On the existence and strength of stable membrane protrusions

The model includes the dense cross-linked actin gel in the bulk and the SR at the leading edge of the lamellipodium (see figure 1). The boundary between the gel and the SR is defined by a critical concentration of cross-linkers bound to the actin filaments. The density of filaments in the SR changes by nucleation of new filaments and capping and severing of existing filaments. Filaments in the SR can attach to the leading edge membrane via linker proteins. Attached filaments can not only push, but also pull the membrane. The leading edge dynamics is determined by the balance of filament forces and forces resisting motion.

Zoom In Zoom Out Reset image size

Semiflexible actin filaments are subject to Brownian motion at the length scale of cells since their persistence length l_p is in the same range. Filaments of contour length l grafted at one end exert an entropic force on an obstacle at distance z. The force has been calculated in [43] as

$$\begin{equation} F_{{\mathrm{d}}}(z,l,l_{\mathrm{p}})=F_{\mathrm{crit}}\tilde{F}(\tilde{\eta}), \end{equation} \tag{ 1 }$$

with the scaling variable

$$\begin{equation} \tilde{\eta} = \frac{l-z}{l_{\parallel}},\quad l_{\parallel}=\frac{l^2}{l_{\rm p}}, \end{equation} \tag{ 2 }$$

and

$$\begin{equation} F_{\rm crit}=\frac{\pi^2}{4}\frac{k_{\mathrm{B}}T}{l_{\parallel}} \end{equation} \tag{ 3 }$$

is the critical force for the Euler buckling instability. In [43], it is shown that for small compression $\tilde {\eta }\lesssim 0.2$, the scaling function of the entropic force can well be approximated as

$$\begin{equation} \tilde{F}^<(\tilde{\eta}) = \frac{4\exp(-\frac{1}{4\tilde{\eta}})}{\pi^{5/2}\tilde{\eta}^{3/2}\left[1-2\,{\rm erfc}\left(1/(2\sqrt{\tilde{\eta}})\right)\right]}. \end{equation} \tag{ 4 }$$

For $\tilde {\eta } \gtrsim 0.2$ the calculation yields

$$\begin{equation} \tilde{F}^>(\tilde{\eta}) = \frac{1-3\exp(-2\pi^2\tilde{\eta})}{1-\frac13\exp(-2\pi^2\tilde{\eta})}. \end{equation} \tag{ 5 }$$

The situation is different for attached filaments, since the tip of the filament is always positioned at the membrane and cannot fluctuate. The proteins linking the filaments to the membrane are assumed to behave like elastic springs. We distinguish three different regimes for the force F_a exerted by the serial arrangement of polymer and linker, depending on the relation between the depth of the semi-flexible region z, the equilibrium end-to-end distance R_∥ = l(1 − l/2l_p) and the contour length l [44]:

$$\begin{equation} F_{\mathrm{a}}(l,z) = \left\{ \begin{array}{@{}ll@{}} -k_{\parallel}(z-R_{\parallel}),\;z\leqslant R_{\parallel}, & \mbox{(i)}\\ -k_{\mathrm{eff}}(z-R_{\parallel}),\;R_{\parallel}<z<l, & \mbox{(ii)}\\ -k_l(z-l)-k_{\mathrm{eff}}(l-R_{\parallel}),\;z\geqslant l. & \mbox{(iii)} \end{array} \right. \end{equation} \tag{ 6 }$$

The three cases correspond to: (i) a compressed filament pushes against the membrane; (ii) filament and linker pull the membrane while being stretched together; and (iii) a filament is fully stretched but the linker continues to pull the membrane by being stretched further. Here, k_∥, k_l and k_eff are the linear elastic coefficients of polymer, linker and serial polymer–linker arrangement, respectively. For k_∥ we use the linear response coefficient of a worm-like chain grafted at both ends k_∥ = 6k_BTl²_p/l⁴ [45, 46].

Detached filaments attach to the membrane with a constant rate k_a. The detachment rate of attached filaments is force dependent since a pulling force facilitates detachment. It can be expressed as

$$\begin{equation} k_{\mathrm{d}}=k_{\mathrm{d}}^0 \exp (-d F_{\mathrm{a}}/k_{\mathrm{B}}T), \end{equation} \tag{ 7 }$$

with the force-free detachment rate k⁰_d. The length increment d added by an actin monomer to the filament is 2.7 nm.

Detached filaments can polymerize and grow. The velocity of polymerization is force dependent because the probability that the filament fluctuates away from the membrane and a gap sufficiently large for insertion of an actin monomer appears decreases with increasing the force [47]. The polymerization velocity reads

$$\begin{equation} v_{\rm p} = v_{\rm p}^{\mathrm{max}} \exp (-dF_{\mathrm{d}}/k_{\mathrm{B}}T). \end{equation} \tag{ 8 }$$

v^max_p is the maximum polymerization velocity depending on the actin monomer concentration.

The rate of filament shortening is contour length dependent. The filaments are shortened by the attachment of cross-linker molecules and incorporation of filament length into the actin gel. The cross-linking velocity is length dependent, since the cross-linker binding probability increases with filament length. It is unlikely that very short filaments get cross-linked. Free cross-linker molecules bind to filaments near the leading edge. Retrograde flow of the actin network transports them as bound molecules to the rear, where they dissociate and diffuse back to the front. Solving the corresponding reaction–diffusion equation, we have shown [48] that the cross-linking velocity can be expressed as

$$\begin{equation} v_{\mathrm{g}}(l,n) = \hat{v}_{\mathrm{g}}^{\mathrm{max}}n\tanh(nl/\bar{l}). \end{equation} \tag{ 9 }$$

It is proportional to the filament density n, because denser filament packing allows cross-linkers to span the inter-filament distance more easily. The characteristic length $\bar {l}$ and the maximum cross-linking rate $\hat {v}_{\mathrm {g}}^{\mathrm {max}}$ are parameters. In the rate of filament shortening $\tilde {v}_{\mathrm {g}}(l,z,n) = \max (1,v_{\mathrm {g}} l/z)$, the additional factor l/z accounts for the fact that a larger portion of filament length is incorporated into the gel during cross-linking when filaments are bent.

