During my Ph.D. with Prof. Yi-Chao Chen at University of Houston, we developed a variational framework to study the equilibrium shape and stability of two charged elastic curves constrained to a surface. This framework was inspired by reverse cholesterol transport, wherein the high-density lipoproteins (HDL) particles constituted by charged Apo-lipoproteins attached to a spherical core are responsible for maintaining the cholesterol level in the human body. Additionally, we worked on the effect of surface tension imbalance on arbitrarily shaped floating bodies, unraveling the mechanism utilized by water-walking insects such as Rove beetles.

I received my Bachelors and Masters degrees in Mechanical Engineering from IIT Kanpur. During my master's, I worked on developing energy-preserving symplectic algorithms for discrete Hamiltonian systems.

Broad research keywords encapsulating my interests include computational mechanics, biophysics, differential geometry, numerical methods, mathematical modeling, and the biomechanics of lipoproteins and viral dynamics. A more detailed description of my research interests can be found in my academic research statement (current as of 1/2024).

Stable Möbius bands from isometrically deformed helicoids Link

Abstract: We consider the problem of producing a ruled Möbius band by subjecting an unstretchable material surface in a circular helicoidal reference configuration to a deformation that is isometric and chirality preserving. We find that such a Möbius band is completely determined by the unit binormal of the Frenet frame of its midline, which must be a geodesic and must have constant torsion inversely proportional to the pitch of the helicoidal reference configuration. For a material surface that is characterized by an energy density which depends quadratically on the mean curvature of its deformed configuration, we show that the total energy stored in producing a Möbius band as described reduces to an integral over the midline of the Möbius band. We formulate and numerically solve a constrained variational problem for finding relative minima of the dimensionally reduced bending energy and constructing corresponding stable Möbius bands. The only input parameter entering our variational problem is the number ν of turns of a referential helicoid. We do not obtain a solution to our problem unless ν exceeds a certain threshold, which we compute to machine precision. Above that threshold, an interplay between the operative constraint leads to a multiplicity of coexisting stable solutions with n ≥ 3 half twists. For each n ≥ 3, we construct an energetically optimal Möbius band which exhibits n-fold rotational symmetry. All other energy minima yield Möbius bands with broken symmetry.



Key finding: Our study provides a guideline to design reference helicoid templates which can be folded, without stretching or contraction, into Möbius bands with given n half twists and n-fold rotational symmetry (optimal symmetry). The pitch of the reference helicoid of axis length ℓ which can be folded into an optimal band with n half twists is given by
\[ p = \frac{n}{2\ell} - \frac{2}{\pi\ell}\sin^{-1}\frac{1}{n}, \qquad n= 3, 5, 7,\dots . \]
The optimal bands with n = 3, 5,7,9,11, and 13 half twists are show in the top-panel of the above figure. These unique bands have minimal ring strains desired in synthesizing ring compounds such as Möbius annulenes and DNA origami.

(Joint with Prof. Eliot Fried. )

Everting motion of Möbius and orientable binormal scrolls Under construction

Abstract: Eversion, traditionally explored in the context of simple structures like spheres and cylinders, takes a leap forward in this study, which extends the concept to more complex geometries. We introduce a novel method for the continuous, isometric, and iso-energetic eversion of Möbius and orientable binormal scrolls with midlines of uniform torsion. Our approach, rooted in a kinematically consistent time-evolution equation, ensures a seamless transformation that preserves both geometry and energy throughout the eversion process. This methodology marks a notable advancement from previous studies that limited their focus to the initial and final states of eversion. By capturing the intermediate stages, our research provides a comprehensive understanding of the eversion dynamics in these intricate structures. The analytical solution derived here not only offers a closed-form expression for the everting motion but also paves the way for practical applications. These include the development of adaptive materials and soft robotic systems with dynamically tunable properties, as well as innovations in medical, aerospace, and architectural engineering. Bridging theoretical concepts with real-world applications, our work presents a paradigm shift in eversion research, opening a new realm of possibilities across a diverse array of scientific and engineering disciplines. Please see the everting motion of Möbius bands and Orientable bands using our framework.



