Modeling of the axon membrane skeleton structure and implications for its mechanical properties
[['It is known for some time that microtubules and neurofilaments are the predominant structural filamentous proteins in the axon [1 , 2] .', 'However , how these filaments are arranged to generate the structure of the axon plasma membrane skeleton was only very recently discovered [3] .', 'Super-resolution fluorescence microscopy [4] revealed that the membrane skeleton of an unmyelinated axon consists of actin filaments , capped with adducin at one end , that form ring-like structures along the circumference of the axon .', 'The actin rings are connected via spectrin tetramers oriented along the longitudinal direction of the axon ( Fig 1A ) .', 'This cytoskeletal structure is extended across the entire axon .', 'The distance between the periodic actin rings is approximately 180 to 190 nm [3 , 5 , 6] .', 'Each spectrin tetramer is formed by the association of two identical heterodimers comprising an α-chain and a β-chain with 22 and 17-triple-helical segments , respectively [7] .', 'In the distal axon , each heterodimer consists of two intertwined αII- and βII-spectrin chains running antiparallel to one another ( Figure A in S1 Fig ) .', 'The axon initial segment ( AIS ) of mature neurons , however , appears to contain the subtype βIV-spectrin instead of βII-spectrin [5] .', 'Spectrin tetramers are associated with the lipid bilayer [3] in a manner similar to that occurring at the red blood cell ( RBC ) membrane [7 , 8] ( Figure B in S1 Fig ) .', 'In RBCs , ankyrin plays a major role in anchoring the lipid bilayer to the spectrin network by associating a spectrin filament with the anion exchanger integral membrane protein band-3 [9 , 10] ( Figure A in S1 Fig ) .', 'Ankyrin binds to the middle of the spectrin tetramer , at the 15th repeat of β-spectrin near its carboxyl terminus [7 , 11] , and at the same time it binds to the cytoplasmic domain of band-3 [9] .', 'In axons , the spatial distribution of ankyrin-G and ankyrin-B is highly periodic in the proximal and distal area of the axon , respectively [3 , 5] .', 'Ankyrin , besides anchoring the spectrin network to the lipid bilayer , is critical for the organization of the axonal plasma membrane because it binds to several molecules .', 'Voltage-gated Na ( Nav ) channels can bind to subdomains 3 and 4 of ankyrin [12] .', 'Since ankyrin has a periodic pattern in the axon , the Nav channels also exhibit a periodic distribution pattern in the AIS alternating with the N terminus of βIV-spectrin [3] .', 'The periodic structure of the membrane skeleton is thought to play an important role in the structural durability of the axon [3] .', 'This conjecture comes from the similarity between the structural elements of the plasma membrane skeletons of the RBC and of the neuronal axon .', 'However , we note that their geometrical arrangements are radically different resulting in hyperelastic flexibility for the RBC cytoskeleton and in reduced radial deformability and longitudinal extensibility for the axon .', 'The two most important differences between the geometric configurations of the RBC and of the axon membrane skeletons are the following: first , the RBC cytoskeleton forms an approximately six-fold symmetric two-dimensional network ( Figure B in S1 Fig ) that behaves as an incompressible hyperelastic material [13 , 14] .', 'In the axon , actin filaments form rings along the circumference , connected by spectrin filaments oriented along the axon .', 'In this case , the membrane skeleton assumes the form of a two-dimensional cylindrically symmetric orthotropic network ( Fig 1 ) .', 'The second important difference is that in the case of the RBC membrane , the end-to-end distance of the spectrin tetramers is ~ 75nm , which is close to the end-to-end distance of a free spectrin filament [15] .', 'This suggests that the spectrin network in the RBC is near equilibrium .', 'In the axon , however , the distance between the actin rings was reported to be approximately 180 to 190nm [3 , 6] .', 'It is thought that microtubules stabilize the axon through interactions with neurofilaments , organelles [1 , 2 , 16] and possibly directly or indirectly with the actin rings , holding them apart at a specific distance .', 'Since we do not know the exact configuration of actin rings , we assume that the upper limit of a junction of actin filaments and proteins promoting actin-spectrin binding is approximately 35 nm , which is close to the size of