A continuum model for the growth of dendritic actin networks

Abstract

Polymerization of dendritic actin networks underlies important mechanical processes in cell biology such as the protrusion of lamellipodia, propulsion of growth cones in dendrites of neurons, intracellular transport of organelles and pathogens, among others. The forces required for these mechanical functions have been deduced from mechano-chemical models of actin polymerization; most models are focused on single growing filaments, and only a few address polymerization of filament networks through simulations. Here, we propose a continuum model of surface growth and filament nucleation to describe polymerization of dendritic actin networks. The model describes growth and elasticity in terms of macroscopic stresses, strains and filament density rather than focusing on individual filaments. The microscopic processes underlying polymerization are subsumed into kinetic laws characterizing the change of filament density and the propagation of growing surfaces. This continuum model can predict the evolution of actin networks in disparate experiments. A key conclusion of the analysis is that existing laws relating force to polymerization speed of single filaments cannot predict the response of growing networks. Therefore, a new kinetic law, consistent with the dissipation inequality, is proposed to capture the evolution of dendritic actin networks under different loading conditions. This model may be extended to other settings involving a more complex interplay between mechanical stresses and polymerization kinetics, such as the growth of networks of microtubules, collagen filaments, intermediate filaments and carbon nanotubes.

1. Introduction

Actin polymerization drives the protrusion of lamellipodia and pseudopodia that play important roles in cell sensing and motility [1–3]. Actin polymerization is also the mechanism behind motion of growth cones in dendrites of neurons, organelles in cells, as well as pathogens, such as Listeria and Shigella [4]. As such, actin based motility has been studied using experiment and theory for at least the last three decades [2]. It is now understood that the propulsive force in actin polymerization is a result of the difference in chemical potential of the actin monomers in their bound (to the filament) and unbound (in solution) states [4,5].

Early Brownian ratchet models [6] for the force generated by actin polymerization focused on a single filament polymerizing against a load represented by a bead moving through a viscous medium, and the goal was to relate the force on the filament with its elongation-rate measured by the velocity of the bead [2,7]. In these models, thermal fluctuations of the filament and the bead created gaps between the bead and the polymerizing tip and resulted in an exponential dependence of the velocity V of the bead and the force f on the filament (taken to be positive in compression):

$$V=V_0\frac{\exp (-\zeta f/f_{\text{stall}})-\exp (-\zeta )}{1-\exp (-\zeta )},$$ 1.1

where V₀ is the velocity at f = 0 and f_stall is the force at which polymerization stalls [7]; ζ is related to the size of the monomers a through ζ = a f_stall/(k_BT) where k_B is the Boltzmann constant and T the absolute temperature. The stall force for actin was estimated to be around 1–8 pN [2] and this was confirmed through experiments on growing bundles of a few actin filaments with forces exerted using an optical trap [4]. However, dendritic actin and the actin gel in Listeria ‘comet tails’ is not a bundle of parallel growing filaments; rather, it is branched and cross-linked with filament lengths that are in the range of 1 μm or less [5,8,9]. Thus, in dendritic actin a large number of short filaments exert forces on the lamellipodial membrane or the moving object (pathogen, bead, organelle, etc.) resulting in an elongation rate–force¹ curve that is not necessarily of the form (1.1). This was shown in subsequent models which accounted for the transient nature of the contact between the load surface and growing filaments and the elasticity of the filament network [10,11]. These models and others [12] recognized that polymerization in dendritic actin occurs in a narrow zone next to the load surface (not all over the network), so that the barbed ends of most of the growing actin filaments are pointed toward the load surface.

Experiments to measure the elongation rate–force relation of dendritic actin are scarce. One experiment that yielded intriguing results was that of Parekh et al. [9], who polymerized dendritic actin between two atomic force microscope (AFM) cantilevers. The deflection of the AFMs allowed them to accurately measure both forces and elongation-rates. They were also able to control the conditions under which polymerization occurred (e.g. constant force, a force proportional to network height). They found that the elongation-rate was independent of the force over a range of force and this was followed by a convex curve in which polymerization stalled over a short range of force; this elongation rate–force relation is very different from the exponential form in equation (1.1). Parekh et al. also demonstrated that the elongation-rate was loading history dependent and in particular that there could be two (or more) steady-state elongation-rates at the same force. Some of these findings were confirmed in a later experiment by Brangbour et al. [5], who used magnetic beads to control the force resisting polymerization. The actin polymerized between two beads with short filaments contacting the beads at random angles, much like a dendritic network. Polymerization caused the beads to move away from each other so that their velocity could be measured as a function of the force applied on them by the magnetic field. These authors showed that constraints on the rotational fluctuations of the filaments resulted in an elongation rate–force relation that is different from the exponential form of equation (1.1).

Our objective in this paper is to explain the results of Brangbour et al. [5] and Parekh et al. [9] through a continuum model of growth. Continuum models² of growth fall into two broad categories: volumetric growth and surface growth. In the former, new material is added to existing material points whereas in the latter, new material points are added to the body at its evolving boundary. One way in which to track the newly added material points is through a continuously evolving reference configuration.³ The driving force for surface growth is essentially the configurational force associated with this evolving reference configuration and can be calculated in several different⁴ (roughly equivalent) ways. The bio-physical micro-mechanisms underlying the growth process are captured at the continuum scale by a kinetic law for growth, with the second law of thermodynamics imposing certain restrictions on such a kinetic law.

In our continuum model, the body is treated as a one-dimensional nonlinearly elastic bar whose length at time t is ℓ(t) in physical space and ℓ_R(t) in reference space. The body we are concerned with is comprised of actin filaments (F-actin)—long polymer chains—formed by the assembly of actin monomers (G-actin). It is surrounded by a pool of free actin monomers that, under suitable conditions, can bind to the tips of the existing filaments leading to their growth. The length ℓ_R(t) evolves due to the addition of new material points. In addition, the formation of new filaments leads to a time-dependent filament density ρ(t) [9]. At this juncture, it is worth emphasizing the distinction between the elongation-rate $\dot{\ell }$ and the growth-rate ${\dot{\ell }}_R$. The former is the rate of change of the length of the specimen in physical space, a quantity that can be observed and measured. Figure 2b shows a plot of $\dot{\ell }$ versus the force σA as predicted by our model in a particular setting; note the commonly observed rapidly declining nature of this curve. On the other hand, any change in the length ℓ_R in reference space occurs solely due to growth. This is in contrast to ℓ which changes due to both growth and stress. The relationship between ${\dot{\ell }}_R$ and force is an input into the model, the kinetic law for growth. The particular relation used to predict figure 2b is shown dotted in figure 6, and the qualitative distinction between the curves in the two figures is worth noting.

Figure 2.

(a) Force σA versus length ℓ according to (3.4) when the loading-rate is much faster than the growth-rate allowing ℓ_R to be treated as constant. The figure has been drawn for EA = 2.259 pN and two different values of ℓ_R. The vertical axis is on a logarithmic scale. Compare with fig. 3 of [5]. (b) Elongation-rate $\dot{\ell }$ versus force σA at two different values of E A according to (3.6). The figure has been drawn for V₀ = 0.42 nm s⁻¹. Compare with fig. 5 of [5]. (Online version in colour.)

Figure 6.

Growth speed V/V₀ versus filament force f/f_stall for surface growth. Dotted: maximum dissipation kinetic law (3.3)₁ from §3. Dashed: exponential kinetic law (1.1) with ζ = 5. Solid: power-law kinetics according to (4.9) with m = 1, 2 and 5. (Online version in colour.)

A phenomenon that brings out the distinction between growth and elongation is ‘treadmilling’, commonly observed in growing actin filaments [1,2]. During treadmilling, the filament length ℓ (in physical space) remains constant because of a precise balance between the rates at which monomers attach to one end (growth) of a filament and detach from the other. Thus, growth occurs continuously, but elongation does not evolve. A recent paper [28] analyses the existence and stability of treadmilling solutions in a one-dimensional elastic bar attached to a spring at one end. By contrast, in the setting studied in the present paper, monomers do not detach from either end (actin tends not to depolymerize at physiological ion concentrations and in the presence of capping proteins), so treadmilling does not occur. Despite this, the length of the specimen can stop changing due to the phenomenon of stall which occurs when the compressive force is so large that it prevents the addition of new monomers [1,2].

In the rest of this section, we describe (with little justification) some key aspects of our model. These will be explained in detail in later sections. Let σ and λ denote the compressive stress and stretch in the body. Since the material stiffens with increasing compressive stress, we take a simple stress–stretch relation that captures this phenomenon: σ = E(λ⁻¹ − 1). The tangent modulus of the material, −∂σ/∂λ, then increases with increasing σ. The elastic modulus E could also depend on the filament density ρ. We are concerned with two settings. In the first (pertaining to Brangbour et al. [5]), the number of actin filaments is relatively small and therefore do not form a cross-linked network. Consequently, filament bending is unimportant and so we take the elastic modulus to depend linearly on the filament density: E(ρ) ∼ ρ. In the second setting (pertaining to Parekh et al. [9]), the filaments form a cross-linked network where filament bending plays an important role. Therefore in this setting, we take the modulus to depend quadratically on the filament density: E(ρ) ∼ ρ². Moreover, in the first setting the filament density remains constant, $\dot{\rho }=0$, whereas in the second, it changes over time which we model through a suitable evolution law $\dot{\rho }=\mathsf{R}(\rho )$. In both settings, new material points can be added to one end of the specimen leading to surface growth through polymerization. The growing boundary propagates at a speed V given by a kinetic law $V=\mathsf{V}(f)$ where f = σ/ρ is the filament force. Finally, in the first setting, both ends of the specimen are attached to supports and the length or force on the specimen can be controlled. In the second setting, one end is attached to an AFM cantilever modelled as a Hookean spring so that, in addition to controlling the force, it is also possible to allow the specimen to freely evolve against the spring. Because of the way in which the experiments are set up, polymerization only occurs at one of the ends of the specimen. At the other end, neither polymerization nor depolymerization takes place.

This paper is organized as follows. In §2, we describe the basic aspects of the model that are common to both settings studied in the subsequent sections. Section 3 is concerned with problems where the filament density is time independent. Further details of the model are given in §3a; the response of the model to various loading programmes are examined in §3b and compared with the experimental measurements of Brangbour et al. [5]; and finally in §3c we explain how our model describes the setting in [5] even though they might seem different at first glance. We then turn in §4 to the growth of a network of actin filaments. The detailed constitutive model is presented in §4a and the values of the various parameters are given in §4a(iv). In §4b, we study the response of the specimen when it is growing under the action of the spring, and in §4c we examine its response to loading programmes that involve a sudden change in the force. The results of both sections are compared with the experimental measurements of Parekh et al. [9]. The online electronic supplementary material provides details on how the parameter values were chosen and the calculations in §4c carried out. Moreover, in electronic supplementary material, section S4, we derive the dissipation inequality, identify the driving force, examine the kinetic laws considered in §§3 and 4 in the context of the dissipation inequality, and verify the conformity of the solutions in §§3 and 4 with this inequality.

2. The basic model

Since we will examine two rather different experimental settings in this paper, here we simply write down those constituents of the mathematical model that are common to both. More detailed descriptions will be given in §§3a and 4a.

Imagine a test specimen composed of actin filaments held between two supports. One of the supports will be compliant in §4 as shown schematically in figure 5. Each filament is a polymeric molecule, comprised of a linear assembly of monomers. The filaments are surrounded by a solvent containing free monomers that can attach to one end of each filament (called the barbed end in actin).

Figure 5.

Schematic figure of test specimen depicting cross-linking, load-bearing filaments and free filaments. The specimen is identified with the interval [y₀(t), y₁(t)] in physical space. It is attached to an AFM spring at its left-hand end y₀(t) and to a (movable) support at its right-hand end y₁(t). In reference space, the specimen is identified with the interval [x₀(t), x₁]. Surface growth occurs at the left end of the specimen causing the boundary x = x₀(t) to move leftward.

In physical space, the specimen is identified with the interval [y₀(t), y₁(t)] at time t and so its corresponding length is

$$\ell (t)=y_1(t)-y_0(t).$$

In reference space, it is identified (at the same instant t) with the interval [x₀(t), x₁] where x₀ is permitted to be a function of time because growth may occur at that end. No growth occurs at⁵ x₁. The length of the specimen in reference space at time t is

$${\ell }_{\text{R}}(t)=x_1-x_0(t).$$ 2.1

The length in physical space is affected by both stress and growth, whereas the length in reference space is only affected by growth. When monomers are added to the specimen at its left-hand boundary, the specimen grows through the leftward motion of that surface in reference space at a speed

$$V=-{\dot{x}}_0.$$ 2.2

Thus growth corresponds to ${\dot{x}}_0<0$ (and therefore by (2.1) to ${\dot{\ell }}_{\text{R}}>0$). If there is no growth, x₀(t) is constant. In order not to confuse the two velocities $\dot{\ell }$ and ${\dot{\ell }}_R$, we shall refer to $\dot{\ell }$ as the elongation-rate and ${\dot{\ell }}_R$ as the growth-rate. In the problems of interest to us, the stress and stretch fields are spatially uniform and so the stretch λ of the specimen relates the reference and current lengths:

$$\ell =\lambda \hspace{0.17em}{\ell }_{\text{R}}.$$ 2.3

Let N(t) denote the number of load bearing filaments in the specimen and let A be the (fixed) area over which they are distributed. The filament density ρ(t) is defined as the number of filaments per unit cross-sectional area ρ: = N/A. If the force in each filament is f, the total force in the specimen is fN and so the stress is related to the filament force and filament density by

$$\sigma =\frac{fN}{A}=f\rho .$$ 2.4

We are concerned exclusively with compressive stress and so take σ to be positive in compression.

The key variables in the model are the stress σ(t), stretch λ(t), filament density ρ(t), length of the specimen in physical space ℓ(t) and length of the specimen in reference space ℓ_R(t). The stress depends on the stretch and the filament density through a constitutive relation $\sigma =\hat{\sigma }(\lambda ,\rho )$. This relation is assumed to be invertible so that we can write

$$\lambda =\mathsf{\Lambda }(\sigma ,\rho ).$$ 2.5

Growth occurs by two mechanisms, one leading to an increase in ℓ_R, the other to an increase in ρ. Typically, the specimen involves ‘load-carrying filaments’ that extend from one support to the other. Monomers from the surrounding monomer pool can be added to the tips of such filaments at the functionalized support at x₀. This changes their reference length ℓ_R. There may also be ‘free filaments’ in the specimen, for example, with one end attached to the existing polymer network and the other free. Monomers can be added to the free ends of such filaments. When a free filament grows, it may eventually touch one of the supports and turn into a load-carrying filament of the first type. Such growth changes the density ρ of load-carrying filaments.

We assume that the rate of increase of the number of load-bearing filaments, $\dot{\rho }$, is governed by a kinetic law of the form

$$\dot{\rho }=\mathsf{R}(\rho ),$$ 2.6

where the kinetic response function $\mathsf{R}$ is allowed to depend on ρ but not σ. Since a ‘free filament’ remains stress-free while it grows by polymerization at its free tip, such growth is expected to be unaffected by stress and hence we have taken $\mathsf{R}$ to be independent of σ. As for surface growth due to polymerization, we assume that the speed of the moving boundary in reference space is related to the filament force by a kinetic relation of the form

$$V=\mathsf{V}(f)=\mathsf{V}(\sigma ,\rho ).$$ 2.7

Since the kinetic response function $\mathsf{V}$ is a function of the filament force, it depends on both stress and filament density. Similar to earlier work [11], we write $\mathsf{V}$ as a function of the (average) filament force f = σ/ρ to facilitate comparison with kinetic laws such as equation (1.1) that are given in the literature in terms of f for single growing filaments [2].

The five variables ℓ(t), ℓ_R(t), σ(t), λ(t) and ρ(t) are governed by the equations (2.3), (2.5), (2.6) and (2.7), together with (2.1), (2.2) and a loading condition such as the prescription of the stress history.

3. Growth at constant filament density

In this section, we model the test specimen as involving a fixed number of nearly parallel filaments as in the experiments of Brangbour et al. [5]. The filament density therefore does not change and the constitutive relation of the specimen can be deduced from that of a single filament. Moreover, there is no cross-linking and filament bending is not important.

(a). Further details of the model

We now make the following choices for the constitutive response functions $\mathsf{\Lambda },\mathsf{V}$ and $\mathsf{R}$. The stress σ in the network is a function of the stretch λ that vanishes when λ = 1 and becomes unbounded at extreme compression when λ → 0⁺. Moreover the material stiffens with increasing compressive stress. A simple model capturing this is

$$\sigma =E({\lambda }^{-1}-1),\lambda =\mathsf{\Lambda }(\sigma ,\rho )=\frac{1}{1+\sigma /E}\text{where}\hspace{0.17em}E=E(\rho ):=\rho E_0$$ 3.1

is the effective Young’s modulus of the specimen. Note that the network elastic modulus is chosen to be (proportional to the number N of filaments and therefore) linear in the filament density ρ since the elasticity of the network in [5] is of entropic origin such that E₀ ∝ k_BT. The tangent modulus E_t—the slope of the stress–stretch curve—is given as a function of stress by

$$E_t(\sigma )=-{\frac{\text{d}\sigma }{\text{d}\lambda }|}_{\lambda ={[1+\sigma /E]}^{-1}}\hspace{0.17em}=E{(1+\frac{\sigma }{E})}^2.$$ 3.2

As shown in figure 1, the graph of E_t(σ) versus σ according to (3.2) rises monotonically and is concave upwards which agrees with the trends observed experimentally; e.g. see fig. 3 of Chaudhuri et al. [29].

Figure 1.

Tangent modulus E_t(σ)/E versus stress σ/E according to (3.2). The horizontal axis has been drawn on a logarithmic scale so as to facilitate comparison with fig. 3 of [29]. (Online version in colour.)

In this section, the number N of load-bearing filaments, and therefore the filament density ρ, is assumed to remain constant during each experiment. Thus $\dot{\rho }=\mathsf{R}(\rho )=0$. Next, suppose that growth occurs at a constant rate V₀ provided the filament force f is less that a certain critical value f_stall, or equivalently, when the stress is less than a certain value σ_stall. Accordingly, we take the kinetic law for surface growth (2.7) to be

$$V=\mathsf{V}(f)=\{\begin{array}{lll}{V}_0& \text{for}& f<f_{\text{stall}},\\ 0& \text{for}& f>f_{\text{stall}},\end{array}V=\{\begin{array}{lll}{V}_0& \text{for}& \sigma <{\sigma }_{\text{stall}},\\ 0& \text{for}& \sigma >{\sigma }_{\text{stall}},\end{array}$$ 3.3

where σ = ρf, σ_stall = ρf_stall. This piecewise constant kinetic relation is depicted by the dotted line in figure 6. Such kinetic relations arise in other mechanics settings where they have been associated with a notion of maximum dissipation, e.g. plasticity [30,31], phase transitions [32] and kink band motion [33]. They are referred to as maximally dissipative kinetics. At this time, we do not know what (if any) the physics underlying the maximally dissipative kinetic law is. In some fields, maximum dissipation is connected to ideas related to maximum entropy production.

(b). Predictions of the model

In this section, we shall consider various predictions of the model described above. Most of the results can be readily and conveniently described using non-dimensional variables. However, we shall continue to use the (dimensional) variables introduced above since that allows us to make quantitative comparisons with the experimental measurements of Brangbour et al. [5]. A brief description of their experiments and a discussion of how their apparently different model relates to ours are postponed to §3c. Details on how the various parameter values were chosen is described in electronic supplementary material, section S1.

By (2.3) and (3.1), the length ℓ of the specimen at any instant is related to the stress and unstressed reference length at that instant by

$$\ell (t)=\frac{{\ell }_R(t)}{1+\sigma (t)/E}.$$ 3.4

In the first set of calculations, the filaments are initially permitted to grow freely under zero stress for some time interval [0, t₀). During this stage, the unstressed reference length of the specimen is ℓ_R(t) = V₀ t as required by (3.3), (2.1) and (2.2). At time t₀ the force on the specimen is increased rapidly, and the force σA and specimen length ℓ are measured. If the loading-rate is much faster than the rate of growth, the value of ℓ_R can be assumed to remain constant at the value ℓ_R = V₀ t₀. Figure 2a shows a plot of the force σA versus the specimen length ℓ in such an experiment as predicted by (3.4) for two different fixed values of ℓ_R = V₀ t₀. This figure may be compared with fig. 3 of Brangbour et al. [5].

In the second set of calculations, the filaments are again permitted to grow freely under zero stress for some initial time interval [0, t₀) during which ℓ_R(t) = V₀ t. At time t₀ a force is applied on the specimen and held constant. Then from (3.3) and (3.4), provided σ < σ_stall,

$$\ell (t)=\frac{V_{0}t}{1+\sigma /E},t>t_0.$$ 3.5

Figure 3a shows a plot of the specimen length ℓ(t) versus time t for three different fixed values of the force σA. Observe that the slope decreases as the force increases. This figure may be compared with fig. 1c of Brangbour et al. [5].

Figure 3.

(a) The length of the specimen ℓ versus time t according to (3.5) at two different values of the force σA. In order to compare this with fig. 1c of Brangbour et al. [5], we have plotted ℓ + 2R on the vertical axis where R = 550 nm. The figure has been drawn using V₀ = 0.42 nm s⁻¹, t₀ = 1000 s and EA = 3.764 pN. The three lines correspond to σA = 0.5 pN; σA = 3 pN; and σA = 17 pN. (b) The length of the specimen ℓ versus time t according to (3.9)₁ with σ₁ A = 0.8 pN and σ₂ A = 39 pN. In order to compare this with fig. 1c of Brangbour et al. [5], we have plotted ℓ + 2R on the vertical axis where R = 550 nm. The figure is drawn for V₀ = 0.42 nm s⁻¹ and EA = 3.764 pN. (Online version in colour.)

The elongation rate–force response⁶ plays a central role in the study of actin filament growth. Thus consider a third set of calculations in which the force is again held constant but now examine the elongation-rate $\dot{\ell }$ as a function of force σA and in particular on how it depends on EA. Differentiating (3.5) with respect to t at constant σ gives

$$\dot{\ell }=\frac{V_0}{1+\sigma A/(EA)},$$ 3.6

and figure 2b shows a plot of the elongation-rate $\dot{\ell }$ versus the force σA according to (3.6). The two curves correspond to EA = 3.387 pN and EA = 0.979 pN. This figure may be compared with fig. 5 of Brangbour et al. [5].

Fourth, suppose that the specimen is initially growing under some constant stress σ₁ and that at some instant t₁ the stress is suddenly increased to a much larger value σ₂ (less than the stall stress). The stress is kept constant at this higher value until time t₂ at which instant it is suddenly decreased back to its original value σ₁ and held constant at that value from then on. This is described by the loading history

$$\sigma (t)=\{\begin{array}{lll}{\sigma }_1& \text{for}& 0<t<t_1,\\ {\sigma }_2& \text{for}& t_1<t<t_2,\\ {\sigma }_1& \text{for}& t>t_2,\end{array}\text{where}\hspace{0.17em}0<{\sigma }_1<{\sigma }_2\hspace{0.17em}(<{\sigma }_{\text{stall}}).$$ 3.7

The corresponding response is found by solving (3.4), (3.3), (2.1) and (2.2):

$$\ell (t)=\frac{{\ell }_R(t)}{1+\sigma (t)/E},{\dot{\ell }}_R(t)=V_0.$$ 3.8

In solving (3.8), it is important to keep in mind that the referential length of the filament ℓ_R(t) changes only due to growth. Thus at an instant at which the stress changes discontinuously, the length ