document: in many studies of cell mechanics and dynamics , the cell is characterized as a viscoelastic body @xcite . 
   it is an intriguing question to what extent such mechanical behaviour can be rationalized in terms of comparatively simple polymer physics models . in this respect , 
   the comparison of cell rheological data and minimalistic _ in vitro _ reconstituted constructs of the cytoskeleton , such as pure actin solutions @xcite or crosslinked actin networks @xcite , has recently provided many new insights . 
   important progress has also been achieved in the development of phenomenological mathematical descriptions . 
   this includes approaches related to the tube model @xcite , tensegrity - based approaches @xcite , effective - medium models @xcite , and some others @xcite . in particular 
   , the glassy wormlike chain ( ) model @xcite , a phenomenological extension of the standard  model of semiflexible polymers @xcite has been successful in describing a plethora of rheological data for polymer solutions @xcite and living cells @xcite over many decades in time with a minimum of parameters . 
   however , all these studies were primarily concerned with _ viscoelastic _ 
   behaviour , while the latest investigations have underscored the glassy @xcite fragile @xcite , and inelastic @xcite character of the mechanical response of living cells . 
   even for biopolymer networks _ in vitro _ 
   , experiments operating in the nonlinear regime had so far to resort to special protocols that minimize plastic flow @xcite in order to make contact with dedicated theoretical models .    the aim of the present contribution is to overcome this restriction by extending the  to situations involving inelastic deformations . as a first step , we concentrate onto _ reversible inelastic _ 
   behaviour , where the deformation does not alter the microscopic ground state . 
   the protocol applied by trepat _ 
   et al . _ 
   @xcite provides a paradigmatic example . 
   cells are subjected to a transient stretch such that , after some additional waiting time in the unstretched state , the ( linear ) material properties of the initial state are recovered . the simplification for the theoretical modelling results from the assumption that not only the macro - state but also the micro - state of the system may to a good approximation be treated as reversible under such conditions ; i.e. 
   , we assume that the complete conformation of the polymer network , including the transiently broken bonds between adjacent polymers , is constrained to eventually return to its original equilibrium state . for the time - delayed hysteretic response of the network to such protocols one could thus still speak of a viscoelastic ( `` anelastic '' ) response in an operational sense , but we refrain from doing so in view of the fundamentally inelastic nature of the underlying stochastic process  in contrast to the reversible softening effects observed in @xcite , for example . 
   indeed , by simply allowing bonds to reform in new conformational states , the model developed below can readily be extended to arbitrary irreversible plastic deformations , as will be demonstrated elsewhere @xcite . before entering the discussion of our model 
   , we would also like to point out that the proposed ( inelastic ) extension of the  is strongly constrained by identifying the newly introduced parameters with those of the original ( viscoelastic )  model , where possible . despite its increased complexity , the extended model will therefore enable us to subject the underlying physical picture to a more stringent test than hitherto possible by comparing its predictions to dedicated experiments . moreover , unlike current state - of - the - art simulation studies @xcite it is not limited to rod networks but is firmly routed in a faithful mathematical description of the underlying brownian polymer dynamics .    this paper is organized as follows . 
   first , we review some basic facts about the  in section [ sec : gwlc ] . next , in section [ sec : interaction ] 
   , we introduce our extended reversible inelastic version , which we formulate using the notion of an effective interaction potential as in the original construction of the  in @xcite . 
   ( a preliminary account of the basic procedure and some of its cell - biological motivation including reversible bond - breaking kinetics has recently been given in a conference proceedings @xcite . ) sections [ sec : viscoelastic ] and [ sec : fluidization ] explain the physical mechanism underlying the mechanical response under pulsed and periodically pulsed loading , while section [ sec : remodelling ] illustrates its phenomenology . we demonstrate that the model exhibits the hallmarks of nonlinear cell mechanics : strain / stress stiffening , fluidization , and cyclic softening @xcite . 
   section [ sec : intr_lengths ] investigates the relevance of the lately quantified structural heterogeneities in networks of semiflexible polymers @xcite for the mechanical properties , before we conclude and close with a brief outlook . 
   the glassy wormlike chain ( ) is a phenomenological extension of the wormlike chain ( ) model , the well - established standard model of semiflexible polymers . 
   a broad overview over  and  dynamics can be found elsewhere @xcite . 
   the  describes the mechanics of an isolated semiflexible polymer in an isothermal viscous solvent . in the weakly bending rod approximation , 
   a solution of the stochastic differential equations of motion for the  is possible _ via _ a mode decomposition ansatz for the transverse displacement of the polymer contour from the straight ground state . 
   the individual modes labelled by an index @xmath0 are independent of each other and decay exponentially with rates @xmath1 . for convenience , 
   we set the thermal energy @xmath2 , so that the bending rigidity can be identified with the persistence length @xmath3 , in the following . using this convention , 
   the  expression for the transverse susceptibility of a polymer of contour length @xmath4 ( with respect to a point force ) reads @xcite @xmath5 here , @xmath6 is an optional backbone tension , and @xmath7 the euler buckling force for a hinged rod of length @xmath4 . 