actin junctions in RBCs [17] .', 'Thus , we consider that the end-to-end distance of the spectrin tetramers is approximately 150 nm while their contour length is approximately 200 nm [18 , 19] .', 'This means that the spectrin filaments in the axon membrane skeleton are held under entropic tension suggesting that the flexibility of the network along the axon might be limited .', 'However , because we do not know the exact thickness of the actin rings and consequently the end-to-end distance of the spectrin filaments , we also considered cases where the size of the junctions between actin and spectrin filaments were 25nm , 15nm , and 5nm .', 'The diameter of an actin filament is approximately 8nm [20 , 21] .', 'Based on the particular structure of the axon , we expect that its mechanical properties are different than the mechanical properties of soma and dendrites .', 'Here , we used atomic force microscopy ( AFM ) to measure , via indentation , the stiffness of the plasma membrane of the subcompartments of cultured hippocampal pyramidal neurons .', 'Importantly , we developed a coarse-grain molecular dynamics ( CGMD ) model for the membrane skeleton of the axon that comprises representation and connectivities of its main filaments .', 'Based on the AFM measurements and on geometric and material parameters for the implemented filament models available in literature , we were able to examine the effect of the particular geometric configuration of the membrane skeleton and reproduce the stiffness of the axon plasma membrane .', 'We note that while the model represents only the axon plasma membrane and considers all connectivities between the different components mostly as stable and not as dynamic processes , it provides a clear picture of how spectrin filaments in conjunction with actin rings contribute to the mechanical properties of the axonal membrane .', 'We expect that the model will be used in studies of the mechanical stability of the axon , and generation and propagation of the action potential .'], ['E18 hippocampal tissue obtained from BrainBits ( BrainBits , Springfield , IL ) was treated with trypsin and plated onto poly-D-lysine ( Sigma Aldrich , St . Louis , MO ) -coated glass bottom petri dishes ( Ted Pella , Redding , CA ) in neurobasal media ( Thermo Scientific , Waltham , MA ) supplemented with B27 ( Thermo Scientific , Waltham , MA ) , penicillin streptomycin ( Thermo Scientific , Waltham , MA ) and glutamax ( Thermo Scientific , Waltham , MA ) .', 'The cells were maintained at 37°C in a humidified incubator with 5% CO2 until use .', 'After 8–10 DIV , neurons were transfected with tau-gfp using Lipofectamine 2000 according to manufacturer directions ( Thermo Scientific , Waltham , MA ) .', 'Tau-gfp was used to visualize axons in living neurons .', 'Although tau-gfp tagged both axons and dendrites , axons were identified by their distinct morphology ( S3 Fig ) .', 'Tau-gfp was a gift from Dr . Walikonis , Department of Physiology and Neurobiology , UCONN , Storrs .', 'We obtained the Young’s moduli of the axon plasma membrane , dendrites , and soma via AFM indentation experiments with a maximum displacement of 200 nm .', 'In these cases , the simple Hertz contact model of elastic half-space indentation cannot be used because cells do not behave elastically under large deformations and because of the geometric characteristics of dendrites and axon .', 'Instead , we implemented a finite element model ( FEM ) to compute force-indentation ( F − d ) relationships and used them to obtain the Young’s moduli for soma , dendrites , and axon .', 'The method and results are explained in detail in the S1 Text .', 'Below , we briefly describe the FE approach .', 'Large deformations of cell plasma membranes can be described reasonably well using the nearly incompressible neo-Hookean constitutive model [13 , 22 , 23] .', 'We employ the isochoric deformation gradient F¯=J−1/3F , and similarly the isochoric right Cauchy-Green tensor C¯=J−2/3C , where C = FTF and J is the Jacobian , the determinant of the deformation gradient F . For the neo-Hookean model ψ=1c ( J−1 ) 2+μ2 ( I¯1−3 ) , where c = 6 ( 1−2ν ) /E = 2/κ , E is the initial Young’s modulus , ν is the Poisson’s ratio , and κ is the initial bulk modulus of the material .', 'In the case of incompressibility , c degenerates to a nonphysical , positive penalty parameter used to enforce incompressibility .', 'The parameter μ is the initial shear modulus , and I¯1=trC¯ is the first invariant of the isochoric right Cauchy-Green tensor .', 'From the deformation gradient , we calculate Cauchy stress σ as σ=2JF∂ψ∂CFT .', 'Applying the nearly incompressible neo-Hookean material model , we simulate indentation of both', '( i ) a homogeneous isotropic rectangular cuboid and', '( ii ) a homogeneous and isotropic thin-walled cylinder , by a conical indenter with a blunt tip using FE analyses in ANSYS workbench 14 . 