Molecular Shape Solution for Mesoscopic Remodeling of Cellular Membranes

Cellular membranes self-assemble from and interact with various molecular species. Each molecule locally shapes the lipid bilayer, the soft elastic core of cellular membranes. The dynamic architecture of intracellular membrane systems is based on elastic transformations and lateral redistribution of these elementary shapes, driven by chemical and curvature stress gradients. The minimization of the total elastic stress by such redistribution composes the most basic, primordial mechanism of membrane curvature-composition coupling (CCC). Although CCC is generally considered in the context of dynamic compositional heterogeneity of cellular membrane systems, in this article we discuss a broader involvement of CCC in controlling membrane deformations. We focus specifically on the mesoscale membrane transformations in open, reservoir-governed systems, such as membrane budding, tubulation, and the emergence of highly curved sites of membrane fusion and fission. We reveal that the reshuffling of molecular shapes constitutes an independent deformation mode with complex rheological properties.This mode controls effective elasticity of local deformations as well as stationary elastic stress, thus emerging as a major regulator of intracellular membrane remodeling.

Hydrophobic Mismatch Controls the Mode of Membrane-Mediated Interactions of Transmembrane Peptides

Various cellular processes require the concerted cooperative action of proteins. The possibility for such synchronization implies the occurrence of specific long-range interactions between the involved protein participants. Bilayer lipid membranes can mediate protein-protein interactions via relatively long-range elastic deformations induced by the incorporated proteins. We considered the interactions between transmembrane peptides mediated by elastic deformations using the framework of the theory of elasticity of lipid membranes. An effective peptide shape was assumed to be cylindrical, hourglass-like, or barrel-like. The interaction potentials were obtained for membranes of different thicknesses and elastic rigidities. Cylindrically shaped peptides manifest almost neutral average interactions-they attract each other at short distances and repel at large ones, independently of membrane thickness or rigidity. The hourglass-like peptides repel each other in thin bilayers and strongly attract each other in thicker bilayers. On the contrary, the barrel-like peptides repel each other in thick bilayers and attract each other in thinner membranes. These results potentially provide possible mechanisms of control for the mode of protein-protein interactions in membrane domains with different bilayer thicknesses.

Nonlinear material and ionic transport through membrane nanotubes

Membrane nanotubes (NTs) and their networks play an important role in intracellular membrane transport and intercellular communications. The transport characteristics of the NT lumen resemble those of conventional solid-state nanopores. However, unlike the rigid pores, the soft membrane wall of the NT can be deformed by forces driving the transport through the NT lumen. This intrinsic coupling between the NT geometry and transport properties remains poorly explored. Using synchronized fluorescence microscopy and conductance measurements, we revealed that the NT shape was changed by both electric and hydrostatic forces driving the ionic and solute fluxes through the NT lumen. Far from the shape instability, the strength of the force effect is determined by the lateral membrane tension and is scaled with membrane elasticity so that the NT can be operated as a linear elastic sensor. Near shape instabilities, the transport forces triggered large-scale shape transformations, both stochastic and periodic. The periodic oscillations were coupled to a vesicle passage along the NT axis, resembling peristaltic transport. The oscillations were parametrically controlled by the electric field, making NT a highly nonlinear nanofluidic circuitry element with biological and technological implications.

Additional contributions to elastic energy of lipid membranes: Tilt-curvature coupling and curvature gradient

Lipid bilayer membranes under biologically relevant conditions are flexible thin laterally fluid films consisting of two unimolecular layers (monolayers) each about 2 nm thick. On spatial scales much larger than the bilayer thickness, the membrane elasticity is well determined by its shape. The classical Helfrich theory considers the membrane as an elastic two-dimensional (2D) film, which has no particular internal structure. However, various local membrane heterogeneities can result in a lipids tilt relative to the membrane surface normal. On the basis of the classical elasticity theory of 3D bodies, Hamm and Kozlov [Eur. Phys. J. E 3, 323 (2000)10.1007/s101890070003] derived the most general energy functional, taking into account the tilt and lipid monolayer curvature. Recently, Terzi and Deserno [J. Chem. Phys. 147, 084702 (2017)10.1063/1.4990404] showed that Hamm and Kozlov's derivation was incomplete because the tilt-curvature coupling term had been missed. However, the energy functional derived by Terzi and Deserno appeared to be unstable, thereby being invalid for applications that require minimizations of the overall energy of deformations. Here, we derive a stable elastic energy functional, showing that the squared gradient of the curvature was missed in both of these works. This change in the energy functional arises from a more accurate consideration of the transverse shear deformation terms and their influence on the membrane stability. We also consider the influence of the prestress terms on the stability of the energy functional, and we show that it should be considered small and the effective Gaussian curvature should be neglected because of the stability requirements. We further generalize the theory, including the stretching-compressing deformation modes, and we provide the geometrical interpretation of the terms that were previously missed by Hamm and Kozlov. The physical consequences of the new terms are analyzed in the case of a membrane-mediated interaction of two amphipathic peptides located in the same monolayer. We also provide the expression for director fluctuations, comparing it with that obtained by Terzi and Deserno.

