Kranthi K. Mandadapu

Interaction between surface shape and intra-surface viscous flow on lipid membranes

The theory of intra-surface viscous flow on lipid bilayers is developed by combining the equations for flow on a curved surface with those that describe the elastic resistance of the bilayer to flexure. The model is derived directly from balance laws and augments an alternative formulation based on a variational principle. Conditions holding along an edge of the membrane are emphasized, and the coupling between flow and membrane shape is simulated numerically.

On the cytoskeleton and soft glassy rheology

The cytoskeleton is a complex structure within the cellular corpus that is responsible for the main structural properties and motilities of cells. A wide range of models have been utilized to understand cytoskeletal rheology and mechanics (see e.g. [Mofrad, M., Kamm, R., 2006. Cytoskeletal Mechanics: Models and Measurements. Cambridge University Press, Cambridge]). From this large collection of proposed models, the soft glassy rheological model (originally developed for inert soft glassy materials) has gained a certain traction in the literature due to the close resemblance of its predictions to certain mechanical data measured on cell cultures [Fabry, B., Maksym, G., Butler, J., Glogauer, M., Navajas, D., Fredberg, J., 2001. Scaling the microrheology of living cells. Physical Review Letters 87, 14102]. We first review classical linear rheological theory in a concise fashion followed by an examination of the soft glassy rheological theory. With this background we discuss the observed behavior of the cytoskeleton and the inherent limitations of classical rheological models for the cytoskeleton. This then leads into a discussion of the advantages and disadvantages presented to us by the soft glassy rheological model. We close with some comments of caution and recommendations on future avenues of exploration.

A stabilized finite element formulation for liquid shells and its application to lipid bilayers

This paper presents a new finite element (FE) formulation for liquid shells that is based on an explicit, 3D surface discretization using C 1 -continuous finite elements constructed from NURBS interpolation. Both displacement-based and mixed displacement/pressure FE formulations are proposed. The latter is needed for area-incompressible material behavior, where penalty-type regularizations can lead to misleading results. In order to obtain quasi-static solutions for liquid shells devoid of shear stiffness, several numerical stabilization schemes are proposed based on adding stiffness, adding viscosity or using projection. Several numerical examples are considered in order to illustrate the accuracy and the capabilities of the proposed formulation, and to compare the different stabilization schemes. The presented formulation is capable of simulating non-trivial surface shapes associated with tube formation and protein-induced budding of lipid bilayers. In the latter case, the presented formulation yields non-axisymmetric solutions, which have not been observed in previous simulations. It is shown that those non-axisymmetric shapes are preferred over axisymmetric ones.

A homogenization method for thermomechanical continua using extensive physical quantities

This article proposes a continuum thermomechanical homogenization method inspired by the Irving–Kirkwood procedure relating the atomistic equations of motion to the balance laws of continuum mechanics. This method yields expressions for the macroscopic stress and heat flux in terms of microscopic kinematic and kinetic quantities. The resulting equation for macroscopic stress affords a rational comparison with the widely used Hill–Mandel stress-deformation condition, while the one for heat flux reduces, under certain assumptions, to a Hill–Mandel-like condition involving heat flux and the gradient of temperature.

On the estimation of spatial averaging volume for determining stress using atomistic methods.

The estimation of stress at a continuum point from the atomistic scale requires volume averaging over a region that contains this point. A hypothesis is put forth to obtain a lower bound for the size of this region based on an analogy to the Ising model. This hypothesis is tested on copper using two classical elasticity problems.

Mechanics of torque generation in the bacterial flagellar motor.

Significance Locomotion in many bacterial species is driven by the rotation of one or more long flagellar filaments, each powered by a bacterial flagellar motor (BFM) at its base. The BFM, then, plays a central role in processes such as chemotaxis, bacterial pathogenicity, and biofilm formation. Using information from structural and biophysical experiments on the BFM, we construct a testable model for the mechanism of torque generation. Our model is, to our knowledge, the first to propose and test a specific physical mechanism for this process, and it provides a mechanical explanation for several fundamental properties of the BFM. In addition to fitting current experimental results, model predictions suggest further experiments to shed light on various aspects of motor function. The bacterial flagellar motor (BFM) is responsible for driving bacterial locomotion and chemotaxis, fundamental processes in pathogenesis and biofilm formation. In the BFM, torque is generated at the interface between transmembrane proteins (stators) and a rotor. It is well established that the passage of ions down a transmembrane gradient through the stator complex provides the energy for torque generation. However, the physics involved in this energy conversion remain poorly understood. Here we propose a mechanically specific model for torque generation in the BFM. In particular, we identify roles for two fundamental forces involved in torque generation: electrostatic and steric. We propose that electrostatic forces serve to position the stator, whereas steric forces comprise the actual “power stroke.” Specifically, we propose that ion-induced conformational changes about a proline “hinge” residue in a stator α-helix are directly responsible for generating the power stroke. Our model predictions fit well with recent experiments on a single-stator motor. The proposed model provides a mechanical explanation for several fundamental properties of the flagellar motor, including torque–speed and speed–ion motive force relationships, backstepping, variation in step sizes, and the effects of key mutations in the stator.