0 ( Canonsburg , PA ) .', 'We note that in the actual AFM experiments , the cantilever tip was of a pyramidal shape while in the FEM calculations we used a conical indenter with a blunt tip to avoid complications stemming from the pyramid edges .', 'However , we show in the S1 Text that the F − d relationship valid for a pyramidal indenter , with a blunt tip with a semi-included angle of 20° and a tip radius of 20 nm , is very similar to the F − d relationship for a conical indenter with the same semi-included angle and tip radius ( S4 Fig ) .', 'Because of this , we expect that the FEM results are suitable in the calculation of the Young’s moduli from the AFM indentation experiments as we explain in the section below .', 'We carried out stiffness measurements on living rat hippocampal pyramidal neurons ( 16–18 DIV ) using AFM silicon nitride cantilevers with a nominal spring constant of 0 . 03 N/m ( MLCT , Bruker Probes , Camarillo , CA ) .', 'Exact values for the cantilever spring constants were obtained via a thermal noise based method implemented by the manufacturer and were used in all calculations .', 'Probes were of four-sided pyramidal shape with nominal tip radius of 20 nm and nominal semi-included angle of approximately 20° , as provided by the manufacturer .', 'The tip was indented ~ 200 nm into the cell .', "Only ~ 100 nm of this indentation was used to determine the Young's modulus .", 'The diameter of the axon at the measurement locations was approximately 1 μm .', 'We note that in pyramidal neurons , microtubules , and neurofilaments are located at a distance greater than 200 nm from the axonal membrane [24 , 25] .', 'Because the indentation depth in our experiments was approximately 100 nm , we do not expect that microtubules and neurofilaments will contribute to the measured axon plasma membrane stiffness .', 'The same argument is true for dendrites since microtubules are located at more than 200 nm distance from dendritic plasma membrane [26] .', 'For dendrite stiffness measurements , we tested areas where the dendrite diameter was larger than 2 μm .', 'Measurements of soma stiffness were performed at different regions of the soma excluding the area over the nucleus .', 'All measurements were performed in supplemented neurobasal media at 37°C .', 'To measure the axon plasma membrane stiffness , we performed indentations at 16 x 16 points distributed uniformly in a 500 nm×500 nm area of the axon surface at a loading rate of 10 , 000 pN/s .', 'For each measurement the cantilever displacement was calibrated at the rigid substrate next to the cell .', "Young's moduli E were calculated based on the force-indentation ( F − d ) relationship ( Eq S7 ) for a conical indenter with a blunt tip , with a semi-included angle θ = 20° and a tip radius of R = 20 nm , indenting ( up to 200 nm ) a neo-Hookean thin-walled cylinder of 1 μm diameter and of h = 10 nm wall-thickness .", 'For the assessment of the plasma membrane stiffness of dendrites we followed the same approach as with the axon plasma membrane and used the same equation ( Eq S7 ) .', 'In addition , the Young’s modulus of soma was calculated based on the ( F − d ) relationship ( Eq S5 ) .', 'The soma was simulated as a nearly incompressible ( Poisson’s ratio ν ≃ 0 . 5 ) neo-Hookean rectangular 10 μm×10 μm×5 μm homogeneous and isotropic cuboid .', 'In the S1 Text we show that the ( F − d ) for the thin-walled cylinder is F = ( 4 . 41×10−3 E ) d1 . 37 ( Eq S7 ) , and for a rectangular cuboid is F = ( 7 . 95×10−3 E ) d1 . 46 ( Eq S5 ) .', 'We note that the above ( F − d ) relationships are valid when the indentation d is measured in nm , the initial Young’s modulus E in kPa , and the force F in pN .', "We used an open source program called force review automation environment ( FRAME ) [27] , developed by our lab , to determine the Young's moduli of the soma , dendrites and axon of hippocampal neurons .", "A value of the Young's modulus from each force-displacement curve was determined by fitting the theoretical curve for a pyramidal indenter with a blunt tip mentioned above .", 'The resulting stiffness for the specific neuronal sub-compartment was determined as the median value of the probability distribution plot generated by the individual measurements of stiffness at each point .', 'We tested n neurons from N samples .', 'The median values were determined horizontally across all samples .', "We found that the soma has a Young's modulus of 0 . 