Reconstitution and real-time quantification of membrane remodeling by single proteins and protein complexes

Cellular membrane processes, from signal transduction to membrane fusion and fission, depend on acute membrane deformations produced by small and short-lived protein complexes working in conditions far from equilibrium. Real-time monitoring and quantitative assessment of such deformations are challenging; hence, mechanistic analyses of the protein action are commonly based on ensemble averaging, which masks important mechanistic details of the action. In this protocol, we describe how to reconstruct and quantify membrane remodeling by individual proteins and small protein complexes in vitro, using an ultra-short (80- to 400-nm) lipid nanotube (usNT) template. We use the luminal conductance of the usNT as the real-time reporter of the protein interaction(s) with the usNT. We explain how to make and calibrate the usNT template to achieve subnanometer precision in the geometrical assessment of the molecular footprints on the nanotube membrane. We next demonstrate how membrane deformations driven by purified proteins implicated in cellular membrane remodeling can be analyzed at a single-molecule level. The preparation of one usNT takes ~1 h, and the shortest procedure yielding the basic geometrical parameters of a small protein complex takes 10 h.

Continuum Models of Membrane Fusion: Evolution of the Theory

Starting from fertilization, through tissue growth, hormone secretion, synaptic transmission, and sometimes morbid events of carcinogenesis and viral infections, membrane fusion regulates the whole life of high organisms. Despite that, a lot of fusion processes still lack well-established models and even a list of main actors. A merger of membranes requires their topological rearrangements controlled by elastic properties of a lipid bilayer. That is why continuum models based on theories of membrane elasticity are actively applied for the construction of physical models of membrane fusion. Started from the view on the membrane as a structureless film with postulated geometry of fusion intermediates, they developed along with experimental and computational techniques to a powerful tool for prediction of the whole process with molecular accuracy. In the present review, focusing on fusion processes occurring in eukaryotic cells, we scrutinize the history of these models, their evolution and complication, as well as open questions and remaining theoretical problems. We show that modern approaches in this field allow continuum models of membrane fusion to stand shoulder to shoulder with molecular dynamics simulations, and provide the deepest understanding of this process in multiple biological systems.

Pore formation in lipid membrane II: Energy landscape under external stress

Lipid membranes are extremely stable envelopes allowing cells to survive in various environments and to maintain desired internal composition. Membrane permeation through formation of transversal pores requires substantial external stress. Practically, pores are usually formed by application of lateral tension or transmembrane voltage. Using the same approach as was used for obtaining continuous trajectory of pore formation in the stress-less membrane in the previous article, we now consider the process of pore formation under the external stress. The waiting time to pore formation proved a non-monotonous function of the lateral tension, dropping from infinity at zero tension to a minimum at the tension of several millinewtons per meter. Transmembrane voltage, on the contrary, caused the waiting time to decrease monotonously. Analysis of pore formation trajectories for several lipid species with different spontaneous curvatures and elastic moduli under various external conditions provided instrumental insights into the mechanisms underlying some experimentally observed phenomena.

