microtubules ( mts ) 
 are essential for health and viability of eukaryotic 
 cells . stable mts are fairly rigid , which 
 enables them to serve as important structural and organizing elements . mts form long and durable linear tracks for neuronal transport , and 
 the mechanical properties of mts help to define cell architecture 
 and polarity . the dynamics of mts , i.e. , 
 their ability to undergo stochastic cycles of polymerization and depolymerization , 
 also play a prominent role in many cellular processes . mts play a vital role during cell division , when they form a mitotic 
 spindle ; as a result , different mt disrupting 
 or stabilizing drugs are widely used as chemotherapeutic agents . importantly , the disassembling mts have been 
 proposed to serve as a primary biological motor for poleward chromosome 
 motion during mitosis . however , understanding the underlying 
 mechanisms for different mt functions is impeded by a lack of quantitative 
 knowledge about the thermodynamics and biomechanics of these complex 
 cytoskeletal structures . mts are hollow protein cylinders that 
 contain lateral assemblies 
 of protofilaments : the linear strands of longitudinally arranged -tubulin 
 dimers ( figure 1a ) . a biologically relevant form of mt contains 13 protofilaments that 
 are arranged in a left - hand 3 start helix . such a multi - protofilament 
 structure makes it difficult to establish a direct correspondence 
 between molecular tubulin characteristics and observed mt properties in vitro . theoretical approaches have played an important 
 role in providing such a link , for example , by exploring different 
 mechanisms of mt dynamics and force generation . theoretical studies have identified 
 several microscopic properties that are critically important for understanding 
 the mt behavior . these features include the thermodynamic characteristics 
 of interfaces between adjacent tubulins , i.e. , the energy of lateral 
 and longitudinal bonds , and the mechanical flexural rigidity of individual 
 tubulin protofilaments . to date , it has not been possible to probe 
 these properties via direct experimental measurements , which explains 
 why virtually every aspect of mt thermodynamics and mechanics is still 
 controversial . ( a ) atomic model of -tubulin heterodimer showing the 
 secondary structure : helices ( blue ) , sheets ( red ) , and loops / turns 
 ( gray ) . ( b ) the c-based representation of an mt 
 lattice fragment ( mt8/13 ) with tubulin heterodimers shown in brown 
 ( -tubulin ) and green ( -tubulin ) . in simulations , the 
 cantilever tip ( gray ball ) produces indentations of mt lattice ( arrow 
 shows direction of force ) . ( c ) schematic of mt fragment with locations 
 of 7 specific points for indentation : for points 1 , 2 , and 3 the force 
 is applied onto the surface of the -tubulin , -tubulin , 
 and dimer  dimer interface , respectively ; for points 4 and 5 
 force is applied at the interfaces between two - and two -tubulins , 
 respectively ; for points 6 and 7 force is applied between protofilaments 
 at the junction connecting four tubulin dimers , and within two dimers , 
 respectively . ( d ) atomic structure of the lateral (  
 and  ) interfaces between adjacent protofilaments . ( e ) atomic structure of the longitudinal interfaces : interdimer (  ; 
 left ) and intradimer (  , right ) . tubulin bonds are stronger than the lateral bonds , 
 because during mt disassembly the lateral tubulin bonds dissociate 
 prior to the longitudinal ones . this is evident from the presence 
 of curled  ram horns  at the ends of shortening polymers and is in agreement with computer calculations 
 based on tubulin structure . however , 
 the absolute values of the tubulin  tubulin bond energies are 
 not known . traditional thermodynamic analyses of these bonds have 
 led to ambiguous results mainly because of the complexity of pathways 
 for tubulin assembly and disassembly . indeed , the binding rate constants for tubulin attachment to various 
 sites at the ragged mt tip can vary due to the differences in the 
 number and location of neighboring subunits . additional difficulty concerns the dissociation rate , which can 
 be affected by the number of lateral contacts for a given dimer , as 
 well as by the rigidity of mt protofilaments , which bend concomitantly 
 with tubulin disassembly . the energy of lateral and longitudinal bonds 
 dissociations have previously been estimated using different kinetic 
 and mechanical mt models ; the energy values vary significantly from 
 3 to 15 kbt for the lateral 
 bonds , and from 6 to 20 kbt for the longitudinal bonds . quantum calculations have also been employed , but the obtained estimates 
 are unrealistically large ( up to 186 kbt for the lateral bonds and 158 kbt for the longitudinal bonds ) . the shapes of the free energy profiles and even the geometry and 
 number of the sites for tubulin  tubulin interactions in the 
 mt models are debated . previous theoretical estimates of this quantity vary by an 
 order of magnitude , from 1,500 to 28,000 pn nm , which correspond to energies of 3.7 to 64 kbt per dimer for full protofilament straightening . accurate determination of protofilament rigidity is experimentally 
 difficult because protofilaments are fragile transient structures . 
