Casimir Interactions Between Scatterers in Carbon Nanotubes

In this thesis we calculate interactions between localized scatterers in metallic carbon nanotubes. Backscattering of electrons between localized scatterers mediates long range forces between them. These interactions are mapped to Casimir forces mediated by one-dimensional massless fermions and calculated using a force operator approach. We first study interactions between scatterers described by spinor polarized potentials relevant to the single-valley problem in carbon nanotubes. We obtain the force between two finite width square barriers, and take the limit of zero width and infinite potential strength to study the Casimir force mediated by the fermions. For the case of identical scatterers we recover the conventional attractive one dimensional Casimir force. For the general problem with inequivalent scatterers we find that the magnitude and sign of this force depend on the relative spinor polarizations of the two scattering potentials which can be tuned to give an attractive, a repulsive, or a compensated null Casimir interaction. Next, we generalize our work on the single-valley Casimir problem to study interactions between physically realizable scatterers in nanotubes. We model spatially localized scatterers by local and non-local potentials and treat simultaneously the effects of intravalley and intervalley backscattering. We find that the long range forces between scatterers exhibit the universal power law decay of the Casimir force in one dimension, with prefactors that control the sign and strength of the interaction. These prefactors are nonuniversal and depend on the symmetry and degree of localization of the scattering potentials. We find that local potentials inevitably lead to a coupled valley scattering problem, though by contrast non-local potentials lead to two decoupled single-valley problems. The Casimir effect due to two-valley scattering potentials is characterized by the appearance of spatially periodic modulations of the force

Casimir Interactions Between Scatterers in Metallic Carbon Nanotubes

We study interactions between localized scatterers on metallic carbon nanotubes by a mapping onto a one-dimensional Casimir problem. Backscattering of electrons between localized scattering potentials mediates long-range forces between them. We model spatially localized scatterers by local and nonlocal potentials and treat simultaneously the effects of intravalley and intervalley backscattering. We find that the long-range forces between scatterers exhibit the universal power-law decay of the Casimir force in one dimension, with prefactors that control the sign and strength of the interaction. These prefactors are nonuniversal and depend on the symmetry and degree of localization of the scattering potentials. We find that local potentials inevitably lead to a coupled valley scattering problem, though by contrast nonlocal potentials lead to two decoupled single-valley problems in a physically realized regime. The Casimir effect due to two-valley scattering potentials is characterized by the appearance of spatially periodic modulations of the force

Casimir Effect for Massless Fermions in One Dimension: A Force Operator Approach

We calculate the Casimir interaction between two short range scatterers embedded in a background of one dimensional massless Dirac fermions using a force operator approach. We obtain the force between two finite width square barriers, and take the limit of zero width and infinite potential strength to study the Casimir force mediated by the fermions. For the case of identical scatterers we recover the conventional attractive one dimensional Casimir force. For the general problem with inequivalent scatterers we find that the magnitude and sign of this force depend on the relative spinor polarizations of the two scattering potentials which can be tuned to give an attractive, a repulsive, or a compensated null Casimir interaction.Comment: (4 pages, 3 figures; to appear in Phys. Rev. A, Rapid Communications

Theoretical Analysis of Competing Conformational Transitions in Superhelical DNA

We develop a statistical mechanical model to analyze the competitive behavior of transitions to multiple alternate conformations in a negatively supercoiled DNA molecule of kilobase length and specified base sequence. Since DNA superhelicity topologically couples together the transition behaviors of all base pairs, a unified model is required to analyze all the transitions to which the DNA sequence is susceptible. Here we present a first model of this type. Our numerical approach generalizes the strategy of previously developed algorithms, which studied superhelical transitions to a single alternate conformation. We apply our multi-state model to study the competition between strand separation and B-Z transitions in superhelical DNA. We show this competition to be highly sensitive to temperature and to the imposed level of supercoiling. Comparison of our results with experimental data shows that, when the energetics appropriate to the experimental conditions are used, the competition between these two transitions is accurately captured by our algorithm. We analyze the superhelical competition between B-Z transitions and denaturation around the c-myc oncogene, where both transitions are known to occur when this gene is transcribing. We apply our model to explore the correlation between stress-induced transitions and transcriptional activity in various organisms. In higher eukaryotes we find a strong enhancement of Z-forming regions immediately 5′ to their transcription start sites (TSS), and a depletion of strand separating sites in a broad region around the TSS. The opposite patterns occur around transcript end locations. We also show that susceptibility to each type of transition is different in eukaryotes and prokaryotes. By analyzing a set of untranscribed pseudogenes we show that the Z-susceptibility just downstream of the TSS is not preserved, suggesting it may be under selection pressure