7 ± 0 . 2 kPa ( N = 2 , n = 8 ) ( Fig 2A ) .", 'The term frequency in Fig 2 corresponds to the percentage of measurements that gave a value within the corresponding bin range .', "The Young's modulus of dendrites was found to be 2 . 5 ± 0 . 7 kPa ( N = 2 , n = 8 ) ( Fig 2B ) .", 'Importantly , the axon was the stiffest sub-compartment of the neuron exhibiting a median value of the Young’s modulus of 4 . 6 ± 1 . 5 kPa ( N = 2 , n = 8 ) .', 'To evaluate if actin is critical for the observed high stiffness of the axon plasma membrane , we incubated neurons with Latrunculin B ( 20 μM for 1 hour ) , a compound that inhibits actin polymerization .', 'This in turn , results in disruption of actin filaments .', 'We found that in the presence of Latrunculin B the median value of the Young’s modulus was reduced to 2 . 2 ± 0 . 6 kPa ( N = 1 , n = 6 ) indicating that actin rings play a very significant role in determining the overall axon plasma membrane stiffness .', "We note that our results on the Young's modulus of soma and dendrites of rat hippocampal neurons are in agreement with published results [28 , 29] .", 'However , to our knowledge , there are no published measurements of the stiffness of pyramidal neuron axon plasma membrane .', 'It is clear that the stiffness of the axon is much higher than the stiffness of the soma and the stiffness of the dendrites .', 'Below , we introduce a CGMD model for the axon and we then use it to understand how the axon plasma membrane skeleton structure leads to its elevated stiffness .', 'The proposed model reflects the structure of the membrane skeleton of the proximal and distal unmyelinated axon as previously described [3 , 5] .', 'The model is a representation of actin rings , oriented along the circumference of the axon , that are connected by spectrin tetramers tethered to the lipid bilayer at their middle section ( Fig 1 ) .', 'A spectrin tetramer consists of two identical , intertwined , head-to-head associated heterodimers [18] .', 'Each heterodimer is comprised of an α-spectrin and a β-spectrin chain consisting of 22 and 19 homologous triple helical repeats , respectively [7] .', 'In our model , a spectrin tetramer is represented as a single chain of 41 spherical beads ( gray particles in Figure A in S2 Fig ) connected by 40 harmonic springs .', 'The solid red line in Figure B in S2 Fig reflects the spring potential , USS ( r ) =1/2k0 ( r-reqSS ) 2 , where r is the distance between two consecutive spectrin particles , reqSS=Lc/40=5nm is the equilibrium distance between the spectrin particles ( close to the size of the spectrin repeats ) , Lc ≃ 200nm is the contour length of the spectrin tetramers , and k0 is the spring constant ( defined below ) .', 'All spectrin particles interact via the repulsive Lennard-Jones ( L-J ) potential:', 'UrepSS ( rij ) ={4ε1[ ( σrij ) 12− ( σrij ) 6]+ε1rij<Rcut , LJ=reqSS0rij>Rcut , LJ=reqSS ,', '( 1 )', 'where ε1 = ( 1/16 ) ε with ε being the energy unit , σ the length unit , and rij the distance between spectrin particles .', 'The value ε1 = ( 1/16 ) ε gives a curvature at the equilibrium equal to the spring constant k0 .', 'Setting the diameter of the spectrin particles equal to the equilibrium distance of the L-J potential reqSS=21/6σ=5nm , the length scale is σ = 4 . 45 nm .', 'We discuss the energy scale in the section below related to modeling the plasma membrane network .', 'We chose the cutoff distance of the potential Rcut , LJ to be the equilibrium distance reqSS between two spectrin particles .', 'The potential is plotted as a dashed red line in Figure B in S2 Fig . We note that in order to reduce the number of free parameters of the model the spring constant was set to k0 = 3 . 56 ε/σ2 which is identical to the curvature at equilibrium of the L-J potential used in actin-spectrin interaction ( see the section on Modeling the axon plasma membrane network ) .', 'We computed the end-to-end distance 〈ree2〉1/2 of the spectrin chain model for KBT/ε = 0 . 03 , where KB is the Boltzmann’s constant .', 'We first equilibrated the filament for 105 time steps , and then measured the end-to-end distance for 3×106 time steps during its thermal fluctuations .', 'The recorded distances follow a Gaussian distribution P ( ree ) =1/ ( λ2π ) exp[− ( ree−〈ree〉 ) 2/2λ2] , where λ=〈 ( ree−〈ree〉 ) 2〉 , and with a mean value of 〈ree2〉1/2=74 . 