College of Computer, Mathematical & Natural Sciences
http://hdl.handle.net/1903/12
2016-05-27T06:08:10ZMEDYAN: Mechanochemical Simulations of Contraction and Polarity Alignment in Actomyosin Networks
http://hdl.handle.net/1903/17452
MEDYAN: Mechanochemical Simulations of Contraction and Polarity Alignment in Actomyosin Networks
Popov, Konstantin; Komianos, James; Papoian, Garegin A.
Active matter systems, and in particular the cell cytoskeleton, exhibit complex mechanochemical
dynamics that are still not well understood. While prior computational models of
cytoskeletal dynamics have lead to many conceptual insights, an important niche still
needs to be filled with a high-resolution structural modeling framework, which includes
a minimally-complete set of cytoskeletal chemistries, stochastically treats reaction and
diffusion processes in three spatial dimensions, accurately and efficiently describes mechanical
deformations of the filamentous network under stresses generated by molecular
motors, and deeply couples mechanics and chemistry at high spatial resolution. To
address this need, we propose a novel reactive coarse-grained force field, as well as a
publicly available software package, named the Mechanochemical Dynamics of Active
Networks (MEDYAN) , for simulating active network evolution and dynamics (available at
www.medyan.org). This model can be used to study the non-linear, far from equilibrium
processes in active matter systems, in particular, comprised of interacting semi-flexible
polymers embedded in a solution with complex reaction-diffusion processes. In this work,
we applied MEDYAN to investigate a contractile actomyosin network consisting of actin
filaments, alpha-actinin cross-linking proteins, and non-muscle myosin IIA mini-filaments.
We found that these systems undergo a switch-like transition in simulations from a random
network to ordered, bundled structures when cross-linker concentration is increased
above a threshold value, inducing contraction driven by myosin II mini-filaments. Our
simulations also show how myosin II mini-filaments, in tandem with cross-linkers, can
produce a range of actin filament polarity distributions and alignment, which is crucially
dependent on the rate of actin filament turnover and the actin filament's resulting
super-diffusive behavior in the actomyosin-cross-linker system. We discuss the biological
implications of these findings for the arc formation in lamellipodium-to-lamellum architectural
remodeling. Lastly, our simulations produce force-dependent accumulation of
myosin II, which is thought to be responsible for their mechanosensation ability, also
spontaneously generating myosin II concentration gradients in the solution phase of the
simulation volume.
2016-01-01T00:00:00ZThe Stability of the Schwarzschild Metric
http://hdl.handle.net/1903/17449
The Stability of the Schwarzschild Metric
Vishveshwara, C. V.
The stability of the Schwarzschild exterior metric against small perturbations is investigated. The exterior extending from the Schwarzschild radius r =2m to spatial infinity is visualized as having been produced by a spherically symmetric mass distribution that collapsed into the Schwarzschild horizon in the remote past. As a preamble to the stability analysis, the phenomenon of spherically symmetric gravitational collapse is discussed under the conditions of zero pressure, absence of rotation and adiabatic flow. This is followed by a brief study of the Kruskal coordinates in which the apparent singularity at r = 2m is no longer present; the process of spherical collapse and the consequent production of the Schwarzschild empty space geometry down to the Schwarzschild horizon are depicted on the Kruskal diagram.
The perturbations superposed on the Schwarzschild background metric are the same as those given by Regge and Wheeler consisting of odd and even parity classes, and with the time dependence exp(-ikt), where k is the frequency. An analysis of the Einstein field equations computed to first order in the perturbations away from the Schwarzschild background metric shows that when the frequency is made purely imaginary, the solutions that vanish at large values of r, conforming to the requirement of asymptotic flatness, will diverge near the Schwarzschild surface in the Kruskal coordinates even at the initial instant t = 0. Since the background metric itself is finite at this surface, the above behaviour of the perturbation clearly contradicts the basic assumption that the perturbations are small compared to .the background metric. Thus perturbations with imaginary frequencies that grow exponentially with time are physically unacceptable and hence the metric is stable. In the case of the odd perturbations, the above proof of stability is made rigorous by showing that the radial functions for real values of k form a complete set, by superposition of which any well behaved initial perturbation can be represented so that the time development of such a perturbation is non-divergent, since each of the component modes is purely oscillatory in time. A similar rigorous extension of the proof of stability has not been possible in the case of the even perturbations because the frequency (or k2) does not appear linearly in the differential equation.
A study of stationary perturbations (k = 0) shows that the only nontrivial stationary perturbation that can exist is that due to the rotation of the source which is given by the odd perturbation with the angular momentum £ = 1. Finally, complex frequencies are introduced under the boundary conditions of only outgoing waves at infinity and purely incoming waves at the Schwarzschild surface. The physical significance of this situation is discussed and its connection with phenomena such as radiation damping and resonance scattering, and with the idea of causality is pointed out.
1968-01-01T00:00:00ZAlmost Symmetric Spaces and Gravitational Radiation
http://hdl.handle.net/1903/17448
Almost Symmetric Spaces and Gravitational Radiation
Matzner, Richard Alfred
1967-01-01T00:00:00ZGravitational Radiation in the Limit of High Frequency
http://hdl.handle.net/1903/17447
Gravitational Radiation in the Limit of High Frequency
Isaacson, Richard Allen
This dissertation deals with a technique for obtaining approximate radiative solutions to the Einstein equations of general relativity in situations where the gravitational fields of interest are quite strong. In the first chapter, we review the history of the problem and discuss previous work along related lines. In the second chapter, we assume the radiation to be of high frequency and expand the field equations in powers of the small wavelength this supplies. This assumption provides an approximation scheme valid for all orders of 1/r, for arbitrary velocities up to that of light, and for arbitrary intensities of the gravitational field. To lowest order we obtain a gauge invariant linear wave equation for gravitational radiation, which is a covariant generalization of that for massless spin-two fields in flat space, This wave equation is then solved by the W.K.B. approximation to show that gravitational waves travel on null geodesics with amplitude and frequency modified by gravitational fields in exactly the same way as are those of light waves, and with their polarization parallel transported along the geodesics, again as is the case for light. The metric containing high frequency gravitational waves is shown to be type N to lowest order, and some limits to the methods used are discussed. In the third chapter we go beyond the linear terms in the high frequency expansion, and consider the lowest order non-linear terms. They are shown to provide a natural, gauge invariant, averaged effective stress tensor for the energy localized in the high frequency radiation. By assuming the W.K.B. form for the field, this tensor is found to have the same structure as that for an electromagnetic null field. A Poynting vector is used to investigate the flow of energy and momentum in the gravitational wave field, and it is seen that high frequency waves propagate along null hypersurfaces and are not backscattered off by the curvature of space. Expressions for the total energy and momentum carried by the field to flat null infinity are given in terms of coordinate independent integrals valid within regions of strong field strength. The formalism is applied to the case of spherical gravitational waves where a news function is obtained, and where the source is found to lose exactly the energy and momentum contained in the radiation field.
1967-01-01T00:00:00Z