Bacterial mechanics: FtsZ force generation & Cell wall growth: 
The bacterial cytoskeletal protein FtsZ is a GTPase that is thought to provide mechanical constriction force via an unidentified mechanism. Purified FtsZ polymerizes into filaments with varying structures in vitro: while GTP-bound FtsZ assembles into straight or gently curved filaments, GDP-bound FtsZ forms highly curved filaments, prompting the hypothesis that a difference in the inherent curvature of FtsZ filaments provides mechanical force. However, no nucleotide-dependent structural transition of FtsZ monomers has been observed to support this force generation model. Here, we present a series of all-atom molecular dynamics simulations probing the effects of nucleotide binding on the structure of an FtsZ dimer. We found that the FtsZ-dimer structure is dependent on nucleotide-binding state. While a GTP-bound FtsZ dimer retained a firm monomer-monomer contact, a GDP-bound FtsZ dimer lost some of the monomer-monomer association, leading to a “hinge-opening” event that resulted in a more bent dimer, while leaving each monomer structure largely unaffected. We constructed models of FtsZ filaments and found that a GDP-FtsZ filament is much more curved than a GTP-FtsZ filament, with the degree of curvature matching prior experimental data. FtsZ dynamic were used to estimate the amount of force an FtsZ filament could exert when hydrolysis occurs (20–30 pN per monomer). This magnitude of force is sufficient to direct inward cell-wall growth during division, and to produce the observed degree of membrane pinching in liposomes. Taken together, our data provide molecularscale insight on the origin of FtsZ-based constriction force, and the mechanism underlying prokaryotic cell division.
"Nucleotide-dependent conformations of FtsZ dimers and force generation observed through molecular dynamics simulations" , Jen Hsin, Ajay Gopinathan and K.C. Huang, Proc. Natl. Acad. Sci. USA, 109 (24): 9432 (2012)
"Mechanical Consequences of Cell-Wall Turnover in the Elongation of a Gram-Positive Bacterium", G Misra, ER Rojas, A Gopinathan, KC Huang, Biophysical journal 104 (11), 2342-2352 (2013)


Molecular motor traffic:

Molecular motors convert chemical energy into mechanical motion and power the transport of material within living cells; the motion of a motor is thought to be influenced by stochastic chemical state transitions of the molecule as well as intramolecular diffusion of one motor head seeking the next binding site. Existing models for the motility of single-headed monomeric motors that map the system to a simplified two-state Brownian ratchet have some predictive power, but in general are unable to elucidate the contributions of different molecular level processes to the overall effective parameters. In this work, we build a detailed molecular-level model of monomeric kinesin motility that naturally incorporates conformational changes (power strokes) and biased diffusion. Our results predict that mean velocity is most sensitive to the power-stroke size, while run-length distribution is sensitive primarily to the strength of the microtubule bias potential with a weak dependence on power stroke that can be tuned by the strength of an applied load. In addition, we demonstrate that motor pairs attached to the same cargo can cooperatively function to increase motility in both the plus- and minus-end directions. These findings illustrate the importance of a detailed mechanochemical model for dissecting the contributions of microscopic parameters to monomeric kinesin dynamics.

"Conformational changes, diffusion and collective behavior in monomeric kinesin based motility" K.C. Huang, C. Vega and Ajay Gopinathan,  Journal of Physics: Condensed Matter (2011) accepted


Vicious Levy Flights:
Diffusive processes with long range jumps such as Levy Flights play an important role in many physical, chemical and biological phenomena. Examples include epidemic spreading, transcription factor proteins binding to  DNA, kinetic Ising models with long-range interactions, foraging animals  and light propagation in disordered optical materials. In many cases, the statistics of encounters between entities performing Levy flights is of significant interest. These encounters can, in fact, affect the outcome of these processes. These include not only traditional condensed matter/physical systems such as chemical reactions in turbulent environments or electron hole recombination in disordered materials but also biological/ecological processes including encounters between marine predators and prey as well as foraging animals such as albatrosses and spider monkeys. In this paper we study the statistics of such encounters by introducing a novel concept - Vicious Levy Flights (VLF) - independent groups of Levy flights where the the process is stopped upon the first encounter between members of different groups. We use renormalization group analyses to compute the late time behavior of the survival probability of this system, which directly translates into understanding the time before the first encounter.
"Vicious Levy Flights", Igor Goncharenko and Ajay Gopinathan, Phys. Rev. Lett. 105, 190601 (2010)
"Vicious Walks with Long-range Interactions", Igor Goncharenko and Ajay Gopinathan,  Phys. Rev. E 82, 011126 (2010)
Also see: Physical review Focus story on this.