   the different powers of @xmath0 in the denominator give notice of the competition of bending and stretching forces . 
   in the , interactions of the polymer with the surrounding network ( e.g.  excluded volume interactions or sticky contacts ) are reflected in an altered  mode relaxation spectrum @xcite . the intuitive  albeit not fully microscopic  picture underlying the formulation of the  is illustrated in figure  [ fig : cartoon ] , which depicts a test polymer reversibly bound to the background network _ via _ potential wells at all topological contacts , the so - called entanglement points . 
   consider now a generic point somewhere along the polymer backbone ( not an entanglement point ) . 
   it can relax freely until the constrictions are being felt , which slow down the contributions from long wavelength bending modes . 
   the  translates this intuition into a simple prescription for the mode spectrum : the short - wavelength modes are directly taken over from the wormlike chain model , while modes of wavelength @xmath8 longer than the typical contour length @xmath9 between adjacent bonds are modified to account for the slowdown . motivated by the physical picture illustrated in figure [ fig : cartoon ] , the slowdown of the relaxation of a wavelength @xmath8 is expressed by an arrhenius factor @xmath10 $ ] for breaking @xmath11 potential energy barriers of height @xmath12 simultaneously . 
   accordingly , the phenomenological recipe to turn a  into a reads : @xmath13 & \lambda_n \geq \lambda \end{array}\right . . \ ] ] upon inserting this into ( [ eq : sus_wlc ] ) one obtains the susceptibility @xmath14 . 
   the microscopic `` modulus '' for transverse point excitations of a generic backbone element on a test polymer is then defined as the inverse susceptibility , @xmath15 . 
   an approximate expression for the macroscopic shear modulus is obtained along similar lines @xcite .    in the original equilibrium  theory , @xmath9 was assumed to be a constant on the order of the entanglement length of the network , @xmath16 . 
   note , however , that @xmath17 is a geometric quantity ( which is determined by the polymer concentration and the persistence length ) while the contour length @xmath9 between closed bonds clearly depends on the state of the bond network . 
   one therefore has to allow for an increase of @xmath9 with the number of broken bonds in non - equilibrium applications . 
   this issue is explored in the following . 
   all mechanical quantities calculated within the  model crucially depend on the interaction length @xmath9 . in previous applications of the model @xcite it was assumed that @xmath9 remains constant  equal to its equilibrium value and unaffected by the deformation of the sample . in other words , 
   the equilibrium theory allowed for statistical bond fluctuations but not for a dynamical evolution of the parameters characterizing the thermodynamic state of the bond network . 
   an obvious starting point for generalizations of the model to non - equilibrium situations is therefore to consider the number of closed bonds , and therefore also @xmath9 , as _ dynamic variables _ , dependent on the strain- and stress - history of the network . 
   we now provide a mean - field description to account for such a dynamically evolving bond network . for clarity , we return to the intuitive picture underlying the where the ( possibly crosslinker- or molecular - motor - mediated ) complex interactions between the polymers are summarized into an effective interaction potential for a test segment against the background , as sketched in figure [ fig : potential ] . 
   the same idea has also been used before in many related situations ( e.g.@xcite ) .     between both states . 
   the transition rates @xmath18 and @xmath19 between the bound and unbound state depend on the barrier heights @xmath12 and @xmath20 , respectively . 
   an externally applied force can tilt the potential and favour binding or unbinding events.,width=9 ]    the generic potential exhibits three essential features : a well - defined bound state , a well - defined unbound state , and a barrier in between ( figure [ fig : potential ] ) . 
   the confining well corresponding to the unbound state represents the tube - like caging of the test polymer within the surrounding network @xcite . 
   for ease of notation , we further introduce the dimensionless fraction of closed bonds or , _ bond fraction _ 
   @xmath21 which is simply the ensemble - averaged fraction of sticky contacts that are actually in the bound state . 
   the minimum bond distance @xmath22 is typically on the order of @xmath17 , but may be somewhat larger in situations where the bonds are mediated by crosslinker molecules or by partially sterically inaccessible sticky patches ( as e.g.  for helical molecules @xcite ) .    a quantitatively fully consistent way of calculating the dynamics of @xmath23 would involve solving the full fokker - planck equation for a -contact  a formidable program to be pursued elsewhere @xcite . here , 
   for the sake of our qualitative purposes , we chose to concentrate onto the physical essence of the discussion , and keep the mathematical structure as simple and transparent as possible . 
   we therefore approximate the dynamics by a simple exponential relaxation as familiar from the standard example of reacting brownian particles , conventionally described by kramers theory with a bell - like force dependence @xcite . using this simplification and assuming a schematic interaction potential as depicted in figure  [ fig : potential ] , the value of @xmath23 is determined by a competition of bond breaking and bond formation with reaction rates @xmath18 and @xmath19 , respectively . 
   both rates are represented in the usual adiabatic approximation ( meaning that the equilibration _ inside _ the wells is much faster than the barrier crossing and external perturbations ) by @xcite @xmath24 where @xmath25 is an intrinsic characteristic brownian time scale for bond breaking and formation and @xmath26 are characterized by different brownian time scales depending on the width of the potential wells . at the present stage , we do not bother to distinguish these time scales nor to fine - tune their numerical values , and identify them , for the sake of simplicity , with the entanglement time scale @xmath27 in our numerical calculations . 