Four representative states of the everting motion of a (5,2) knotted Möbius band.

(Joint with Prof. Eliot Fried. )

Reverse cholesterol transport

Interacting elastic loops on a sphere Link

Abstract: A variational approach is used to study the behavior of two closed, inextensible, interacting elastic loops that are constrained to lie on a sphere. In addition to the bending energy of each loop, the total potential energy of the system includes nonlocal contributions that account for intraloop and interloop interactions. Euler--Lagrange equations and energy based stability condition are derived in a coordinate-free setting using the first and second variation of the potential energy functional. As an illustrative application, a problem in which all the interaction potentials are Coulombic and both loops possess the same length, bending rigidity, and positive charge density is considered. To ensure existence of a trivial solution in which the loops are parallel and circular, the length of the loops are taken to be smaller than perimeter of the great circle of the sphere. Detailed bifurcation and linear stability analyses of the trivial solution are conducted. The stability of the trivial solution is governed by three dimensionless parameters a, ζ, and χ, where a is the ratio between the radius of the loops to radius of the sphere and where ζ and χ encompass information about the ratio of intraloop interaction and interloop interaction to the bending rigidity. While the bending energy and the intraloop interaction energy stabilize the trivial solution, the interloop interaction has a destabilizing influence. Moreover, a cross-over phenomenon associated with the nature of the most destabilizing mode is discovered: for 0 < a < a_c, the number of modes represented in the most destabilizing modes varies with ζ and χ; for a_c < a < 1, the most destabilizing mode is always the lowest mode in keeping with results for problems involving only bending energy.


A few possible arrangements of charged curves, shown with red color, on a rigid sphere are shown in the figure above. While the curves represent the Apo-lipoproteins, the rigid sphere models the core of high-density lipoprotein particles.

Key findings: For the given charge, length, and bending rigidity of the proteins, our framework yields the precise equilibrium shape of the proteins confined to a core. We provide a detailed stability analysis of the circular configuration of the proteins. Depending on the parameters, the first unstable mode could be 1, 2, 3, and so on.

(Joint with Prof. Eliot Fried and Prof. Yi-Chao Chen.)

Virus modeling via energy minimization

Coronavirus Peplomer Charge heterogeneity Link

Abstract: Recent advancements in viral hydrodynamics afford the calculation of the transport properties of particle suspensions from first principles, namely, from the detailed particle shapes. For coronavirus suspensions, for example, the shape can be approximated by beading (i) the spherical capsid and (ii) the radially protruding peplomers. General rigid bead-rod theory allows us to assign Stokesian hydrodynamics to each bead. Thus, viral hydrodynamics yields the suspension rotational diffusivity, but not without first arriving at a configuration for the cationic peplomers. Prior work considered identical peplomers charged identically. However, a recent pioneering experiment uncovers remarkable peplomer size and charge heterogeneities. In this work, we use energy minimization to arrange the spikes, charged heterogeneously to obtain the coronavirus spike configuration required for its viral hydrodynamics. For this, we use the measured charge heterogeneity. We consider 20,000 randomly generated possibilities for cationic peplomers with formal charges ranging from 30 to 55. We find the configurations from energy minimization of all of these possibilities to be nearly spherically symmetric, all slightly oblate, and we report the corresponding breadth of the dimensionless rotational diffusivity, the transport property around which coronavirus cell attachment revolves.



Left: probability distribution function (pdf) of the charges on the Covid-19 spike proteins. Right: pdf of the relaxation-time of the Covid-19 particles. As shown, the charge heterogeneity of the spike proteins leads to a pdf of the relaxation-time.


(Joint with Prof. Eliot Fried, Mona Kanso, and Prof. A.J. Giacomin.)