Polymer brush architecture of the nuclear pore complex:
One of the major open questions is : what is the supramolecular organization of the nuclear pore complex proteins and how do they facilitate selestive gating. We are addressing this question using a combination of polymer physics theory and coarse grained MD simulations...

“A Bimodal Distribution of Two Distinct Categories of Natively Unfolded Structures with Separate Functions in FG Nucleoporins”, Justin Yamada; Joshua L Phillips; Samir Patel; Gabriel Goldfien; Alison Calestagne-Morelli; Hans Huang; Ryan Reza; Justin Acheson; Viswanathan Krishnan; Shawn Newsam; Ajay Gopinathan; Edmond Y Lau; Michael Colvin; Vladimir N Uversky; Michael F Rexach, Molecular and Cellular Proteomics (2010) doi:10.1074/mcp.M000035-MCP201
 Also see: A preview commentary on the article in Cell: Structure
"Physical Motif Clustering within Intrinsically Disordered Nucleoporin Sequences Reveals Universal Functional Features", D Ando, M Colvin, M Rexach, A Gopinathan, PloS one 8 (9), e73831 (2013)




 Optimizing nanocarriers for targeted drug delivery: One of the main problems in cancer therapeutics is to be able to target cancer cells specifically with minima damage to normal tissue. We examine how a particular class of nano-carriers with multiple ligand bearing arms can be designed to optimize this selectivity.
"Design of High-Specificity Nanocarriers by Exploiting Non-Equilibrium Effects in Cancer Cell Targeting", K Tsekouras, I Goncharenko, ME Colvin, KC Huang, A Gopinathan, PloS one 8 (6) (2013)


 Novel forms of Chemotaxis:  Chemotaxis usually involves cellular motion towards an exogenous, often static, source of chemoattractant. We demonstrate an enhanced mechanism in which a single particle (the cell) releases a chemical that diffuses to fixed particles (targets) and signals the production of another chemical by the targets. This secondary chemical acts as the cell’s chemoattractant. To reach a single target, we describe optimal strategies that arise for the release of a fixed amount of probe chemical. In the presence of multiple targets, the one selected by the cell depends on the strength and, more interestingly, the frequency of probe chemical release. Although it involves one more step than standard chemotaxis, our simple chemical “pinging” mechanism can offer far greater flexibility in regulating target selection seen in a number of physical or biological realizations.
"Frequency-dependent Chemotactic Target Selection" Sarah A. Nowak, Buddhapriya Chakrabarti, Tom Chou, and Ajay Gopinathan, Physical Biology, 7, 026003 (2010)