Detached filaments may get capped. The binding rate of capping proteins is force dependent, similar to the attachment of actin monomers to the filament barbed ends during polymerization. We find an Arrhenius factor in the capping rate

$$\begin{equation} k_{\mathrm{c}} = k_{\mathrm{c}}^{\mathrm{max}} \exp(- dF_{\mathrm{d}}/k_{\mathrm{B}} T). \end{equation} \tag{ 10 }$$

New filament branches are nucleated by Arp2/3 off attached filaments with a nucleation rate k_n. We assume nucleation from attached filaments since activation of the Arp2/3 complex by nucleation-promoting factors involves membrane binding. New filaments enter the force balance and thus the SR dynamics when they reach the support by the actin gel. Since the branching point vanishes into the gel quickly, newly nucleated filaments have the same length as the mother filament in our model. The total number n of filaments is undefined without a feedback to the nucleation process [42]. Such a feedback could be caused by a limited number of Arp2/3 proteins. The effective nucleation rate reads

$$\begin{equation} k_{\mathrm{n}} = k_{\mathrm{n}}^0 -k_{\mathrm{n}}^N n. \end{equation} \tag{ 11 }$$

The rates k⁰_n and k^N_n are constants.

We also want to include the disassembly of actin filaments by ADF/cofilin into our model. ADF/cofilin binds to ADP-actin within filaments and promotes its dissociation by severing and depolymerization of filaments [33]. We hypothesize that filaments to which cofilin is bound vanish from the SR because they cannot exert a force any longer once they are severed. Actin filaments bind ATP-actin monomers at their plus ends and quickly hydrolyze ATP to ADP-Pi but it takes longer to lose the y-phosphate. Cofilin only binds to ADP-actin when y-phosphate has dissociated [49, 50]. We can describe the dissociation by an exponential decay. The half lifetime for y-phosphate dissociation within the filament is 6 min [33]. We neglect that y-phosphate dissociation is probably accelerated by cofilin. The probability of cofilin binding is proportional to the probability of finding an ADP-actin monomer at a given site x from the tip of the filament:

$$\begin{equation} p_{\mathrm{ADP}} = 1 - \mathrm{e}^{-t\ln(2)/T_{1/2}} = 1 - \mathrm{e}^{-x\ln(2)/(v_{\rm p}^{\mathrm{max}}T_{1/2})}. \end{equation} \tag{ 12 }$$

We assume that the polymerization velocity is constant v^max_p. The probability of filament severing is found by integrating over the whole filament length

$$\begin{equation} p_{\mathrm{sev}} = \int_0^l 1 - \mathrm{e}^{-x\ln(2)/(v_{\rm p}^{\mathrm{max}}T_{1/2})} \mathrm{d}x = l + \frac{v_{\rm p}^{\mathrm{max}}T_{1/2}}{\ln(2)}\left(\mathrm{e}^{-\frac{\ln(2)l}{v_{\rm p}^{\mathrm{max}}T_{1/2}}} -1\right). \end{equation}\noindent \tag{ 13 }$$

That leads to l-dependent terms in the dynamics of the number of attached and detached filaments.

We have calculated the retrograde flow velocity in the actin gel [48] using the theory of the active polar gel by Kruse et al [51, 52]. It depends linearly on the force acting on the leading edge membrane. Solving the gel equations leads to the expressions for the coefficients of this linear equation as a function of the gel parameters. We obtain for the gel velocity

$$\begin{eqnarray} u &\approx& v_{\mathrm{link}} - \frac{\mu L}{4\eta}g_1 + \frac{f_0}{L\xi}g_2,\nonumber\\ g_1 &=& \frac{1}{2.0+0.12\frac{\xi L^2}{4\eta h_0}},\\ g_2 &=& \left(1.0 +0.92 \frac{\xi L^2}{h_0 4\eta}\right)^{1/2}\left(1.0 + 0.03\frac{\mu L}{4\eta v_{\mathrm{link}}}\right).\nonumber \end{eqnarray}\noindent \tag{ 14 }$$

Here, v_link is the cross-linking velocity, hence the velocity at which the actin gel is produced. The gel boundary does not move forward with the velocity of cross-linking since there is a backwards-directed retrograde flow in the actin gel. The term proportional to g₁ expresses the retrograde flow arising from contractile stress μ in the actin gel. Contractions may be caused by myosin motors or actin depolymerization [53]. L = 10 μm is the width of the gel part of the lamellipodium and η is the viscosity of the actin gel. The g₂-term describes a retrograde flow due to filaments in the SR pushing the gel backwards. The total force that they exert on the gel boundary is denoted by f₀. Adhesions between the gel and the substrate are described by the friction coefficient ξ. h₀ is the height of the lamellipodium at the boundary between the gel and the SR. We have fit g₁, g₂ for $0 \leqslant \frac {\xi L^2}{4\eta h_0} \leqslant 50$. Equations (14) are valid on the condition that $\frac {\mu L}{4\eta v_{\mathrm {link}}} < 1$.

The processes considered so far determine the length distribution of attached N_a(l,t), detached N_d(l,t) and capped filaments N_c(l,t) in the SR. Their dynamics are described by the following linear equations (see also [42]):

$$\begin{eqnarray} &&\fl \frac{\partial}{\partial t}N_{\mathrm{d}}(l,t) = \frac{\partial}{\partial l}((\tilde{v}_{\mathrm{g}}-v_{\rm p})N_{\mathrm{d}}(l,t)) + k_{\mathrm{d}} N_{\mathrm{a}}(l,t) - k_{\mathrm{a}} N_{\mathrm{d}}(l,t) - k_{\mathrm{c}} N_{\mathrm{d}}(l,t)\nonumber\\ &&\ms - k_{\mathrm{sev}}N_{\mathrm{d}}(l,t) \left[l + \frac{v_{\rm p}^{\mathrm{max}}T_{1/2}}{\ln(2)}\left(\mathrm{e}^{-\frac{l\ln(2)}{v_{\rm p}^{\mathrm{max}}T_{1/2}}} -1\right)\right], \end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray} &&\fl \frac{\partial}{\partial t}N_{\mathrm{a}}(l,t) = \frac{\partial}{\partial l}(\tilde{v}_{\mathrm{g}} N_{\mathrm{a}}(l,t)) - k_{\mathrm{d}} N_{\mathrm{a}}(l,t) + k_{\mathrm{a}} N_{\mathrm{d}}(l,t) + k_{\mathrm{n}} N_{\mathrm{a}}(l,t)\nonumber\\ &&- k_{\mathrm{sev}}N_{\mathrm{a}}(l,t) \left[l + \frac{v_{\rm p}^{\mathrm{max}}T_{1/2}}{\ln(2)}\left(\mathrm{e}^{-\frac{l\ln(2)}{v_{\rm p}^{\mathrm{max}}T_{1/2}}} -1\right)\right], \end{eqnarray} \tag{ 16 }$$

