Mechanical Regulation of Neurite Polarization and Growth: A Computational Study.

The densely packed microtubule (MT) array found in neuronal cell projections (neurites) serves two fundamental functions simultaneously: it provides a mechanically stable track for molecular motor-based transport and produces forces that drive neurite growth. The local pattern of MT polarity along the neurite shaft has been found to differ between axons and dendrites. In axons, the neurons' dominating long projections, roughly 90% of the MTs orient with their rapidly growing plus end away from the cell body, whereas in vertebrate dendrites, their orientations are locally mixed. Molecular motors are known to be responsible for cytoskeletal ordering and force generation, but their collective function in the dense MT cytoskeleton of neurites remains elusive. We here hypothesized that both the polarity pattern of MTs along the neurite shaft and the shaft's global extension are simultaneously driven by molecular motor forces and should thus be regulated by the mechanical load acting on the MT array as a whole. To investigate this, we simulated cylindrical bundles of MTs that are cross-linked and powered by molecular motors by iteratively solving a set of force-balance equations. The bundles were subjected to a fixed load arising from actively generated tension in the actomyosin cortex enveloping the MTs. The magnitude of the load and the level of motor-induced connectivity between the MTs have been varied systematically. With an increasing load and decreasing motor-induced connectivity between MTs, the bundles became wider in cross section and extended more slowly, and the local MT orientational order was reduced. These results reveal two, to our knowledge, novel mechanical factors that may underlie the distinctive development of the MT cytoskeleton in axons and dendrites: the cross-linking level of MTs by motors and the load acting on this cytoskeleton during growth.

Theoretical Analysis of Stress Distribution and Cell Polarization Surrounding a Model Wound.

A growing amount of experimental evidence shows that the local elastic field acting on cells governs their spatial organization and polarity in a tissue. Interestingly, experiments on wound healing reveal a universal formation of thick actomyosin bundles around the margins of epithelial gaps. Although the forces involved in this process have been measured, the mechanisms governing cellular alignment and contractile ring formation are still not fully understood. To theoretically investigate this process, we have carried out a self-consistent calculation of the elastic field that is actively generated around a circular gap in a contractile cell monolayer that is adhered to an elastic substrate, taking into account the responsiveness of actomyosin activity to the locally generated stress. We model actomyosin contractility by a radial distribution of point force dipoles that may alter in magnitude and orientation in response to the local elastic stress. In addition, the model takes into account the forces exerted by leader cells on the margins of the cell monolayer. Our model suggests that the presence of a hole in the center of a contractile cell monolayer creates a mechanical tendency for actomyosin forces to polarize tangentially around the hole margin. In addition, it predicts that this tendency optimizes with substrate rigidity, thickness, and strength of cell adhesion to the substrate. Our calculations support the view that the universal formation of a peripheral contractile ring is a consequence of actomyosin contractility in the bulk and its inherent responsiveness to the local stress.

Dynamics of force generation by spreading platelets.

In order to gain more insight into the role of human platelets for blood clot formation, here we investigate the dynamics of force generation by platelet spreading onto elastic substrates of variable stiffness. Despite their small size, platelets generate high and rapidly varying traction forces on their extracellular environment, which we reconstruct with adapted implementations of Fourier transform traction cytometry. We find that while the final spread area is reached within a few minutes, the build-up of forces typically takes 10-30 minutes. In addition, we identify two distinct behaviors of individual cells, namely oscillating and non-oscillating platelets. An eigenvalue analysis of the platelet dipole tensor reveals a small anisotropy of the exerted force, which is compatible with a random distribution of a few force transmitting centers, in agreement with the observed shapes and traction patterns. We find a correlation between the maximum force level a platelet reaches and its spread area, which we explain by a thin film model for the actively contracting cell. The model reveals a large internal stress of hundreds of kPa. Experimentally we do not find any statistically relevant relation between the force level reached and the substrate stiffness within the stiffness range from 19 to 83 kPa, which might be related to the high platelet activation level used in our study. In addition, our model suggests that due to the uniquely small thickness of platelets, their mechanosensitivity might be limited to a lower stiffness range.