 knowing flexural rigidity , however , is important , because it has direct 
 implications for mechanisms of force generation during mt depolymerization . indeed , mt depolymerization can generate a large force in 
 vitro and in vivo , but the underlying mechanism is controversial . in the power stroke based mechanism of force generation , 
 the bending protofilaments are thought to transmit a large force available 
 from tubulin tubulin energetics , but this energy can be used 
 to move the associated cargo only if protofilaments are fairly rigid . the various functional roles played by 
 mts in eukaryotic cells 
 necessitate rigorous quantitative analysis of their thermodynamic 
 and mechanical properties . recently , considerable progress has been 
 achieved in the mechanical testing of biological protein assemblies , including the mt response to compressive force . such dynamic force measurements present a unique methodology to 
 deform or rupture the noncovalent bonds of the mt lattice , opening 
 an experimental avenue to determine the underlying microscopic characteristics . the published experimental force  indentation spectra for the 
 mt reveal a complex multistep deformation mechanism . however , a detailed structure - based interpretation of these results 
 has been lacking . here , we have carried out the controlled in silico nanoindentations of the mt by combining molecular 
 dynamics ( md ) simulations accelerated on graphics processing units 
 ( gpus ) of the atomic tubulin structure and the 
 c-based self - organized polymer ( sop ) model of the mt fragment , which contains 13 protofilaments , each 8 tubulin 
 dimers in length ( figure 1 ) . the computational 
 acceleration on gpus has enabled us to apply the experimentally relevant 
 force - loading rate ( cantilever velocity vf= 1.0 m / s ) and to span the experimental 
 time scale ( 50 ms ) . close agreement between experimental and 
 simulated force spectra has allowed us to resolve structural transitions 
 in the mt lattice that underpin the mt lattice biomechanics in the 
 experimentally inaccessible sub - nanometer scale of length . importantly , 
 using our novel methodology of nanoindentation in silico we were able to directly calculate the energies of lateral and longitudinal 
 tubulin  tubulin contacts and to obtain an independent estimate 
 of the flexural rigidity of single tubulin protofilaments . i.e. , the profiles of the indentation force f vs the cantilever tip displacement ( indentation depth ) x ( the fx curves ) and the profiles of f vs the virtual cantilever base ( or piezo ) displacement z ( the fz curves ) , are presented in figure 2a . importantly , these curves are very similar to 
 the corresponding experimental spectra ( see figure 1c in ref ( 32 ) ) . the fz and fx curves exhibit the single - step transitions , 
 characterized by a single force peak , and multistep transitions with 
 several force peaks . although the force spectra show some variability 
 depending on the location of indentation points , each spectrum reveals 
 three distinct regimes ( see supporting movie s1 in the supporting information ) : ( 1 ) the linear - like 
 regime of continuous elastic deformation ( z < 
 1520 nm ; x < 68 nm ) ; ( 2 ) the transition 
 regime where the mt lattice undergoes discrete structural transitions 
 ( 1520 nm < z < 2530 nm ; 68 
 nm < x < 1113 nm ) ; and ( 3 ) the postcollapse 
 regime ( z > 2530 nm ; x > 
 1113 nm ) ( figure 2 ) . we estimated the 
 spring constant kmt from the initial slope 
 of the fz curves ( linear - like regime ) , and extracted 
 the values of critical force f * ( peak force in fz curves ) and the critical distance z * , 
 at which the transition to the collapsed state occurs . these values 
 are in good quantitative agreement with their experimental counterparts 
 ( compared in table 1 ) . the slightly higher 
 theoretical values of f * and z * 
 are due to our using a faster cantilever velocity ( vf = 1.0 m / s vs vf  
 0.2 m / s used in refs ( 32 , 33 ) ) . thus , the sop model provides a very good description of the physicochemical 
 properties of the mt lattice . in the simulations described above , 
 we imposed hard constraints at the ends of the mt fragment ( see materials and methods ) to mimic the long persistence 
 length of the mt polymers ( microns to millimeters ) . however , the force 
 spectra were very similar when the soft ( harmonic ) constraints were 
 applied , and the relative differences between the values of f * , z * , and x * from simulations 
 with soft constraints and hard constraints were within the standard 
 deviations ( data not shown ) . ( a ) the force  deformation 
 ( fz ) curves for 7 indentation points ( figure 1c ) , each depicted with different color . results 
 were obtained with vf = 1.0 m / s 
 and rtip = 10 nm ( see figure s1 in the supporting information for rtip = 15 nm ) . z is the displacement of 
 the virtual cantilever base ( piezo in afm ) . dashed black curves represent 
 the fz profiles for the tip retraction simulations , 
 which followed the forward indentations ( solid black curve ) with z = 17 , 24 , and 35 nm as initial conditions . the top inset 
 shows the corresponding fx curves for the forward 
 indentation ( solid curves ; colors as in a ) and backward tip retraction 
 ( dashed black curves ) . the bottom inset shows the time profiles of the structure 
 overlap  for mt lattice restructuring during tip retraction 
 ( starting from z = 17 , 24 , and 35 nm indentations ) . ( b ) the mt structure snapshots 1 , 2a , 2b , and 3 illustrating the mechanism 
 of mt deformation and collapse . structure 1 : continuous deformation 
 ( z < 1520 nm ; elastic regime ) . structures 
 2a and 2b : disruption of lateral and longitudinal interfaces , respectively 
 ( 2025 nm < z < 2530 nm ; transition 
 regime ) . these structures correspond to the accordingly numbered 
 regions in fz and fx curves in panel 
 a. values are means with standard 
 deviations : mt spring constant , kmt ; critical 
 force , f * ; and critical distance , z * ( cantilever velocity vf = 1.0 m / s ) . experimental data in vitro are from the results of 
 de pablo et al . and shaap et al . , who used vf  