Pore formation in lipid membrane I: Continuous reversible trajectory from intact bilayer through hydrophobic defect to transversal pore

Lipid membranes serve as effective barriers allowing cells to maintain internal composition differing from that of extracellular medium. Membrane permeation, both natural and artificial, can take place via appearance of transversal pores. The rearrangements of lipids leading to pore formation in the intact membrane are not yet understood in details. We applied continuum elasticity theory to obtain continuous trajectory of pore formation and closure, and analyzed molecular dynamics trajectories of pre-formed pore reseal. We hypothesized that a transversal pore is preceded by a hydrophobic defect: intermediate structure spanning through the membrane, the side walls of which are partially aligned by lipid tails. This prediction was confirmed by our molecular dynamics simulations. Conversion of the hydrophobic defect into the hydrophilic pore required surmounting some energy barrier. A metastable state was found for the hydrophilic pore at the radius of a few nanometers. The dependence of the energy on radius was approximately quadratic for hydrophobic defect and small hydrophilic pore, while for large radii it depended on the radius linearly. The pore energy related to its perimeter, line tension, thus depends of the pore radius. Calculated values of the line tension for large pores were in quantitative agreement with available experimental data.

Undulations Drive Domain Registration from the Two Membrane Leaflets

Phase separation in biological membranes plays an important role in protein targeting and transmembrane signaling. Its occurrence in both membrane leaflets commonly gives rise to matching liquid or liquid-ordered domains in the opposing monolayers. The underlying mechanism of such co-localization is not fully understood. The decrease of the line tension around the thicker ordered domain constitutes an important driving force. Yet, robust domain coupling requires an additional energy source, which we have now identified as thermal undulations. Our theoretical analysis of elastic deformations in a lipid bilayer shows that stiffer lipid domains tend to distribute into areas with lower fluctuations of monolayer curvature. These areas naturally align in the opposing monolayers. Thus, coupling requires both membrane leafs to display a heterogeneity in splay rigidities. The heterogeneity may either originate from intrinsic lipid properties or be acquired by adsorption of peripheral molecules. Undulations and line tension act synergistically: the gain in energy due a minimized line tension is proportional to domain radius and thus primarily fuels the registration of smaller domains; whereas the energetic contribution of undulations increases with membrane area and thus primarily acts to coalesce larger domains.

Elastic deformations of bolalipid membranes

Archaeal membranes have unique mechanical properties that enable these organisms to survive under extremely aggressive environmental conditions. The so-called bolalipids contribute to this exceptional stability. They have two polar heads joined by two hydrocarbon chains. The two headgroups can face different sides of the membrane (O-shape conformation) or the same side (U-shape conformation). We have developed an elasticity theory for bolalipid membranes and show that the energetic contributions of (i) tilt deformations, (ii) area compression/stretching deformations, (iii) as well as those of Gaussian splay from the two membrane surfaces are additive, while splay deformations yield a cross-term. The presence of a small fraction of U-shaped molecules resulted in spontaneous membrane curvature. We estimated the tilt modulus to be approximately equal to that of membranes in eukaryotic cells. In contrast to conventional lipids, the bolalipid membrane possesses two splay moduli, one of which is estimated to be an order of magnitude larger than that of conventional lipids. The projected values of elastic moduli act to hamper pore formation and to decelerate membrane fusion and fission.

Behavior of cells in rotating electric fields with account to surface charges and cell structures

The behavior of a single biological cell in a rotating electric field is investigated both theoretically and experimentally. The torque acting on the cell is calculated. The dependence of the torque on electric cell properties (the dielectric constants, the conductivities, and the surface charges of the cell components) and the field frequency is discussed. The dependence of the rotation velocity on the field frequency shows a typical resonance behavior. It is discussed in which manner the single rotation extrema are related to the different homogeneous cell compartments (cytoplasm, cell membrane, and cell wall). It is shown that the cell surface charge shifts the resonance frequency and influences the absolute value of rotation velocity.

Domain formation in membranes caused by lipid wetting of protein

Formation of rafts and other domains in cell membranes is considered as wetting of proteins by lipids. The membrane is modeled as a continuous elastic medium. Thermodynamic functions of the lipid films that wet proteins are calculated using a mean-field theory of liquid crystals as adapted to biomembranes. This approach yields the conditions necessary for a macroscopic wetting film to form; its thickness could also be determined. It is shown that films of macroscopic thicknesses form around large (tens nanometers in diameter) lipid-protein aggregates; only thin adsorption films form around single proteins or small complexes. The means by which wetting films can facilitate the merger of these aggregates is considered. It is shown that a wetting film prevents a protein from leaving an aggregate. Using experimentally derived values of elastic moduli and spontaneous curvatures as well as height mismatch between aggregates and bulk membrane, we obtained numerical results, which can be compared with the experimental data.