Contribution of myosin II activity to cell spreading dynamics.

Myosin II activity and actin polymerization at the leading edge of the cell are known to be essential sources of cellular stress. However, a quantitative account of their separate contributions is still lacking; so is the influence of the coupling between the two phenomena on cell spreading dynamics. We present a simple analytic elastic theory of cell spreading dynamics that quantitatively demonstrates how actin polymerization and myosin activity cooperate in the generation of cellular stress during spreading. Consistent with experiments, myosin activity is assumed to polarize in response to the stresses generated during spreading. The characteristic response time and the overall spreading time are predicted to determine different evolution profiles of cell spreading dynamics. These include, a (regular) monotonic increase of cell projected area with time, a non-monotonic (overshooting) profile with a maximum, and damped oscillatory modes. In addition, two populations of myosin II motors are distinguished based on their location in the lamella; those located above the major adhesion zone at the cell periphery are shown to facilitate spreading whereas those in deeper regions of the lamella are shown to oppose spreading. We demonstrate that the attenuation of myosin activity in the two regions may result in reciprocal effects on spreading. These findings provide important new insight into the function of myosin II motors in the course of spreading.

Active mechanical coupling between the nucleus, cytoskeleton and the extracellular matrix, and the implications for perinuclear actomyosin organization.

Experimental and theoretical studies have demonstrated that the polarization of actomyosin forces in the cytoskeleton of adherent cells is governed by local elastic stresses. Based on this phenomenon, and the established observation that the nucleus is mechanically connected to the extracellular matrix (ECM) via the cytoskeleton, we theoretically analyze here the active mechanical coupling between the nucleus, cytoskeleton and the ECM. The cell is modeled as an active spherical inclusion, containing a round nucleus at its center, and embedded in a 3D elastic matrix. We investigate three sources of cellular stress: spreading-induced stress, actomyosin contractility and chromatin entropic forces. Formulating the coupling of actomyosin contractility to the local stress we predict the consequences that the nucleus, cytoskeleton and ECM mechanical properties may have on the overall force-balance in the cell and the perinuclear acto-myosin polarization. We demonstrate that the presence of the nucleus induces symmetry breaking of the elastic stress that, we predict, elastically tends to orient actomyosin alignment tangentially around the nucleus; the softer the nucleus or the matrix, the stronger is the preference for tangential alignment. Spreading induced stresses may induce radial actomyosin alignment near stiff nuclei. In addition, we show that in regions of high actomyosin density myosin motors have an elastic tendency to orient tangentially as often occurs near the cell periphery. These conclusions highlight the role of the nucleus in the regulation of cytoskeleton organization and may provide new insight into the mechanics of stem cell differentiation involving few fold increase in nucleus stiffness.

Dynamics of cell area and force during spreading.

Experiments on human pulmonary artery endothelial cells are presented to show that cell area and the force exerted on a substrate increase simultaneously, but with different rates during spreading; rapid-force increase systematically occurred several minutes past initial spreading. We examine this theoretically and present three complementary mechanisms that may accompany the development of lamellar stress during spreading and underlie the observed behavior. These include: 1), the dynamics of cytoskeleton assembly at the cell basis; 2), the strengthening of acto-myosin forces in response to the generated lamellar stresses; and 3), the passive strain-stiffening of the cytoskeleton.

Early-time dynamics of actomyosin polarization in cells of confined shape in elastic matrices.