 0.2 m / s and rtip  1520 
 nm the experimental values of kmt were 
 extracted from the experimental histogram ( figure 2 in ref ( 33 ) ) ; the values of f * and z * were taken from the experimental 
 force  indentation curves ( figure 1 in ref ( 32 ) ) . theoretical values of kmt , f * , and z * were obtained by averaging over 3 simulation runs for each indentation 
 point 17 . the summarized 
 description of all observed transitions is provided in table s1 in 
 the supporting information . comparison 
 of the fz and fx curves for a 10 
 nm tip ( figure 2 ) vs a 15 nm tip ( figure s1 
 in the supporting information ) shows that 
 the force spectra are similar , although the values of f * , x * ( critical indentation depth ) , and kmt increase slightly with tip size ( table 1 ) . consider examples of the forward indentation 
 for the 10 nm tip followed by tip retraction at the surface of a protofilament 
 ( indentation points 2 and 3 ; figures 3a and 3b ) and between protofilaments ( indentation points 
 6 and 7 ; figures 3c and 3d ) . the critical force f * and critical indentation 
 depth x * depend on where the compressive force is 
 applied : f * = 0.650.7 nn and x *  12 nm for compressing a protofilament ( figure 3a ) are larger than f * = 0.50.55 
 nn and x *  10 nm for compressing the interface 
 between protofilaments ( figure 3c ) . we also 
 profiled the slope of the fx curve ( df / dx ) , a measure of mechanical compliance of the 
 mt . we found that df / dx varies largely 
 with x ( figures 3b and 3d ) : steep increases interrupted by sudden drops 
 of df / dx indicate that the mt lattice 
 behaves as a soft material . the heights of the peaks of df / dx mark the limits of deformability of the mt cylinder . the mt resists the mechanical collapse longer ( strong last peak of 
 df / dx ) when indented on the protofilament 
 rather than between the protofilaments ( weak last peak of df / dx ) . this indicates that the lateral 
 interfaces between tubulins are softer ( more compliant mechanically ) 
 than between longitudinal tubulins within a protofilament . we arrive 
 at similar conclusions when considering the results obtained with 
 a 15 nm tip for indentation points 1 , 3 , and 5 , 7 ( figure s2 in the supporting information ) . the profiles of structure 
 overlap  ( defined in the supporting information ) show that the collapsed mt lattice remains 8090% 
 similar to the uncompressed state ( the insets to figures 3b and 3d , and figures s2b 
 and s2d in the supporting information ) . this implies that stress - dependent changes are mainly localized to 
 the lateral and longitudinal interfaces . hence , the network of lateral 
 and longitudinal bonds is the origin of elasticity for the mt lattice . force 
 spectra for indentation and retraction and deformation - induced 
 mt structure alterations . shown are results for indentation points 
 2 ( black ) and 3 ( red ) in panels a and b , and for indentation points 
 7 ( black ) and 6 ( red ) in panels c and d obtained with vf = 1.0 m / s and rtip = 10 nm ( see figure s2 in the supporting information for results obtained with rtip = 15 
 nm ) . ( a , c ) the fx curves for forward indentation 
 ( solid black and red lines ) . curves for backward tip retraction ( dashed red lines ) were 
 generated using the structures obtained from forward indentation for x = 7 , 11 , and 21 nm ( indicated on the graphs ) . ( b , d ) the 
 slope df / dx for force spectra from 
 panels a and c. snapshots show the side views of the mt structure 
 before dissociation of the lateral bonds and after dissociation of 
 the longitudinal bonds . the insets show the profiles of structure 
 overlap  vs x. our study demonstrates a qualitative similarity between the fx curves and profiles of df / dx and  for different indentation points ( figures 2 and 3 for 10 nm tip , and figures s1 
 and s2 in the supporting information for 
 15 nm tip ) . analysis of structures generated under different indentation 
 conditions has revealed that the mt transition to the collapsed state 
 occurs by a surprisingly conserved pathway ( supporting movie s1 in 
 the supporting information ) , which we illustrate 
 for two examples of mt indentation ( figure 4 ) . initially , the mt lattice resists deformation , as seen from the 
 increase of df / dx ( the inset in 
 figure 4 ) , which results in small variations 
 in the local curvature of the mt cylinder under the tip ( figure s3a 
 in the supporting information ) . this is 
 the linear - like regime of ( continuous ) elastic deformation as evidenced 
 from the quasi - linear dependence of f on x ( white region in figure 4 ) ; this 
 regime persists until x  68 nm ( structure 
 1 in figure 2 , figure s1 in the supporting information , and figure 4 ; see figure s3a in the supporting information ) . the compressive force loads an increasingly larger portion of 
 the mt surface leading to the mt cylinder flattening (  buckling  ) . indentation beyond x  68 nm can no 
 longer be accommodated by the mt bending alone . at this point , the 
 mt system enters the transition regime ( gray region in figure 4 ) in which discrete structural changes occur . in 
 this regime , mechanical tension exceeds the strength of lateral and 
 longitudinal bonds , which results in their sequential rupture : the 
 lateral bonds dissociate first at x  68 
 nm ( structure 2a , figure 4 ) , and the longitudinal 
 bonds dissociate second at x  1113 
 nm ( structure 2b , figure 4 ) . the latter event 
 triggers the mt lattice rapid transitioning to the collapsed state , 
 which results in a sharp force drop ( force peak in figure 4 ) . this crossover from the continuous deformation 
 to the multistep discrete dissociation transitions was observed in 
 all indentation simulations ( 21 runs ) , regardless of where the compressive 
 force was applied . importantly , we detected the dissociation of the 
 longitudinal interdimer bonds but not the intradimer bonds , consistent 
 with tubulin heterodimer being a major structural unit for mt disassembly . disruption of the lateral interfaces between the -tubulins 
 occurred simultaneously with loss of lateral contacts between the -tubulins . 
