From solution to surface to filament: actin flux into branched networks

Abstract

The actin cytoskeleton comprises a set of filament networks that perform essential functions in eukaryotic cells. The idea that actin filaments incorporate monomers directly from solution forms both the “textbook picture” of filament elongation and a conventional starting point for quantitative modeling of cellular actin dynamics. Recent work, however, reveals that filaments created by two major regulators, the formins and the Arp2/3 complex, incorporate monomers delivered by nearby proteins. Specifically, actin enters Arp2/3-generated networks via binding sites on nucleation-promoting factors clustered on membrane surfaces. Here, we describe three functions of this surface-associated actin monomer pool: (1) regulating network density via product inhibition of the Arp2/3 complex, (2) accelerating filament elongation as a distributive polymerase, and (3) converting profilin-actin into a substrate for the Arp2/3 complex. These linked functions control the architecture of branched networks and explain how capping protein enhances their growth.

Appendices

Appendix 1. Definition of key parameters and dynamical variables

Models discussed in this work assume that the concentrations of all soluble components—actin, profilin-actin, capping protein, and the Arp2/3 complex—are approximately constant. For this reason, they have been combined with the appropriate n-th order rate constants to form pseudo (n-1)-th order rate constants. Note that, since the dynamical variables in our model are surface densities (expressed in μm⁻²) rather than concentrations (expressed in μM), the values of rate constants determined in solution must be corrected (typically by a factor < 4) before the results can be compared to experimental data.

Dynamical variables

E :

is the density of free barbed ends of actin filaments proximal to the membrane surface. The proximity requirement is defined as the distance within which the filaments can productively interact with membrane-associated WASP family nucleation-promoting factors. This variable is treated as a surface density (in μm⁻²).

W _A :

is the surface density (in μm⁻²) of WH2 domains that are bound to monomeric actin.

P _A :

is the surface density (in μm⁻²) of poly-proline domains that are bound to profilin-actin complexes.

Parameters

E ₀ :

is the initial density (in μm⁻²) of free barbed ends of actin filaments in proximity to the membrane.

W _tot :

is the total surface density (in μm⁻²) of WASP family proteins associated with the membrane.

k _N :

is a pseudo third-order rate constant (in μm⁴ s⁻¹) that describes Arp2/3-dependent filament nucleation. This parameter is the product of a fourth-order rate constant, k_n+ (in μM⁻¹ μm⁴ s⁻¹), with the solution concentration of the Arp2/3 complex, [Arp2/3], which is assumed to remain constant. For low concentrations of the Arp2/3 complex, this parameter can be approximated by k_n+[Arp2/3]. Because this is a high-order reaction that requires formation of a multi-component complex, however, the range over which this approximation holds is unclear.

k _C :

is a pseudo first-order rate constant (in s⁻¹) that describes filament capping. This parameter is the product of the second-order rate constant for capping protein binding to free barbed ends, k_c+ (in μM⁻¹ s⁻¹), with the solution concentration of capping protein [C], which is assumed to remain constant. This can also be expressed as: k_c+[C].

k_wload and k_pload:

are pseudo first-order rate constants (in sec⁻¹) that describe, respectively, the binding of actin to WH2 domains and the binding of profilin-actin complexes to poly-proline sequences. These parameters are products of second-order rate constants for protein-protein interaction, k_wa+ and k_ppa+ (in μM⁻¹ s⁻¹), with the solution concentrations of monomeric actin [A] and profilin-actin complexes [PA], which are assumed to remain constant. These parameters can also be expressed as: k_wa+[A] and k_ppa+[PA].

k_wd and k_pd:

are first-order rate constants (in s⁻¹) that describe, respectively, dissociation of actin from a WH2 domain and dissociation of a profilin-actin complex from a poly-proline sequence.

k_wpol and k_ppol:

are second-order rate constants (in μm⁻² s⁻¹) that describe, respectively, the transfer of actin from WH2 domains and profilin-actin complexes on poly-proline sequences to the ends of nearby actin filaments.

K _trans :

is the rate (in μm⁻² s⁻¹) of intramolecular transfer of an actin monomer from a profilin-actin complex bound to a proline-rich domain to an adjacent, empty WH2 domain.