Lateral tension increases the line tension between two domains in a lipid bilayer membrane

The effect of an external applied lateral tension on the line tension between two domains of different thickness in a lipid bilayer membrane is calculated. The thick domain is treated as a liquid-ordered phase in order to model a raft in a biological membrane; the thin domain is considered a liquid-disordered phase to model the surrounding region. In our model, the monolayers elastically distort at the boundary to create a smooth rather than steplike boundary to avoid exposure of the hydrophobic interior of the thick raft to water. The energy of this distortion is described by the fundamental deformations of splay and tilt. This energy per unit length of boundary yields the line tension of the raft. Applying lateral tension alters the fundamental deformations such that line tension increases. This increase in line tension is larger when the spontaneous curvature of a raft is greater than that of the surround; if the spontaneous curvature of the raft is less than that of the surround, the increase of the line tension due to application of the lateral tension is more modest.

Line tension and interaction energies of membrane rafts calculated from lipid splay and tilt

Membrane domains known as rafts are rich in cholesterol and sphingolipids, and are thought to be thicker than the surrounding membrane. If so, monolayers should elastically deform so as to avoid exposure of hydrophobic surfaces to water at the raft boundary. We calculated the energy of splay and tilt deformations necessary to avoid such hydrophobic exposure. The derived value of energy per unit length, the line tension gamma, depends on the elastic moduli of the raft and the surrounding membrane; it increases quadratically with the initial difference in thickness between the raft and surround; and it is reduced by differences, either positive or negative, in spontaneous curvature between the two. For zero spontaneous curvature, gamma is approximately 1 pN for a monolayer height mismatch of approximately 0.3 nm, in agreement with experimental measurement. Our model reveals conditions that could prevent rafts from forming, and a mechanism that can cause rafts to remain small. Prevention of raft formation is based on our finding that the calculated line tension is negative if the difference in spontaneous curvature for a raft and the surround is sufficiently large: rafts cannot form if gamma < 0 unless molecular interactions (ignored in the model) are strong enough to make the total line tension positive. Control of size is based on our finding that the height profile from raft to surround does not decrease monotonically, but rather exhibits a damped, oscillatory behavior. As an important consequence, the calculated energy of interaction between rafts also oscillates as it decreases with distance of separation, creating energy barriers between closely apposed rafts. The height of the primary barrier is a complex function of the spontaneous curvatures of the raft and the surround. This barrier can kinetically stabilize the rafts against merger. Our physical theory thus quantifies conditions that allow rafts to form, and further, defines the parameters that control raft merger.

A quantitative model for membrane fusion based on low-energy intermediates

The energetics of a fusion pathway is considered, starting from the contact site where two apposed membranes each locally protrude (as "nipples") toward each other. The equilibrium distance between the tips of the two nipples is determined by a balance of physical forces: repulsion caused by hydration and attraction generated by fusion proteins. The energy to create the initial stalk, caused by bending of cis monolayer leaflets, is much less when the stalk forms between nipples rather than parallel flat membranes. The stalk cannot, however, expand by bending deformations alone, because this would necessitate the creation of a hydrophobic void of prohibitively high energy. But small movements of the lipids out of the plane of their monolayers allow transformation of the stalk into a modified stalk. This intermediate, not previously considered, is a low-energy structure that can reconfigure into a fusion pore via an additional intermediate, the prepore. The lipids of this latter structure are oriented as in a fusion pore, but the bilayer is locally compressed. All membrane rearrangements occur in a discrete local region without creation of an extended hemifusion diaphragm. Importantly, all steps of the proposed pathway are energetically feasible.

Stalk dynamics and lipid flow upon membrane hemifusion

A hydrodynamic theory describing the stalk dynamics and lipid flows upon BLM hemifusion was developed. The value of intermonolayer viscosity, etar, for membranes formed from azolectin mixed with lysophosphatidylcholine (7.10(-4) mg/ml) in n-decane, etar approximately 10(-9) g/s, was determined from comparison of the theoretical calculations and literature data. For membranes formed in squalene, the values etar approximately 10(-7) g/s for phosphatidylethanolamine and etar approximately 2-10(-7) g/s for azolectin were obtained. The calculated values are close to the published results of independent experiments which shows that the developed theory describes well the stalk growth and lipid flow.