 beyond x  20 nm indentation , which corresponds 
 to the postcollapse regime , the tip indented the lower portion of 
 the mt cylinder ( not shown ) , and the resulting events were not analyzed . the fx curves for 
 force application between protofilaments at the junction 
 connecting four tubulin dimers ( indentation point 6 , red curve ) and 
 within the adjacent dimers ( point 7 , black curve ) illustrate that 
 the sequence of transitions is overall similar . images of the mt lattice 
 and tip on the main graph are shown in an mt cross - section view . white 
 area indicates the linear - like regime for elastic deformations , represented 
 by the buckled mt cylinder ( structure 1 , x  
 34 nm ) . gray area corresponds to the transition regime , characterized 
 by the rupture of lateral ( structure 2a , x  
 68 nm ) , then longitudinal contacts ( structure 2b , x  911 nm ) . the inset shows the slope df / dx vs x , so the peaks 
 correspond to the activated system states for the buckling and dissociation 
 transitions . images in the inset are the same structures 1 , 2a , and 
 2b ( with no tip ) viewed from the top ; areas circled in orange contain 
 the disrupted lateral and longitudinal interfaces ( see also figure 
 s3 in the supporting information ) . we carried out simulations of the force - quenched 
 tip retraction with 10 nm tip ( figures 2 and 3 ) and 15 nm tip ( figures s1 and s2 in the supporting information ) , in which we reversed 
 the direction of cantilever motion thereby gradually decreasing to 
 zero the amplitude of compressive force ( see supporting movie s2 in 
 the supporting information ) . we used the 
 mt structures from the simulations of forward deformation for x = 7 , 11 , and 21 nm indentation . these structures correspond 
 to the buckled mt ( x  7 nm ) , the mt with 
 disrupted lateral bonds ( x  11 nm ) , and the 
 mt with disrupted lateral and longitudinal bonds ( x  21 nm ) . to monitor the progress of mt lattice remodeling , 
 we analyzed the structure overlap  ( the inset to figures 2 and s1 in the supporting information ) . in full agreement with experiment ( figures 1 and 5 in ref ( 33 ) ) , we found that the deformation is fully reversible for small indentations 
 ( x < 7 nm ) , partially reversible with small hysteresis 
 for larger indentations ( 7 nm < x < 11 nm ) , 
 and irreversible with large hysteresis for indentations larger than 
 critical ( x *  1113 nm ) . these findings 
 are also supported by the results from the dynamics of mt lattice 
 remodeling , which show that mt restructuring is 100% complete (  
  1 ) over 1020 ms for x = 7 and 11 
 nm indentation , but is incomplete (   0.900.93 ) 
 for x = 21 nm indentation ( the bottom inset in figures 2 and s1 in the supporting information ) . this demonstrates that ruptured lateral contacts between the adjacent 
 protofilaments can be efficiently restored over the time scale of 
 a few tens of microseconds . the disruption of longitudinal bonds , 
 however , inflicts irreparable damage on the mt lattice . our finding that the mt lattice 
 structure in the collapsed state is 8590% similar 
 to the native state ( figures 3 and s2 in the supporting information ) strongly suggests that 
 the compression - induced alterations in the mt lattice are mostly localized 
 to the lateral and longitudinal interfaces , consistent with recent 
 results from other groups . we analyzed 
 the fx curves for forced indentation and force - quenched 
 retraction ( supporting movie s2 in the supporting 
 information ) to determine the enthalpy change h , reversible work wrev , and 
 free energy change g for the mt transitioning 
 from the native state ( x = 0 ) to the collapsed state 
 ( x = 20 nm ) ( see the supporting 
 information for more detail ) . next , we calculated the enthalpy 
 change and free energy change for disruption of the contacts at the 
 lateral interface ( hlat and glat ) and longitudinal interface ( hlong and glong ) . the number of lateral / longitudinal contacts was determined using 
 a conservative estimate of the distance characteristic of contact 
 disruption ( see the supporting information ) . a comparison of hlat , hlong , and glat , glong demonstrates that these 
 state functions show little variation with tip size and tip position 
 on the mt ( table s2 in the supporting information ) . the changes in enthalpy , free energy , and entropy for the disruption 
 of one lateral bond and one longitudinal bond are summarized in table 2 . importantly , the obtained values of glat = 6.9  0.4 kcal / mol and glong = 14.9  1.5 kcal / mol indicate that 
 the intra - protofilament longitudinal tubulin tubulin bonds 
 are roughly twice as strong as the lateral inter - protofilament tubulin  tubulin 
 bonds . this is consistent with our finding that the lateral bonds 
 dissociate prior to the longitudinal bonds . interestingly , the difference 
 in entropy for the rupture of longitudinal bonds vs lateral bonds 
 is roughly 4-fold ( table 2 ) . this large ts difference can be understood 
 from the increased flexibility of the newly created protofilament 
 ends . we also estimated the range for tubulin  tubulin interactions 
 ( see supporting information ) , and found 
 that the longitudinal bonds are characterized by the longer interaction 
 range ( ylong  1.251.5 
 nm ) compared to the lateral bonds ( ylat  0.851.1 nm ) . both of these values lie within the 
 1.52 nm interaction range characteristic of protein complexes . averaging is performed over all 
 indentation points ( 17 , figure 1c ) 
 and for 10 and 15 nm tip . we analyzed mechanical deformations 
 of the mt cylinder using a thin - shell approximation ( see supporting information ) . for the average slope of the fx curves of kmt = 51.8  2.8 pn / nm ( for simulations 
 with 10 nm tip ; table 1 ) , the mt flexural rigidity 
 ei comes to ( 25,400  1,500 )  10 pn nm , which corresponds to the mt persistence length lp = ei / kbt of 6.18  0.36 mm . to estimate the flexural rigidity of tubulin 
 strands , we performed simulations of bending deformation of single 
 protofilaments formed by 8 ( pf8/1 ) , 16 ( pf16/1 ) , 24 ( pf24/1 ) , and 
 32 ( pf32/1 ) dimers ( see supporting movie s3 in the supporting information for pf16/1 ) . in these simulations , 
 the protofilament ends were clamped and the bending in response to 
 forced indentations was examined . the simulated force deformation 
 spectra , i.e. , the profiles of the deformation force f vs the cantilever tip displacement ( deformation ) x ( the fx curves ) , and the corresponding profiles 
 of the deformation energy ( obtained by calculating the area under 
 the fx curve ) vs x are presented 
 in figure s5 in the supporting information . using the harmonic approximation valid for small 23 nm 
 deformations we find that the values of ei for these protofilament 
 fragments are in the range of 18,00026,000 pn nm ( figure s5 in the supporting information , table 3 ) , which corresponds to the 4.56.6 
 m range for the persistence length ( table 3 ) . values are averages with standard 
 deviations of the flexural rigidity ( ei ) and persistence length ( lp ) obtained from 5 bending simulation runs for 