Appendix 2. Simple autocatalytic activation of the Arp2/3 complex

We assume that (1) the availability of soluble Arp2/3 complexes is not rate limiting and (2) the nucleation reaction is fast enough that the density of WASP family proteins available to interact with Arp2/3 complexes is approximately equal to the total surface density, W_tot. We can then write the rate of nucleation as k_NW_tot²E, where E is the number density of growing filaments near the NPF-coated surface and k_N is a nucleation rate constant. Strictly speaking, the nucleation rate depends on the density of filamentous polymer near the surface (F) rather than the density of free barbed ends (E). For a branched actin network growing against a surface, however, the density of polymer in proximity to the surface is proportional to E and inversely proportional to the cosine of the average angle (θ) between the filaments and a vector normal to the surface. We can, therefore, define the nucleation rate constant (k_N) to account for the difference between E and P.

The density W_tot is squared to account for the fact that two WASP family proteins are required to activate an Arp2/3 complex. In growing networks, capped filaments rapidly lose contact with NPF-coated surfaces and cease contributing to the nucleation reaction, so the change in filament number depends on both nucleation and capping:

$$ \frac{dE}{dt}={k}_N{W}_{tot}^2E-{k}_CE $$

(4)

Here, k_C is a pseudo-first-order capping rate constant that assumes capping protein concentration (2) does not change. Note that this equation has steady-state solutions only when k_NW_tot² ≤ k_C.

$$ E(t)={E}_0{e}^{\left({k}_N{W}_{tot}^2-{k}_C\right)t} $$

(5)

For k_NW_tot² < k_C capping outpaces nucleation and the density of the network monotonically decreases with time, from its initial value (E₀) toward zero. When k_NW_tot² > k_C, the exponent is positive and any filaments present at the beginning of the reaction (E₀) will spark unbounded, exponential growth. Finally, when k_NW_tot² = k_C (i.e. when W_tot = [k_C/k_N]^1/2), capping and nucleation are balanced, and filament density remains constant at E₀. This is not a stable steady-state, however, and tiny fluctuations in nucleation or capping will cause the filament density to either collapse or to grow without bound.

Appendix 3. The effect of WH2 polymerase activity on Arp2/3 activation

A stable steady-state solution to Eq. (4) requires some sort of negative feedback by which an increase in the number of growing filaments could decrease the rate of new filament formation. The nucleation reaction itself provides some negative feedback, as recognized by Weichsel et al. (2017). These authors suggested that nucleation uses up “active” nucleation-promoting factors which much be “re-activated.” Although they did not elaborate on the putative “active” and “inactive” states, we know from previous work (Machesky, 1999; Zalevsky, 2001; Dayel, 2004) that NPFs with monomeric actin bound to their WH2 domains can activate the Arp2/3 complex while NPFs lacking actin cannot. We can rewrite Eq. (4), taking into account the fact that at least one of the WASP-family proteins that activates an Arp2/3 complex must be charged with actin.

$$ \frac{dE}{dt}={k}_N{W}_{tot}{W}_AE-{k}_CE $$

(6)

When a nucleation-promoting factor transfers actin to the Arp2/3 complex or to the end of a nearby filament, it becomes temporarily inactive until “recharged” with actin. We must, therefore, also take account of the charging and depletion of WH2 domains, which can be (4) described by the following equation:

$$ \frac{d{W}_A}{dt}={k}_{wload}\left({W}_{tot}-{W}_A\right)-{k}_{wd}{W}_A-{k}_NE{W}_{tot}{W}_A-{k}_{wpol}E{W}_A $$

(7)

Equation 7 assumes (1) that the concentration of soluble actin monomers remains relatively constant and (2) that the total surface density of nucleation-promoting factors (W_tot) is simply the sum of the densities of actin-charged (W_A) and actin-depleted (W) molecules. In addition to nucleation and capping, we introduce rate constants for association and dissociation of actin and WH2 domains (k_wload and k_wd) and a rate constant for the polymerase activity of the WH2 domain (k_wpol).

We can now set the rates of change in Eqs. 6 and 4 equal to zero and solve for both the steady-state occupancy of the WH2 domain by monomeric actin, W_A, and the density of free barbed ends. The steady-state occupancy of WH2 domains turns out to be:

According to this equation, WH2 occupancy decreases with nucleation and increases with capping, reflecting the fact that free barbed ends in the actin network leech away bound actin.

$$ {\left.{W}_A\right|}_{SS}=\frac{k_C}{k_N{W}_{tot}} $$

(8)

Note that this equation is physically meaningful over a limited range of parameter values. If the nucleation or capping rates are zero, no stable network forms. We can also show (see below) that no network forms when k_C is equal to greater than k_NW₀². Satisfyingly, when k_C is exactly equal to k_NW₀², and the steady-state filament density is zero, occupancy of the WH2 domain is simply W_tot, the total surface density of WASP family proteins.