Dynamics of fusion pores connecting membranes of different tensions

The energetics underlying the expansion of fusion pores connecting biological or lipid bilayer membranes is elucidated. The energetics necessary to deform membranes as the pore enlarges, in some combination with the action of the fusion proteins, must determine pore growth. The dynamics of pore growth is considered for the case of two homogeneous fusing membranes under different tensions. It is rigorously shown that pore growth can be quantitatively described by treating the pore as a quasiparticle that moves in a medium with a viscosity determined by that of the membranes. Motion is subject to tension, bending, and viscous forces. Pore dynamics and lipid flow through the pore were calculated using Lagrange's equations, with dissipation caused by intra- and intermonolayer friction. These calculations show that the energy barrier that restrains pore enlargement depends only on the sum of the tensions; a difference in tension between the fusing membranes is irrelevant. In contrast, lipid flux through the fusion pore depends on the tension difference but is independent of the sum. Thus pore growth is not affected by tension-driven lipid flux from one membrane to the other. The calculations of the present study explain how increases in tension through osmotic swelling of vesicles cause enlargement of pores between the vesicles and planar bilayer membranes. In a similar fashion, swelling of secretory granules after fusion in biological systems could promote pore enlargement during exocytosis. The calculations also show that pore expansion can be caused by pore lengthening; lengthening may be facilitated by fusion proteins.

Lipid flow through fusion pores connecting membranes of different tensions

When two membranes fuse, their components mix; this is usually described as a purely diffusional process. However, if the membranes are under different tensions, the material will spread predominantly by convection. We use standard fluid mechanics to rigorously calculate the steady-state convective flux of lipids. A fusion pore is modeled as a toroid shape, connecting two planar membranes. Each of the membrane monolayers is considered separately as incompressible viscous media with the same shear viscosity, etas. The two monolayers interact by sliding past each other, described by an intermonolayer viscosity, etar. Combining a continuity equation with an equation that balances the work provided by the tension difference, Deltasigma, against the energy dissipated by flow in the viscous membrane, yields expressions for lipid velocity, upsilon, and area of lipid flux, Phi. These expressions for upsilon and Phi depend on Deltasigma, etas, etar, and geometrical aspects of a toroidal pore, but the general features of the theory hold for any fusion pore that has a roughly hourglass shape. These expressions are readily applicable to data from any experiments that monitor movement of lipid dye between fused membranes under different tensions. Lipid velocity increases nonlinearly from a small value for small pore radii, rp, to a saturating value at large rp. As a result of velocity saturation, the flux increases linearly with pore radius for large pores. The calculated lipid flux is in agreement with available experimental data for both large and transient fusion pores.

Skin appendageal macropores as a possible pathway for electrical current

The electrical properties of the outermost layer of skin are described by lipid-corneocyte (Zm) and appendageal (Za) impedance, which are connected in parallel. Appendageal macropores are considered as long tubes with distributed electrical parameters. It has been shown that not only Za, but also the macropore resistance Ra and capacitance Ca are frequency dependent. The input of Za in the overall impedance (Z) depends on the space density of active (conductive) macropores n(i), which increase with current density (i) and the duration of iontophoresis. Skin impedance has been demonstrated to decrease under the influence of iontophoretic treatment. Application of the theoretical model to these data provides an estimate of the increase in macropore density during iontophoresis. A comparison of these results with n(i), which was measured directly, shows a strong correlation supporting this unique model.

Changes in the electrical properties of the skin outermost layer during pulse electrotreatment

The kinetics of changes in the electric current I(t) passing through human skin samples of full thickness in vitro during rectangular voltage pulses (amplitude, 10-60 V; duration, 5-8 ms) was investigated. The function I(t) was shown to rapidly decrease, to pass through its minimum, and then to increase slowly. With the increase in voltage, the minimal current grew; the dropping branch became less pronounced (up to its complete disappearance at 40 V); and the position of the minimum shifted to short times. All these features of the current response were explained in the assumption that the electrical properties of the skin at a voltage less than 30 V are determined by macropores of skin appendages (hair follicles, sweat glands, etc.). The dropping branch of the current was a superposition of the charging current of the macropore wall capacity and the conductive current through the electroporated walls. At voltages over 30 V, increases in current and conductivity are determined by electroporation of the lipid-corneocyte matrix of the skin outermost layer (stratum corneum). The kinetics of skin resistance restoration after pulse electrotreatment was also investigated. The characteristic time of restoration, which did not exceed 1 min at 10 V, increased up to dozens of minutes at voltages above 30 V. The investigated phenomena were found to be very much similar to the electroporation of plane lipid bilayers and plasma membranes of isolated cells.