3nm ( S14 Fig ) .', 'For flexible filaments with lp << Lc , the end-to-end distance is correlated with persistence length and contour length via the expression 〈ree2〉1/2≅2lpLc .', 'Taking into consideration that the spectrin contour length is approximately 200 nm [18 , 19] , we calculated the persistence length to be 13 . 8nm .', 'This result is close to experimentally reported values of approximately 20 nm [30] and 10 nm [31] .', 'The actin rings consist of short actin filaments arranged along the circumference of the axon [3] .', 'The exact configuration of the aligned actin filaments and how they are connected to form the actin rings is not known .', 'It has however been shown that adducin is present in the actin rings [3] probably capping and stabilizing the plus end of actin filaments .', 'We then expect that the minus end of F-actin is stabilized by another protein , probably tropomodulin [21] , while additional cross-linking proteins are possibly involved in the formation of the actin rings [21 , 32 , 33] ( Fig 1A ) .', 'Because the exact molecular structure of the actin rings and whether actin filaments are connected in a side-by-side or an end-to-end arrangement is not known , we adopted a coarse-grain particle model that produces stable actin rings but ignores their specific molecular structure .', 'In this particle model , an actin ring is represented as a collection of 39 beads ( red particles in Fig 1B and insert and in Figure A in S2 Fig ) with a diameter of approximately 35 nm .', 'These beads form a circle with a diameter of approximately 434 nm , which lies within the range of experimental results [3 , 5 , 34] .', 'We chose the diameter of the actin particles to be 35 nm based on values for the RBC membrane skeleton , which comprises short actin oligomers ( consisting of approximately 13 to 15 subunits ) with a length of 33 ± 5 nm [17 , 35 , 36] .', 'Two adjacent actin particles in the same ring connect via a spring potential UAA=1/2kA ( r−reqAA ) 2 , with equilibrium distance reqAA=35nm , and a repulsive L-J potential UrepAA ( S1 Table ) , with Rcut , LJ=reqAA ( shown as purple lines in Figure B in S2 Fig ) .', 'The value of the spring constant kA = 38 . 0 ε/σ2 is determined in computational results in conjunction with the AFM stiffness measurement of the axon plasma membrane .', 'In addition , we employed a finitely deformable nonlinear bending potential that behaves as a finitely extendable nonlinear elastic ( FENE ) potential to maintain the circular shape of the actin rings .', 'The potential has the form Ub=−12kbΔθmaxln[1− ( θ−θ0Δθmax ) 2] , where kb = 3 , 500 KBT is the parameter that directly regulates the bending stiffness of the actin filament and it is determined in S1 Text .', 'θ is the angle formed by three consecutive particles of the same ring .', 'θ0=180° ( 39−2 ) 39=170 . 77° is the equilibrium angle and Δθmax = 0 . 3θ0 is the maximum allowed bending angle .', 'We note that the resistance of the actin ring to small deformations depends on kb/Δθmax since for small deformations the FENE potential is approximated by Ub=12 ( kb/Δθmax ) ( θ−θ0 ) 2 , which corresponds to a harmonic potential with a spring constant kbs = kb/Δθmax .', 'This means that the exact value of Δθmax does not uniquely determines the stiffness of the structure close to equilibrium but in combination with kb .', 'Δθmax defines the maximum deformation of the actin rings but its exact value does not affect the behavior of the system near equilibrium .', 'The employed value of kb produces a bending rigidity of κbend = 7 . 1 × 10−26 Nm2 for a straight stiff filament based on numerical calculations shown in S1 Text and in [37] .', 'The obtained value is similar to the experimentally measured bending rigidity of actin filaments 7 . 