coefficient, sedimentation coefficient, and the DNA hydrodynamic radius strongly depended on circularity, loop length, and degree of negative supercoiling. Because AUC cannot ascertain shape beyond degree of non-globularity, we applied linear elasticity theory to predict DNA shapes, and combined these with hydrodynamic calculations to interpret the AUC data, with reasonable agreement between theory and experiment. These complementary approaches, together with earlier electron cryotomography data, provide a framework for understanding and predicting the effects of supercoiling on the shape and hydrodynamic properties of DNA.

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Correction for ‘Stokesian dynamics of sedimenting elastic rings’ by Magdalena Gruziel-Słomka et al., Soft Matter, 2019, 15, 7262–7274, https://doi.org/10.1039/C9SM00598F.

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Inclusion of hydrodynamic interactions is essential for a quantitatively accurate Brownian dynamics simulation of colloidal suspensions or polymer solutions. We use the generalized Rotne–Prager–Yamakawa (GRPY) approximation, which takes into account all long-ranged terms in the hydrodynamic interactions, to derive the complete set of hydrodynamic matrices in different geometries: unbounded space, periodic boundary conditions of Lees–Edwards type, and vicinity of a free surface. The construction is carried out both for non-overlapping as well as for overlapping particles. We include the dipolar degrees of freedom, which allows one to use this formalism to simulate the dynamics of suspensions in a shear flow and to study the evolution of their rheological properties. Finally, we provide an open-source numerical package, which implements the GRPY algorithm in Lees–Edwards periodic boundary conditions.

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Two oppositely charged surfaces separated by a dielectric medium attract each other. In contrast we observe a strong repulsion between two plates of a capacitor that is filled with an aqueous electrolyte upon application of an alternating potential difference between the plates. This long-range force increases with the ratio of diffusion coefficients of the ions in the medium and reaches a steady state after a few minutes, which is much larger than the millisecond timescale of diffusion across the narrow gap. The repulsive force, an order of magnitude stronger than the electrostatic attraction observed in the same setup in air, results from the increase in osmotic pressure as a consequence of the field-induced excess of cations and anions due to lateral transport from adjacent reservoirs.

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We consider elastic microfilaments which form closed loops. We investigate how the loops change shape and orientation while settling under gravity in a viscous fluid. Loops are circular at the equilibrium. Their dynamics are investigated numerically based on the Stokes equations for the fluid motion and the bead–spring model of the microfilament. The Rotne–Prager approximation for the bead mobility is used. We demonstrate that the relevant dimensionless parameter is the ratio of the bending resistance of the filament to the gravitation force corrected for buoyancy. The inverse of this ratio, called the elasto-gravitation number [scr B, script letter B], is widely used in the literature for sedimenting elastic linear filaments. We assume that [scr B, script letter B] is of the order of 10^4–10^6, which corresponds to easily deformable loops. We find out that initially tilted circles evolve towards different sedimentation modes, depending on [scr B, script letter B]. Very stiff or stiff rings attain almost planar, oval shapes, which are vertical or tilted, respectively. More flexible loops deform significantly and converge towards one of several characteristic periodic motions. These sedimentation modes are also detected when starting from various shapes, and for different loop lengths. In general, multi-stability is observed: an elastic ring converges to one of several sedimentation modes, depending on the initial conditions. This effect is pronounced for very elastic loops. The surprising diversity of long-lasting periodic motions and shapes of elastic rings found in this work gives a new perspective for the dynamics of more complex deformable objects at micrometer and nanometer scales, sedimenting under gravity or rotating in a centrifuge, such as red blood cells, ring polymers or circular DNA.

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We study the dynamics of knotted deformable closed chains sedimenting in a viscous fluid. We show experimentally that trefoil and other torus knots often attain a remarkably regular horizontal toroidal structure while sedimenting, with a number of intertwined loops, oscillating periodically around each other. We then recover this motion numerically and find out that it is accompanied by a very slow rotation around the vertical symmetry axis. We analyze the dependence of the characteristic timescales on the chain flexibility and aspect ratio. It is observed in the experiments that this oscillating mode of the dynamics can spontaneously form even when starting from a qualitatively different initial configuration. In numerical simulations, the oscillating modes are usually present as transients or final stages of the evolution, depending on chain aspect ratio and flexibility, and the number of loops.

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Two main problems that arise in the context of hydrodynamic bead modeling are an inaccurate treatment of bead overlaps and the necessity of using volume corrections when calculating intrinsic viscosity. We present a formalism based on the generalized Rotne-Prager-Yamakawa approximation that successfully addresses both of these issues. The generalized Rotne-Prager-Yamakawa method is shown to be highly effective for the calculation of transport properties of rigid biomolecules represented as assemblies of spherical beads of different sizes, both overlapping and nonoverlapping. We test the method on simple molecular shapes as well as real protein structures and compare its performance with other computational approaches.

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We develop a generalized Rotne–Prager–Yamakawa approximation for the dipolar components of the inverse friction matrix and use it for calculating the intrinsic viscosity of rigidly connected bead conglomerates. Such bead models are commonly used in the calculation of hydrodynamic properties of macromolecules. We consider both the case of non-overlapping constituent beads as well as overlapping beads of different sizes. We demonstrate the accuracy of the approximation in two test cases and show that it performs well even if the distances between the beads are small or if the beads overlap. Robust performance of this approximation in the case of overlapping beads stems from its correct limiting behaviour at a complete overlap, with one sphere fully immersed in the other. The generalized Rotne–Prager–Yamakawa approximation is thus well suited for evaluation of intrinsic viscosity, which is a key quantity in characterizing molecular conformations of biological macromolecules.