   ] , and @xmath6 is the force pulling on the bond . noting that the fraction of open bonds is @xmath28 , we can then write down the following rate equation for the fraction of closed bonds @xmath29 the time dependence of @xmath30 leads via ( [ eq : nu_lambda ] ) to an implicit time dependence of the -parameter @xmath31 and thereby of all observables derived from the . 
   the time - dependent force @xmath32 in ( [ eq : rate ] ) may be thought to result from an externally imposed stress protocol or from internal dynamical elements such as molecular motors setting the network under dynamic stress . 
   hence , via an appropriately chosen set of slowly changing state parameters @xmath32 , @xmath33 , @xmath34 ,   the model can in principle accommodate for the active biological remodelling in living cells and tissues @xcite . 
   ( for a discussion of the relation between the microscopic @xmath6 and the macroscopic stress , see @xcite . ) 
   note that for constant force , the stationary value of @xmath23 , @xmath35\right)^{-1 } , \ ] ] obtained by setting @xmath36 , does not depend on @xmath12 , as it should be ( the steady state is independent of the transition state ) . 
   on the bond fraction @xmath23 for various values of @xmath12 ( brown : @xmath37 , red : @xmath38 , blue : @xmath39 , green : @xmath40 , black : @xmath41 ) at a frequency @xmath42 and a force @xmath43 ( @xmath44 is the relaxation time of a mode of length @xmath22 and @xmath45 is the euler buckling force for a rod of length @xmath22 ) ; _ left : _ 
   normalized real part @xmath46 ; _ right : _ loss angle.,title="fig:",width=7 ]   on the bond fraction @xmath23 for various values of @xmath12 ( brown : @xmath37 , red : @xmath38 , blue : @xmath39 , green : @xmath40 , black : @xmath41 ) at a frequency @xmath42 and a force @xmath43 ( @xmath44 is the relaxation time of a mode of length @xmath22 and @xmath45 is the euler buckling force for a rod of length @xmath22 ) ; _ left : _ normalized real part @xmath46 ; _ right : _ loss angle.,title="fig:",width=7 ]    whenever the bond kinetics can be disregarded ( @xmath47 ) , the viscoelastic properties are simply those of the bare @xcite , which can basically be characterized as  short time dynamics followed by a slow , highly stretched logarithmic relaxation resembling power - law rheology with a non - universal exponent within the typical experimental time windows . 
   strong perturbations ( stress or strain ) then give rise to a pronounced stiffening due to the steeply nonlinear response of semiflexible polymers under elongation @xcite . 
   this well established behaviour can be understood as the canvas against which we aim to discern the characteristic mechanical signatures of the bond kinetics . 
   an observable that will be of particular interest in the following is the complex microscopic `` modulus '' @xmath48 , introduced in the previous section . 
   it is used to determine the linear as well as ( via a nonlinear update scheme , see [ sec : technical ] ) the nonlinear force response of the system . to understand how the time - dependence ( [ eq : rate ] ) of @xmath23 affects this important quantity , 
   it is instructive to first examine the dependence of @xmath49 on @xmath23 . to this end , we formally consider @xmath23 temporarily as an independent variable instead of determining it from ( [ eq : rate ] ) . 
   note that this approach nevertheless makes sense as for a fixed set of parameters , @xmath23 can still take any value between zero and one . 
   this is due to the freedom of choosing an initial state , which can be imagined to have evolved from the prior deformation history .    for fixed other parameters , both the real part @xmath46 ( figure [ fig : mod_lam]a ) and the imaginary part @xmath50 of @xmath49 increase monotonically with @xmath23 . 
   we emphasize that @xmath50 is not simply proportional to @xmath46 and therefore the loss angle @xmath51 also depends on @xmath23 ( figure [ fig : mod_lam]b ) . for small @xmath23 , the loss angle is large , corresponding to fluid - like behaviour . with increasing @xmath23 , 
   the system becomes more solid - like . 
   increasing the barrier height @xmath12 makes the dependence of @xmath49 on @xmath23 more pronounced ( figure [ fig : mod_lam ] ) . 
   conversely , as can be expected , the dependence on the barrier height @xmath12 vanishes with decreasing bond fraction @xmath23 . 
   note that due to the boundary conditions the limit of a newtonian fluid ( @xmath52 ) is not recovered when formally taking the limit of vanishing bond fraction ( @xmath53 , @xmath54 ) . 
   finally , the potential difference @xmath55 solely determines the dynamics of @xmath23 via ( [ eq : rate ] ) , and therefore influences the modulus @xmath49 solely through the equilibrium value for @xmath23 . 
   after these general considerations , we now concentrate on the nonlinear and non - equilibrium dynamic response of the extended inelastic  model , which results from the coupled relaxation of the viscoelastic polymer network _ and _ the transient bond network .      a convenient way to characterize the mechanical properties of an inelastic material is a force - displacement curve . for a perfectly linear elastic ( hookean ) body , it would simply consist of a straight line , whereas a perfectly plastic body would feature as a rectangle delineated by a yield force @xmath56 and an arbitrary plastic strain value . for our qualitative purpose 
   , we identify the average transverse displacement of the test polymer segment ( which is used to determine the force response , see [ sec : technical ] ) with the reaction coordinate @xmath57 of the schematic potential sketched in figure [ fig : potential ] ; and we identify the force @xmath6 entering the reaction rates with the backbone tension of the test polymer . as a characteristic length scale for the transverse displacements we use the width @xmath58 of the effective confinement potential , which is a measure of a typical mean - square displacement of the polymer in the unbound state . using this convention 
   , we now consider the effect of a time - symmetric displacement pulse on the evolution of the force @xmath32 and bond fraction @xmath30 . for technical convenience 
   , we use a gaussian shape for the displacement pulses , but the qualitative conclusions to be drawn are largely independent of the precise protocol . 
   the duration of the displacement pulse , which sets the time scale for the dynamic response , is used as the unit of time in the following . here , we are not interested in short - time tension - propagation and single - polymer dynamics @xcite , hence a lower bound for pertinent pulse durations is provided by the interaction time scale @xmath59 . on the other hand , for pulse durations longer than @xmath60 
   the system deformation would be quasistatic so that no genuinely dynamic effects of the bond kinetics could be observed .    for a small gaussian displacement pulse of relative amplitude @xmath61 ( c.f . 
   figure  [ fig : force_displacement]a and [ sec : technical ] ) , the shape of the curve predicted by our inelastic  model shows all features of a _ viscoelastic _ medium ( figure [ fig : force_displacement]b , blue dashed curve ) . 
   it starts with a nearly linear regime for relative displacements @xmath62 , where a weak stiffening sets in . 
   due to dissipation in the medium , the path back to the initial state takes its course at lower force , which causes a weak hysteresis . 
   this is essentially the viscoelastic response that already the bare  model would have predicted .    : 
   gaussian deformation pulses ( @xmath63 ) with relative displacement amplitudes of @xmath61 ( dashed ) and @xmath64 ( solid ) ; corresponding force - displacement curves and bond fraction evolutions ( @xmath65 , @xmath66 ; dotted curve in @xmath65 : response of a  without bond breaking ) ; order - of - magnitude estimate for the rate - dependence of the yield force , ( [ eq : rate_fc ] ) with const.@xmath67 , versus the numerically obtained maximum forces of the force - displacement curves for various mean force rates ( @xmath68 ) ; the minimum bond fraction ( @xmath68 , inset ) depends non - monotonically on the force rate ; @xmath69 , @xmath70 , @xmath71 , @xmath72 , @xmath73 . , 
   title="fig:",width=7 ] : gaussian deformation pulses ( @xmath63 ) with relative displacement amplitudes of @xmath61 ( dashed ) and @xmath64 ( solid ) ; corresponding force - displacement curves and bond fraction evolutions ( @xmath65 , @xmath66 ; dotted curve in @xmath65 : response of a  without bond breaking ) ; order - of - magnitude estimate for the rate - dependence of the yield force , ( [ eq : rate_fc ] ) with const.@xmath67 , versus the numerically obtained maximum forces of the force - displacement curves for various mean force rates ( @xmath68 ) ; the minimum bond fraction ( @xmath68 , inset ) depends non - monotonically on the force rate ; @xmath69 , @xmath70 , @xmath71 , @xmath72 , @xmath73 . , 
   title="fig:",width=7 ] : gaussian deformation pulses ( @xmath63 ) with relative displacement amplitudes of @xmath61 ( dashed ) and @xmath64 ( solid ) ; corresponding force - displacement curves and bond fraction evolutions ( @xmath65 , @xmath66 ; dotted curve in @xmath65 : response of a  without bond breaking ) ; order - of - magnitude estimate for the rate - dependence of the yield force , ( [ eq : rate_fc ] ) with const.@xmath67 , versus the numerically obtained maximum forces of the force - displacement curves for various mean force rates ( @xmath68 ) ; the minimum bond fraction ( @xmath68 , inset ) depends non - monotonically on the force rate ; @xmath69 , @xmath70 , @xmath71 , @xmath72 , @xmath73 . 
   , title="fig:",width=7 ] : gaussian deformation pulses ( @xmath63 ) with relative displacement amplitudes of @xmath61 ( dashed ) and @xmath64 ( solid ) ; corresponding force - displacement curves and bond fraction evolutions ( @xmath65 , @xmath66 ; dotted curve in @xmath65 : response of a  without bond breaking ) ; order - of - magnitude estimate for the rate - dependence of the yield force , ( [ eq : rate_fc ] ) with const.@xmath67 , versus the numerically obtained maximum forces of the force - displacement curves for various mean force rates ( @xmath68 ) ; the minimum bond fraction ( @xmath68 , inset ) depends non - monotonically on the force rate ; @xmath69 , @xmath70 , @xmath71 , @xmath72 , @xmath73 . , title="fig:",width=7 ]    for a larger relative deformation amplitude of @xmath64 , however , the predictions of the bare  and the inelastic  diverge . 