Coronavirus pleomorphism Link

Abstract: The coronavirus is always idealized as a spherical capsid with radially protruding spikes. However, histologically, in the tissues of infected patients, capsids in cross section are elliptical, and only sometimes spherical [Neuman et al., “Supramolecular architecture of severe acute respiratory syndrome coronavirus revealed by electron cryomicroscopy,” J Virol, 80, 7918 (2006)]. This capsid ellipticity implies that coronaviruses are oblate or prolate or both. We call this diversity of shapes, pleomorphism. Recently, the rotational diffusivity of the spherical coronavirus in suspension was calculated, from first principles, using general rigid bead-rod theory [Kanso et al., “Coronavirus rotational diffusivity,” Phys Fluids 32, 113101 (2020)]. We did so by beading the spherical capsid and then also by replacing each of its bulbous spikes with a single bead. In this paper, we use energy minimization for the spreading of the spikes, charged identically, over the oblate or prolate capsids. We use general rigid bead-rod theory to explore the role of such coronavirus cross-sectional ellipticity on its rotational diffusivity, the transport property around which its cell attachment revolves. We learn that coronavirus ellipticity drastically decreases its rotational diffusivity, be it oblate or prolate.



Left: Oblate and prolate cores of Covid-19 particles. Right: Shear viscosity of ellipsoidal core covid particles relative to the spherical core. As shown, the sphericity of the core affects its viscosity.


(Joint with Prof. Eliot Fried, Mona Kanso, and Prof. A.J. Giacomin.)

Peplomer bulb shape and coronavirus rotational diffusivity Link

Abstract: Recently, the rotational diffusivity of the coronavirus particle in suspension was calculated, from first principles, using general rigid bead-rod theory [M. A. Kanso, Phys. Fluids 32, 113101 (2020)]. We did so by beading the capsid and then also by replacing each of its bulbous spikes with a single bead. However, each coronavirus spike is a glycoprotein trimer, and each spike bulb is triangular. In this work, we replace each bulbous coronavirus spike with a bead triplet, where each bead of the triplet is charged identically. This paper, thus, explores the role of bulb triangularity on the rotational diffusivity, an effect not previously considered. We thus use energy minimization for the spreading of triangular bulbs over the spherical capsid. The latter both translates and twists the coronavirus spikes relative to one another, and we then next arrive at the rotational diffusivity of the coronavirus particle in suspension, from first principles. We learn that the triangularity of the coronavirus spike bulb decreases its rotational diffusivity. For a typical peplomer population of 74, bulb triangularity decreases the rotational diffusivity by 1/3.



Left: Triangular spike proteins of Covid-19 particles. Right: Dimensionless rotational diffusivity of covid particles with triangular proteins relative to single-bead shape. As shown, the triangularity of spike proteins decrease the rotational diffusivity of covid particles.


(Joint with Prof. Eliot Fried, Mona Kanso, and Prof. A.J. Giacomin.)

Surface tension mediated shape changes

Effect of a surface tension imbalance on a partly submerged cylinder Link

Abstract: We perform a static analysis of a circular cylinder that forms a barrier between surfactant-laden and surfactant-free portions of a liquid-gas interface. In addition to determining the general implications of the balances for forces and torques, we quantify how the imbalance Δγ = γ_a - γ_b between the uniform surface tension γ_a of the surfactant-free portion of the interface and the uniform surface tension γ_b of the surfactant-laden portion of the interface influences the load-bearing capacity of a hydrophobic cylinder. Moreover, we demonstrate that the difference between surface tensions on either side of a cylinder with a cross-section of arbitrary shape induces a horizontal force component f^h equal to Δ γ in magnitude, when measured per unit length of the cylinder. With an energetic argument, we show that this relation also applies to a rod-like barrier with cross-sections of variable shape. In addition, we apply our analysis to amphiphilic Janus cylinders and we discuss practical implications of our findings for Marangoni propulsion and surface pressure measurements.


Left: picture of a rove beetle showing deformation of the water surface around its legs. Center: effect of surface tension imbalance on a floating cylinder. Right: generalization of our model to arbitrary shaped floating objects.


(Joint with Prof. Eliot Fried and S. D. Janssens (lead author)).