$$\begin{equation} \fl \frac{\partial}{\partial t}N_{\mathrm{c}}(l,t) = \frac{\partial}{\partial l}(\tilde{v}_{\mathrm{g}} N_{\mathrm{c}}(l,t)) + k_{\mathrm{c}} N_{\mathrm{d}}(l,t). \end{equation} \tag{ 17 }$$

k_sev is the binding rate of cofilin.

As shown in [41, 42], a δ-function-ansatz for N_d(l) and N_a(l) used in equations (15) and (16) leads to ordinary differential equations for the total density of attached and detached filaments n_a and n_d and for their mean lengths l_a and l_d:

$$\begin{eqnarray} &&\dot{n}_{\mathrm{d}}= k_{\mathrm{d}} n_{\mathrm{a}}- (k_{\mathrm{a}}+k_{\mathrm{c}})n_{\mathrm{d}} -k_{\mathrm{sev}}n_{\mathrm{d}} \left[l_{\mathrm{d}} + \frac{v_{\rm p}^{\mathrm{max}}T_{1/2}}{\ln(2)}\left(\mathrm{e}^{-\frac{l_{\mathrm{d}}\ln(2)}{v_{\rm p}^{\mathrm{max}}T_{1/2}}} -1\right)\right], \end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray} &&\dot{n}_{\mathrm{a}}= k_{\mathrm{a}} n_{\mathrm{d}}- (k_{\mathrm{d}}-k_{\mathrm{n}})n_{\mathrm{a}} -k_{\mathrm{sev}}n_{\mathrm{a}} \left[l_{\mathrm{a}} + \frac{v_{\rm p}^{\mathrm{max}}T_{1/2}}{\ln(2)2}\left(\mathrm{e}^{-\frac{l_{\mathrm{a}}\ln(2)}{v_{\rm p}^{\mathrm{max}}T_{1/2}}} -1\right)\right], \end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray} &&\dot{l}_{\mathrm{d}} = -(\tilde{v}_{\mathrm{g}}(l_{\mathrm{d}},z,n)-v_{\rm p}(l_{\mathrm{d}},z))+ k_{\mathrm{d}}(l_{\mathrm{a}},z)\frac{n_{\mathrm{a}}(t)}{n_{\mathrm{d}}(t)}(l_{\mathrm{a}}(t)-l_{\mathrm{d}}(t)), \end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray} &&\dot{l}_{\mathrm{a}} = -\tilde{v}_{\mathrm{g}}(l_{\mathrm{a}},z,n)+ k_{\mathrm{a}}\frac{n_{\mathrm{d}}(t)}{n_{\mathrm{a}}(t)}(l_{\mathrm{d}}(t)-l_{\mathrm{a}}(t)). \end{eqnarray} \tag{ 21 }$$

The dynamics of the SR width reads

$$\begin{equation} \dot{z} = \frac{1}{\kappa}(f_0 -f_{\mathrm{ext}}) - u(v_{\mathrm{link}}, -f_0), \end{equation} \tag{ 22 }$$

with the total filament force

$$\begin{equation} f_0 = \left(F_{\mathrm{d}}(l_{\mathrm{d}},z)n_{\mathrm{d}}(t) + F_{\mathrm{a}}(l_{\mathrm{a}},z)n_{\mathrm{a}}(t) + f_{\mathrm{c}}(n_{\mathrm{d}},l_{\mathrm{d}},z,n)\right). \end{equation} \tag{ 23 }$$

We have also included a constant external force f_ext that acts on the leading edge. Expression (14) is used for the gel boundary velocity u. The total force of capped filaments f_c, the average cross-linking rate v_link and the total filament density n will be calculated below.

The monodisperse approximation is not valid for the distribution of capped filaments N_c(l,t). That renders the calculation of N_c(l,t) much more complicated than it was for the other densities. However, we need N_c(l,t) for its contributions f_c to the force, n_c to the total number of filaments and v^c_g to the average cross-linking velocity v_link (see equation (42)).

Equation (17) is solved using the method of characteristics. Here, we assume that filaments are long when they get capped. We neglect the length dependence of v_g and only account for $\tilde {v}_{\mathrm {g}} = \max (1,l/z)\hat {v}_{\mathrm {g}}^{\mathrm {max}}n$. As before, we write $v_{\mathrm {g}}^{\mathrm {max}}(n)=\hat {v}_{\mathrm {g}}^{\mathrm {max}}n$. Furthermore, we are only interested in N_c(l,t) for z ⩽ l ⩽ l_d, since for l < z, capped filaments exert no force. Hence, $\tilde {v}_{\mathrm {g}} = \frac {l}{z} v_{\mathrm {g}}^{\mathrm {max}}$. Using the monodisperse approximation for the detached filaments N_d(l,t) = n_d(t)δ(l − l_d(t)), equation (17) reads

$$\begin{equation} \frac{\partial}{\partial t}N_{\mathrm{c}} = \frac{v_{\mathrm{g}}^{\mathrm{max}}}{z}N_{\mathrm{c}} + \frac{l}{z}v_{\mathrm{g}}^{\mathrm{max}}\frac{\partial}{\partial l} N_{\mathrm{c}} + k_{\mathrm{c}}(l_{\mathrm{d}})n_{\mathrm{d}}(t)\delta(l-l_{\mathrm{d}}). \end{equation} \tag{ 24 }$$

