\relax 
\providecommand\hyper@newdestlabel[2]{}
\@nameuse{bbl@beforestart}
\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
\global\let\oldcontentsline\contentsline
\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
\global\let\oldnewlabel\newlabel
\gdef\newlabel#1#2{\newlabelxx{#1}#2}
\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
\AtEndDocument{\ifx\hyper@anchor\@undefined
\let\contentsline\oldcontentsline
\let\newlabel\oldnewlabel
\fi}
\fi}
\global\let\hyper@last\relax 
\gdef\HyperFirstAtBeginDocument#1{#1}
\providecommand\HyField@AuxAddToFields[1]{}
\providecommand\HyField@AuxAddToCoFields[2]{}
\citation{Cao2015}
\citation{Bershadsky,Discher2005,Janmey2020,Schwarz2013}
\citation{Ingber2014,FRALDI2019,Palumbo2018}
\citation{Wang2009}
\citation{Geiger2001,Geiger2009}
\citation{FAS}
\citation{Balaban2001}
\citation{Geiger2011,Deshpande2008}
\citation{Riveline2001}
\citation{Prager2011,Trichet2012,Fusco2017}
\citation{Deshpande2006}
\citation{Vernerey2014}
\citation{C2IB20159C,martin1,reorientationexp}
\citation{Durotaxis,Trichet2012,Lazopoulos2008,Espina}
\citation{mechanotropism,Bischofs2003}
\citation{Discher2005,Nelson2005,Engler2006,Zhang2013}
\citation{Discher2005}
\citation{Mattila2008}
\citation{mechanotropism}
\citation{Cao2015}
\citation{Discher2005}
\citation{Mattila2008}
\citation{mechanotropism}
\citation{Cao2015}
\citation{ReviewFAs,Ronan2014,Deshpande2006,Vernerey2016,Deshpande2008,Vernerey2011,Vernerey2014,Schwarz2013,chen2015,Nicolas,Shemesh,HE2014,Cao2015,Cao2}
\citation{Nicolas}
\citation{Shemesh}
\citation{Cao2015}
\citation{Cao2015}
\citation{Cao2015}
\babel@aux{english}{}
\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{1}{section.1}\protected@file@percent }
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces \textbf  {a)} A synoptic scheme reporting some key cellular processes mediated by the mechanosensing and mechanotransduction functions of FAs, such as: \textbf  {a1)} differential adhesion over substrates with different deformability \cite  {Discher2005}; \textbf  {a2)} directional migration from soft to stiff regions of an elastic substrate, propelled by actin-dependent protrusions of the cell leading edge, i.e. filopodia and lamellipodia (the related image has been re-adapted from the work \cite  {Mattila2008}); \textbf  {a3)} cell reorientation under the action of exogenous loads, along optimal directions depending on the mechanical properties of the underlying medium and on the features of the applied forces, e.g. on their static or dynamic nature \cite  {mechanotropism}. \textbf  {b)} Sketch of an adherent cell comprising the nucleus, the cytosketetal compartment, made of an actomyosin SF and a MT, and the FA complex, comprising the adhesion plaque and integrin receptors binding to the ECM by crossing the cell membrane. \textbf  {c)} Mechanical model of the adherent cell in its stress-free reference state and \textbf  {d)} in its current configuration, deformed --with possible MT buckling-- as a consequence of the activation of actomyosin contraction in the SF. \textbf  {e)} Focus on the structural scheme adopted for the FA-ECM complex, whose overall equivalent stiffness is given by $k_{eff}=F_a / u_a$ (borrowed from \cite  {Cao2015}). \relax }}{2}{figure.caption.1}\protected@file@percent }
\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