Theoretical Analysis of the Stress Induced B-Z Transition in Superhelical DNA

We present a method to calculate the propensities of regions within a DNA molecule to transition from B-form to Z-form under negative superhelical stresses. We use statistical mechanics to analyze the competition that occurs among all susceptible Z-forming regions at thermodynamic equilibrium in a superhelically stressed DNA of specified sequence. This method, which we call SIBZ, is similar to the SIDD algorithm that was previously developed to analyze superhelical duplex destabilization. A state of the system is determined by assigning to each base pair either the B- or the Z-conformation, accounting for the dinucleotide repeat unit of Z-DNA. The free energy of a state is comprised of the nucleation energy, the sequence-dependent B-Z transition energy, and the energy associated with the residual superhelicity remaining after the change of twist due to transition. Using this information, SIBZ calculates the equilibrium B-Z transition probability of each base pair in the sequence. This can be done at any physiologically reasonable level of negative superhelicity. We use SIBZ to analyze a variety of representative genomic DNA sequences. We show that the dominant Z-DNA forming regions in a sequence can compete in highly complex ways as the superhelicity level changes. Despite having no tunable parameters, the predictions of SIBZ agree precisely with experimental results, both for the onset of transition in plasmids containing introduced Z-forming sequences and for the locations of Z-forming regions in genomic sequences. We calculate the transition profiles of 5 kb regions taken from each of 12,841 mouse genes and centered on the transcription start site (TSS). We find a substantial increase in the frequency of Z-forming regions immediately upstream from the TSS. The approach developed here has the potential to illuminate the occurrence of Z-form regions in vivo, and the possible roles this transition may play in biological processes

Non-equilibrium phenomena implemented at a mesoscopic time scale

The purpose of this project is to develop an algorithm that speeds up large scale simulations of many-body systems. A numerical method is implemented that simulates non-equilibrium phenomena on a mesoscopic time scale. A system is perturbed by an external force, and time averages of variables renormalized in space are calculated numerically, using results of linear response theory, as the system relaxes to equilibrium. The coarse-grained variables evolve slowly in time, allowing one to advance them on a mesoscopic time scale.The algorithm was tested on two physical systems: a lattice confined ferromagnetic Ising model and an off-lattice Argon-like molecular system. The method simulated accurately the non-equilibrium phenomena studied. It was found that the algorithm is most efficient when it is applied to a process occurring on at least two time scales. This allows one to integrate out the fast, microscopic time scale in order to study long-time, macroscopic behaviour. Through the study of diffusion in a molecular system, it was concluded that the proposed method is computationally faster than solving the microscopic equations of motion and more accurate than solving the macroscopic equations

Transition profiles around gene start and end sites.

<p>The average probability for (a) denaturation and (b) Z-DNA formation are shown as functions of base pair position for human gene sequences centered at their transcription start sites (TSSs) and at their transcript end positions (TESs). These calculations were performed at T = 305 K and  = −0.07.</p

The average probabilities of transition calculated as functions of negative superhelicity (here plotted as ) for a 5 kb plasmid containing a single denaturation-susceptible site and a single Z-susceptible site in an otherwise transition-resistant background.

<p>The analysis is done for the melting region and the Z-susceptible segment st  = 300 K using (a) BDZ<i>trans</i> (b) SIDD for denaturation and SIBZ for Z-DNA.</p

The average probability of B-Z transition is plotted as a function of base pair location, the average being taken over 12,841 mouse genes.

<p>The sequences were aligned so their TSSs are all located at position 2500, indicated by the vertical dashed line. The solid and dotted lines are the results for −0.07 and −0.055, respectively.</p

Onset of transition: Comparison of experimental results with SIBZ.

<p>Critical superhelical densities for the onset of a B-Z transition are obtained experimentally for three sequences in which Z-forming regions are inserted <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1001051#pcbi.1001051-Ellison1" target="_blank">[37]</a>, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1001051#pcbi.1001051-Peck2" target="_blank">[56]</a>. These results are compared with SIBZ, where we define a region to be in Z-form when its probability of transition is 80%.</p