3 × 10−26 Nm2 reported in [38 , 39] .', 'To build a mechanically stable network , we first connected each spectrin filament at its two ends to actin particles belonging to consecutive actin rings via a breakable L-J potential ULJAS ( rij ) =4ε[ ( 4σ/rij ) 12− ( 4σ/rij ) 6] , where rij is the distance between actin and spectrin particles ( blue dashed line in Figure B in S2 Fig ) .', 'The equilibrium distance between actin and spectrin is 21/6 ( 4σ ) ≃ 20 nm resulting to an actin junction size of approximately 40 nm [40] .', 'This equilibrium distance corresponds to an end-to-end distance of 145 nm for the spectrin filaments .', 'However , because the exact thickness of the actin rings and consequently the equilibrium distance between actin and spectrin particles are not known , we also considered the cases where reqAS=15nm , 10nm , and5nm ( approximately the diameter of a G-actin monomer [20] ) , which correspond to end-to-end distance of ree = 155nm , 165nm , and 175nm respectively for the spectrin filaments .', 'We note that in this model the actin-spectrin association can break and reform .', 'The association breaks , by setting the attractive force to zero , when the distance between the particles crosses the inflexion point of the L-J potential at rinflexion = ( 26/7 ) 1/6 4σ , indicated by the red circle ( blue dashed line in Figure B in S2 Fig ) .', 'It can reform as the distance between actin and spectrin particles becomes smaller than the capture distance of 2 . 5×4σ .', 'For a stable membrane skeleton , the spectrin-actin junction association energy was chosen to be ε ≃ kBT/0 . 03 ≃ 0 . 86 eV for T = 300°K .', 'Equilibrium measurements have shown that the association energy for the spectrin-actin-protein 4 . 1 complex in normal RBCs is about 17 Kcal/mole = 0 . 74eV [41] .', 'However , for this value the membrane skeleton would be partially broken when the distance between actin rings is set equal to the experimental value of 185 nm .', 'This means that the association energy of the spectrin-actin complex in the axon is most likely larger than in normal RBCs , to maintain stable membrane skeleton with spectrin filaments under tension .', 'Microtubules and neurofilaments are thought to play an important role in maintaining the polarity and structure of the AIS through interactions with the axonal cytoskeleton .', 'Pyramidal neurons , which we used to experimentally determine the axonal membrane stiffness , have microtubules that are not structured in bundles but rather like a string of beads ( in cross-section ) [24 , 25 , 42] and they are at a distance greater than 200 nm from the axonal membrane [25] .', 'Similarly , neurofilaments in pyramidal neurons are not arranged in bundles and are much farther away from the axonal membrane than microtubules [24] .', 'Since the indentation depth of the AFM probe was ~ 100–150nm , microtubules and neurofilaments are not expected to contribute to axonal membrane stiffness in our measurements .', 'In our model , the effect of microtubules and neurofilaments on the structural integrity of the axon was implemented implicitly .', 'We considered that microtubules interact with actin to maintain the equilibrium distance of actin rings at 185 nm .', 'To achieve this , we applied the FENE potential Umt=−12kmtΔdmaxln[1− ( d−deqRRΔdmax ) 2] on all actin particles belonging to consecutive rings .', 'kmt is the parameter that determines the stiffness of the nonlinear spring between two actin rings , d and deqRR=185nm are the distance and the equilibrium distance between the centers of the two actin rings , respectively [3 , 5] , and Δdmax=0 . 3deqRR is the maximum allowed deformation .', 'The position of the center of each ring is calculated by utilizing the mean value of the z–coordinate of the actin particles .', 'The choice of kmt is justified based on the following rationale: for small deformations , the FENE potential is approximated by Umt=12 ( kmt/Δdmax ) ( d−deqRR ) 2 , which corresponds to a harmonic potential with a spring constant ksp = kmt/Δdmax [37] .', 'In this case , we can assume that τ = ELh , where τ is the stress , EL is the longitudinal Young’s modulus of the axon , and h is the strain .', 'The final equation is F/A = EL ( ΔL/L ) , where F=ksptΔL is the force applied on the cross-section of the axon A = πR2 , where kspt=ksp/ ( N−1 ) is the spring constant for the total axon , R = 217 nm is the radius of the axon , ΔL is the elongation of the axon , L= ( N−1 ) deqRR is the length of the axis , and N is the number of springs .', 'Combining the equations above , we determined that ksp=ELπR2/deqRR and consequently kmt = ksp Δdmax = 0 . 