\newlabel{fig.sketch}{{1}{2}{\textbf {a)} A synoptic scheme reporting some key cellular processes mediated by the mechanosensing and mechanotransduction functions of FAs, such as: \textbf {a1)} differential adhesion over substrates with different deformability \cite {Discher2005}; \textbf {a2)} directional migration from soft to stiff regions of an elastic substrate, propelled by actin-dependent protrusions of the cell leading edge, i.e. filopodia and lamellipodia (the related image has been re-adapted from the work \cite {Mattila2008}); \textbf {a3)} cell reorientation under the action of exogenous loads, along optimal directions depending on the mechanical properties of the underlying medium and on the features of the applied forces, e.g. on their static or dynamic nature \cite {mechanotropism}. \textbf {b)} Sketch of an adherent cell comprising the nucleus, the cytosketetal compartment, made of an actomyosin SF and a MT, and the FA complex, comprising the adhesion plaque and integrin receptors binding to the ECM by crossing the cell membrane. \textbf {c)} Mechanical model of the adherent cell in its stress-free reference state and \textbf {d)} in its current configuration, deformed --with possible MT buckling-- as a consequence of the activation of actomyosin contraction in the SF. \textbf {e)} Focus on the structural scheme adopted for the FA-ECM complex, whose overall equivalent stiffness is given by $k_{eff}=F_a / u_a$ (borrowed from \cite {Cao2015}). \relax }{figure.caption.1}{}}
\citation{Palumbo2018}
\citation{DEGUCHI2006,Vernerey2011}
\citation{martin2,martin3,Vernerey2016}
\citation{Besser2007sfs}
\citation{holzapfel2}
\citation{Cao2015,Besser2007sfs,Cao2}
\citation{DEGUCHI2006,puglisi1,puglisi2}
\citation{holzapfel1}
\@writefile{toc}{\contentsline {section}{\numberline {2}Mechanical modelling of an adherent single-cell}{3}{section.2}\protected@file@percent }
\citation{holzapfel1,gorielybook}
\citation{tens1,FRALDI2019,puglisi3,tens2}
\citation{Palumbo2018}
\citation{fraldi2021,Brodland1990,Brangwynne2006}
\citation{tens1}
\citation{Cao2015}
\citation{Cao2015}
\newlabel{lamf}{{2.1}{4}{Mechanical modelling of an adherent single-cell}{equation.2.1}{}}
\newlabel{2lf}{{2.2}{4}{Mechanical modelling of an adherent single-cell}{equation.2.2}{}}
\newlabel{lt}{{2.3}{4}{Mechanical modelling of an adherent single-cell}{equation.2.3}{}}
\citation{Palumbo2018,holzapfel2000}
\citation{euler1744}
\citation{Brangwynne2006,Brodland1990,FRALDI2019}
\newlabel{lflt}{{2.5}{5}{Mechanical modelling of an adherent single-cell}{equation.2.5}{}}
\newlabel{lamfe}{{2.6}{5}{Mechanical modelling of an adherent single-cell}{equation.2.6}{}}
\newlabel{delta}{{2.7}{5}{Mechanical modelling of an adherent single-cell}{equation.2.7}{}}
\newlabel{ept}{{2.11}{5}{Mechanical modelling of an adherent single-cell}{equation.2.11}{}}
\citation{Cao2015}
\citation{Cao2015}
\citation{FRALDI2019}
\citation{FRALDI2019,DEGUCHI2006,Pampaloni2006,Kurachi1995}
\citation{DEGUCHI2006}
\citation{Cao2015}
\citation{Cao2015}
\citation{Cao2015}
\citation{Cao2015}
\citation{FRALDI2019}
\citation{DEGUCHI2006}
\citation{FRALDI2019,Brangwynne2006}
\citation{Cao2015}
\citation{timoshenko1961,Bigoni2012}
\newlabel{eq:1}{{2.14a}{6}{Mechanical modelling of an adherent single-cell}{equation.2.14a}{}}
\newlabel{eq:2}{{2.14b}{6}{Mechanical modelling of an adherent single-cell}{equation.2.14b}{}}