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complex fluids, low-Reynolds-number flows, mathematical foundations

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A simple, coarse-grained model of chiral, helical filaments is used to study the polymorphism of fibrous aggregates. Three generic morphologies of the aggregates are observed: ribbons, in which the filaments are joined side-by-side, twisted, helicoidal fibrils, in which filaments entwine along each other and tubular forms, with filaments wound together around a hollow core of the tube. A relative simplicity of the model allows us to supplement numerical simulations with an analytic description of the elastic properties of the aggregates. The model is capable of predicting geometric and structural characteristics of the composite structures, as well as their relative stabilities. We also investigate in detail the transitions between different morphologies of the aggregates.

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The Rotne–Prager–Yamakawa (RPY) approximation is a commonly used approach to model the hydrodynamic interactions between small spherical particles suspended in a viscous fluid at a low Reynolds number. However, when the particles overlap, the RPY tensors lose their positive definiteness, which leads to numerical problems in the Brownian dynamics simulations as well as errors in calculations of the hydrodynamic properties of rigid macromolecules using bead modelling. These problems can be avoided by using regularizing corrections to the RPY tensors; so far, however, these corrections have only been derived for equal-sized particles. Here we show how to generalize the RPY approach to the case of overlapping spherical particles of different radii and present the complete set of mobility matrices for such a system. In contrast to previous ad hoc approaches, our method relies on the direct integration of force densities over the sphere surfaces and thus automatically provides the correct limiting behaviour of the mobilities for the touching spheres and for a complete overlap, with one sphere immersed in the other one. This approach can then be used to calculate hydrodynamic properties of complex macromolecules using bead models with overlapping, different-sized beads, which we illustrate with an example.

Słowa kluczowe:

complex fluids, computational methods, low-Reynolds-number flows, mathematical foundations, suspensions

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Rotne-Prager-Yamakawa approximation is a commonly used approach to model hydrodynamic interactions between particles suspended in fluid. It takes into account all the long-range contributions to the hydrodynamic tensors, with the corrections decaying at least as fast as the inverse fourth power of the interparticle distances, and results in a positive definite mobility matrix, which is fundamental in Brownian dynamics simulations. In this communication, we show how to construct the Rotne-Prager-Yamakawa approximation for the bulk system under shear flow, which is modeled using the Lees-Edwards boundary conditions.

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Tensor methods, Hydrodynamics, Shear flows, Brownian dynamics, Boundary value problems

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The Rotne–Prager–Yamakawa approximation is one of the most commonly used methods of including hydrodynamic interactions in modelling of colloidal suspensions and polymer solutions. The two main merits of this approximation are that it includes all long-range terms (i.e. decaying as R−3 or slower in interparticle distances) and that the diffusion matrix is positive definite, which is essential for Brownian dynamics modelling. Here, we extend the Rotne–Prager–Yamakawa approach to include both translational and rotational degrees of freedom, and derive the regularizing corrections to account for overlapping particles. Additionally, we show how the Rotne–Prager–Yamakawa approximation can be generalized for other geometries and boundary conditions.

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computational methods, low-Reynolds-number flows, suspensions

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The formation of aggregates of helical fibrils is analyzed numerically. The aggregate morphology, chirality and stability are studied as a function of temperature and helical pitch of individual fibrils. The simulations show the existence of a critical pitch above which the handedness of the aggregates is opposite to that of the constituting fibrils. We also observe and analyze the process of spontaneous chirality inversion of individual fibrils within the aggregates. This inversion is accompanied by a helical wave propagating along the fibril axis, with a kink separating left-handed and right-handed regions moving along the fibril. The frequency of this process is strongly dependent on the initial pitch of the fibrils with a local maximum near the critical pitch.

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The analysis of the TEM images of indium tin oxide (ITO) clusters in ink solutions deposited from ink dispersions reveals that their geometry arises from a diffusion limited cluster aggregation (DLCA) process. We model films of ITO clusters as built through deposition of DLCA clusters made of primary spherical nanoparticles of 13 nm in diameter.
The deposition is then followed by a further compactification process that imitates sintering. We determine the conductivity of the sintered films by mapping the problem to that of the resistor network in which the contact regions between the touching spheres provide the dominant electric resistance. For a given volume fraction, conductivity of the sintered films is shown to be larger than that for the randomly packed spheres. However, the larger a typical radius of gyration of the clusters the smaller the enhancement. We also provide numerical tests for the routines used in the interpretation of the TEM images.

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cluster aggregation, ITO

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Stokesian dynamics simulations of close particles are reported, taking into account lubrication forces and many-body hydrodynamic interactions between spheres. A periodic trajectory of three particles maintaining a permanent proximity to each other has been found and analyzed. This solution is used as a benchmark to study the accuracy and stability of various numerical integration schemes. In particular, different methods of preventing unphysical overlaps of the particles are considered and potential artifacts discussed.

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Kinematics, Renormalization, Surface dynamics, Lubrication, Hydrodynamics

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