Similarly, we find that the steady-state density of free barbed ends is given by:

$$ {\left.E\right|}_{ss}=\frac{k_{wload}{W}_{tot}^2\frac{k_N}{k_C}-\left({k}_{wload}+{k}_{wd}\right)}{k_N{W}_{tot}+{k}_{wpol}} $$

Activation of the Arp2/3 complex may leech some bound actin from nucleation-promoting factors, but this reaction is limited by a slow, first-order step (Zalevsky, 2001; Marchand, 2001). This slow step makes nucleation much slower than transfer of actin from WH2 domains to free barbed ends (Bieling et al. 2018). From the measured rates of these reactions, we estimate that, given identical local densities of barbed ends and Arp2/3 complexes, the rate of actin flow from WH2 domains into filament elongation will be almost two orders of magnitude greater than that caused by nucleation. Neglecting the effects of nucleation and dissociation on the availability of actin-charged WH2 domains yields a simpler equation.

$$ {\left.E\right|}_{ss}=\frac{k_{wload}}{k_{wpol}}\left({W}_{tot}^2\frac{k_N}{k_C}-1\right) $$

(9)

The parenthetical expression in Eq. 6 defines a fundamental stability criterion for formation of an actin network in contact with a surface. Our physical interpretation is that, when this expression is less than or equal to zero, no network forms. The condition that must be satisfied for network formation is, therefore:

$$ {W}_{tot}>\sqrt{\frac{k_C}{k_N}} $$

In other words, the surface density of WASP family proteins required to generate a stable actin network scales as the square root of the ratio of capping and nucleation rates. This relationship likely explains failure of network formation at high concentrations of capping protein or low concentrations of the Arp2/3 complex (e.g., Akin, 2008).

Appendix 4. Estimating actin contributed by the surface-associated monomer pool

We assume that monomers from solution add to filaments at a rate of K_soln while surface-associated actin incorporates at a rate of K_surf. The composite rate of elongation is, therefore, given by:

$$ {K}_{comp}=f{K}_{surf}+\left(1-f\right){K}_{so\ln } $$

where f is the fraction of monomers incorporated from the surface. If we express the overall filament growth rate and the surface-mediated component of elongation in terms of the diffusion-limited growth from soluble monomers, we can rewrite the above equation as:

$$ \alpha {K}_{so\ln }=f\left(\beta {K}_{so\ln}\right)+\left(1-f\right){K}_{so\ln } $$

where α and β are the ratios of the composite and surface-driven elongation rates to the diffusion-limited growth rate. Solving for f, the fraction of monomers that incorporate from the PWCA-coated surface yields:

$$ f=\left(\frac{\alpha -1}{\beta -1}\right) $$

At PWCA surface densities of ~ 1800 μm⁻², which are more than an order of magnitude lower than those estimated for PWCA-coated membranes in living cells, Bieling et al. (2018) placed a upper bound on β of ~ 6 and a lower bound on α of ~ 4. Substituting these values into the equation for f indicates that, under these conditions, more than 60% of the actin in a branched network enters from the surface, and less than 40% enters from solution. Note that the filaments in a branched network are not all aligned with the direction of motion, and if the average angle between filaments and the direction of motion is φ then our estimate for β decreases by cosφ, increasing the lower bound on f. Under most conditions φ ~ 35°, so a more accurate estimate for the fraction of actin incorporated from these PWCA-coated surfaces is > 75%. Finally, the bounds on α and β provided by Bieling et al. (2018) were measured at different profilin-actin concentrations, 5 and 1 μM, respectively. Correcting the value of β for the differences in elongation rate and surface occupancy at the two different concentrations pushes the estimate of the surface contribution up even further, to > 90%.