Electrical properties of skin at moderate voltages: contribution of appendageal macropores

The electrical properties of human skin in the range of the applied voltages between 0.2 and 60 V are modeled theoretically and measured experimentally. Two parallel electric current pathways are considered: one crossing lipid-corneocyte matrix and the other going through skin appendages. The appendageal ducts are modeled as long tubes with distributed electrical parameters. For both transport systems, equations taking into account the electroporation of lipid lamella in the case the lipid-corneocyte matrix or the walls of the appendageal ducts in the case of the skin appendages are derived. Numerical solutions of these nonlinear equations are compared with published data and the results of our own experiments. The current-time response of the skin during the application of rectangular pulses of different voltage amplitudes show a profound similarity with the same characteristics in model and plasma membrane electroporation. A comparison of the theory and the experiment shows that a significant (up to three orders of magnitude) drop of skin resistance due to electrotreatment can be explained by electroporation of different substructures of stratum corneum. At relatively low voltages (U < 30 V) this drop of skin resistance can be attributed to electroporation of the appendageal ducts. At higher voltages (U > 30 V), electroporation of the lipid-corneocyte matrix leads to an additional drop of skin resistance. These theoretical findings are in a good agreement with the experimental results and literature data.

A theoretical study of outermost skin layer electroporation

Two models of electrohydration of stratum corneum (SC) have been developed. In the first model, the hydration of one interbilayer region is considered on the assumption that water molecules are adsorbed on the inhomogeneous surface of a bilayer and can interact, thereby lowering their energy on the surface. The dependence of the hydration degree on the voltage across the skin had been found. At certain parameter values the degree of hydration rapidly grows at certain voltage up to the magnitudes at which continuous water pathways appear. The second model has used the macroscopic approach which presumes water to be present in the interbilayer region as microdrops. The dependence of the hydration degree on voltage has been also found. At voltages of the order of tens of volts the obtained hydrations of interbilayer regions are sufficient to generate electroinduced hydrophilic pores in the SC lipid phase. Formation of tortuous continuous pathways for the transport of small ions is little probable because it requires voltages much higher than 100 V. We suggest that small ions pass the skin by the straight way through corneocytes and lipid bilayers at voltages of the order of tens of volts and higher.

A mechanism of skin appendage macropores electroactivation during iontophoresis

A physical mechanism of activation of skin appendage macropores under the influence of an electric field is considered theoretically. The macropore is considered as a long cylinder tube which is closed and flattened out before the electric field is applied. The charging of the capacitance of the macropore walls is the driving force of electroactivation. During this process the free energy of the system decreases, which is energetically favourable and results in water pulled into the tube and to a gradual opening of the tube. It is shown that consideration of the macropore wall conductance leads to a considerable slowdown of electroactivation. The opening time of a separate macropore is estimated. It is equal to 30 min for a macropore 4 mm in length. The dependence of the surface density of activated macropores on time is calculated theoretically. The obtained theoretical results are in a good agreement with the literature data.

Formation of cell protrusions by an electric field: a thermodynamic analysis

This work gives a thermodynamic analysis of outgrowth extraction from the cell body by a pulling force. The results are applied for a case when the pulling force is generated by an external high-frequency electric field. Two equilibrium conditions are analyzed: internal equilibrium of an outgrowth and equilibrium between the outgrowth and the cell body. In both cases the stability of feasible equilibrium states was studied. The work shows that the curvature of an outgrowth equilibrated with a pulling electric force depends on the squared amplitude of the electric field E0(2), on the outgrowth length l and on the transmembrane pressure differential delta P, and that at a sufficiently large transmembrane pressure differential the cylindrical form of the outgrowth loses its stability. Long outgrowths are more stable than short ones. The minimal value of critical pressure differential was estimated. The work also shows that outgrowth extraction from the cell body requires that the applied force exceeds a critical value below which no outgrowth is formed. The value of the electric field at which outgrowth formation is feasible was estimated.