3ELπR2 .', "A reasonable value for the axon's longitudinal Young's modulus is EL ≃ 10 kPa [43] , resulting in kmt≃477KBT/σ≃19 , 822KBT/deqRR , at T = 300°K .", 'The final aspect of the model is the association between the axon membrane skeleton and the lipid bilayer .', 'In RBCs , the membrane skeleton is anchored to the lipid bilayer via glycophorin at the actin junction complexes and via the integral membrane protein band-3 and ankyrin at the middle section of spectrin tetramers [9 , 10] ( S1 Fig ) .', 'Regarding the association between a spectrin filament and the lipid bilayer in RBCs , ankyrin binds at the 15th repeat of β-spectrin near its carboxyl terminus , at the middle section of the spectrin tetramer [7] .', 'At the same time , it binds to the cytoplasmic domain of band-3 [9] , mediating the anchoring of spectrin filaments to the lipid bilayer .', 'In the case of the axon , we considered the following experimental findings:', '( i ) The spatial distribution of ankyrin-G is highly periodic in the proximal area of the axon , while ankyrin-B also exhibits a periodic pattern in distal axons [3 , 5] ,', '( ii ) Nav channels exhibit a periodic distribution pattern in the AIS alternating with actin rings [3] ,', '( iii ) Nav can bind to subdomains 3 and 4 of ankyrin [12] , and', '( iv ) Ankyrin-G and sodium channels are in 1:1 molar ratio in the brain .', 'Based on these findings and on the fact that ankyrin binds near the carboxyl terminus of β-spectrin it is reasonable to assume that Nav channels are arranged in a periodic pattern along the axon via their association with ankyrin in a manner similar to band-3 association with spectrin in the RBC membrane .', 'We also note that by assigning one Nav channel per ankyrin molecule , and consequently per spectrin tetramer , the Nav channel density is approximately 150 channels per μm2 , which lies within the range of 110 to 300 channels per μm2 in AIS [44] .', 'To represent the anchoring of spectrin tetramers to the lipid bilayer , we used the following approach: We connected an ankyrin particle ( depicted as a green particle in Fig 1B and Figure A in S2 Fig ) to the 20th particle of the spectrin filament by the spring potential USK ( rij ) =1/2k0 ( rij−reqSK ) 2 , where the radial equilibrium distance is reqSK=15nm , ( black solid line in Figure B in S2 Fig ) .', 'This distance corresponds to the radius of a spectrin particle ( 2 . 5 nm ) and the effective radius of the cytoplasmic domain of the ankyrin complex connected to an Nav channel ( ~ 12 . 5 nm ) [45] .', 'We also implemented a repulsive L-J potential UrepSK ( S1 Table ) , with Rcut , LJ=reqSK ( dashed black line in Figure B in S2 Fig ) to simulate a steric repulsion between the particles that represent spectrin and ankyrin .', 'For simplicity , we did not use a representation of the lipid bilayer in this model .', 'Instead , we used a spring potential to represent the confinement applied on the motion of ankyrin particles and spectrin filaments by the lipid bilayer .', 'The harmonic confining potential is given by UC ( r ) = 1/2kc ( r − r0 ) 2 , where r is the radial distance of spectrin and ankyrin particles from the central axis of the axon , and r0 is the equilibrium distance from the central axis .', 'We considered r0 to be 217 nm and 232 nm for the spectrin and ankyrin particles , respectively .', 'Because the ankyrin particles are attached to the bilayer , the confinement potential acts on both radial directions ( inwards and outwards ) .', 'However , only the outward motion of the spectrin particles is confined , since the spectrin filament cannot cross the lipid bilayer .', 'In contrast , the inward motion will not face additional constrain .', 'The confinement mild stiffness in this model is arbitrarily chosen to be kc = 0 . 1k0 since it is due to the bending rigidity of the lipid bilayer which is in the range of ( 10–20KBT ) .', 'We finally note that an important consideration in RBC membrane modeling is that mutations can cause disruption of the association between ankyrin and spectrin resulting in stiffer skeleton , local membrane instabilities , and vesiculation [10 , 46–48] .', 'Here , we opted to focus on establishing the membrane model and explore possible membrane skeleton defects and their effect on the axon plasma membrane in a later work .', 'The configuration used in this paper consists of N = 16 , 029 particles , corresponding to an axon length of approximately 1 . 85 μm .', 'The numerical integrations of the equations of motion are performed using the Beeman algorithm .', 'The temperature of the system is maintained at KBT/ε = 0 . 03 by employing the Berendsen’s thermostat [49] , where KB is Boltzmann’s constant and T is the temperature .', 'The model is implemented in the NVT ensemble ( constant number of particles N , constant volume V , and constant temperature T ) .', 'The time scale is ts=mσ2/ε , the time step is dt = 0 . 