Appendix 5. The effect of profilin-actin bound to proline-rich domains

In the presence of profilin and actin, the composite motor activity of branched actin networks can be described by three state equations: one for free barbed ends (E), a second for actin-charged WH2 domains (W_A), and a third describing occupancy of poly-l-proline sequences by profilin-actin complexes (P_A). For simplicity, we will assume that the proline-rich domain contains only one profilin-binding site.

$$ \frac{dE}{dt}={k}_N{W}_{tot}{W}_AE-{k}_CE $$

(10)

$$ \frac{d{W}_A}{dt}={K}_{trans}+{k}_{wload}\left({W}_{tot}-{W}_A\right)-{k}_{wd}{W}_A-{k}_{wpol}E{W}_A-{k}_NE{W}_{tot}{W}_A $$

(11)

$$ \frac{d{P}_A}{dt}=-{K}_{trans}+{k}_{pload}\left({W}_{tot}- PA\right)-{k}_{pd} PA-{k}_{ppol}E{P}_A $$

(12)

The above expressions for E and W_A come from Eqs. 6 and 7 of Appendix 3, with one additional term (K_trans) added to Eq. 7 to account for the transfer of actin onto the WH2 domain from profilin-actin complexes bound to the proline-rich region. We call this modified expression Eq. 11. In the expression for P_A (Eq. 12) the rate constants, k_pload and k_pd, govern binding and dissociation of profilin-actin and the proline-rich region. As usual, k_pload is a pseudo first-order rate constant that assumes the concentration of soluble profilin-actin complexes does not change. The second-order rate constant, k_ppol, governs polymerase activity, in which profilin-actin is transferred from the poly-proline sequence to the end of a nearby filament.

To better understand the connection between kinetics and network architecture, we can set all three differential equations above to zero and solve for the steady-state values of E, W_A, and P_A. To simplify the math, we assume that soluble actin monomers and profilin-actin complexes are both present at high enough concentrations so that loading WH2 and proline-rich domains from solution is much faster than transfer of actin between these sites. Given these assumptions, E|_ss and W_A|_ss, are given by Eqs. 8 and 9 (see Appendix 3), and the steady-state value of P_A is given by:

$$ {\left. PA\right|}_{ss}={W}_{tot}\left(\frac{k_{pload}}{k_{ppol}}\right)\frac{k_C{k}_{wpol}}{k_C\left[{k}_{wpol}\left(\frac{k_{pload}}{k_{ppol}}\right)-{k}_{wload}\right]+{k}_{wload}{k}_N{W}_{tot}^2} $$

(13)

This equation reveals that P_A|_ss has a hyperbolic dependence on capping rate. We can see this more easily by lumping the other parameters into three composites: α, β, and γ.

$$ {\left. PA\right|}_{ss}=\frac{\alpha {k}_C}{\beta {k}_C+\upgamma} $$

At low capping rates (i.e., low concentrations of capping protein), the occupancy of the proline-rich domain increases linearly with capping rate, but at higher capping rates, PRD occupancy begins to saturate. It is important to note that this equation is valid only over a limited range of capping rates. When the capping rate matches the maximum possible nucleation rate, W_tot²k_N, Eq. 9 (see Appendix 3) says that the steady-state density of free barbed ends will fall to zero and Eq. 13 says that the proline-rich domains will be maximally occupied by profilin-actin complexes (i.e., P_A|_ss = W_tot). If we limit ourselves to the biologically relevant regime of Eq. 13, where the capping rate is low enough to permit formation of a stable, branched actin network, we can simplify the expression significantly:

$$ {\left.{P}_A\right|}_{ss}=\frac{k_C}{W_{tot}{k}_N}\left(\frac{k_{pload}}{k_{ppol}}\right)\left(\frac{k_{wpol}}{k_{wload}}\right) $$

(14)

To assess the overall polymerase activity of surface-associated PWCA domains, we must combine contributions from both WH2 and proline-rich regions. The rate at which a filament in proximity to a PWCA-coated surface elongates is the sum of the WH2 and proline-rich domain occupancies multiplied by their respective polymerase rate constants:

$$ \frac{dL}{dt}={k}_{wpol}{W}_A{\left|{}_{ss}+{k}_{ppol}{P}_A\right|}_{ss} $$

Substituting from Eqs. 8 (see Appendix 3) and 14, we obtain:

$$ \frac{dL}{dt}={k}_{wpol}\frac{k_C}{k_N{W}_{tot}}\left(1+\frac{k_{pload}}{k_{wload}}\right) $$

(15)