 single protofilaments of 8 ( pf8/1 ) , 16 ( pf16/1 ) , 24 ( pf24/1 ) , and 
 32 ( pf32/1 ) tubulin dimers . dynamic , mechanical , and force - generating properties of mts play 
 important roles in many cellular processes , but little is known about 
 the thermodynamics of tubulin  tubulin interactions and mechanics 
 of individual protofilaments that form the mt lattice . previously , 
 the energies of lateral and longitudinal bonds dissociations 
 were estimated with the help of molecular - mechanical models , in which 
 the tubulin monomer / dimer was the smallest unit . the major drawback 
 of this approach is that tubulin energies are derived from the dynamic 
 parameters of mt assembly and disassembly , which report on the thermodynamics 
 of tubulin  tubulin interactions only indirectly . in contrast , 
 the afm - based dynamic force measurements provide a more straightforward 
 experimental avenue , because in these experiments the protofilaments 
 deformation and tubulin  tubulin bond rupture events are recorded 
 with high spatial and temporal resolution . however , due to the complexity of the multi - protofilament mt structure , 
 the molecular interpretation of experimental force  indentation 
 spectra at the level of protein protein bonds is not trivial , 
 as it requires the structure - based understanding of fine features 
 of the experimental spectra . we have overcome this limitation 
 by carrying out the dynamic force measurements in silico using the amino acid as the smallest structural unit . our computer - based 
 experiments mimic the afm based dynamic force experiments in vitro . the full control over the system we have during 
 the entire course of forced deformation ( contact point and direction 
 of force application , constrained residues , indenter size ) and the 
 structural resolution ( intact versus disrupted lateral / longitudinal 
 interfaces ) allows us to directly correlate the energy changes with 
 the structure alterations at the residue level . our approach to forced 
 indentation in silico involves following stochastic 
 dynamics of mechanical deformation of a biological particle , which 
 is microscopically reversible when a force loading is sufficiently 
 slow . in this regime of compressive force application , the rate of 
 force increase is slower than the rate of system re - equilibration 
 at each point along the deformation reaction path ( quasi - equilibrium ) . this can be gleaned , e.g. , from the comparison of fx curves for the 24 dimer long protofilament fragment pf24/1 obtained 
 using varying cantilever velocities ( see figure s6 in the supporting information ) . we see that as vf decreases , the fx curves 
 become less and less different . for example , the fx curves obtained for pf24/1 with vf = 
 1.0 and 0.5 m / s look almost identical , implying similar mechanical 
 responses ( figure s6 in the supporting information ) . these results show that in silico indentation 
 experiments reported here are carried out under near - equilibrium force - loading 
 conditions . the dynamic force spectroscopy in silico was previously 
 applied by us to examine the forced unfolding of fibrin polymers and to map the free energy landscape for deformation 
 of the cowpea chlorotic mottle virus capsid . this approach is made possible by combining the atomic - level and 
 c-based coarse - grained modeling with nanomanipulation 
 of the mt lattice in silico . by taking advantage of the computational acceleration on a gpu we 
 were able to carry out detailed exploration of mt biomechanics on 
 a long time scale ( 50 ms ) using the experimentally relevant 
 conditions of force application . with this 
 approach , the atomic - level details underlying the lateral and longitudinal 
 interactions are implicit in the sop model of the mt cylinder . the 
 next - neighbor interactions that stabilize mt structure and the lattice 
 confinement for individual dimers are explicitly described . within 
 the context of these advances , although this model can be applied 
 to describe processes that occur on a millisecond time scale , given 
 current computational limitations it can not be applied to much slower 
 processes , such as the rupture of tubulin bonds and protofilament 
 bending during mt depolymerization . here , we have determined 
 the thermodynamic and mechanical characteristics 
 of the mt that are difficult to access experimentally by analyzing 
 the force  deformation curves from in silico nanoindentation simulations . the area under the fx curve is the total work performed on the system , and the reversible 
 part of work can be linked to the gibbs free energy change . hence , 
 the obtained free energies for the tubulin bonds dissociation 
 are based on theoretical analyses of in silico experiments , 
 in which these bonds are directly manipulated . the only free parameter 
 of the sop model is h , but the values of this quantity 
 for each group of contacts between amino acids were calculated using 
 the all - atom md simulations ( table s3 in the supporting 
 information ) . the sop model based nanoindentation assays provided 
 the force  indentation curves which agree well with the experimental 
 afm spectra . thus , the agreement between the 
 experiment and simulations was achieved without model fitting and 
 without adjustment of free parameters . the detailed understanding 
 of the mechanisms of mt deformation and structural collapse that we 
 have achieved offers unique insights into the mechanochemistry of 
 the mt lattice . first , we found that the compressed mt 
 behaves as a system of rigid elements interconnected through a network 
 of lateral and longitudinal elastic bonds . large rigidity of tubulin 
 monomers agrees well with the results of prior computational modeling 
 study , in which tubulin monomers were found to have a stable central 
 core . accordingly , under small deformations 
 the mt cylinder responds elastically , while undergoing continuous 
 deformation characteristic of long wavelength modes . the importance 
 of slow global modes in mechanical deformation of the mt lattice has 
 also been revealed in a recent elastic network modeling study . beyond 34 nm compression , the -tubulin 
 dimers buckle , leading to flattening of the mt cylinder . with further 
 compression the lateral contacts between the adjacent protofilaments dissociate ( both  and  
 lateral bonds behave similarly ) , which is then followed by the rupture 
 of longitudinal interdimer but not intradimer bonds ( figure s4 in 
 the supporting information ) ; these discrete 
 structure changes are characteristic of short wavelength modes . importantly , 
 the sequence of microscopic events during mechanical mt compression 
 defined here is likely to provide a blueprint for a pathway of normal 
 mt disassembly . indeed , in the course of mechanical compression tubulin 
 bonds dissociate in the same order as during normal mt disassembly 
 ( lateral bonds prior to the longitudinal ones ) . since the gibbs free 
 energy change for any bond dissociation is a state function and , therefore , does not depend on the exact cause of transition , our findings from 