   the bare  predicts a strong stiffening ( figure [ fig : force_displacement]b , red dotted curve ) , whereas the full model exhibits an initial stiffening regime followed by a pronounced softening ( figure [ fig : force_displacement]b , solid red curve ) . 
   a very interesting observation is that `` softening '' in this case does not only mean a decrease of the slope of the force - extension curve , but that the slope actually becomes _ negative _ over an extended parameter region , once an operationally defined threshold force @xmath56 is exceeded . 
   this `` flow state '' bears more resemblance to plastic flow than to viscoelastic relaxation . 
   the reason for this qualitatively new phenomenon is the force - induced bond breaking . 
   theoretically this is best exemplified by the time - dependence of the ( experimentally not easily accessible ) fraction @xmath30 of closed bonds ( figure [ fig : force_displacement]c ) . 
   a glance at the bond fraction during the weak deformation scenario ( figure  [ fig : force_displacement]c , blue dashed curve ) reveals how the limit of a bare  is recovered , namely whenever the deformation is not sufficiently violent and persistent to significantly decrease the fraction of closed bonds . for the large deformation scenario , in contrast , 
   the bond fraction stays only initially constant ( figure  [ fig : force_displacement]c , solid red curve ) . as a consequence of the large strain 
   the force rises steeply , as can be seen from the strong stiffening in figure  [ fig : force_displacement]b ( solid red curve ) . at the yield force 
   , the bond fraction suddenly decreases to nearly half of its initial value during a very short time . 
   the decrease in the bond fraction is accompanied by a somewhat slower drop in the force , which is reflected in the sudden softening of the force - displacement curve . 
   the bond fraction eventually recovers on a much slower time scale , which is roughly given by @xmath74 . 
   note that even though the effects of the inelastic response dominate the stress - strain curve , the viscoelastic relaxations from the underlying  model are still present . 
   they could hardly be disentangled from the inelastic contributions , though , without an underlying faithful model of the viscoelastic response at hand . in summary , we observe a competition between force - amplitude dependent stress - stiffening and force - rate dependent yielding events . 
   if the backbone force stays much smaller than the yield force , the adaptation of the bond network requires an adiabatically long time , on the order of the bond lifetime @xmath75 , so that plain viscoelastic  behaviour is observed on the pulse time scale . in contrast , if the backbone tension @xmath6 reaches the yield force @xmath56 , the bond fraction declines sharply , thereby switching the response from the /-typical stiffening to a pronounced softening . 
   the rate dependence of the yield force @xmath56 can be estimated by setting the time it takes to reach the yield force , @xmath76 , equal to the force - dependent time scale of bond opening , @xmath77 , from ( [ eq : kpm ] ) , @xmath78 here , lw@xmath79 is the positive real branch of the lambert w function . in figure 
   [ fig : force_displacement]d , this estimate is compared with results from numerical evaluations for gaussian displacement pulses at different average rates . 
   apart from the numerical errors , the slight deviations from the estimate may be attributed to the fact that the force rate is not constant for the gaussian protocol . 
   they can be mostly eliminated by using force ramp protocols of various slopes instead of the gaussian displacement pulses . for not too low rates , 
   the rate dependence can be approximated by the even simpler relation @xmath80 where the force rate has to be normalized by a suitable force scale . 
   the minimum bond fraction reached during the application of the pulse , which we interpret as a measure of the degree of fluidization , depends _ non - monotonically _ on the rate ( figure [ fig : force_displacement]d , inset ) . 
   qualitatively , this can be understood by noticing that two different factors influence the fluidization , namely the maximum force attained during deformation and the time over which the force is applied . 
   the maximum force is simply the yield force @xmath56 , with the rate dependence in ( [ eq : rate_fc ] ) and figure [ fig : force_displacement]d , while the reciprocal rate itself sets the time scale . for high rates @xmath81 ( in our example ) 
   the rate - dependence of the maximum force wins and the minimum @xmath23 reached decreases with increasing rate . for low rates , where the rate - dependence of the maximum force is much weaker ( figure [ fig : force_displacement]d ) the minimum @xmath23 decreases with increasing pulse duration , _ viz . 
   _  decreasing rate . note that for slow pulses ( low rates ) , the bond fraction after the pulse may be quite different from the minimum @xmath23 , as significant recovery may already take place during the pulse . 
   the substantial changes in the material properties accompanying bond breaking can be exemplified by monitoring the linear elastic modulus @xmath82 ( measured at a fixed frequency @xmath83 ) in response to a strain pulse ( figure [ fig : modulus]a ) . 
   apart from the usual /-typical stress - stiffening below the threshold force @xmath84 , the modulus is seen to drop _ below _ the value it had before the deformation pulse , where it apparently saturates ( remember that the deformation vanishes for @xmath85 and that the recovery takes roughly a time @xmath86 ) . 
   we thus observe what we call a `` passive '' , `` physical '' remodelling of the bond network as opposed to the `` active '' , `` biological '' remodelling of the cytoskeleton of living cells in response to external stimuli . 
   these passive remodelling effects have recently been observed for human airway smooth muscle cells @xcite . 
   ( a more quantitative discussion of the relation to the experimental observations will be given elsewhere . ) 
   the fact that the deformation pulse leads to a decrease in @xmath46 and an increase in the loss angle @xmath87 ( figure [ fig : modulus]c ) suggests the notion of _ fluidization _ @xcite .     and 
   the bond fraction @xmath23 largely determine the storage modulus @xmath46 and the loss angle @xmath87 : normalized storage modulus @xmath88 ( @xmath63 ) and loss angle @xmath87 ( @xmath66 ) during displacement pulses of large and small amplitudes , 2.2 ( solid ) , 0.73 ( dashed ) ; the stronger deformation pulse causes a strong and persistent overshoot of the loss angle , indicating _ fluidization _ ; slow recovery after the pulse , ( @xmath65 , @xmath68 ) ; parameters as in figure [ fig : force_displacement ] , @xmath89.,title="fig:",width=7 ]   and the bond fraction @xmath23 largely determine the storage modulus @xmath46 and the loss angle @xmath87 : normalized storage modulus @xmath88 ( @xmath63 ) and loss angle @xmath87 ( @xmath66 ) during displacement pulses of large and small amplitudes , 2.2 ( solid ) , 0.73 ( dashed ) ; the stronger deformation pulse causes a strong and persistent overshoot of the loss angle , indicating _ fluidization _ ; slow recovery after the pulse , ( @xmath65 , @xmath68 ) ; parameters as in figure [ fig : force_displacement ] , @xmath89.,title="fig:",width=7 ]   and the bond fraction @xmath23 largely determine the storage modulus @xmath46 and the loss angle @xmath87 : normalized storage modulus @xmath88 ( @xmath63 ) and loss angle @xmath87 ( @xmath66 ) during displacement pulses of large and small amplitudes , 2.2 ( solid ) , 0.73 ( dashed ) ; the stronger deformation pulse causes a strong and persistent overshoot of the loss angle , indicating _ fluidization _ ; slow recovery after the pulse , ( @xmath65 , @xmath68 ) ; parameters as in figure [ fig : force_displacement ] , @xmath89.,title="fig:",width=7 ]   and the bond fraction @xmath23 largely determine the storage modulus @xmath46 and the loss angle @xmath87 : normalized storage modulus @xmath88 ( @xmath63 ) and loss angle @xmath87 ( @xmath66 ) during displacement pulses of large and small amplitudes , 2.2 ( solid ) , 0.73 ( dashed ) ; the stronger deformation pulse causes a strong and persistent overshoot of the loss angle , indicating _ fluidization _ ; slow recovery after the pulse , ( @xmath65 , @xmath68 ) ; parameters as in figure [ fig : force_displacement ] , @xmath89.,title="fig:",width=7 ]    the passive remodelling described here is a _ transient _ phenomenon , because , after cessation of the external perturbations , the bond fraction will eventually recover its equilibrium value . 
   this indicates that also the change in the system properties is a transient effect , which is demonstrated by the recovery from fluidization in figure  [ fig : modulus ] b & d. thus , while the fluidization resembles a plastic process on short time scales , on long times scales , the phenomenology is more similar to pseudo- and superelasticity as observed in shape - memory alloys @xcite .    to demonstrate that the transient passive remodelling also affects the nonlinear material properties 
   , we consider a series of three pulses , next ( figure [ fig : stress_strain ] ) . 
   the force response to such a protocol is depicted in figure [ fig : stress_strain]b . 
   the force - displacement curve for the second stretching ( second left branch of the solid curve ) is less steep than for the first stretching ( very left branch of the solid curve ) and the yield force is lower . 
   this indicates a cyclic softening , or viscoelastic shake - down effect , closely related to the fluidization of the network by strain . 
   the strength of the shake - down depends on the fraction of transiently broken bonds , and hence on the rate as well as on the amplitude of the applied deformation ( c.f . 
   figure [ fig : stress_strain]c , solid red curve ) . upon repeated application of deformation pulses 
   the force - displacement curves settle on a limit - curve that is essentially preconditioned by the initial deformation pulse and the inelastic work it performed on the sample . for more gentle protocols that only break a smaller fraction of bonds , 
   the initial fluidization would be not as pronounced and one would obtain a gradual shake - down , which converges to a limit curve only after many cycles .    ) and response ( @xmath65 , @xmath66 ) for three subsequent gaussian strain pulses ; force - displacement curves ( @xmath65 ) for a single minimum interaction distance ( @xmath90 , solid ) and for a distribution of @xmath22 according to @xcite with mean @xmath91 ( dashed ) ; the bond fraction @xmath23 ( @xmath66 ) for @xmath92 ( dash - dotted ) , @xmath93 ( solid ) , and @xmath94 ( dashed ) ; other parameters as in figure [ fig : force_displacement ] . 
   , title="fig:",width=14 ] ) and response ( @xmath65 , @xmath66 ) for three subsequent gaussian strain pulses ; force - displacement curves ( @xmath65 ) for a single minimum interaction distance ( @xmath90 , solid ) and for a distribution of @xmath22 according to @xcite with mean @xmath91 ( dashed ) ; the bond fraction @xmath23 ( @xmath66 ) for @xmath92 ( dash - dotted ) , @xmath93 ( solid ) , and @xmath94 ( dashed ) ; other parameters as in figure [ fig : force_displacement ] . , title="fig:",width=7 ] ) and response ( @xmath65 , @xmath66 ) for three subsequent gaussian strain pulses ; force - displacement curves ( @xmath65 ) for a single minimum interaction distance ( @xmath90 , solid ) and for a distribution of @xmath22 according to @xcite with mean @xmath91 ( dashed ) ; the bond fraction @xmath23 ( @xmath66 ) for @xmath92 ( dash - dotted ) , @xmath93 ( solid ) , and @xmath94 ( dashed ) ; other parameters as in figure [ fig : force_displacement ] . , 
   title="fig:",width=7 ]      so far , we have assumed that there is one well - defined characteristic length scale @xmath22 for the polymer interactions , which is on the order of the entanglement length @xmath17 and interpreted as the minimum average contour distance between adjacent bonds of the test polymer with the background network . 
   this is clearly a mean - field assumption . 
   recent combined experimental and theoretical studies have established that the local tube diameter and entanglement length in pure semiflexible polymer solutions actually exhibit a skewed leptocurtic distribution with broad tails @xcite . to include this information in the above analysis is not straightforward , since the elastic interactions between regions of different entanglement length are not known , _ a priori_. for the sake of a first qualitative estimate , the simplest procedure seems to be to average the above results over a distribution of @xmath22 , corresponding to a parallel connection of independent entanglement elements . 
   qualitatively , this renders the abrupt transition from stiffening to softening somewhat smoother ( c.f . 
   figure [ fig : stress_strain]b , dashed curve ) . 
   moreover , the initial stiffening is slightly more pronounced . 
   this is due to the contributions of @xmath95 , which exhibit both stronger stiffening and fluidization 
   . nevertheless , all qualitative features  like stiffening and remodelling effects and a `` flow state '' indicating fluidization or inelastic shakedown  can still be well discerned . 
   this is consistent with the assumption that pure semiflexible polymer solutions are at least qualitatively well described by a single entanglement length @xcite . 
   we have presented a theoretical framework for a polymer - based description of the inelastic nonlinear mechanical behaviour of sticky biopolymer networks . 
   we represented the polymer network on a mean - field level by a test polymer described by the phenomenologically highly successful  model . beyond the equilibrium statistical fluctuations captured by the original model 
   , we additionally allowed for a dynamically evolving thermodynamic state variable @xmath31 characterizing the network of thermo - reversible sticky bonds , which is updated dynamically upon bond breaking according to an appropriate rate equation . within our approximate treatment 
   we could obtain a number of robust and qualitatively interesting results , which are not sensitive to the precise parameter values chosen , nor to the choice of the transverse rather than the longitudinal susceptibility in ( [ eq : sus_wlc ] ) . 
   for sufficiently strong deformations , changes in the mean fraction of closed bonds , reflected in @xmath31 , can influence material properties to a degree at which the material behaves qualitatively different from the usual viscoelastic paradigm . 
   in particular , our theory predicts a pronounced fluidization response , which develops upon strong deformations on top of the intrinsic viscoelastic stiffening response provided by the individual polymers . 
   after cessation of loading , the system slowly recovers its initial state . 
   these observations are in qualitative agreement with recently published experimental results obtained for living cells @xcite , and a more quantitative comparison with dedicated measurement results for cells and _ in vitro _ 
   biopolymer networks will be the subject of future work . 
   we found the yield force for the onset of the fluidization to be sensitive to the deformation rate . 
   moreover , the fluidization response was shown to be accompanied by a cyclic softening or shake - down effect . 
   taking into account the spatial heterogeneity of biopolymer solutions by a distribution of entanglement lengths leads to a smoothing of the force - displacement curves without affecting their qualitative characteristics . 
   one may still expect qualitatively new effects in situations with unusually broad distributions of entanglement lengths , such as for strongly heterogeneous ( e.g.  phase separating ) networks . 
   a parameter that was found to be very important for cells @xcite but has not been discussed much in this contribution is the _ prestressing force _ 
   @xmath96 ( see appendix [ sec : technical ] ) , which is present in adhering cells even in the absence of external driving . experimentally , it was established that increasing prestress is correlated with a higher stiffness of adhering cells @xcite which in turn is correlated with a higher stiffness of the substrate @xcite . 
   when naively treating the prestressing force just as an additive contribution to the overall force , our model predicts the opposite : the additional force breaks more bonds and the network becomes softer and more fluid - like . to reconcile these apparently contradicting trends , one can appeal to the notion of _ homeostasis _ , which essentially amounts to postulating that the cell actively adapts its structure such that a certain set of thermodynamic state variables remain in a physiologically sensible range @xcite . 
   this basically implies that the cell will respond to stiff substrates or persistent external stresses by a biological remodelling that corresponds to an increase of @xmath12 and @xmath55 , and possibly to a decrease of @xmath22 . by adapting also the internal prestress @xmath96 , 
   the cell may then avoid , apart from the structural collapse , an equally undesirable loss of flexibility . 