 In-vitro Filopodia: Filopodia are bundles of actin filaments that extend out ahead of the leading edge of a crawling cell to probe its upcoming environment. In vitro experiments [D. Vignjevic et al., J. Cell Biol.  160, 951 (2003)] have determined the minimal ingredients required for the formation of filopodia from the dendritic-like morphology of the leading edge. We model these experiments using kinetic aggregation equations for the density of growing bundle tips. In mean field in the absence of bending effects, we determine the bundle size distribution to be broad for bundle sizes smaller than a characteristic bundle size after which the distribution decays exponentially. Two-dimensional simulations incoporating both bundling and cross-linking measure a bundle size distribution that agrees qualitatively with mean field. The presence of bending effects yields a bundle size distribution that is peaked at a particular bundle size. Finally, two-dimensional simulations demonstrate a nonmonotonicity in the radial extent of the dendritic region as a function of capping protein concentration due to the interplay between percolation and the ratcheting of growing filaments off a spherical obstacle.
 "Modeling the formation of in vitro filopodia", K.-C. Lee, Ajay Gopinathan, J. M. Schwarz, (2009), Journal of Mathematical Biology 63(2), 229 (2011)
Optimizing Yield Rates : The problem of optimizing the yield rate in reactions that are catalyzed by enzymes is of fundamental importance. In the case of protein folding catalyzed by chaperonins, there is a distinct possibility of the formation of misfolded proteins within the chaperonin nano-cage which can be considered as an undesirable intermediate state of the reaction. Chaperonins are known to nonspecifically release their contents in an ATP consuming reaction which suggests a mechanism for releasing undesirable intermediates. However, this also results in the release of proteins that are on the correct folding pathway. Balancing these two conflicting outcomes suggests the existence of an optimal release rate. This paper shows the existence of maxima in the yield rate at specific optimal values of the release rate of the undesirable intermediate state. We discuss the conditions for the existence of such maxima and their dependence on the other parameters of the system.
“Optimal Kinetics for Chaperonin Assisted Protein Folding”, Igor Goncharenko and Ajay Gopinathan,IAENG Transactions on Engineering Technologies 2, p13 (2009)
“Optimal Yield Rates in Enzymatic Reactions with Undesirable Intermediate States”, Igor Goncharenko and Ajay Gopinathan, Proceedings of the International Conference on Computational Biology (WCECS 2008), p24 (2008) 
 Diffusion and binding of finite-size particles in confined geometries: Describing the diffusion of particles through crowded, confined environments with which they can interact is of considerable biological and technological interest. Under conditions where the confinement dimensions become comparable to the particle dimensions, steric interactions between particles, as well as particle-wall interactions, will play a crucial role in determining transport properties. To elucidate the effects of these interactions on particle transport, we consider the diffusion and binding of finite-size particles within a channel whose diameter is comparable to the size of the particles. Using a simple lattice model of this process, we calculate the steady-state current and density profiles of both bound and free particles in the channel. We show that the system can exhibit qualitatively different behavior depending on the ratio of the channel width to the particle size. We also perform simulations of this system, and find excellent agreement with our analytic results.
“Diffusion and binding of finite-size particles in confined geometries”, Mark L. Henle, Brian DiDonna, Christian D. Santangelo, and Ajay Gopinathan, Phys. Rev. E 78, 031118 (2008)


Polymer Translocation in Crowded Environments : Polymer translocation is an extensively studied topic and is biologically an important process that occurs in a variety of circumstances where biopolymers (like DNA say from a virus) are transported across a membrane into a different environment (say the cell interior). An important question that arises is : How does the crowded nature of the cellular cytoplasm affect this process? While this has been addressed in the context of protein folding and biochemical rates in vivo, one would expect this to have a dramatic effect on translocation. We systematically treat the entropic penalty due to the crowded environment and find new power law scalings of the translocation time with polymer length. We also find that the crowding inflicts a significant barrier and that adding a chemical potential gradient in order to overcome this results in very interesting translocation regimes as a function of crowding, chemical potential and polymer length.
“Polymer Translocation in Crowded Environments”, Ajay Gopinathan and Y.W. Kim, Phys. Rev. Lett., 99, 228106 (2007)


Actin based propulsion :  A key process in motility is the production of a protrusive force that drives the cell’s leading edge forward. It has been well established that the cytoskeletal protein actin forms a dense, branched and crosslinked network of filaments at the leading edge and it is the polymerization of these filaments that gives rise to the force. The discovery that intracellular pathogens like listeria use the cell’s actin machinery to propel themselves played a vital role in establishing the biochemical basis of motility.  However, there still remains a hotly debated question: How does the polymerization activity at the molecular scale translate into a macroscopic force?  This led to the development of both the microscopic view that considers the protrusive force as arising from the monomer-by-monomer growth of a population of these filaments  and macroscopic models that treat the filaments as a continuum gel. Both of these describe the phenomenon albeit at very different length and time scales and are unable to satisfactorily account for all aspects of the motility (see for example the work by Kuo Lab. To get a correct and unified picture, we introduced a “dynamic gel” picture where we treated the actin network as an elastic gel modeled by a finite element mesh while allowing for spatial variations in polymerization activity .We treat the actin comet tail as an elastic continuum tethered to the rear of the bacterium. The interplay of polymerization and tethering gives rise to inhomogeneous stresses calculated with a finite element analysis. We quantitatively reproduce many distinctive features of actin propulsion that have been observed experimentally, including stepped motion, hopping, tail shape and the propulsion of flat surfaces. 
 
