Optimal control in pandemics.

During a pandemic, there are conflicting demands that arise from public health and socioeconomic costs. Lockdowns are a common way of containing infections, but they adversely affect the economy. We study the question of how to minimize the socioeconomic damage of a lockdown while still containing infections. Our analysis is based on the SIR model, which we analyze using a clock set by the virus. This use of the "virus time" permits a clean mathematical formulation of our problem. We optimize the socioeconomic cost for a fixed health cost and arrive at a strategy for navigating the pandemic. This involves adjusting the level of lockdowns in a controlled manner so as to minimize the socioeconomic cost.

Quantum Brownian motion: Drude and Ohmic baths as continuum limits of the Rubin model.

The motion of a free quantum particle in a thermal environment is usually described by the quantum Langevin equation, where the effect of the bath is encoded through a dissipative and a noise term, related to each other via the fluctuation dissipation theorem. The quantum Langevin equation can be derived starting from a microscopic model of the thermal bath as an infinite collection of harmonic oscillators prepared in an initial equilibrium state. The spectral properties of the bath oscillators and their coupling to the particle determine the specific form of the dissipation and noise. Here we investigate in detail the well-known Rubin bath model, which consists of a one-dimensional harmonic chain with the boundary bath particle coupled to the Brownian particle. We show how in the limit of infinite bath bandwidth, we get the Drude model, and a second limit of infinite system-bath coupling gives the Ohmic model. A detailed analysis of relevant equilibrium correlation functions, such as the mean squared displacement, velocity autocorrelation functions, and response function are presented, with the aim of understanding the various temporal regimes. In particular, we discuss the quantum-to-classical crossover time scales where the mean square displacement changes from a ∼lnt to a ∼t dependence. We relate our study to recent work using linear response theory to understand quantum Brownian motion.

Nonclassical paths in quantum interference experiments.

In a double slit interference experiment, the wave function at the screen with both slits open is not exactly equal to the sum of the wave functions with the slits individually open one at a time. The three scenarios represent three different boundary conditions and as such, the superposition principle should not be applicable. However, most well-known text books in quantum mechanics implicitly and/or explicitly use this assumption that is only approximately true. In our present study, we have used the Feynman path integral formalism to quantify contributions from nonclassical paths in quantum interference experiments that provide a measurable deviation from a naive application of the superposition principle. A direct experimental demonstration for the existence of these nonclassical paths is difficult to present. We find that contributions from such paths can be significant and we propose simple three-slit interference experiments to directly confirm their existence.

Statistical mechanics of ribbons under bending and twisting torques.

We present an analytical study of ribbons subjected to an external torque. We first describe the elastic response of a ribbon within a purely mechanical framework. We then study the role of thermal fluctuations in modifying its elastic response. We predict the moment-angle relation of bent and twisted ribbons. Such a study is expected to shed light on the role of twist in DNA looping and on bending elasticity of twisted graphene ribbons. Our quantitative predictions can be tested against future single molecule experiments.

Bending elasticity of macromolecules: analytic predictions from the wormlike chain model.

We present a study of the bend angle distribution of semiflexible polymers of short and intermediate lengths within the wormlike chain model. This enables us to calculate the elastic response of a stiff molecule to a bending moment. Our results go beyond the Hookean regime and explore the nonlinear elastic behavior of a single molecule. We present analytical formulas for the bend angle distribution and for the moment-angle relation. Our analytical study is compared against numerical Monte Carlo simulations. The functional forms derived here can be applied to fluorescence microscopic studies on actin and DNA. Our results are relevant to recent studies of "kinks" and cyclization in short and intermediate length DNA strands.

Biopolymer elasticity: Mechanics and thermal fluctuations.

We present an analytical study of the role of thermal fluctuations in shaping molecular elastic properties of semiflexible polymers. Our study interpolates between mechanics and statistical mechanics in a controlled way and shows how thermal fluctuations modify the elastic properties of biopolymers. We present a study of the minimum-energy configurations with explicit expressions for their energy and writhe and plots of the extension versus link for these configurations and a study of fluctuations around the local minima of energy and approximate analytical formulas for the free energy of stretched twisted polymers. The central result of our study is a closed-form expression for the leading thermal fluctuation correction to the free energy around the nonperturbative writhing family solution for the configuration of a biopolymer. From the derived formulas, the predictions of the wormlike chain model for molecular elasticity can be worked out for a comparison against numerical simulations and experiments.

Comment on "Writhe formulas and antipodal points in plectonemic DNA configurations".

We point out that the disagreement between the paper by Neukirch and Starostin [S. Neukirch and E. L. Starostin, Phys. Rev. E 78, 041912 (2008)] and ours [J. Samuel, S. Sinha, and A. Ghosh, J. Phys.: Condens. Matter 18, S253 (2006)] is only apparent and stems from a difference in approach. Neukirch and Starostin are concerned with classical elasticity and individual curves while we focus on statistical averages over curves.