The cell shape and the rigidity of the extracellular matrix have been shown to play an important role in the regulation of cytoskeleton structure and force generation. Elastic stresses that develop by actomyosin contraction feedback on myosin activity and govern the anisotropic polarization of stress fibers in the cell. We theoretically study the consequences that the cell shape and matrix rigidity may have on the dynamics and steady state polarization of actomyosin forces in the cell. Actomyosin forces are assumed to polarize in accordance with the stresses that develop in the cytoskeleton. The theory examines this self-polarization process as a relaxation response determined by two distinct susceptibility factors and two characteristic times. These reveal two canonical polarization responses to local variations in the elastic stress: an isotropic response, in which actomyosin dipolar stress isotropically changes in magnitude, and an orientational response, in which actomyosin forces orient with no net change in magnitude. Actual polarization may show up as a superimposition of the two mechanisms yielding different phases in the polarization response as observed experimentally. The cell shape and elastic moduli of the surroundings are shown to govern both the dynamics of the process as well as the steady-state. We predict that in the steady-state, beyond a critical matrix rigidity, spherical cells exert maximal force, and below that rigidity, elongated or flattened cells exert more force. Similar behaviors are reflected in the rate of the polarization process. The theory is also applicable to study the elastic response of whole cell aggregates in a gel.

Active mechanics and dynamics of cell spreading on elastic substrates.

The spreading area of cells has been shown to play a central role in the determination of cell fate and tissue morphogenesis; however, a clear understanding of how spread cell area is determined is still lacking. The observation that cell area and force generally increase with substrate rigidity suggests that cell area is dictated mechanically, by means of a force-balance between the cell and the substrate. A simple mechanical model, corroborated by experimental measurements of cell area and force is presented to analyze the temporal force balance between the cell and the substrate during spreading. The cell is modeled as a thin elastic disc that is actively pulled by lamellipodia protrusions at the cell front. The essential molecular mechanisms of the motor activity at the cell front, including, actin polymerization, adhesion kinetics, and the actin retrograde flow, are accounted for and used to predict the dynamics of cell spreading on elastic substrates; simple, closed-form expressions for the evolution of cell size and force are derived. Time-resolved, traction force microscopy, combined with measurements of cell area are performed to investigate the simultaneous variations of cell size and force. We find that cell area and force increase simultaneously during spreading but the force develops with an apparent delay relative to the increase in cell area. We demonstrate that this may reflect the strain-stiffening property of the cytoskeleton. We further demonstrate that the radial cell force is a concave function of spreading speed and that this may reflect the strengthening of cell-substrate adhesions during spreading.

Unrestrained growth of correctly oriented microtubules instructs axonal microtubule orientation.

In many eukaryotic cells, directed molecular transport occurs along microtubules. Within neuronal axons, transport over vast distances particularly relies on uniformly oriented microtubules, whose plus-ends point towards the distal axon tip (anterogradely polymerizing, or plus-end-out). However, axonal microtubules initially have mixed orientations, and how they orient during development is not yet fully understood. Using live imaging of primary Drosophila melanogaster neurons, we found that, in the distal part of the axon, catastrophe rates of plus-end-out microtubules were significantly reduced compared to those of minus-end-out microtubules. Physical modelling revealed that plus-end-out microtubules should therefore exhibit persistent long-term growth, while growth of minus-end-out microtubules should be limited, leading to a bias in overall axonal microtubule orientation. Using chemical and physical perturbations of microtubule growth and genetic perturbations of the anti -catastrophe factor p150, which was enriched in the distal axon tip, we confirmed that the enhanced growth of plus-end-out microtubules is critical for achieving uniform microtubule orientation. Computer simulations of axon development integrating the enhanced plus-end-out microtubule growth identified here with previously suggested mechanisms, that is, dynein-based microtubule sliding and augmin-mediated templating, correctly predicted the long-term evolution of axonal microtubule orientation as found in our experiments. Our study thus leads to a holistic explanation of how axonal microtubules orient uniformly, a prerequisite for efficient long-range transport essential for neuronal functioning.