With

$$\begin{equation} \frac{\mathrm{d}N}{\mathrm{d}s} = \frac{\partial N}{\partial t}\frac{\mathrm{d}t}{\mathrm{d}s} + \frac{\partial N}{\partial l}\frac{\mathrm{d}l}{\mathrm{d}s}, \end{equation} \tag{ 25 }$$

we can identify the characteristics

$$\begin{equation} \frac{\mathrm{d}t}{\mathrm{d}s} = 1\,,\quad \frac{\mathrm{d}l}{\mathrm{d}s}= -v_{\mathrm{g}}^{\mathrm{max}}\frac{l}{z} \end{equation} \tag{ 26 }$$

and

$$\begin{equation} \frac{\mathrm{d}N_{\mathrm{c}}}{\mathrm{d}s}= \frac{v_{\mathrm{g}}^{\mathrm{max}}}{z}N_{\mathrm{c}} + k_{\mathrm{c}}(l_{\mathrm{d}})n_{\mathrm{d}}(t)\delta(l-l_{\mathrm{d}}). \end{equation} \tag{ 27 }$$

The first equation (the first of equations (26)) gives s = t and therefore we obtain

$$\begin{equation} \frac{\mathrm{d}l}{\mathrm{d}t} = -\frac{v_{\mathrm{g}}^{\mathrm{max}}}{z}l \end{equation} \tag{ 28 }$$

with the solution (obtained by separation of variables)

$$\begin{equation} l(t) = l(t^*)\exp\left(-\int_{t^*}^t \frac{v_{\mathrm{g}}^{\mathrm{max}}(t')}{z(t')}\mathrm{d}t'\right). \end{equation} \tag{ 29 }$$

The time point of capping is denoted by t*. Solving

$$\begin{equation} \frac{\mathrm{d}N_{\mathrm{c}}}{\mathrm{d}t}= \frac{v_{\mathrm{g}}^{\mathrm{max}}}{z}N_{\mathrm{c}} + k_{\mathrm{c}}(l_{\mathrm{d}})n_{\mathrm{d}}(t)\delta(l-l_{\mathrm{d}}) \end{equation} \tag{ 30 }$$

requires a little more effort. The general solution of the inhomogeneous equation equals the sum of the solution of the homogeneous equation and a special solution of the inhomogeneous equation. The solution of the homogeneous equation reads

$$\begin{equation} N_{\mathrm{c}}^{\mathrm{h}} = C\exp\left(\int_{t^*}^{t}\frac{v_{\mathrm{g}}^{\mathrm{max}}}{z}\mathrm{d}t'\right). \end{equation} \tag{ 31 }$$

The special solution of the inhomogeneous equation is found by the variation of constants:

$$\begin{eqnarray*} &&\fl N_{\mathrm{c}}^{\mathrm{sp}} = \left[\int_{t^*}^t \mathrm{d}t' k_{\mathrm{c}}(l_{\mathrm{d}}(t'))n_{\mathrm{d}}(t')\delta(l(t')-l_{\mathrm{d}}(t')\exp\left(-\int_{t^*}^{t'}\frac{v_{\mathrm{g}}^{\mathrm{max}}}{z}\mathrm{d}t''\right)\right]\exp\left(\int_{t^*}^{t}\frac{v_{\mathrm{g}}^{\mathrm{max}}}{z}\mathrm{d}t'\right)\\ &&= \int_{t^*}^t \mathrm{d}t' k_{\mathrm{c}}(l_{\mathrm{d}}(t'))n_{\mathrm{d}}(t')\delta(l(t')-l_{\mathrm{d}}(t')\exp\left(\int_{t'}^{t}\frac{v_{\mathrm{g}}^{\mathrm{max}}}{z}\mathrm{d}t''\right)\\ &&=\frac{k_{\mathrm{c}}(l_{\mathrm{d}}(t^*))n_{\mathrm{d}}(t^*)}{|\frac{\mathrm{d}}{\mathrm{d}t'}(l(t')-l_{\mathrm{d}}(t'))|_{t'=t^*}}\exp\left(\int_{t^*}^{t}\frac{v_{\mathrm{g}}^{\mathrm{max}}}{z}\mathrm{d}t''\right). \end{eqnarray*}$$

In the last line, we have used $\delta (g(x))=\sum _{i=1}^n\frac {\delta (x-x_i)}{|g'(x_i)|}$, where x_i are the roots of g(x). Note that l(t*) = l_d(t*) at the time of capping t*. Equation (29) yields

$$\begin{equation} \exp\left(\int_{t^*}^{t}\frac{v_{\mathrm{g}}^{\mathrm{max}}}{z}\mathrm{d}t''\right) = \frac{l_{\mathrm{d}}(t^*)}{l(t)}. \end{equation} \tag{ 32 }$$

We find that

$$\begin{equation*} \frac{\mathrm{d}}{\mathrm{d}t}l(t)|_{t=t^*} = -\frac{v_{\mathrm{g}}^{\mathrm{max}}(t^*)}{z(t^*)}l_{\mathrm{d}}(t^*) \end{equation*}$$

using (29). Furthermore,

$$\begin{equation*} \frac{\mathrm{d}}{\mathrm{d}t}l_{\mathrm{d}}(t)|_{t=t^*} = -\frac{v_{\mathrm{g}}^{\mathrm{max}}(t^*)}{z(t^*)}l_{\mathrm{d}}(t^*)+v_{\rm p}(l_{\mathrm{d}}(t^*))+k_{\mathrm{d}}(l_{\mathrm{a}}(t^*))\frac{n_{\mathrm{a}}(t^*)}{n_{\mathrm{d}}(t^*)}(l_{\mathrm{a}}(t^*)-l_{\mathrm{d}}(t^*)). \end{equation*}$$