 nanoindentation experiments are directly applicable to describe and 
 model the molecular events during mt disassembly . second , our 
 work provides direct estimates of the free energies of dissociation 
 of the lateral and longitudinal tubulin  tubulin bonds . we show 
 that interfaces between tubulin subunits in the mt wall are characterized 
 by strong noncovalent interactions . structural analyses revealed that 
 ( i ) the  and  lateral 
 interfaces are formed by a total of 19 and 21 stable residue residue 
 contacts , respectively , and that ( ii ) the longitudinal intradimer 
 bonds and interdimer bonds are stabilized by 78 and 38 total contacts , 
 respectively . in both the lateral and longitudinal interfaces , the 
 most stable residue residue contacts are hydrophobic bonds 
 and salt bridges ( figure s4a  d in the supporting 
 information ) , consistent with a recent molecular modeling study ( major structural determinants in tubulin monomers 
 involved in the intermonomer contacts formation are accumulated 
 in table s4 in the supporting information ) . the difference between the obtained energies of the lateral and 
 longitudinal bonds , g = glong  glat = 8.0 kcal / mol ( 13.3 kbt ) , is close to the 11 kbt estimate reported earlier . importantly , the large strength of the tubulin  tubulin bonds 
 reported here should prompt a re - evaluation of current molecular models 
 of mt dynamics and stability , where these characteristics play a significant 
 role . this finding provides an important clue 
 for understanding the molecular mechanisms of mt rescue , an abrupt 
 switch from mt depolymerization to polymerization . we suggest that 
 high reversibility of lateral bond dissociation can promote  sealing  
 of the cracks between shortening protofilaments in the mt wall , thereby 
 inhibiting the depolymerization . another insight from our study concerns 
 the geometry of contacts in the lateral tubulin interfaces . previously , 
 it has been suggested that the steplike  gaps  in the 
 force  indentation curves obtained with afm reflect the existence 
 of two additional interaction sites between the dimers in adjacent 
 protofilaments . our results demonstrate 
 that a single pair of lateral interaction sites can account for all 
 features of the experimental force spectrum , and the steplike gaps 
 appear to result from the dissociation of lateral bonds under 68 
 nm deformations ( figure 4 ) . interestingly , 
 subjecting the mt lattice to mechanical stress can lead to the formation 
 of small defects at the junction points connecting - and -tubulins ; 
 these defects grow in size with increasing force load ( figure s3 in 
 the supporting information ) . such defects 
 could also emerge due to the mechanical  fatigue  of 
 an mt lattice that repeatedly experiences large deformations , similar 
 to the behavior of carbon nanotubes . however , since 
 the local force to cause such a crack is large ( > 300 pn ) , thermal 
 vibrations of the mts are highly unlikely to lead to the mt lattice 
 fatigue , so they can not explain the mt 
  aging  in vitro . yet , crack formation might be pertinent to the activity 
 of mt - severing enzymes , which are thought to exert large local forces 
 while pulling the tubulins out of the mt wall . tubulin 
 bond is irreversible on the time scale of a few tens of microseconds . the corresponding irreversibility of the mt lattice restructuring 
 is directly related to the long time scale required for the re - formation 
 of all the longitudinal bonds and recovery of the mt lattice structure 
 upon large indentation , a feature that was also detected in the original 
 afm experiments . the authors found that it would 
 take 4 min for the mt lattice to fully self - heal ( in the absence 
 of free tubulin in solution ) following an indentation beyond 10 nm . hence , this finding shows that even after disruption of lateral and 
 longitudinal bonds the lattice is able to recover , but only on a long 
 time scale of a few minutes . to further explore this ( ir)reversibility 
 aspect , we carried out simulations of force - induced forward indentation 
 followed by force - quenched backward tip retraction , but for the short 
 protofilament fragment pf8/1 . interestingly , 
 we found that , following the initial dissociation of the longitudinal 
 bond between the 4th and 5th dimers , when the direction of tip motion 
 was reversed , the bond re - formed and the protofilament unbent completely 
 over a 1020 ms time scale ( figure s7 in the supporting information ) . hence , the longitudinal bond dissociation 
 in short protofilament fragments ( such as pf8/1 ) is fully reversible 
 on the millisecond time scale . this can be explained by lower entropic 
 barriers for reassociation of tubulin dimers in single protofilaments 
 compared to the mt lattice . the large flexural rigidity of individual protofilaments 
 reported here implies that tubulin protofilaments are fairly rigid . our estimates ( 18,00026,000 pn nm , table 3 ) come close to the experimental values from several 
 studies ( table 1 in ref ( 18 ) ) and agree with those from a recent all - atom md simulation 
 study ( ei = 27,740 pn nm ) . this large protofilament s rigidity supports the idea that 
 almost the entire energy from gtp hydrolysis is stored as mechanical 
 tension in a straightened protofilament . the obtained value of the persistence length for the mt cylinder lp = 6.18 mm agrees well with the experimentally 
 measured value 3.45 mm ( ref ( 1 ) ) and with the estimates from other computation studies , demonstrating the validity of 
 sop modeling . importantly , our results imply that tubulin protofilaments 
 are about 3 orders of magnitude more flexible than the intact mt cylinder 
 ( of comparable length ) , which is in tune with some but not all previous 
 estimates . we also obtained a similar 3 orders of magnitude difference for 
 the persistence length , i.e. , 6.18 mm for the mt cylinder vs 4.56.6 
 m for the tubulin protofilaments ( table 3 ) , implying a stabilizing role played by the lateral tubulin our work has important implications for the mechanism 
 of force generation by the disassembling mts . the ability of shortening 
 mts to transport a large cargo in vitro ( up to 30 
 pn ) has been proposed to result from 
 different mechanisms , including the biased - diffusion and power stroke 
 based models . only the latter mechanism takes 
 a direct advantage of the tubulin dissociation pathway during which 
 the lateral tubulin bonds dissociate prior to the longitudinal ones , 
 causing a splaying of the protofilaments into the  ram s 
 horns  structures . this structural transition 
 has been proposed to exert a power stroke , capable of moving the appropriately 
 attached cargo . although sop modeling does not allow one to calculate directly the 
 mt disassembly due to the prohibitively large computational time , 
 it is interesting to discuss the estimates obtained here in the framework 
 of the current models for mt force generation . the work to straighten 
 the 832 dimer long protofilament with the intradimer bending 
 angle of 0.4 rad can be estimated from the protofilament s 
 rigidity , assuming that the protofilament behaves as a hookean spring : 
 3958 kbt . this large bending energy suggests that a significant portion of 
 this chemical energy can be converted into useful work . tubulin dissociation energies obtained here 
 support the proposal that the disassembling mt can serve as a strong 
 depolymerization motor . we hope that future 
 advances in computational molecular modeling and availability of high - resolution , 
 nucleotide specific tubulin structures will enable a direct testing 
 of these conclusions . the modeling tools we have developed here can 
 also be applied to study other complex biological assemblies when 
 their detailed physicochemical characteristics can not be resolved 
 using modern experimental approaches . the structure of a finite - length 
 fragment of the mt lattice was obtained from the 13 subunit ring structure 
 of -tubulin dimers , as in ref ( 60 ) . this ring structure utilizes 
 atomic coordinates of the -tubulin dimer ( pdb code : 1jff ) , in which the e - site in -tubulin contains gdp . the 
 finite - length fragment of gdp - tubulin mt lattice ( mt8/13 , figure 1a ) was constructed by replicating the ring structure 
 8 times using the shift distance of 85  to obtain an mt construct 
 of 8 dimers in length ( mt8/13 ; see figure 1b ) . we used the all - atom molecular dynamics simulations in implicit 
 water ( sasa and gb models of implicit solvation ) to obtain an accurate 
 parametrization of the sop model , as described below ( see table s3 
 in the supporting information ) . the structures 
 of 8 , 16 , 24 , and 32 dimer long single protofilaments ( pf8/1 , pf16/1 , 
 pf24/1 , and pf32/1 ) were extracted starting from the structure of 
 mt8/13 . we used the sop 
 model of the polypeptide chain to describe 
 each monomer ( -tubulin and -tubulin ) . in the topology - based 
 sop model , each amino acid is represented by a single interaction 
 center ( c-atom ) , and the cc covalent bond with the bond distance a = 3.8  ( peptide bond length ) . the potential energy function 
 of the protein conformation usop specified 
 in terms of the coordinates { ri } = r1 , r2 , ... , rn ( n is the total number of amino acid residues ) is given by usop = ufene + unbatt + unbrep . the finite extensible nonlinear elastic potential ufene = i=1n1(k/2)(r02){log[1  
 ( ri , i+1  ri , i+10)/r02 ] } with the spring constant k = 14 n / m and 
 the tolerance in the change of a covalent bond distance r0 = 2  describes the backbone chain connectivity . 
 the distance between residues i and i + 1 is ri , i+1 , and ri , i+10 is its value 
 in the native ( pdb ) structure . we used the lennard - jones potential unbatt = i=1n3j = i+3nh[(rij0/rij ) 2(rij0/rij)]ij to account for the noncovalent ( nonbonded attractive ) interactions 
 that stabilize the native folded state . we assumed that if the noncovalently 
 linked residues i and j ( |i  j| > 2 ) are within the cutoff 
 distance rc = 8  in the native state , 
 then ij = 1 , and it is zero otherwise . the non - native ( nonbonded repulsive ) interactions unbrep = i=1n2l(l / ri , i+1 ) + i=1n3j = i+3nl(rij0/rij)(1ij ) are treated as repulsive . an additional constraint was imposed 
 on the bond angle formed by residues i , i+1 , and i+2 by including the repulsive potential 
 with parameters l = 1 kcal / mol and l = 3.8  . a more detailed description of the sop model parametrization 
 is presented 
 in the supporting information . in short , 
 in the sop model , the parameter h defines the average 
 strength of noncovalent residue importantly , the values of h were 
 calculated directly using md simulations of an atomic structure model 
 mt8/13 of the mt lattice at t = 300 k. the atomic - level 
 details that determine the type and number of binary contacts between 
 amino acids and their energies were ported to the sop model of the 
 mt lattice . three 10 ns simulation runs were performed to calculate 
 for each group of contacts the average nonbonded energy ( enb ) , given by the sum of the van der waals energy ( lennard - jones 
 potential ) and the electrostatic energy ( coulomb potential ) , and the 
 average number of binary contacts between amino acids ( nnb ) that stabilize the native mt structure ( native contacts ) . we assumed that a pair of residues formed a contact if the distance 
 between their c-atoms in the native state does not 
 exceed the cutoff distance rc . all the native contacts were divided into five groups 
 ( contact types ) : ( 1 ) the intramonomer contacts in the -tubulin 
 monomers ; ( 2 ) the intramonomer contacts in the -tubulin monomers ; 
 ( 3 ) the intradimer contacts that stabilize the dimer s structure ; 
 ( 4 ) the longitudinal interdimer contacts between any two dimers along 
 the mt cylinder axis ; and ( 5 ) the lateral interdimer contacts between 
 the -tubulin monomers and between the -tubulin monomers 
 in adjacent protofilaments . to calculate the energy for nonbonded 
 interactions , we employed the solvent accessible surface area ( sasa ) and generalized born ( gb ) models of implicit solvation , which are based on the charmm19 
 force - field . we used the output from 
 sasa model based simulations ( coordinate and energy files ) to calculate 
 the values of enb and nnb for the contact groups 13 . since electrostatic 
 interactions are important for the formation of longitudinal and lateral 
 tubulin tubulin bonds , we used a more accurate gb model to 
 calculate enb and nnb for the contact groups 4 and 5 . finally , dividing enb by the corresponding value of nnb for each contact group , we obtained the value of h ( see table s3 in the supporting information ) , which were used in all simulations reported here . nanoindentation measurements 
 were performed at different points on 
 the mt surface using the spherical tip of radius rtip = 10 and 15 nm ( figure 1c ) , 
 similar to the 15 nm tip used in atomic force microscopy ( afm ) experiments . in the simulations of mechanical indentation of the mt cylinder 
 and deformation of single protofilament fragments , the tip mt 
 lattice interactions and the tip  protofilament interactions 
 were modeled by the repulsive potential , utip = tip[tip/(ri rtip ) ] , where ri is the position of the ith particle , tip = 4.18 kj / mol , and tip = 1.0  . in 
 the forward indentation measurement , the tip exerted the time - dependent 
 compressive force f(t ) = f(t)n in the direction n perpendicular to the mt or protofilament surface ( figure 1b ) . the force magnitude f(t ) = rft was 
 increased with the force - loading rate rf = vf , where vf = 1.0 m / s ( for mt indentations ) and vf = 0.2 m / s ( for protofilament deformations ) is 
 the velocity of the cantilever base ( piezo ) represented by a virtual 
 particle , and  = 0.05 n / m is the cantilever spring constant . the force f(t ) is transmitted to 
 the tip through the cantilever spring , and the resisting indentation 
 force ( for mt lattice ) or deformation force ( for the protofilament ) f is calculated using the energy output from simulations . 