For humans to be able to wiggle their toes, messages need to travel from the brain to the foot, a distance well over a meter in many adults. This is made possible by neurons, the cells that form the nervous system, which transmit electrical signals along long extensions called 'axons'. Axons can only transmit signals if all the required molecules, which are produced in a part of the neuron known as the cell body, are ferried to the ends of the axons. This ferrying around of molecules is carried out by long, filamentous molecules called microtubules, which act as a directed carrier system, shuttling molecules along the axon, either towards or away from the cell body. Microtubules can be thought of as asymmetrical rods. One end ­ known as the plus end ­ is dynamic and can undergo growth or shrinkage, while the other end ­ called the minus end ­ is stable. For transport along the axon to happen efficiently, microtubules in the neuron need to be oriented with their plus end pointing towards the ends of the axon. Microtubules in growing neurons develop this orientation, but how that is achieved is not fully understood. To understand the basis of this cellular phenomenon, Jakobs, Zemel and Franze examined the behaviour of microtubules in developing neurons from fruit fly larvae. A fluorescent protein, which emits light when the microtubules are growing, helped the researchers visualise the plus end of microtubules, the microtubule orientation, and their growth in developing axons. This experiment showed that microtubules that had their plus end pointing towards the axon end shrank more slowly than those with the opposite orientation, leading them to grow longer. This resulted in a higher proportion of the correctly-oriented microtubules in the axon. Treating the neurons with Nocodazole, a chemical that disrupts microtubule growth, or with sodium chloride, which changes the osmotic pressure, caused the microtubules that were oriented with their plus end towards the axon to grow less, and disrupted the uniform orientation of the microtubules in the axon. The next step was to determine whether specific axonal proteins such as p150 ­ a protein that is enriched at the tip of the axon and decreases microtubule shrinkage rates ­ are involved in this process. Reducing the levels of p150 in fruit flies using molecular and genetic methods resulted in microtubules with their plus end pointing towards the axon tip shrinking faster, reducing the proportion of microtubules with this orientation in the axon. This role of proteins enriched in the axonal tip, along with previously discovered mechanisms, explains how microtubules align unidirectionally in axons. These findings open new avenues of research into neurodegenerative diseases like Alzheimer's and Parkinson's, which might manifest due to a breakdown of transport along microtubules in neurons.

Do cells sense stress or strain? Measurement of cellular orientation can provide a clue.

We predict theoretically the steady-state orientation of cells subject to dynamical stresses that vary more quickly than the cell relaxation time. We show that the orientation is a strong function of the Poisson's ratio, nu, of the matrix when cell activity is governed by the matrix strain; if cell activity is governed by the matrix stress, the orientation depends only weakly on nu. These results can be used to differentiate systems in which the strain or the stress determine the setpoint for the mechanosensitivity of cells.

Modulation of the spontaneous curvature and bending rigidity of lipid membranes by interfacially adsorbed amphipathic peptides.

Amphipathic alpha-helical peptides are often ascribed an ability to induce curvature stress in lipid membranes. This may lead directly to a bending deformation of the host membrane, or it may promote the formation of defects that involve highly curved lipid layers present in membrane pores, fusion intermediates, and solubilized peptide-micelle complexes. The driving force is the same in all cases: peptides induce a spontaneous curvature in the host lipid layer, the sign of which depends sensitively on the peptide's structural properties. We provide a quantitative account for this observation on the basis of a molecular-level method. To this end, we consider a lipid membrane with peptides interfacially adsorbed onto one leaflet at high peptide-to-lipid ratio. The peptides are modeled generically as rigid cylinders that interact with the host membrane through a perturbation of the conformational properties of the lipid chains. Through the use of a molecular-level chain packing theory, we calculate the elastic properties, that is, the spontaneous curvature and bending stiffness, of the peptide-decorated lipid membrane as a function of the peptide's insertion depth. We find a positive spontaneous curvature (preferred bending of the membrane away from the peptide) for small penetration depths of the peptide. At a penetration depth roughly equal to half-insertion into the hydrocarbon core, the spontaneous curvature changes sign, implying negative spontaneous curvature (preferred bending of the membrane toward the peptide) for large penetration depths. Despite thinning of the membrane upon peptide insertion, we find an increase in the bending stiffness. We discuss these findings in terms of how the peptide induces elastic stress.