01ts , and m is the unit mass of the spectrin particles .', 'We selected the temperature to render the conformation time of the spectrin filaments close to expected theoretical values [50] .', 'We gradually brought the model to the equilibrium length and temperature , and then equilibrated it for 15×104 time steps .', 'We performed the measurements during a period of 10×106 time steps after equilibration .'], ['To measure the stiffness of the simulated axon plasma membrane skeleton , we defined a cylindrical Lennard-Jones repulsive potential ULJS ( rij ) =4ε[ ( σ/ ( ri− ( rc−rS ) ) ) 12− ( σ/ ( ri− ( rc−rS ) ) ) 6] , for the spectrin particles and ULJA ( rij ) =4ε[ ( 7σ/ ( ri− ( rc−rA ) ) ) 12− ( 7σ/ ( ri− ( rc−rA ) ) ) 6] , for the actin particles .', 'rc is the distance between the center-line of the axon and the surface of an imaginary cylinder ( Fig 3A ) , ri is the distance between particle i and the center-line of the axon , rS is the radius of the spectrin particle and rA is the radius of the actin particle .', 'The cutoff distances of the L-J potent
['Introduction', 'Materials and models', 'Results and discussion', 'Conclusion']
['Super-resolution microscopy recently revealed that , unlike the soma and dendrites , the axon membrane skeleton is structured as a series of actin rings connected by spectrin filaments that are held under tension .', 'Currently , the structure-function relationship of the axonal structure is unclear .', 'Here , we used atomic force microscopy ( AFM ) to show that the stiffness of the axon plasma membrane is significantly higher than the stiffnesses of dendrites and somata .', 'To examine whether the structure of the axon plasma membrane determines its overall stiffness , we introduced a coarse-grain molecular dynamics model of the axon membrane skeleton that reproduces the structure identified by super-resolution microscopy .', 'Our proposed computational model accurately simulates the median value of the Young’s modulus of the axon plasma membrane determined by atomic force microscopy .', 'It also predicts that because the spectrin filaments are under entropic tension , the thermal random motion of the voltage-gated sodium channels ( Nav ) , which are bound to ankyrin particles , a critical axonal protein , is reduced compared to the thermal motion when spectrin filaments are held at equilibrium .', 'Lastly , our model predicts that because spectrin filaments are under tension , any axonal injuries that lacerate spectrin filaments will likely lead to a permanent disruption of the membrane skeleton due to the inability of spectrin filaments to spontaneously form their initial under-tension configuration .']
['Super-resolution microscopy has suggested that the actin cytoskeleton structure differ between various neuronal subcompartments .', 'To determine the possible implication of the differing actin cytoskeleton structure , we determined the stiffness of the plasma membrane of neuronal subcompartments using atomic force microscopy ( AFM ) .', 'We found that axons are almost ~6 fold stiffer than the soma and ~2 fold stiffer than dendrites .', 'By using a particle-based model for the surface membrane skeleton of the axon that comprises actin rings connected with spring filaments to represent the axonal structure , we show that regions neighboring actin rings are stiffer than areas between these rings .', 'In these in between sub-regions , the spectrin filaments determine stiffness .', 'Our modeling also shows that because the spectrin filaments are under tension , the thermal jitter of the actin-associated ankyrin particles , connected to the middle area of spectrin filaments , is minimal .', 'As a result , we propose that the sodium channels bound to ankyrin particles will maintain an ordered distribution along the axon .', 'We also predict that laceration of the spectrin filaments due to injury will cause a permanent damage to the axon since spontaneous repair of the spectrin network is not possible as spectrin filaments are under entropic tension .']
['stiffness', 'mechanical properties', 'cell motility', 'medicine and health sciences', 'actin filaments', 'membrane potential', 'electrophysiology', 'neuroscience', 'spectrins', 'materials science', 'nerve fibers', 'cellular structures and organelles', 'neuronal dendrites', 'animal cells', 'axons', 'proteins', 'cell membranes', 'biochemistry', 'cytoskeletal proteins', 'cellular neuroscience', 'cell biology', 'ankyrins', 'physiology', 'neurons', 'biology and life sciences', 'cellular types', 'physical sciences', 'material properties']