Ajay Gopinathan and Andrea Liu, "Elastic Actin Tails: Shape, Stresses and Propulsion" in preparation


Cell Membrane Dynamics :
During cell motility, the leading edge of the cell exhibits a range of dynamic structures such as  lamellipodia (flat disk-like extensions), filopodia (thin needle like protrusions) and membrane ruffles (wave-like instability). These structures are like the “feet” of the cell and are put out in an effort to feel and make attachments to the surroundings. They are hence vital to the process of motility. These dynamic surface patterns of moving cells are usually observed to have length-scales in the micron range, and appear in many different cell types.  They occur usually when the cell is stimulated by an external signal, indicating that the receptors on the cell membrane have a role to play. The receptors actually turn on in response to the signal and in turn initiate actin polymerization activity in their vicinity, which leads to force production. Our recent work showed that the dynamic membrane structures described above can arise from a simple analytic model which couples the membrane dynamics to the dynamics of receptors that reside on the membrane and the actin based protrusive forces they generate. Our model predicted membrane ruffling and the onset of filopodia and also gave explanations for several experimentally observed puzzling features such as the increased microviscosity at the leading edges of motile cells and the temperature dependence of the ruffle velocities.

“Dynamics of Membranes Driven by Actin Polymerization”, Nir Gov, and Ajay Gopinathan, Biophys .J , 90(2), 454 (2006)


Cytoskeletal Kinetics : Controlling the polymerization activity of cytoskeletal actin network plays an important role in cell motility. Even in the absence of motility, the actin network is not static but evolves via kinetic processes such as actin polymerization, depolymerization, capping, branching and severing which are regulated by various proteins in the cell . 
Abnormal levels of expression of these regulatory proteins lead to diseased states characterized by drastic morphological changes in the cytoskeleton and loss of function. It is therefore imperative to understand how the regulatory protein concentrations act in concert to maintain a normal cytoskeletal morphology. Previous work treated one or more but not all of the above mentioned processes. We recently studied the steady-state morphology of such networks and derived simple expressions for characteristics such as the length distribution of filaments and branches, branch spacing, and monomer to filamentous actin ratio as functions of regulatory protein concentrations. We found that these characteristics exhibit several scaling regimes with respect to the different protein concentrations and that the severing and branching activities are optimally coupled in the cell. 
 
“Branching, Capping, and Severing in Dynamic Actin Structures”, Ajay Gopinathan, J. Schwarz, K.C. Lee and A.J. Liu, Phys. Rev. Lett. 99, 058103 (2007)



 The Non-equilibrium Behavior of Fluctuation Induced Forces:  While techniques to compute thermal fluctuation induced, or pseudo-Casimir, forces in equilibrium systems are well established, the same is not true for non-equilibrium cases. We present a general formalism that allows us to unambiguously compute non-equilibrium fluctuation induced forces by specifying the energy of interaction of the fluctuating fields with the boundaries. For a general class of classical fields with dissipative dynamics, we derive a very general relation between the Laplace transform of the time-dependent force and the static partition function for a related problem with a different Hamiltonian. In particular, we demonstrate the power of our approach by computing, for the first time, the explicit time dependence of the non-equilibrium pseudo-Casimir force induced between two parallel plates, upon a sudden change in the temperature of the system. We also show how our results can be used to determine the steady-state behavior of the non-equilibrium force in systems where the fluctuations are driven by colored noise.
"Out-of-equilibrium behavior of Casimir-type fluctuation induced forces for free classical fields", David S. Dean and Ajay Gopinathan, Phys. Rev. E 81, 041126 (2010)
"The Non-equilibrium Behavior of Pseudo-Casimir Forces", David S. Dean and Ajay Gopinathan, J. Stat. Mech. L08001 (2009)