\newlabel{eq:3}{{2.14c}{6}{Mechanical modelling of an adherent single-cell}{equation.2.14c}{}}
\newlabel{alpha}{{2.15}{6}{Mechanical modelling of an adherent single-cell}{equation.2.15}{}}
\citation{tens1,Shemesh}
\citation{HILL1982,Shemesh}
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces \textbf  {a)} Variation of $\lambda _{f,cr}^c$ as a function of the normalized plaque's length $L_p/d_i$ and definition of pre- and post-buckling domains in the related phase-space. \textbf  {b)} Equilibrium bifurcation path followed by the system for growing (from right to left) levels of SF inelastic contraction. The inclination angle $\phi $ of the MT is plotted as a function of the contractile stretch $\lambda _f^c$ normalized with respect to its critical value $\lambda _{f,cr}^c$, for three different lengths of the adhesion plaque, i.e. $L_p=3\tmspace  +\thinmuskip {.1667em} d_i,\tmspace  +\thinmuskip {.1667em} 10\tmspace  +\thinmuskip {.1667em} d_i,\tmspace  +\thinmuskip {.1667em} 30\tmspace  +\thinmuskip {.1667em} d_i$. Herein, solid tracts indicate (either straight or deviated) stable configurations while the dashed line identifies the unstable (straight) ones. \textbf  {c)} Elastic aliquot $\lambda _f^e$ of the stretch born by the SF and \textbf  {d)} purely elastic stretch $\lambda _t$ in the MT as functions of the actomyosin contraction $\lambda _f^c$. All the plots refer to values of the model's parameters reported in table \ref  {table}, by in particular setting: $L=20\tmspace  +\thinmuskip {.1667em} \mu m$, $k_s=10 \tmspace  +\thinmuskip {.1667em} pN/nm$, $k_n= 20 \tmspace  +\thinmuskip {.1667em} pN/nm$, $B_t = 215 \tmspace  +\thinmuskip {.1667em} nN \cdot \mu m^2 $. \relax }}{7}{figure.caption.2}\protected@file@percent }
\newlabel{fig.bifurcation_diag}{{2}{7}{\textbf {a)} Variation of $\lambda _{f,cr}^c$ as a function of the normalized plaque's length $L_p/d_i$ and definition of pre- and post-buckling domains in the related phase-space. \textbf {b)} Equilibrium bifurcation path followed by the system for growing (from right to left) levels of SF inelastic contraction. The inclination angle $\phi $ of the MT is plotted as a function of the contractile stretch $\lambda _f^c$ normalized with respect to its critical value $\lambda _{f,cr}^c$, for three different lengths of the adhesion plaque, i.e. $L_p=3\, d_i,\, 10\, d_i,\, 30\, d_i$. Herein, solid tracts indicate (either straight or deviated) stable configurations while the dashed line identifies the unstable (straight) ones. \textbf {c)} Elastic aliquot $\lambda _f^e$ of the stretch born by the SF and \textbf {d)} purely elastic stretch $\lambda _t$ in the MT as functions of the actomyosin contraction $\lambda _f^c$. All the plots refer to values of the model's parameters reported in table \ref {table}, by in particular setting: $L=20\, \mu m$, $k_s=10 \, pN/nm$, $k_n= 20 \, pN/nm$, $B_t = 215 \, nN \cdot \mu m^2 $. \relax }{figure.caption.2}{}}
\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces Values employed for the geometrical and constitutive parameters of the cell equivalent structural scheme. \relax }}{8}{table.caption.3}\protected@file@percent }
\newlabel{table}{{1}{8}{Values employed for the geometrical and constitutive parameters of the cell equivalent structural scheme. \relax }{table.caption.3}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3}Influence of mechanics on the growth rate of FAs}{8}{section.3}\protected@file@percent }