Leukocytes Breach Endothelial Barriers by Insertion of Nuclear Lobes and Disassembly of Endothelial Actin Filaments.

The endothelial cytoskeleton is a barrier for leukocyte transendothelial migration (TEM). Mononuclear and polymorphonuclear leukocytes generate gaps of similar micron-scale size when squeezing through inflamed endothelial barriers in vitro and in vivo. To elucidate how leukocytes squeeze through these barriers, we co-tracked the endothelial actin filaments and leukocyte nuclei in real time. Nuclear squeezing involved either preexistent or de novo-generated lobes inserted into the leukocyte lamellipodia. Leukocyte nuclei reversibly bent the endothelial actin stress fibers. Surprisingly, formation of both paracellular gaps and transcellular pores by squeezing leukocytes did not require Rho kinase or myosin II-mediated endothelial contractility. Electron-microscopic analysis suggested that nuclear squeezing displaced without condensing the endothelial actin filaments. Blocking endothelial actin turnover abolished leukocyte nuclear squeezing, whereas increasing actin filament density did not. We propose that leukocyte nuclei must disassemble the thin endothelial actin filaments interlaced between endothelial stress fibers in order to complete TEM.

Force Generation by Molecular-Motor-Powered Microtubule Bundles; Implications for Neuronal Polarization and Growth.

The heavily cross-linked microtubule (MT) bundles found in neuronal processes play a central role in the initiation, growth and maturation of axons and dendrites; however, a quantitative understanding of their mechanical function is still lacking. We here developed computer simulations to investigate the dynamics of force generation in 1D bundles of MTs that are cross-linked and powered by molecular motors. The motion of filaments and the forces they exert are investigated as a function of the motor type (unipolar or bipolar), MT density and length, applied load, and motor connectivity. We demonstrate that only unipolar motors (e.g., kinesin-1) can provide the driving force for bundle expansion, while bipolar motors (e.g., kinesin-5) oppose it. The force generation capacity of the bundles is shown to depend sharply on the fraction of unipolar motors due to a percolation transition that must occur in the bundle. Scaling laws between bundle length, force, MT length and motor fraction are presented. In addition, we investigate the dynamics of growth in the presence of a constant influx of MTs. Beyond a short equilibration period, the bundles grow linearly in time. In this growth regime, the bundle extends as one mass forward with most filaments sliding with the growth velocity. The growth velocity is shown to be dictated by the inward flux of MTs, to inversely scale with the load and to be independent of the free velocity of the motors. These findings provide important molecular-level insights into the mechanical function of the MT cytoskeleton in normal axon growth and regeneration after injury.

Mechanical control of stem cell differentiation.

Numerous studies have focused on identifying the chemical and biological factors that govern the differentiation of stem cells; however, recent research has shown that mechanical cues may play an equally important role. Mechanical forces such as shear stresses and tensile loads, as well as the rigidity and topography of the extracellular matrix were shown to induce significant changes in the morphology and fate of stem cells. We survey experimental studies that focused on the response of stem cells to mechanical and geometrical properties of their environment and discuss the mechanical mechanisms that accompany their response including the remodeling of the cytoskeleton and determination of cell and nucleus size and shape.

Hyaluronic acid matrices show matrix stiffness in 2D and 3D dictates cytoskeletal order and myosin-II phosphorylation within stem cells.

Physical features of microenvironments such as matrix elasticity E can clearly influence cell morphology and cell phenotype, but many differences between model matrices raise questions as to whether a standard biological scale for E exists, especially in 3D as well as in 2D. An E-series of two distinct types of hydrogels are ligand-functionalized here with non-fibrous collagen and used to elucidate wide-ranging cell and cytoskeletal responses to E in both 2D and 3D matrix geometries. Cross-linked hyaluronic acid (HA) based matrices as well as standard polyacrylamide (PA) hydrogels show that, within hours of initial plating, the adhesion, asymmetric shape, and cytoskeletal order within mesenchymal stem cells generally depend on E nonmonotonically over a broad range of physiologically relevant E. In particular, with overlays of a second matrix the stiffer of the upper or lower matrix dominates key cell responses to 3D: the cell invariably takes an elongated shape that couples to E in driving cytoplasmic stress fiber assembly. In contrast, embedding cells in homogeneous HA matrices constrains cells to spherically symmetric shapes in which E drives the assembly of a predominantly cortical cytoskeleton. Non-muscle myosin II generates the forces required for key cell responses and is a target of a phospho-Tyrosine signaling pathway that likely regulates contractile assemblies and also depends nonmonotonically on E. The results can be understood in part from a theory for stress fiber polarization that couples to matrix elasticity as well as cell shape and accurately predicts cytoskeletal order in 2D and 3D, regardless of polymer system.

Theoretical concepts and models of cellular mechanosensing.

Recent discoveries have established that mechanical properties of the cellular environment such as its rigidity, geometry, and external stresses play an important role in determining the cellular function and fate. Mechanical properties have been shown to influence cell shape and orientation, regulate cell proliferation and differentiation, and even govern the development and organization of tissues. In recent years, many theoretical and experimental investigations have been carried out to elucidate the mechanisms and consequences of the mechanosensitivity of cells. In this review, we discuss recent theoretical concepts and approaches that explain and predict cell mechanosensitivity. We focus on the interplay of active and passive processes that govern cell-cell and cell-matrix interactions and discuss the role of this interplay in the processes of cell adhesion, regulation of cytoskeleton mechanics and the response of cells to applied mechanical stresses.

Motor-induced sliding of microtubule and actin bundles.

Interactions of multiple molecular motors with bundles of actin and microtubule filaments form the basis for many cytoskeletal processes including axonal growth, muscle contraction, cell division and platelet formation. Continuum models based on generalized diffusion equations have been suggested to quantify the dynamics of such active bundles. In highly cross-linked and densely packed filament bundles, however, a major complication arises due to the multiple interactions that each filament forms with its neighbors. To explore the effects of these interactions, we used detailed computer simulations and studied the bundles with different types of motors at different densities and boundary conditions. We found that highly cross-linked bundles exhibit effects of long-ranged interactions that are sensitive to the boundary conditions. In open bundles, these give rise to 'telescopic' patterns resulting in significant acceleration of the filaments at the edges. In contrast, in ringed bundles, the long-ranged interactions 'lock' filaments and slow down their movements. The filaments in loosely connected bundles, on the other hand, undergo local diffusion-drift dynamics consistent with previous continuum models. Our simulations also demonstrate the sorting phenomena in the mixed-polarity bundles and reveal characteristic scales and conditions for spontaneous pattern formation in the bundle. We discuss the relevance of our results for cytoskeleton systems such as microtubules in axons, platelet formation, kinetochore fibers and actin bundles in motile cells.

Perturbation of a lipid membrane by amphipathic peptides and its role in pore formation.