 in the simulations of backward ( tip ) retraction , the direction of 
 tip ( or piezo ) motion was reversed , which resulted in a gradual decrease 
 of the compressive force to zero . these simulations 
 were performed using the sop model of mt8/13 and langevin simulations 
 accelerated on a gpu . to account for the long persistence 
 length of mts ( microns to millimeters ) , positions of the c-atoms for 
 a total of 9 residues 248 , 253 , 257 , 262 , 325 , 326 , 329 , 348 , 349 
 in each tubulin monomer at the plus mt end , and positions of the c-atoms for 9 residues 98 , 176 , 177 , 180 , 
 221 , 224 , 225 , 403 , 407 in each tubulin monomer at the plus mt end , 
 were constrained ( figure 1 ) . we implemented 
 hard constraint conditions , in which the positions of all 9 constrained 
 residues were fixed , and soft harmonic constraints , in which we connected 
 these same 9 residues to a virtual wall through a harmonic spring 
 with the spring constant of  = 0.1 nn / nm . a total of 42 indentation 
 runs were generated ( using hard constraints ) : 3 runs per indentation 
 point 17 ; 21 runs for each rtip value . we profiled the dependence of the indentation force f in dynamic force experiments on the cantilever tip displacement x ( indentation depth ) and z ( the piezo 
 displacement ) for all indentation points ( figure 1c ) . the fx curves show higher sensitivity 
 to the mt deformation dynamics than fz curves , and 
 so x is a better reaction coordinate . however , for 
 the purpose of comparing the results of experiments with simulations , 
 we analyzed the fz and fx curves 
 ( figure 2a ) . these simulations were performed 
 using the sop models of protofilament 
 fragments pf8/1 , pf16/1 , pf24/1 , and pf32/1 and langevin simulations 
 on a gpu . to constrain a protofilament , we fixed the same positions 
 of the c-atoms at the n - terminus 
 of the first monomer and at the c - terminus of the last monomer . the 
 cantilever tip was set to move in the direction perpendicular to the 
 protofilament axis , as shown in figure s5 in the supporting information . a total of 12 indentation runs for 
 indentation point 3 were generated for all four protofilament fragments : 
 3 runs per fragment ( rtip = 10 nm ) . we 
 profiled the dependence of the deformation force f on the cantilever tip displacement x for the protofilament 
 fragments ( figure s5 in the supporting information ) . flexural rigidity of the protofilament fragments was calculated 
 in the harmonic approximation as described in the supporting information . because the harmonic approximation 
 is valid only in the regime of small deformations , we analyzed the 
 initial quadratic 23 nm portion of the fx curves ( figure s5 in the supporting information ) . dimer 
 bending angle of 12. to obtain accurate estimates 
 of the gibbs free energy change associated with the disruption of 
 single lateral interface and longitudinal interface ( glat and glong ) , we determined the mechanical work performed on the mt ( area under 
 the fx curve ) . since it is not possible to apply 
 infinitely slow force loading , which would correspond to the equilibrium 
 conditions of mechanical force application ( and reversible work ) , 
 the total work ( w ) in our indentation cycle contains 
 the reversible part wrev , which is spent 
 to deform the mt lattice , and to dissociate the lateral and longitudinal 
 bonds , as well as the irreversible part wirrev ( see hysteresis in the fx curves in figures 2 and 3 ; see also figures 
 s1 and s2 in the supporting information ) . we calculated the reversible part of work wrev using the crooks theorem ( see the supporting 
 information ) . these analyses for 
 mechanical compression of mt8/13 , and bending deformation of pf8/1 , 
 pf16/1 , pf24/1 , and pf32/1 , including the structure visualization 
 and determination of the thermodynamic quantities ( g , h , and ts ) , flexural rigidity ( ei ) , persistence length ( lp ) , and the range of lateral and longitudinal 
 bonds ( y ) , are described in detail in the supporting information .
microtubules , 
 the primary components of the chromosome segregation 
 machinery , are stabilized by longitudinal and lateral noncovalent 
 bonds between the tubulin subunits . however , the thermodynamics of 
 these bonds and the microtubule physicochemical properties are poorly 
 understood . here , we explore the biomechanics of microtubule polymers 
 using multiscale computational modeling and nanoindentations in silico of a contiguous microtubule fragment . 
 a close 
 match between the simulated and experimental force 
 deformation 
 spectra enabled us to correlate the microtubule biomechanics with 
 dynamic structural transitions at the nanoscale . 
 our mechanical testing 
 revealed that the compressed mt behaves as a system of rigid elements 
 interconnected through a network of lateral and longitudinal elastic 
 bonds . 
 the initial regime of continuous elastic deformation of the 
 microtubule is followed by the transition regime , during which the 
 microtubule lattice undergoes discrete structural changes , which include 
 first the reversible dissociation of lateral bonds followed by irreversible 
 dissociation of the longitudinal bonds . 
 we have determined the free 
 energies of dissociation of the lateral ( 6.9  0.4 kcal / mol ) 
 and longitudinal ( 14.9  1.5 kcal / mol ) tubulin 
 tubulin 
 bonds . 
 these values in conjunction with the large flexural rigidity 
 of tubulin protofilaments obtained ( 18,00026,000 pnnm2 ) support the idea that the disassembling microtubule is capable 
 of generating a large mechanical force to move chromosomes during 
 cell division . 
 our computational modeling offers a comprehensive quantitative 
 platform to link molecular tubulin characteristics with the physiological 
 behavior of microtubules . 
 the developed in silico nanoindentation method provides a powerful tool for the exploration 
 of biomechanical properties of other cytoskeletal and multiprotein 
 assemblies .