\citation{Cao2015}
\citation{Cao2015}
\citation{Cao2015}
\citation{Cao2015}
\citation{Cao2015}
\citation{Cao2015}
\citation{Cao2015}
\citation{Cao2015}
\newlabel{J}{{3.5}{9}{Influence of mechanics on the growth rate of FAs}{equation.3.5}{}}
\@writefile{toc}{\contentsline {section}{\numberline {4}Actomyosin contraction as active tuner of the FAs' assembly}{9}{section.4}\protected@file@percent }
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces \textbf  {a)} The isolated SF element from the work by Cao et al. \cite  {Cao2015} (on the top) and the SF-MT tensegrity system extracted from the cell structural description considered in the present work (on the bottom): scheme for the kinematical derivation of the contractile force $f_0$ as a function of the actomyosin contraction level $\lambda _f^c$ through the equivalence of the two models. \textbf  {b)} Variation of $f_0$ in terms of $\lambda _f^c$ and (in green) identification of $\lambda _f^c=0.987$ as corresponding to a contractile force $f_0=100\tmspace  +\thinmuskip {.1667em}pN$ \cite  {Cao2015}. \textbf  {c)} Normalized growth rate of the adhesion plaque $J/D$ and \textbf  {d)} pulling axial force $F_a$, both obtained for $\lambda _f^c=0.987$ at the current configuration in figure \ref  {fig.sketch}\hyperref  [fig.sketch]{c}, as functions of the normalized plaque's length $L_p/d_i$. All the plots refer to values of the model's parameters reported in table \ref  {table}, by in particular setting: $L=E_f A_f/k_a $, $k_s=10 \tmspace  +\thinmuskip {.1667em} pN/nm$, $k_n= 20 \tmspace  +\thinmuskip {.1667em} pN/nm$, $B_t = 215 \tmspace  +\thinmuskip {.1667em} nN \cdot \mu m^2 $, $\Delta \mu _0=30\tmspace  +\thinmuskip {.1667em} k_B T$ and $k_a= 50 \tmspace  +\thinmuskip {.1667em} pN/nm $ \cite  {Cao2015}.\relax }}{9}{figure.caption.4}\protected@file@percent }
\newlabel{fig.f0_equivalence}{{3}{9}{\textbf {a)} The isolated SF element from the work by Cao et al. \cite {Cao2015} (on the top) and the SF-MT tensegrity system extracted from the cell structural description considered in the present work (on the bottom): scheme for the kinematical derivation of the contractile force $f_0$ as a function of the actomyosin contraction level $\lambda _f^c$ through the equivalence of the two models. \textbf {b)} Variation of $f_0$ in terms of $\lambda _f^c$ and (in green) identification of $\lambda _f^c=0.987$ as corresponding to a contractile force $f_0=100\,pN$ \cite {Cao2015}. \textbf {c)} Normalized growth rate of the adhesion plaque $J/D$ and \textbf {d)} pulling axial force $F_a$, both obtained for $\lambda _f^c=0.987$ at the current configuration in figure \ref {fig.sketch}\hyperref [fig.sketch]{c}, as functions of the normalized plaque's length $L_p/d_i$. All the plots refer to values of the model's parameters reported in table \ref {table}, by in particular setting: $L=E_f A_f/k_a $, $k_s=10 \, pN/nm$, $k_n= 20 \, pN/nm$, $B_t = 215 \, nN \cdot \mu m^2 $, $\Delta \mu _0=30\, k_B T$ and $k_a= 50 \, pN/nm $ \cite {Cao2015}.\relax }{figure.caption.4}{}}
\citation{Cao2015}
\citation{Cao2015}
\citation{Cao2015}
\citation{Cao2015}
\citation{Cao2015,Shemesh}
\citation{Cao2015,Ronan2014,Vernerey2011}
\citation{Prager2011,Trichet2012,Fusco2017}