 Non-trivial geometries in shrinking plastic sheets: Biaxially oriented polystyrene thermoplastic sheets (shrinky dinks) have been recently used by one of us (Khine, Lab on a Chip, 2008) as a template for rapid and non-photolithographic microfluidic pattern generation. This method utilizes the shrinkage properties of the shrinky dinks upon heating to generate microscale structures. During the heating process the sheets show a variety of non-trivial three dimensional intermediate structures before returning to a shrunken flat state upon completion of the process. We show that these structures arise due to the imposition of a non-uniform spatial metric on the sheet which in turn is governed by the dynamic temperature gradients generated in the sheet. Our results allow us to quantitatively describe the dynamic sequence of structures generated and suggest routes to the design and fabrication of different structures in a controllable fashion.
Anthony Grimes, Michelle Khine, Arnold Kim and Ajay Gopinathan (in preparation)


 Controllable biaxial and uniaxial nanowrinkles (see figure) can also be fabricated by a simple two-step approach  using the shrinking plastic substrate -  metal deposition and subsequent heating. The wavelengths of the wrinkles can be tuned by controlling the thickness of deposited metal. Ready integration of the nanowrinkles into microchannels and their effectiveness in surface enhanced sensing indicates a host of potential applications.
Bimetallic nanopetals for thousand-fold fluorescence enhancements", Chi-Cheng Fu, Giulia Ossato, Maureen Long, Michelle A. Digman, Ajay Gopinathan, Luke P. Lee, Enrico Gratton, and Michelle Khine, Appl. Phys. Lett. 97, 203101 (2010); doi:10.1063/1.3495773 
"Tunable Nanowrinkles on Shape Memory Polymer Sheets", Chi-Cheng Fu, Anthony Grimes, Maureen Long, Christopher G. L. Ferri, Brent D. Rich, Somnath Ghosh, Sayantani Ghosh, Luke P. Lee, Ajay Gopinathan, Michelle Khine, Advanced Materials, 21(44) 4472 (2009)


 Charge Transport in Fractal Networks:  We examine how the fractal nature of the interpenetrating networks in bulk heterojunction materials affects the transport of charges in such systems. Such systems can be potentially used in next generation solar cells and understanding in such systems is of critical importance and point to directions for the improvement of such materials.
“Nanostructure of the Interpenetrating Networks in Poly(3-hexylthiophene)/fullerene Bulk Heterojunction Materials: Implications for Charge Transport”, Wanli Ma, Ajay Gopinathan and Alan. J. Heeger, Advanced Materials, 19(21), 3656 (2007)




Statistically locked-in transport : Measurements of colloidal transport through arrays of micrometer-scale potential wells created with holographic optical tweezers were performed at the Grier Lab . Varying the orientation of the trap array relative to the external driving force resulted in a hierarchy of lock-in transitions analogous to symmetry-selecting processes in a wide variety of systems with implications for immediate applications for continuously fractionating particles, biological cells, and macromolecules. Classical particles driven through periodically modulated potential energy landscapes are predicted to follow a Devil's staircase hierarchy of commensurate trajectories depending on the orientation of the driving force.The experiments did indeed reveal such a hierarchy, but not with the predicted structure. The microscopic trajectories, moreover, appeared to be random, with commensurability emerging only in a statistical sense. We introduced an idealized model for periodically modulated transport in the presence of randomness that captures both the structure and statistics of such statistically locked-in trajectories.

“Statistically Locked-in Transport through Periodic Potential Arrays”, Ajay Gopinathan and D.G. Grier,Phys. Rev. Lett., 92, 130602 (2004)


Self Assembly of Nanowires : Experiments at the Jaeger lab have shown that when metals are evaporated and deposited on a templated substrate, like a phase separated diblock copolymer surface, certain metals show a marked preference for one phase over the other and in certain cases form continuous wires of nanometer scale. What surprised us was the stability of these wires which by surface energy considerations should exhibit the pearling instability (a liquid cylinder breaking up into drops ). We proposed an explanation based on the rate limiting step of nanocluster coalescence being nucleation of new terraces. We show that the different morphologies obtained can be understood in terms of the relative importance of the energetics and kinetics. We also show the existence of ``non-trivial'' correlations  between adjacent wires that can be understood based on a purely kinetic mechanism. We also compare these correlations quantitatively to those obtained from simulations done with the relevant experimental parameters and find them in good agreement.

