Osman Kahraman

Fluid membrane vesicles in confinement

We numerically study the morphology of fluid membrane vesicles with the prescribed volume and surface area in confinement. For spherical confinement, we observe axisymmetric invaginations that transform into ellipsoidal invaginations when the area of the vesicle is increased, followed by a transition into stomatocyte-like shapes. We provide a detailed analysis of the axisymmetric shapes and investigate the effect of the spontaneous curvature of the membrane as a possible mechanism for shape regulation. We show that the observed morphologies are stable under small geometric deformations of the confinement. The results could help us to understand the role of mechanics in the complex folding patterns of biological membranes.

Morphogenesis of membrane invaginations in spherical confinement

We study the morphology of a fluid membrane in spherical confinement. When the area of the membrane is slightly larger than the area of the outer container, a single axisymmetric invagination is observed. For higher area, self-contact occurs: the invagination breaks symmetry and deforms into an ellipsoid-like shape connected to its outer part via a small slit. For even higher areas, a second invagination forms inside the original invagination. The folding patterns observed could constitute basic building blocks in the morphogenesis of biological tissues and organelles.

Dipoles in thin sheets

A flat elastic sheet may contain pointlike conical singularities that carry a metrical “charge” of Gaussian curvature. Adding such elementary defects to a sheet allows one to make many shapes, in a manner broadly analogous to the familiar multipole construction in electrostatics. However, here the underlying field theory is non-linear, and superposition of intrinsic defects is non-trivial as it must respect the immersion of the resulting surface in three dimensions. We consider a “charge-neutral” dipole composed of two conical singularities of opposite sign. Unlike the relatively simple electrostatic case, here there are two distinct stable minima and an infinity of unstable equilibria. We determine the shapes of the minima and evaluate their energies in the thin-sheet regime where bending dominates over stretching. Our predictions are in surprisingly good agreement with experiments on paper sheets.Graphical abstract

Bilayer-thickness-mediated interactions between integral membrane proteins.

Hydrophobic thickness mismatch between integral membrane proteins and the surrounding lipid bilayer can produce lipid bilayer thickness deformations. Experiment and theory have shown that protein-induced lipid bilayer thickness deformations can yield energetically favorable bilayer-mediated interactions between integral membrane proteins, and large-scale organization of integral membrane proteins into protein clusters in cell membranes. Within the continuum elasticity theory of membranes, the energy cost of protein-induced bilayer thickness deformations can be captured by considering compression and expansion of the bilayer hydrophobic core, membrane tension, and bilayer bending, resulting in biharmonic equilibrium equations describing the shape of lipid bilayers for a given set of bilayer-protein boundary conditions. Here we develop a combined analytic and numerical methodology for the solution of the equilibrium elastic equations associated with protein-induced lipid bilayer deformations. Our methodology allows accurate prediction of thickness-mediated protein interactions for arbitrary protein symmetries at arbitrary protein separations and relative orientations. We provide exact analytic solutions for cylindrical integral membrane proteins with constant and varying hydrophobic thickness, and develop perturbative analytic solutions for noncylindrical protein shapes. We complement these analytic solutions, and assess their accuracy, by developing both finite element and finite difference numerical solution schemes. We provide error estimates of our numerical solution schemes and systematically assess their convergence properties. Taken together, the work presented here puts into place an analytic and numerical framework which allows calculation of bilayer-mediated elastic interactions between integral membrane proteins for the complicated protein shapes suggested by structural biology and at the small protein separations most relevant for the crowded membrane environments provided by living cells.

Confotronic dynamics of tubular filaments

Tubular lattices are ubiquitous in nature and technology. Microtubules and nanotubes of all kinds act as important pillars of biological cells and the man-made nano-world. We show that when prestress is introduced in such structures, localized conformational quasiparticles emerge and govern the collective shape dynamics of the lattice. When coupled via cooperative interactions these quasiparticles form larger-scale quasipolymer superstructures exhibiting collective dynamic modes and giving rise to a hallmark behavior radically different from semiflexible beams.

Architecture and Function of Mechanosensitive Membrane Protein Lattices

Experiments have revealed that membrane proteins can form two-dimensional clusters with regular translational and orientational protein arrangements, which may allow cells to modulate protein function. However, the physical mechanisms yielding supramolecular organization and collective function of membrane proteins remain largely unknown. Here we show that bilayer-mediated elastic interactions between membrane proteins can yield regular and distinctive lattice architectures of protein clusters, and may provide a link between lattice architecture and lattice function. Using the mechanosensitive channel of large conductance (MscL) as a model system, we obtain relations between the shape of MscL and the supramolecular architecture of MscL lattices. We predict that the tetrameric and pentameric MscL symmetries observed in previous structural studies yield distinct lattice architectures of MscL clusters and that, in turn, these distinct MscL lattice architectures yield distinct lattice activation barriers. Our results suggest general physical mechanisms linking protein symmetry, the lattice architecture of membrane protein clusters, and the collective function of membrane protein lattices.