\citation{Durotaxis,Trichet2012}
\citation{CellClusters,Fraldi2015,Nebuloni2016,Cross2007,Goetz2011}
\citation{FRALDI2019}
\newlabel{f0lamfc}{{4.2}{10}{Actomyosin contraction as active tuner of the FAs' assembly}{equation.4.2}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces \textbf  {a)} Pulling axial force $F_a$ and \textbf  {b)} normalized growth rate $J/D$ plotted as functions of the adhesion plaque length $L_p$ normalized with respect to the integrin spacing $d_i$. The curves refer to three different values of inelastic contraction such to keep the MT straight ($\lambda _f^c= 0.9$: red dashed curve), to induce MT buckling independently from the plaque length ($\lambda _f^c= 0.8$: blue dashed curve) or to cause instability if outside a certain range of FA size ($\lambda _f^c= 0.84$: solid curve, red-coloured for the straight configurations and blue-coloured for the deviated states). \textbf  {c)} Pulling axial force $F_a$ and \textbf  {d)} normalized growth rate $J/D$ plotted as functions of the actomyosin contraction level occurring in the SF before (red tracts) and after (blue tracts) buckling of the MT, for a fixed magnitude of the plaque length $L_p=20 \tmspace  +\thinmuskip {.1667em}d_i$. All the plots refer to values of the model's parameters reported in table \ref  {table}, by in particular setting: $L=20\tmspace  +\thinmuskip {.1667em} \mu m$, $k_s=10 \tmspace  +\thinmuskip {.1667em} pN/nm$, $k_n= 20 \tmspace  +\thinmuskip {.1667em} pN/nm$, $B_t = 215 \tmspace  +\thinmuskip {.1667em} nN \cdot \mu m^2 $, $\Delta \mu _0=250\tmspace  +\thinmuskip {.1667em} k_B T$.\relax }}{11}{figure.caption.5}\protected@file@percent }
\newlabel{fig.JFa}{{4}{11}{\textbf {a)} Pulling axial force $F_a$ and \textbf {b)} normalized growth rate $J/D$ plotted as functions of the adhesion plaque length $L_p$ normalized with respect to the integrin spacing $d_i$. The curves refer to three different values of inelastic contraction such to keep the MT straight ($\lambda _f^c= 0.9$: red dashed curve), to induce MT buckling independently from the plaque length ($\lambda _f^c= 0.8$: blue dashed curve) or to cause instability if outside a certain range of FA size ($\lambda _f^c= 0.84$: solid curve, red-coloured for the straight configurations and blue-coloured for the deviated states). \textbf {c)} Pulling axial force $F_a$ and \textbf {d)} normalized growth rate $J/D$ plotted as functions of the actomyosin contraction level occurring in the SF before (red tracts) and after (blue tracts) buckling of the MT, for a fixed magnitude of the plaque length $L_p=20 \,d_i$. All the plots refer to values of the model's parameters reported in table \ref {table}, by in particular setting: $L=20\, \mu m$, $k_s=10 \, pN/nm$, $k_n= 20 \, pN/nm$, $B_t = 215 \, nN \cdot \mu m^2 $, $\Delta \mu _0=250\, k_B T$.\relax }{figure.caption.5}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces \textbf  {a)} Pulling axial force $F_a$, \textbf  {b)} normalized growth rate $J/D$, \textbf  {c)} elastic stretch aliquot in the SF $\lambda _f^e$ and \textbf  {d)} stretch in the MT $\lambda _t$, all plotted for varying actomyosin contractile stretch $\lambda _f^c$ at two different values of the ECM stiffness, i.e $k_s=5\tmspace  +\thinmuskip {.1667em} pN/nm$ and $k_s=30\tmspace  +\thinmuskip {.1667em} pN/nm$, compatible with ranges measured in healthy and tumour environments, respectively. The colours red and blue are adopted for indicating curves' tracts related to pre-buckling and post-buckling configurations, respectively. All the plots refer to values of the model's parameters reported in table \ref  {table}, by in particular setting: $L=20\tmspace  +\thinmuskip {.1667em} \mu m$, $L_p=20 \tmspace  +\thinmuskip {.1667em} d_i$, $k_n= 20 \tmspace  +\thinmuskip {.1667em} pN/nm$, $B_t = 215 \tmspace  +\thinmuskip {.1667em} nN \cdot \mu m^2 $, $\Delta \mu _0=250\tmspace  +\thinmuskip {.1667em} k_B T$.\relax }}{12}{figure.caption.6}\protected@file@percent }
\newlabel{fig.varstiff}{{5}{12}{\textbf {a)} Pulling axial force $F_a$, \textbf {b)} normalized growth rate $J/D$, \textbf {c)} elastic stretch aliquot in the SF $\lambda _f^e$ and \textbf {d)} stretch in the MT $\lambda _t$, all plotted for varying actomyosin contractile stretch $\lambda _f^c$ at two different values of the ECM stiffness, i.e $k_s=5\, pN/nm$ and $k_s=30\, pN/nm$, compatible with ranges measured in healthy and tumour environments, respectively. The colours red and blue are adopted for indicating curves' tracts related to pre-buckling and post-buckling configurations, respectively. All the plots refer to values of the model's parameters reported in table \ref {table}, by in particular setting: $L=20\, \mu m$, $L_p=20 \, d_i$, $k_n= 20 \, pN/nm$, $B_t = 215 \, nN \cdot \mu m^2 $, $\Delta \mu _0=250\, k_B T$.\relax }{figure.caption.6}{}}
\citation{Cao2015}
\@writefile{toc}{\contentsline {section}{\numberline {5}Conclusions}{13}{section.5}\protected@file@percent }
\bibstyle{ieeetr}
\bibdata{biobiblio}
\bibcite{Cao2015}{1}
\bibcite{Bershadsky}{2}
\bibcite{Discher2005}{3}
\bibcite{Janmey2020}{4}
\bibcite{Schwarz2013}{5}
\bibcite{Ingber2014}{6}
\bibcite{FRALDI2019}{7}
\bibcite{Palumbo2018}{8}
\bibcite{Wang2009}{9}
\bibcite{Geiger2001}{10}
\bibcite{Geiger2009}{11}
\bibcite{FAS}{12}
\bibcite{Balaban2001}{13}
\bibcite{Geiger2011}{14}
\bibcite{Deshpande2008}{15}
\bibcite{Riveline2001}{16}
\bibcite{Prager2011}{17}
\bibcite{Trichet2012}{18}
\bibcite{Fusco2017}{19}
\bibcite{Deshpande2006}{20}
\bibcite{Vernerey2014}{21}
\bibcite{C2IB20159C}{22}
\bibcite{martin1}{23}
\bibcite{reorientationexp}{24}
\bibcite{Durotaxis}{25}
\bibcite{Lazopoulos2008}{26}
\bibcite{Espina}{27}
\bibcite{mechanotropism}{28}
\bibcite{Bischofs2003}{29}
\bibcite{Nelson2005}{30}
\bibcite{Engler2006}{31}
\bibcite{Zhang2013}{32}
\bibcite{Mattila2008}{33}
\bibcite{ReviewFAs}{34}
\bibcite{Ronan2014}{35}
\bibcite{Vernerey2016}{36}
\bibcite{Vernerey2011}{37}
\bibcite{chen2015}{38}
\bibcite{Nicolas}{39}
\bibcite{Shemesh}{40}
\bibcite{HE2014}{41}
\bibcite{Cao2}{42}
\bibcite{DEGUCHI2006}{43}
\bibcite{martin2}{44}
\bibcite{martin3}{45}
\bibcite{Besser2007sfs}{46}
\bibcite{holzapfel2}{47}
\bibcite{puglisi1}{48}
\bibcite{puglisi2}{49}
\bibcite{holzapfel1}{50}
\bibcite{gorielybook}{51}
\bibcite{tens1}{52}
\bibcite{puglisi3}{53}
\bibcite{tens2}{54}
\bibcite{fraldi2021}{55}
\bibcite{Brodland1990}{56}
\bibcite{Brangwynne2006}{57}
\bibcite{holzapfel2000}{58}
\bibcite{euler1744}{59}
\bibcite{Pampaloni2006}{60}
\bibcite{Kurachi1995}{61}
\bibcite{timoshenko1961}{62}
\bibcite{Bigoni2012}{63}
\bibcite{HILL1982}{64}
\bibcite{CellClusters}{65}
\bibcite{Fraldi2015}{66}
\bibcite{Nebuloni2016}{67}
\bibcite{Cross2007}{68}
\bibcite{Goetz2011}{69}