“Kinetic Self-Assembly of Metals on Co-Polymer Templates”, Ajay Gopinathan, Phys. Rev. E, 71(4) 041601 (2005)




 Non-equilibrium kinetics :  Deliberately miscutting a crystal surface can  produce a regular array of monoatomic steps. Under suitable conditions (temperature, oxygen dosage) these steps can be made to merge to form double height double width steps. Experiments performed at the Sibener lab have been instrumental in elucidating the mechanism of this process. It is found that step doubling proceeds via a nucleation step where two adjacent step edges come together at a "point" and then the two steps "zipper" together irreversibly. Theoretical effort has gone into describing the mechanism for a pair of steps. However when there is a large array of steps, as in reality, the dynamical process of nucleation and zippering gives rise to a non-equilibrium evolution of the surface morphology. One also expects defect structures of various types.  We study the time evolution of the surface morphology by making an approximate mapping to the parking lot problem. This allows us to predict the number and nature of the defects as well as the time evolution. We also suggest protocols that can help generate surfaces with fewer defects in less time.

“Defect Formation and Kinetics of Atomic Terrace Merging”, Ajay Gopinathan and T.A. Witten, Phys. Rev. E., 70, 041603 (2004)
Crumpling - Dynamics : A crumpled sheet has certain characteristic features that we are all familiar with. There are the sharply curved places - ridges and the almost flat places- the facets. Extensive work characterizing the static properties of the ridges and facets has been done by Witten and coworkers. But we knew of no characterization of the dynamics. We were interested in the question : what happens if we tap a certain point on a crumpled sheet and listen at another? What changes in elastic wave propagation arise due to the unique structures in a crumpled sheet ? To answer this we first derived the wave equation governing transverse elastic waves on an arbitratily curved and strained surface using a Lagrangian formalism. Our analysis led us to the conclusion that the ridges act as barriers leading to the trapping of certain modes within the facets!!

“Trapping of Vibrational Energy in Crumpled Sheets”, Ajay Gopinathan, T.A. Witten and S.C. Venkataramani, Phys. Rev. E., 65, 036613 (2002)


Charged Colloidal Systems : There is now a growing body of evidence that like charge colloidal spheres dispersed in water need not simply repel each other. Under certain conditions they actually attract. Experiments like those performed at the Grier Lab are helping us gain insights into this phenomenon.In this work we investigated the influence of geometric confinement on the free energy of an idealized model for charge-stabilized colloidal suspensions. The mean-field Poisson-Boltzmann formulation for this system predicts pure repulsion among macroionic colloidal spheres. However fluctuations in the simple ions distribution provide a mechanism for the macroions to attract each other at large separations.Although this Casimir interaction long-ranged,we found it was too weak to influence colloidal crystals dynamics.
“Weak Long Range Casimir Attraction in Colloidal Crystals”, Ajay Gopinathan, T. Zhou, S.N. Coppersmith, L.P. Kadanoff and D.G. Grier, Europhys. Lett., 57 (3), 451 (2002) 


Phase Ordering Kinetics :  Consider a collection of spins at a high temperature that is suddenly quenched to zero temperature. What follows is domain coarsening where domains of up and down spins grow and grow. An interesting question is : given a spin what is the probability that after time t it still retains its original state without ever having flipped? An exponent characterizing how this probability scales with the typical length scale in the system is called the persistence exponent  beta. We found an exact expression for this quantity for 1D q-state Potts' system (spin with q states) with a suitably chosen model of coarsening.

“Scaling Exponent β for Coarsening in a 1D q-state Potts System”, Ajay Gopinathan, J. Phys. A. 31, 5499 (1998)