Free energy of twisted semiflexible polymers.

We investigate the role of fluctuations in single-molecule measurements of torque-link (t-lk) curves. For semiflexible polymers of finite persistence length (i.e., polymers with contour length L comparable to the persistence length LP), the torque versus link curve in the constant-torque (isotorque) ensemble is distinct from the one in the constant-link (isolink) ensemble. Thus, one encounters the conceptually interesting issue of a "free energy of transition" in switching ensembles while making torque-link measurements. We predict the dependence on the semiflexibility parameter beta=L/LP of this extra contribution to the free energy, which shows up as an area in the torque-link plane. This can be tested against future torque-link experiments with single biopolymers. We bring out the inequivalence of torque-link curves for a stiff polymer and present explicit analytical expressions for the distinct torque-link relations in the two ensembles and the free-energy difference in switching ensembles in this context. The predictions of our work can be tested against single-molecule experiments on torsionally constrained biopolymers.

Elasticity of stiff biopolymers.

We present a statistical mechanical study of stiff polymers, motivated by experiments on actin filaments and the considerable current interest in polymer networks. We obtain simple, approximate analytical forms for the force-extension relations and compare these with numerical treatments. We note the important role of boundary conditions in determining force-extension relations. The theoretical predictions presented here can be tested against single molecule experiments on neurofilaments and cytoskeletal filaments like actin and microtubules. Our work is motivated by the buckling of the cytoskeleton of a cell under compression, a phenomenon of interest to biology.

Surface tension and the cosmological constant.

One of the few predictions from quantum gravity models is Sorkin's observation that the cosmological constant has quantum fluctuations originating in the fundamental discreteness of spacetime at the Planck scale. Here we present a compelling analogy between the cosmological constant of the Universe and the surface tension of fluid membranes. The discreteness of spacetime on the Planck scale translates into the discrete molecular structure of a fluid membrane. We propose an analog quantum gravity experiment which realizes Sorkin's idea in the laboratory. We also notice that the analogy sheds light on the cosmological constant problem, suggesting a mechanism for dynamically generating a vanishingly small cosmological constant. We emphasize the generality of Sorkin's idea and suggest that similar effects occur generically in quantum gravity models.

Euler buckling-induced folding and rotation of red blood cells in an optical trap.

We investigate the physics of an optically driven micromotor of biological origin. When a single, live red blood cell (RBC) is placed in an optical trap, the normal biconcave disc shape of the cell is observed to fold into a rod-like shape. If the trapping laser beam is circularly polarized, the folded RBC rotates. A model based on geometric considerations, using the concept of buckling instabilities, captures the folding phenomenon; the rotation of the cell is rationalized using the Poincaré sphere. Our model predicts that (i) at a critical power of the trapping laser beam the RBC shape undergoes large fluctuations, and (ii) the torque that is generated is proportional to the power of the laser beam. These predictions are verified experimentally. We suggest a possible mechanism for the emergence of birefringent properties in the RBC in the folded state.

Inequivalence of statistical ensembles in single molecule measurements.

We study the role of fluctuations in single molecule experimental measurements of force-extension (f-zeta) curves. We use the worm-like chain (WLC) model to bring out the connection between the Helmholtz ensemble characterized by the free energy [F(zeta)] and the Gibbs ensemble characterized by the free energy [G(f)] . We consider the rigid rod limit of the WLC model as an instructive special case to bring out the issue of ensemble inequivalence. We point out the need for taking into account the free energy of transition when one goes from one ensemble to another. We also comment on the "phase transition" noticed in an isometric setup for semiflexible polymers and propose a realization of its thermodynamic limit. We present general arguments which rule out nonmonotonic force-extension curves in some ensembles and note that these do not apply to the isometric ensemble.

Writhe distribution of stretched polymers.

Motivated by experiments in which single deoxyribose nucleic acid molecules are stretched and twisted we consider a perturbative approach around very high forces, where we determine the writhe distribution in a simple, analytically tractable model. Our results are in agreement with recent simulations and experiments.

Molecular elasticity and the geometric phase.

We present a method for solving the wormlike-chain (WLC) model for twisting semiflexible polymers to any desired accuracy. We show that the WLC free energy is a periodic function of the applied twist with period 4pi. We develop an analogy between WLC elasticity and the geometric phase of a spin-1 / 2 system. These analogies are used to predict elastic properties of twist-storing polymers. We graphically display the elastic response of a single molecule to an applied torque. This study is relevant to mechanical properties of biopolymers such as DNA.

Elasticity of semiflexible polymers.

We present a method for solving the wormlike chain model for semiflexible polymers to any desired accuracy over the entire range of polymer lengths. Our results are in excellent agreement with recent computer simulations and reproduce important qualitatively interesting features observed in simulations of polymers of intermediate lengths. We also make a number of predictions that can be tested in a variety of concrete experimental realizations. The expected level of finite size fluctuations in force-extension curves is also estimated. This study is relevant to mechanical properties of biological molecules.