Comparison of Molecular and Primitive Solvent Models for Electrical Double Layers in Nanochannels.

In a recent article (Lee et al. J. Comput. Theor. Chem., 2012, 8, 2012-2022.), it was shown that an electrolyte solution can be modeled in molecular dynamics (MD) simulations using a uniform dielectric constant in place of a polar solvent to validate Fluid Density Functional Theory (f-DFT) simulations. This technique can be viewed as a coarse-grained approximation of the polar solvent and reduces computational cost by an order of magnitude. However, the consequences of replacing the polar solvent with an effective permittivity are not well characterized, despite its common usage in f-DFT, Monte Carlo simulation, and Poisson-Boltzmann theory. In this paper, we have examined two solvent models of different fidelities with MD simulation of nanochannels. We find that the models produce qualitatively similar ion density profiles, but physical quantities such as electric field, electric potential, and capacitance differ by over an order of magnitude. In all cases, the bulk is explicitly modeled so that surface properties can be evaluated relative to a reference state. Moreover, quantities that define the reference state, such as bulk ion density, bulk solvent density, applied electric field, and temperature, are measurable, so cases with the same thermodynamic state can be compared. Insights into the solvent arrangement, most of which can not be determined from the coarse-grained model, are drawn from the model with an explicitly described polar solvent.

Polarization as a field variable from molecular dynamics simulations

A theoretical and computational framework for systematically calculating the macroscopic polarization density as a field variable from molecular dynamics simulations is presented. This is done by extending the celebrated Irving and Kirkwood [J. Chem. Phys. 18, 817 (1950)] procedure, which expresses macroscopic stresses and heat fluxes in terms of the atomic variables, to the case of electrostatics. The resultant macroscopic polarization density contains molecular dipole, quadrupole, and higher-order moments, and can be calculated to a desired accuracy depending on the degree of the coarse-graining function used to connect the molecular and continuum scales. The theoretical and computational framework is verified by recovering the dielectric constant of bulk water. Finally, the theory is applied to calculate the spatial variation of the polarization vector in the electrical double layer of a 1:1 electrolyte solution. Here, an intermediate asymptotic length scale is revealed in a specific region, which validates the application of mean field Poisson-Boltzmann theory to describe this region. Also, using the existence of this asymptotic length scale, the lengths of the diffuse and condensed/Stern layers are identified accurately, demonstrating that this framework may be used to characterize electrical double layers over a wide range of concentrations of solutions and surface charges.

Irreversible thermodynamics of curved lipid membranes

The theory of irreversible thermodynamics for arbitrarily curved lipid membranes is presented here. The coupling between elastic bending and irreversible processes such as intramembrane lipid flow, intramembrane phase transitions, and protein binding and diffusion is studied. The forms of the entropy production for the irreversible processes are obtained, and the corresponding thermodynamic forces and fluxes are identified. Employing the linear irreversible thermodynamic framework, the governing equations of motion along with appropriate boundary conditions are provided.

Topological localization in out-of-equilibrium dissipative systems

Significance Topological insulators and their analogs in mechanical materials support conducting states only on their surface. We show that such topologically protected edge modes can also occur as the steady states of classical systems driven out of equilibrium. As proof of principle of the generic applicability of such notions, we show the existence of topologically localized states in a collection of interacting particles described by a hydrodynamic theory and discuss a general procedure to establish them in stochastic networks. In both cases, dissipative processes that break time-reversal symmetry are key to topological protection. Our results provide design principles for robust edge modes in synthetic systems as well as for the localization of flow of matter and information in biology. In this paper, we report that notions of topological protection can be applied to stationary configurations that are driven far from equilibrium by active, dissipative processes. We consider two physically disparate systems: stochastic networks governed by microscopic single-particle dynamics, and collections of driven interacting particles described by coarse-grained hydrodynamic theory. We derive our results by mapping to well-known electronic models and exploiting the resulting correspondence between a bulk topological number and the spectrum of dissipative modes localized at the boundary. For the Markov networks, we report a general procedure to uncover the topological properties in terms of the transition rates. For the active fluid on a substrate, we introduce a topological interpretation of fluid dissipative modes at the edge. In both cases, the presence of dissipative couplings to the environment that break time-reversal symmetry are crucial to ensuring topological protection. These examples constitute proof of principle that notions of topological protection do indeed extend to dissipative processes operating out of equilibrium. Such topologically robust boundary modes have implications for both biological and synthetic systems.