Why nanorods assemble

In a recent paper in Phys. Rev. E, Brandeis postdoc Yasheng Yang and Assistant Professor of Physics Michael Hagan developed a theory to describe the assembly behavior of colloidal rods (i.e. nanorods) in the presence of inert polymer (which induces attractions between the nanorods). The nanorods assemble into several kinds of structures, including colloidal membranes, which  are two dimensional membrane-like structures composed of a one rod-length thick monolayer of aligned rods.  The theory shows that colloidal membranes are stabilized against stacking on top of each other by an entropic force arising from protrusions of rods from the membranes and that there is a critical aspect ratio (rod length/rod diameter) below which membranes are never stable. Understanding the forces that stabilize colloidal membranes is of practical importance since these structures could enable the manufacture of inexpensive and easily scalable optoelectronic devices. This work was part of a collaboration with the experimental lab of Zvonimir Dogic at Brandeis, where colloidal membranes are developed and studied.

Yang YS, Hagan MF. Theoretical calculation of the phase behavior of colloidal membranes. Phys Rev E. 2011;84(5).

Collective behaviors in active matter

Active matter is describes systems whose constituent elements consume energy and are thus out-of-equilibrium. Examples include flocks or herds of animals, collections of cells, and components of the cellular cytoskeleton. When these objects interact with each other, collective behavior can emerge that is unlike anything possible with an equilibrium system. The types of behaviors and the factors that control them however, remain incompletely understood. In a recent paper in Physical Review Letters, “Excitable patterns in active nematics“, Giomi and coworkers develop a continuum theoretical description motivated by recent experiments from the Dogic group at Brandeis in which microtubules (filamentous cytoskeletal molecules) and clusters of kinesin (a molecular motor) exhibit dramatic spatiotemporal fluctuations in density and alignment. Specifically, they consider a hydrodynamic description for density, flow, and nematic alignment. In contrast to previous theories of this type, the degree of nematic alignment is allowed to vary in space and time.  Remarkably, the theory predicts that the interplay between non-uniform nematic order, activity and flow results in spatially modulated relaxation oscillations, similar to those seen in excitable media and biological examples such as the cardiac cycle. At even higher activity the dynamics is chaotic and leads to large-scale swirling patterns which resemble those seen in recent experiments. An example of the flow pattern is shown below left, and the nematic order parameter, which describes the degree of alignment of the filaments, as shown for the same configuration below right. These predictions can be tested in future experiments on systems of microtubules and motor proteins.

The system behavior for an active nematic at high activity. (left) The velocity field (arrows) is superimposed on a plot of the concentration of active nematogens (green=large concentration, red=small concentration). (right) A plot of the nematic order parameter, S,  (blue=large S, brown=small S) is superimposed on a plot of the nematic director (arrows). The flow under high activity is characterized by large vortices that span lengths of the order of the system size and the director field is organized in grains.

 

Prolonging assembly through dissociation

Microtubules are semiflexible polymers that serve as structural components inside the eukaryotic cell and are involved in many cellular processes such as mitosis, cytokinesis, and vesicular transport. In order to perform these functions, microtubules continually rearrange through a process known as dynamic instability, in which they switch from a phase of slow elongation to rapid shortening (catastrophe), and from rapid shortening to growth (rescue). The basic self-assembly mechanism underlying this process, assembly mediated by nucleotide phosphate activity, is omnipresent in biological systems.  A recent paper, Prolonging assembly through dissociation: A self-assembly paradigm in microtubules ,  published in the May 3 issue of Physical Review E,  presents a new paradigm for such self-assembly in which increasing depolymerization rate can enhance assembly.  Such a scenario can occur only out of equilibrium. Brandeis Physics postdoc Sumedha, working with Chakraborty and Hagan, carried out theoretical analysis of a stochastic hydrolysis model to demonstrate the effect and predict features of growth fluctuations, which should be measurable in experiments that probe microtubule dynamics at the nanoscale.

Model for microtuble dynamics. All activity is assumed to occur at the right end of the microtubule (denoted as ">")

The essential features of the model that leads to the counterintuitive result of depolymerization helping assembly are (a) stochastic hydrolysis that allows GTP to transform into GDP  in any part of the microtubule, and (b) a much higher rate of GTP attachment if the end of the microtubule has a GTP-bound tubulin dimer, compared to a GDP-bound tubulin dimer.    Process (a) leads to islands of GTP-bound tubulins to be buried deep in the microtubule.   Depolymerization from the end reveals these islands and enhances assembly because of the biased attachment rate (b).  The simplicity of the model lent itself to analytical results for various aspects of the growth statistics in particular parameter regimes.   Simulations of the model supported these analytical results, and extended them to regimes where it was not possible to solve the model analytically.  The statistics of the growth fluctuations in this stochastic hydrolysis model are very different from “cap models” which do not have GTP remnants buried inside a growing microtubule.   Testing the predictions in experiments could, therefore, lead to a better understanding of the processes underlying dynamical instability in-vivo and in-vitro.   An interesting question to explore is whether the bias in the attachment rates is different under different conditions of microtubule growth.

Chirality leads to self-limited self-assembly

Simple building blocks that self-assemble into ordered structures with controlled sizes are essential for nanomaterials applications, but what are the general design principles for molecules that undergo self-terminating self-assembly? The question is addressed in a recent paper in Physical Review Letters by Yasheng Yang, graduate student in Physics, working together with Profs. Meyer and Hagan,  The paper considers molecules that self assemble to form filamentous bundles, and shows that chirality, or asymmetry with respect to a molecule’s mirror image, can result in stable self-limited structures. Using modern computational techniques, the authors demonstrate that chirality frustrates long range order and thereby terminates assembly upon formation of regular self-limited bundles.  With strong interactions, however, the frustration is relieved by defects, which give rise to branched networks or irregular bundles.

Figure: (a) Snapshots of regular chiral bundles. Free energy calculations and dynamics demonstrate that the optimal diameter decreases with increasing chirality. (b) Branched bundles form with strong interactions

New in Pubmed

Have no time to write News and Views, but there are a few new papers from our labs that have recently popped up in Pubmed.

Simulating viral capsid assembly

Viral capsids assemble into complex structures with high fidelity, but also can adapt when given other nucleic acids cargoes to package. In a recent paper in Nano Letters, Brandeis physics grad student Oren Elrad and Professor Michael Hagan used computer simulations to investigate the mechanisms by which this occurs. These simulations were done on the Brandeis High Performance Computing cluster.

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