We study the structural and energetic consequences of (alpha-helical) amphipathic peptide adsorption onto a lipid membrane and the subsequent formation of a transmembrane peptide pore. Initially, each peptide binds to the membrane surface, with the hydrophobic face of its cylinder-like body inserted into the hydrocarbon core. Pore formation results from subsequent peptide crowding, oligomerization, and eventually reorientation along the membrane normal. We have theoretically analyzed three peptide-membrane association states: interfacially-adsorbed monomeric and dimeric peptides, and the multi-peptide transmembrane pore state. Our molecular-level model for the lipid bilayer is based on a combination of detailed chain packing theory and a phenomenological description of the headgroup region. We show that the membrane perturbation free energy depends critically on peptide orientation: in the transmembrane pore state the lipid perturbation energy, per peptide, is smaller than in the adsorbed state. This suggests that the gain in conformational freedom of the lipid chains is a central driving force for pore formation. We also find a weak, lipid-mediated, gain in membrane perturbation free energy upon dimerization of interfacially-adsorbed peptides. Although the results pertain mainly to weakly-charged peptides, they reveal general properties of the interaction of amphipathic peptides with lipid membranes.

Membrane perturbation induced by interfacially adsorbed peptides.

The structural and energetic characteristics of the interaction between interfacially adsorbed (partially inserted) alpha-helical, amphipathic peptides and the lipid bilayer substrate are studied using a molecular level theory of lipid chain packing in membranes. The peptides are modeled as "amphipathic cylinders" characterized by a well-defined polar angle. Assuming two-dimensional nematic order of the adsorbed peptides, the membrane perturbation free energy is evaluated using a cell-like model; the peptide axes are parallel to the membrane plane. The elastic and interfacial contributions to the perturbation free energy of the "peptide-dressed" membrane are evaluated as a function of: the peptide penetration depth into the bilayer's hydrophobic core, the membrane thickness, the polar angle, and the lipid/peptide ratio. The structural properties calculated include the shape and extent of the distorted (stretched and bent) lipid chains surrounding the adsorbed peptide, and their orientational (C-H) bond order parameter profiles. The changes in bond order parameters attendant upon peptide adsorption are in good agreement with magnetic resonance measurements. Also consistent with experiment, our model predicts that peptide adsorption results in membrane thinning. Our calculations reveal pronounced, membrane-mediated, attractive interactions between the adsorbed peptides, suggesting a possible mechanism for lateral aggregation of membrane-bound peptides. As a special case of interest, we have also investigated completely hydrophobic peptides, for which we find a strong energetic preference for the transmembrane (inserted) orientation over the horizontal (adsorbed) orientation.

Energetics and self-assembly of amphipathic peptide pores in lipid membranes.

We present a theoretical study of the energetics, equilibrium size, and size distribution of membrane pores composed of electrically charged amphipathic peptides. The peptides are modeled as cylinders (mimicking alpha-helices) carrying different amounts of charge, with the charge being uniformly distributed over a hydrophilic face, defined by the angle subtended by polar amino acid residues. The free energy of a pore of a given radius, R, and a given number of peptides, s, is expressed as a sum of the peptides' electrostatic charging energy (calculated using Poisson-Boltzmann theory), and the lipid-perturbation energy associated with the formation of a membrane rim (which we model as being semitoroidal) in the gap between neighboring peptides. A simple phenomenological model is used to calculate the membrane perturbation energy. The balance between the opposing forces (namely, the radial free energy derivatives) associated with the electrostatic free energy that favors large R, and the membrane perturbation term that favors small R, dictates the equilibrium properties of the pore. Systematic calculations are reported for circular pores composed of various numbers of peptides, carrying different amounts of charge (1-6 elementary, positive charges) and characterized by different polar angles. We find that the optimal R's, for all (except, possibly, very weakly) charged peptides conform to the "toroidal" pore model, whereby a membrane rim larger than approximately 1 nm intervenes between neighboring peptides. Only weakly charged peptides are likely to form "barrel-stave" pores where the peptides essentially touch one another. Treating pore formation as a two-dimensional self-assembly phenomenon, a simple statistical thermodynamic model is formulated and used to calculate pore size distributions. We find that the average pore size and size polydispersity increase with peptide charge and with the amphipathic polar angle. We also argue that the transition of peptides from the adsorbed to the inserted (membrane pore) state is cooperative and thus occurs rather abruptly upon a change in ambient conditions.