   we note , however , that internal stresses are indeed observed to disrupt the cytoskeleton , as implied by the simple physical theory presented here , if they are not permanently balanced by a substrate @xcite . 
   even though the nonlinear effects presented in this contribution depend crucially on the bond kinetics , it should not be overlooked that the  is an equally indispensable ingredient of the complete theory . 
   first , it provides the constitutive relation that gives the force in response to an infinitesimal displacement , which in turn governs the bond kinetics . 
   secondly , the time - dependent linear susceptibility strongly filters the dynamic remodelling of the bond network in practical applications , in which one rarely has access to the microstate of the underlying bond network . in summary , 
   the extended model thus integrates experimentally confirmed features of the  response such as slow relaxation , power - law rheology , viscoelastic hysteresis , and shear stiffening with a simple bond kinetics scheme , which allows less intuitive complex nonlinear physical remodelling effects like fluidization and inelastic shake - down to be addressed . 
   finally , we want to point out that it is straightforward to extend the above analysis in a natural way to account for irreversible plastic deformations @xcite . to this end 
   , one has to allow for the possibility that the transiently broken bonds reform somewhere else than at their original sites ( as always assumed in the foregoing discussion ) after strong deformations with finite residual displacement 
   . this can be included by accommodating an additional term acting as a `` slip rate '' in ( [ eq : rate ] ) @xcite . 
   we acknowledge many fruitful discussions with j.  glaser , s.  sturm , and k.  hallatschek , and financial support from the deutsche forschungsgemeinschaft ( dfg ) through for 877 and , within the german excellence initiative , the leipzig school of natural sciences  building with molecules and nano - objects . 
   the deformation pulse protocol used in this paper is @xmath97 to obtain the full nonlinear response of the ( extended )  to the displacement protocol ( [ eq : strain_gauss ] ) , we make use of the superposition principle : we know that the fourier transform of @xmath98 is the linear response to a small displacement step . 
   after decomposing a finite displacement into infinitesimal displacement steps and a partial integration , we write @xmath99 note that the right hand side of ( [ eq : force ] ) depends on @xmath100 , rendering it a highly nonlinear implicit equation . using ( [ eq : strain_gauss ] ) and integrating by parts , we obtain @xmath101    the next step is to identify a connection between the force in ( [ eq : rate ] ) and @xmath102(t ) . 
   a finite prestressing force @xmath96 is introduced mainly for technical reasons , namely to avoid the unphysical region of negative ( i.e.  compressive ) backbone stress , which would buckle the polymers . 
   while the physics of the prestressed network can essentially be mapped back to that of a network without prestress by a renormalization of the parameters @xmath12 and @xmath55 , prestress may also be seen as an important feature : indeed , the cytoskeleton of adhered cells is well known to be under permanent prestress , and suspended cells seem prone to spontaneous shape oscillations that could be indicative of a propensity of the cells to set themselves under prestress @xcite . in the context and on the longer time scales of the biological remodelling of the cytoskeleton 
   it would therefore make sense to think of @xmath96 as a dynamic force generated by molecular motors and polymerization forces . for the following , however , we take the prestress to be constant so that the force entering ( [ eq : rate ] ) is @xmath103 we now face the problem that ( [ eq : force ] ) is an implicit equation for @xmath100 , which depends on @xmath30 , which in turn depends via @xmath32 on @xmath104 . 
   we therefore use a two - step euler scheme : as initial values for @xmath6 and @xmath23 , we choose the prestressing force , @xmath105 , and the steady - state value under the prestressing force , @xmath106 , respectively . 
   we then choose a sufficiently small time step @xmath107 and apply the following iterative rule : @xmath108 where @xmath109 . 
   in the limit @xmath110 , this procedure converges to the exact solution . 
   ke kasza , f  nakamura , s  hu , p  kollmannsberger , n  bonakdar , b  fabry , tp stossel , n  wang and da weitz 2009 filamin a is essential for active cell stiffening but not passive stiffening under external force * 96 43264335 . *                                          eh zhou , x  trepat , cy park , g  lenormand , mn oliver , sm mijailovich , c  hardin , da weitz , jp butler and jj fredberg 2009 universal behavior of the osmotically compressed cell and its analogy to the colloidal glass transition * 106 10632 . * 
   n  wang , i m  tolic - norrelykke , jx  chen , sm  mijailovich , jp  butler , jj  fredberg and d  stamenovic 2002 cell prestress . 
   i. stiffness and prestress are closely associated in adherent contractile cells * 282 c606c616 . *
summary: we propose a physical model for the nonlinear inelastic mechanics of sticky biopolymer networks with potential applications to inelastic cell mechanics . 
   it consists in a minimal extension of the glassy wormlike chain ( ) model , which has recently been highly successful as a quantitative mathematical description of the viscoelastic properties of biopolymer networks and cells . to extend its scope to nonequilibrium situations , where the thermodynamic state variables may evolve dynamically , 
   the is furnished with an explicit representation of the kinetics of breaking and reforming sticky bonds . in spite of its simplicity 
   the model exhibits many experimentally established non - trivial features such as power - law rheology , stress stiffening , fluidization , and cyclic softening effects .