Signatures of protein structure in the cooperative gating of mechanosensitive ion channels

Membrane proteins deform the surrounding lipid bilayer, which can lead to membrane-mediated interactions between neighboring proteins. Using the mechanosensitive channel of large conductance (MscL) as a model system, we demonstrate how the observed differences in protein structure can affect membrane-mediated interactions and cooperativity among membrane proteins. We find that distinct oligomeric states of MscL lead to distinct gateway states for the clustering of MscL, and predict signatures of MscL structure and spatial organization in the cooperative gating of MscL. Our modeling approach establishes a quantitative relation between the observed shapes and the cooperative function of membrane proteins.

Searching for empirical evidence on traffic equilibrium

Cities around the world are inundated by cars and suffer traffic congestion that results in excess delays, reduced safety and environmental pollution. The interplay between road infrastructure and travel choices defines the level and the spatio-temporal extent of congestion. Given the existing infrastructure, understanding how the route choice decisions are made and how travellers interact with each other is a crucial first step in mitigating traffic congestion. This is a problem with fundamental importance, as it has implications for other limited supply systems where agents compete for resources and reach an equilibrium. Here, we observe the route choice decisions and the traffic conditions through an extensive data set of GPS trajectories. We compare the actual paths followed by travellers to those implied by equilibrium conditions (i) at a microscopic scale, where we focus on individual path similarities, and (ii) at a macroscopic scale, where we perform network-level comparison of the traffic loads. We present that non-cooperative or selfish equilibrium replicates the actual traffic (to a certain extent) at the macroscopic scale, while the majority of individual decisions cannot be reproduced by neither selfish nor cooperative equilibrium models.

Stochastic lattice model of synaptic membrane protein domains

Neurotransmitter receptor molecules, concentrated in synaptic membrane domains along with scaffolds and other kinds of proteins, are crucial for signal transmission across chemical synapses. In common with other membrane protein domains, synaptic domains are characterized by low protein copy numbers and protein crowding, with rapid stochastic turnover of individual molecules. We study here in detail a stochastic lattice model of the receptor-scaffold reaction-diffusion dynamics at synaptic domains that was found previously to capture, at the mean-field level, the self-assembly, stability, and characteristic size of synaptic domains observed in experiments. We show that our stochastic lattice model yields quantitative agreement with mean-field models of nonlinear diffusion in crowded membranes. Through a combination of analytic and numerical solutions of the master equation governing the reaction dynamics at synaptic domains, together with kinetic Monte Carlo simulations, we find substantial discrepancies between mean-field and stochastic models for the reaction dynamics at synaptic domains. Based on the reaction and diffusion properties of synaptic receptors and scaffolds suggested by previous experiments and mean-field calculations, we show that the stochastic reaction-diffusion dynamics of synaptic receptors and scaffolds provide a simple physical mechanism for collective fluctuations in synaptic domains, the molecular turnover observed at synaptic domains, key features of the observed single-molecule trajectories, and spatial heterogeneity in the effective rates at which receptors and scaffolds are recycled at the cell membrane. Our work sheds light on the physical mechanisms and principles linking the collective properties of membrane protein domains to the stochastic dynamics that rule their molecular components.

Distribution of randomly diffusing particles in inhomogeneous media

Diffusion can be conceptualized, at microscopic scales, as the random hopping of particles between neighboring lattice sites. In the case of diffusion in inhomogeneous media, distinct spatial domains in the system may yield distinct particle hopping rates. Starting from the master equations (MEs) governing diffusion in inhomogeneous media we derive here, for arbitrary spatial dimensions, the deterministic lattice equations (DLEs) specifying the average particle number at each lattice site for randomly diffusing particles in inhomogeneous media. We consider the case of free (Fickian) diffusion with no steric constraints on the maximum particle number per lattice site as well as the case of diffusion under steric constraints imposing a maximum particle concentration. We find, for both transient and asymptotic regimes, excellent agreement between the DLEs and kinetic Monte Carlo simulations of the MEs. The DLEs provide a computationally efficient method for predicting the (average) distribution of randomly diffusing particles in inhomogeneous media, with the number of DLEs associated with a given system being independent of the number of particles in the system. From the DLEs we obtain general analytic expressions for the steady-state particle distributions for free diffusion and, in special cases, diffusion under steric constraints in inhomogeneous media. We find that, in the steady state of the system, the average fraction of particles in a given domain is independent of most system properties, such as the arrangement and shape of domains, and only depends on the number of lattice sites in each domain, the particle hopping rates, the number of distinct particle species in the system, and the total number of particles of each particle species in the system. Our results provide general insights into the role of spatially inhomogeneous particle hopping rates in setting the particle distributions in inhomogeneous media.