Hence,

$$\begin{equation} N_{\mathrm{c}}^{\mathrm{sp}}(t,t^*)=\frac{k_{\mathrm{c}}(l_{\mathrm{d}}(t^*))n_{\mathrm{d}}(t^*)}{v_{\rm p}(l_{\mathrm{d}}(t^*))+k_{\mathrm{d}}(l_{\mathrm{a}}(t^*))\frac{n_{\mathrm{a}}}{n_{\mathrm{d}}}(t^*)(l_{\mathrm{a}}(t^*)-l_{\mathrm{d}}(t^*))}\frac{l_{\mathrm{d}}(t^*)}{l(t)} \end{equation} \tag{ 33 }$$

for $k_{\mathrm {d}}(l_{\mathrm {a}}(t^*))\frac {n_{\mathrm {a}}}{n_{\mathrm {d}}}(t^*)(l_{\mathrm {a}}(t^*)-l_{\mathrm {d}}(t^*))>-v_{\mathrm {p}}(l_{\mathrm {d}}(t^*))$. To find the length distribution of capped filaments, for every length l, t* has to be calculated by solving l = l_d(t*). The number of capped polymers is determined by the number of detached polymers and the capping rate at the time of capping.

For calculating the total number, force and cross-linking rate, we need

$$\begin{eqnarray*} &&\fl \frac{\partial l}{\partial t^*} = \frac{\partial}{\partial t^*}\left[l_{\mathrm{d}}(t^*)\exp\left(-\int_{t^*}^t \frac{v_{\mathrm{g}}^{\mathrm{max}}(t')}{z(t')}\mathrm{d}t'\right)\right]\\ &&= \left[\dot{l}_{\mathrm{d}}(t^*)-\left(-\frac{v_{\mathrm{g}}^{\mathrm{max}}}{z}(t^*)\right)l_{\mathrm{d}}(t^*)\right]\exp\left(-\int_{t^*}^t \frac{v_{\mathrm{g}}^{\mathrm{max}}(t')}{z(t')}\mathrm{d}t'\right)\\ &&= \left[-\frac{v_{\mathrm{g}}^{\mathrm{max}}}{z}l_{\mathrm{d}}(t^*)+v_{\rm p}(t^*)+k_{\mathrm{d}}\frac{n_{\mathrm{a}}}{n_{\mathrm{d}}}(l_{\mathrm{a}}-l_{\mathrm{d}})(t^*)+\frac{v_{\mathrm{g}}^{\mathrm{max}}}{z}l_{\mathrm{d}}(t^*)\right]\exp\left(-\int_{t^*}^t \frac{v_{\mathrm{g}}^{\mathrm{max}}}{z}\mathrm{d}t'\right)\\ &&= \left[v_{\rm p}\left(l_{\mathrm{d}}(t^*)\right)+k_{\mathrm{d}}\left(l_{\mathrm{a}}(t^*)\right)\frac{n_{\mathrm{a}}}{n_{\mathrm{d}}}(t^*)\left(l_{\mathrm{a}}(t^*)-l_{\mathrm{d}}(t^*)\right)\right]\exp\left(-\int_{t^*}^t \frac{v_{\mathrm{g}}^{\mathrm{max}}(t')}{z(t')}\mathrm{d}t'\right). \end{eqnarray*}$$

The total density of capped filaments with z ⩽ l ⩽ l_d is then given by

$$\begin{equation} \fl n_{\mathrm{c}} = \int_{z(t)}^{l_{\mathrm{d}}(t)}\mathrm{d}l\,N_{\mathrm{c}}(l,t) = \int_{t^*_z}^t \mathrm{d}t^*\frac{\partial l}{\partial t^*}N_{\mathrm{c}}(t^*,t) = \int_{t^*_z}^t \mathrm{d}t^* k_{\mathrm{c}}\left(l_{\mathrm{d}}(t^*),z(t^*)\right)n_{\mathrm{d}}(t^*). \end{equation} \tag{ 34 }$$

The lower integral boundary is z(t) since shorter filaments do not exert any force and are therefore not considered in the model. The time of capping of filaments with length z at time t corresponds to t*_z. We again use equation (29)

$$\begin{equation} z(t) = l_{\mathrm{d}}(t_z^*)\exp\left(-\int_{t_z^*}^t \frac{v_{\mathrm{g}}^{\mathrm{max}}(t')}{z(t')}\mathrm{d}t'\right) \end{equation} \tag{ 35 }$$

and apply a root finding algorithm to determine t*_z. The total number of all filaments is given by n = n_c + n_a + n_d.

Along these lines, we can also calculate the total force of capped filaments

$$\begin{equation} \fl f_{\mathrm{c}} = \int_{z(t)}^{l_{\mathrm{d}}(t)}\mathrm{d}lN_{\mathrm{c}}(l,t)F_{\mathrm{d}}(l,z) = \int_{t^*_z}^t \mathrm{d}t^* k_{\mathrm{c}}\left(l_{\mathrm{d}}(t^*),z(t^*)\right)n_{\mathrm{d}}(t^*)F_{\mathrm{d}}\left(l(t^*),z(t)\right). \end{equation} \tag{ 36 }$$

Note that we have to calculate l(t*) according to expression (29) for every t*. The average cross-linking rate yields

$$\begin{equation} \fl v_{\mathrm{g}}^{\mathrm{c}} = \frac{1}{n}\int_{z(t)}^{l_{\mathrm{d}}(t)}\mathrm{d}lN_{\mathrm{c}}(l,t)v_{\mathrm{g}}(l,n) = \frac{1}{n}\int_{t^*_z}^t \mathrm{d}t^* k_{\mathrm{c}}\left(l_{\mathrm{d}}(t^*),z(t^*)\right)n_{\mathrm{d}}(t^*)v_{\mathrm{g}}\left(l(t^*),n(t)\right). \end{equation} \tag{ 37 }$$

The total number of filaments (attached, detached and capped) n enters v^max_g, and n_c is therefore already required for determining t*_z (equation (35)). The total number changes according to

$$\begin{equation} \frac{\mathrm{d}n}{\mathrm{d}t}=k_{\mathrm{n}} n_{\mathrm{a}}(t) - v_{\mathrm{g}}^{\mathrm{max}}(t)N_{\mathrm{c}}(l=z,t). \end{equation} \tag{ 38 }$$

It increases by nucleation and decreases because capped filaments are eaten by the gel. We only consider capped filaments longer than z that exert a force in order to simplify the calculations. Capped filaments with length l = z vanish at the rate of gel cross-linking from this filament population. Their number is determined by equation (33) for l(t) = z(t).

