# Research

## Coarse-grained modelling of DNA and RNA

In our group, I work on developing and applying coarse-grained models of nucleic acids. Our DNA model, oxDNA, is parameterized to reproduce quantitatively structural, mechanical and thermodynamical properties of DNA. It has been successfully used to study both fundamental physics of DNA molecules as well as DNA nanotechnological systems. DNA nanotechnology is a rapidly developing field which uses DNA molecules as basic building blocks to assemble nanoscale structures as well as active molecular devices. Computer simulations using our coarse-grained model provide insight into the functioning of such systems and give us better understanding of underlying physical processes. Among other things, we studied pulling of single-stranded DNA, burnt-bridges DNA motor, association of DNA strands and toehold-mediated strand displacement. I work in collaboration with Tom Ouldridge and Ard Louis from Theoretical Physics and Flavio Romano and Jon Doye from Physical and Theoretical Chemistry Department and Lorenzo Rovigatti from Sapienza University, Rome. The code implementing our model, oxDNA, is available as a free software. The code website also provides examples and tutorials.

A coarse-grained model for RNA, oxRNA, was also recently introduced, along with examples of its application to hairpin unzipping, pseudoknot folding and kissing complex thermodynamics. The oxRNA model is also available for download.

## Smartgrid: Control of reactive power from photovoltaic generators

Smartgrid refers to power grid which incorporates modern computer communication technology to improve efficiency and reliability of production, distribution and consumption of electricity. One of the big challenges that the power grid faces at the moment is the integration of renewable sources of energy. Together with Kostya Turitsyn, Misha Chertkov and Scott Backhaus from Los Alamos National Laboratory, we investigated ([1], [2], [3]) the possibilities of controlling reactive power flow in a distribution network that contains photovoltaic generators (typical example would be an urban neighborhood with houses with photovoltaic cells on roofs). We considered both global and local control of the generated reactive power and found optimization schemes that can reduce thermal losses in the distribution line and improve voltage stability.

## Particle tracking: protein diffusion in membrane

We simulated a diffusion of a protein in biological membrane with compartments. We investigated how the compartmentalization of the membrane affects diffusion of the protein and proposed a statistical analysis technique which uses kurtosis of the displacement distribution to observe non-Gaussian behavior. The work was done with Kostya Turitsyn from MIT and Pavel Lushnikov from UNM.

## Statistical physics and graphs

In a work with Olivier Martin, we looked at the random walks on the largest connected component of sparse Erdos-Renyi graphs. We derived a probability distribution function of hitting time (i.e. the number of steps random walker takes before reaching a specific node of of the graph) by exploiting a relation between the hitting probability and probability of return to the original node.

In a different project with Lenka Zdeborová we studied the application of belief propagation for graph bisectioning problem. Bisectioning of a graph means dividing its nodes into two equally sized groups such that the number of edges between nodes in different groups is minimal. This optimization problem is NP-hard. We developed a heuristic algorithm that is linear in number of nodes. We further provide phase diagram of bisectioning problem on Erdos-Renyi graphs.

## Mutually unbiased bases

Orthonormal bases in a finite-dimensional Hilbert space of dimension N are called mutually unbiased if the square of absolute value of inner product of any two vectors from different bases is equal to 1/N. This means is that if a base state from one base is measured in a different unbiased base, each of the N possible base vectors from the second base have equal probability to be observed. The concept of mutually unbiased bases is used in quantum cryptography protocols. The maximum number of mutually unbiased bases is N+1, but they are known explicitly only if N is prime or a power of a prime. It is conjectured that for all other Ns, N+1 mutually unbiased bases cannot be constructed. In a work with prof. Jiří Tolar, we gave a new constructive proof of existence of N+1 mutually unbiased bases when N is a prime number.