To calculate t*_z in every time step and integrate over F_d and v_g is computationally very demanding. We also want to avoid tracking the history of l_a, l_d, z, n_a, n_d and n. To simplify the calculation, we assume a stationary distribution N_c(l). In the stationary case, we obtain

$$\begin{eqnarray} &&\fl N_{\mathrm{c}}(l) = \int_{t^*}^t \mathrm{d}t' k_{\mathrm{c}}(l_{\mathrm{d}}(t'))n_{\mathrm{d}}(t')\delta(l(t')-l_{\mathrm{d}}(t')\exp\left(\int_{t'}^{t}\frac{v_{\mathrm{g}}^{\mathrm{max}}}{z}\mathrm{d}t''\right)\nonumber\\ &&= -\int_{l_{\mathrm{d}}}^l \mathrm{d}l' \frac{z}{v_{\mathrm{g}}^{\mathrm{max}}l'} k_{\mathrm{c}} n_{\mathrm{d}} \delta(l'-l_{\mathrm{d}}) \frac{l'}{l} = \frac{z k_{\mathrm{c}} n_{\mathrm{d}}}{l v_{\mathrm{g}}^{\mathrm{max}}}. \end{eqnarray} \tag{ 39 }$$

We have changed the integration variable according to equation (28).

The total density of capped filaments that exert a force (i.e. with length z < l < l_d) reads

$$\begin{equation} n_{\mathrm{c}} = \int_z^{l_{\mathrm{d}}} \mathrm{d}l N_{\mathrm{c}}(l) = \frac{z k_{\mathrm{c}} n_{\mathrm{d}}}{v_{\mathrm{g}}^{\mathrm{max}}} \ln\left(\frac{l_{\mathrm{d}}}{z}\right). \end{equation} \tag{ 40 }$$

With $v_{\mathrm {g}}^{\mathrm {max}}= \hat {v}_{\mathrm {g}}^{\mathrm {max}}(n_{\mathrm {a}}+n_{\mathrm {d}}+n_{\mathrm {c}})$, we obtain

$$\begin{equation} n_{\mathrm{c}} = -\frac{n_{\mathrm{a}}+n_{\mathrm{d}}}{2}+\sqrt{\left(\frac{n_{\mathrm{a}}+n_{\mathrm{d}}}{2.0}\right)^2 + \ln\left(\frac{l_{\mathrm{d}}}{z}\right)\frac{k_{\mathrm{c}}n_{\mathrm{d}}z}{\hat{v}_{\mathrm{g}}^{\mathrm{max}}}}. \end{equation} \tag{ 41 }$$

To calculate the average cross-linking rate,

$$\begin{equation} v_{\mathrm{link}} = \frac{1}{n}\left(n_{\mathrm{a}}v_{\mathrm{g}}(l_{\mathrm{a}}) + n_{\mathrm{d}}v_{\mathrm{g}}(l_{\mathrm{d}}) + v_{\mathrm{g}}^{\mathrm{c}} \right), \end{equation} \tag{ 42 }$$

we again consider only capped filaments with z ⩽ l ⩽ l_d. We neglect the length dependence of v_g and set v_g = v^max_g to obtain analytic expressions

$$\begin{equation} \fl v_{\mathrm{g}}^{\mathrm{c}} = \int_z^{l_{\mathrm{d}}}v_{\mathrm{g}}(l)N_{\mathrm{c}}(l)\,\mathrm{d}l = zk_{\mathrm{c}}n_{\mathrm{d}}\int_z^{l_{\mathrm{d}}}\frac{v_{\mathrm{g}}(l)}{lv_{\mathrm{g}}^{\mathrm{max}}}\mathrm{d}l \approx zk_{\mathrm{c}}n_{\mathrm{d}}\int_z^{l_{\mathrm{d}}}\frac{1}{l}\mathrm{d}l = zk_{\mathrm{c}}n_{\mathrm{d}}\ln\left(\frac{l_{\mathrm{d}}}{z}\right). \end{equation} \tag{ 43 }$$

The force of capped filaments is given by

$$\begin{equation} f_{\mathrm{c}} = \int_{z}^{l_{\mathrm{d}}}\mathrm{d}l N_{\mathrm{c}}(l)F_{\mathrm{d}}(l,z) = \frac{k_{\mathrm{c}}n_{\mathrm{d}}z}{v_{\mathrm{g}}^{\mathrm{max}}}\int_z^{l_{\mathrm{d}}}\frac{F_{\mathrm{d}}}{l}\mathrm{d}l, \end{equation} \tag{ 44 }$$

with $F_{\mathrm {d}}(l,z) = \frac {\pi ^2}{4}\frac {k_{\mathrm {B}}Tl_{\mathrm {p}}}{l^2} \tilde {F}(\tilde {\eta })$ (see equation (1)). The scaling function $\tilde {F}(\tilde {\eta })$ (equations (4) and (5)) cannot be integrated analytically. It increases monotonically to 1 with increasing the compression $\tilde {\eta }$ of the filament. When simulating the dynamical system we see that the 1/l³-dependence of the integrand of f_c dominates over the increasing part of $\tilde {F}$ (see figure 2). Therefore, we approximate F_d by the Euler buckling force F_crit and obtain

$$\begin{equation} f_{\mathrm{c}} = \frac{k_{\mathrm{c}}n_{\mathrm{d}}z}{v_{\mathrm{g}}^{\mathrm{max}}}\int_z^{l_{\mathrm{d}}} \frac{\pi^2}{4}\frac{k_{\mathrm{B}}Tl_{\rm p}}{l^3}\mathrm{d}l = \frac{k_{\mathrm{c}}n_{\mathrm{d}}z}{v_{\mathrm{g}}^{\mathrm{max}}}\frac{\pi^2}{8}k_{\mathrm{B}}Tl_{\rm p}\left(\frac{1}{z^2}-\frac{1}{l_{\mathrm{d}}^2}\right). \end{equation}\noindent \tag{ 45 }$$

Zoom In Zoom Out Reset image size

In figure 3, we compare the solution of our time-dependent model with the solution of the model with the approximations introduced in this section.

Zoom In Zoom Out Reset image size

Our model defines criteria for the existence of stable lamellipodia. In figure 4, we examine how the stationary filament density changes with some model parameters. The black areas indicate regions in the parameter space where n = 0 (no filaments in the SR) is the only stable fixed point. Since n = 0 means that there is no protrusion, the conditions for the existence of attractors with n > 0 (fixed points or limit cycles) describe the conditions for the existence of stable protrusions. Different sets of parameters in our model correspond to different cell types or different levels of expression or activation of signaling molecules within one cell type.

Zoom In Zoom Out Reset image size

No stable lamellipodium exists for low nucleation rates in figure 4(A), because the creation of new filaments by nucleation cannot compensate for filament extinction by capping and severing. The stable protrusion vanishes for small cross-linking rates $\hat {v}_{\mathrm {g}}^{\mathrm {max}}$ since the filaments are long. That entails large severing rates and renders the filaments floppy, which increases the capping rate. Similarly, the filament density decreases with increasing capping rate k^max_c (figure 4(B)). A larger external force has among others the consequence of decreasing the capping rate via its force dependence (equation (10)). Furthermore, filaments shorten to adapt to the external force, which decreases the severing rate. In this way, applying an external force may cause protrusion formation in the parameter regime shown in figure 4(B). Nucleation is proportional to the number of attached filaments. Consequently, filament binding may cause protrusion generation in figure 4(C).

Besides stable fixed points with a certain filament density, our model also exhibits stable limit cycles. Those limit cycles also correspond to stable lamellipodia. However, the leading edge shows oscillatory motion and varying protrusion velocities. In figure 5, we show two examples of oscillatory solutions of the model. As already discussed in [41], the membrane velocity can either stay at an intermediate value most of the time and periodically drop to lower values during short 'stops' (figure 5(F)), or the membrane periodically jerks forward during short 'jumps' (figure 5(E)). During the phase of slow movement, attached filaments are shorter than z and pull the membrane. The effective cross-linking velocity $\tilde {v}_{\mathrm {g}}$ is larger than the effective polymerization velocity v_p. Consequently, filaments shorten until the force is sufficiently large to disrupt the attached filaments from the membrane and to push it forward. Now the filaments can grow longer again, exert weaker forces and attach to the membrane (see [41] for a detailed description of the oscillation mechanism). The retrograde flow increases or decreases with the membrane velocity since both are proportional to the total filament force. New filaments are nucleated from attached filaments in our model. Due to the nucleation, the total filament density increases when forces are low and the number of attached filaments goes up (figures 5(A) and (B)). The number of capped filaments also increases, because the capping rate is high at low forces, and the filaments are long and it takes longer until they vanish into the gel.

Zoom In Zoom Out Reset image size

Many mathematical models equate the leading edge velocity with the polymerization rate or a monotonically increasing algebraic function of it. That excludes a phase difference between the maxima of the polymerization rate and leading edge velocity in oscillations. However, such a phase difference has been observed [54]. The leading edge velocity increases first and subsequently the polymerization rate increases. Figure 6 shows the two oscillation types as limit cycles in the phase plane spanned by polymerization velocity v_p and leading edge velocity. The system cycles clockwise in both the cases. The red limit cycle corresponds to oscillations shown in figures 5(A), (C) and (E), and the blue one to figures 5(B), (D) and (F)). The phase difference between both velocities in the red limit cycle is obvious. It is about 9 s expressed in time. That is less than the 20 s observed in [54], but we did not search for parameters with longer delays since the important message is the existence of a clear phase difference. The blue limit cycle exhibits almost no phase difference between the two maxima, but the leading edge velocity decreases earlier than the polymerization rate.

Zoom In Zoom Out Reset image size

In figure 7, we show two examples of bifurcation diagrams where increasing the nucleation rate (figures 7(A), (C) and (E)) or external force (figures 7(B), (D) and (F) leads to a transition from no lamellipodium to a stable, stationarily protruding lamellipodium (a stable fixed point, solid line) to a stable, oscillating lamellipodium (an unstable fixed point, dashed line, a stable limit cycle). In figures 7(A), (C) and (E), the fixed points of the dynamical system are plotted against the nucleation rate k⁰_n. At k⁰_n = 0.25 s⁻¹, the nucleation rate becomes large enough to generate a stable fixed point. Attached and detached filaments are both longer than the SR depth z here. The filament density, and therefore also the cross-linking rate, increases with increasing the nucleation rate k⁰_n. Consequently, filaments get shorter and exert higher forces. Eventually, attached filaments are shorter than z and the effective cross-linking rate may get larger than the effective polymerization rate, leading to a transition to the oscillatory regime. A stable fixed point different from n = 0 can also be generated by increasing the external force (figures 7(B), (D) and (F)). Again, to balance the increasing external force, filaments shorten and their number increases until the fixed point loses stability. Since now the filament density decreases again and the filaments are very short, in the range of the saturation length of the cross-linking velocity, the cross-linking velocity drops below the polymerization velocity and the fixed point becomes stable again upon further growth of force. Note that in those examples attachment rates k_a and detachment rates k⁰_d are lower than in figure 4, which is necessary for observing the oscillations.

Zoom In Zoom Out Reset image size

Cells that exhibit stable lamellipodia have to be able to withstand substantial forces from their surroundings. The force–velocity relation describes the lamellipodium protrusion velocity as a function of the force exerted on the leading edge. It is usually measured with a scanning force microscope (SFM) cantilever (see [30]). In [30], we simulated the experimentally measured force–velocity curves with a model with a constant filament density. We now repeat our fit with the model including capping, nucleation and severing. The result is shown in figure 8 and the parameters in table 1. The measured data are still very well reproduced by the model. When the cell touches the cantilever, the velocity of the leading edge drops from ∼250 nm s⁻¹ to less than 1 nm s⁻¹ (figure 8(C)), followed by a concave force–velocity relation (figure 8(B)). As already described in [30], the force–velocity relation mainly gets its characteristic shape due to an initial bending of long filaments and a subsequent adaptation of filament lengths to the increasing external force (figure 8(E)). Retrograde flow slowly increases until it compensates for polymerization in the stalled state (figure 8(C)). The total number of filaments first increases during the concave phase because the capping rate decreases with increasing the force, and severing decreases with shrinking the filament length. Later, the filament number decreases again, because the ratio of attached to detached filaments decreases and therefore also the nucleation rate. The value in the stalled state is slightly higher than in the freely running cell.

Zoom In Zoom Out Reset image size

Table 1. List of model parameters and their values in figure 8. See also [30].

Symbol	Meaning	Value	Units	Reference
ka	Attachment rate of filaments to the membrane	10.0	s−1	10 s−1 in [55]
k0d	Detachment constant	25.0	s−1	Fitted
vmaxp	Saturation value of polymerization velocity	46.2	μm min−1	30 μm min−1 in [47]
$\hat {v}_{\mathrm {g}}^{\mathrm {max}}$	Saturation value of the gel cross-linking rate	0.075	μm2 min−1	Fitted
k0n	Nucleation rate	0.6	s−1	[24]
kNn	Limiting factor of the nucleation rate	0.0016	μm s−1	Fitted
kmaxc	Capping rate	0.065	s−1	[24]
ksev	Binding rate of cofilin	2.0	s−1 μm−1	Assumed
T1/2	Half-life of ATP-actin within filament	6.0	min	[33]
$\bar {l}$	Saturation length of the cross-linking rate	10		Assumed
κ	Drag coefficient of the plasma membrane	0.113	nN s μm−2	[56]
k	Elastic modulus of SFM cantilever	291	nN μm−2	[30]
d	Actin monomer radius	2.7	nm	[57]
lp	Persistence length of actin	15	μm	[58]
kl	Spring constant of linker protein	1	nN μm−1	[47, 59]
η	Viscosity of actin gel	0.833	nN s μm−2	[60, 61]
ξ	Friction coefficient of actin gel to adhesion sites	0.175	nN s μm−3	[62]
μ	Active contractile stress in actin gel	8.33	pN μm−2	Fitted
h0	Height of lamellipodium at the leading edge	0.25	μm	[63, 64]
L	Length of the gel part of lamellipodium	10	μm	[21, 64]
	Contact length with beads	4.4	μm	[30]

Our model fits the measured protrusion and retrograde flow velocities of the freely running cell before cantilever contact, the velocity measured with the cantilever during the concave phase, the value of the stall force and the shape of the force–velocity relation. Due to the good agreement between the measured data and the simulation, we assume that the model parameters determined by the fit of the force–velocity relation represent the 'default' values for the stable keratocyte lamellipodium in parameter space. We can also conclude some features of the structure of the lamellipodium such as filament length and branch point density. The capping, nucleation and severing rates are relatively low. With the filament density of about 280 μm⁻¹ (see figure 8(D)), the effective nucleation rate k_n = k⁰_n − k^N_nn is approximately 9 min⁻¹. Since filaments polymerize in the freely running cell with about 31μm min⁻¹ (the rate of filament elongation equals the rate of filament shortening $n\hat {v}_{\mathrm {g}}^{\mathrm {max}}l/z$), we should find a branching point approximately every 3.5 μm along the filament. Filaments in the SR are less than 2 μm long and consequently the branch point density is low. If we keep in mind that new branches grow from attached filaments only, we find about 50 branch points in the SR per μm lateral width. The capping rate is also low. The model result for the density of capped filaments in the freely running cell is approximately 10 μm⁻¹ (figure 8(D)). We should bear in mind that this is only the number of capped filaments with lengths between z and l_d. Hence, the total number of capped filaments in the SR amounts to 30 μm⁻¹ (see figure 8(E)). Consequently, to accomplish a stationary filament number, the newly nucleated filaments are partly compensated for by capping, partly by severing. In [24], the authors find on average one branch point every 0.8 μm along a filament by evaluating electron microscopy tomograms. However, this value was measured in NIH 3T3 cells and treadmilling is much slower in those cells than in keratocytes, which entails also a smaller branch point distance given a comparable branching rate. The capping and nucleation rates in their simulations (k_cap = 0.03 s⁻¹, k_br = 0.042 s⁻¹) are slightly lower but in the same range as in our fit (table 1). Moreover, the actual branch point density should be higher because we only account for filament branches that have already grown to the length of the mother filament in our model.

We would like to emphasize that the force–velocity relation measured with the SFM cantilever is not the relation between a constant external force and stationary velocity values. It gets its characteristic shape due to the adaptation of filament length, SR depth and retrograde flow to the increasing external force. The stationary force–velocity relation with the parameter values from our fit of the keratocyte data is shown in figure 9(A). The stationary force–velocity relation is defined as the stationary protrusion velocity at a given constant external force. It is dominated by the force–velocity relation of the gel (equation (14)) and is almost linear. The slight change in slope occurs because the filament density, and therefore also the maximum cross-linking rate, first increases and then decreases (figure 9(C)). Filaments in the SR shorten to balance the increasing external force (figure 9(E)). However, they remain long enough that the effective cross-linking rate does not drop below its maximum value. Otherwise, we could observe a change in slope (see [48]). We find that the stationary force–velocity relation allows for much faster motion for forces below the stall force than the dynamic force–velocity relation. Cells adapt to the constantly applied external force and become faster by that adaptation. It is not possible that a stationary relation exhibits the initial velocity drop seen in the experiment. If we increase the capping and nucleation rates, the maximum in the filament density is more pronounced (figure 9(D)). Consequently, the stationary force–velocity relation has a concave shape (figure 9(B)). The velocity first increases with increasing force, reaches a maximum and then drops. As we see in figures 7(B), (D) and (F), the force–velocity relation can get much more complex for lower attachment and detachment rates. Here, we enter the oscillatory regime by increasing the external force and observe a drop in the stationary velocity.

Zoom In Zoom Out Reset image size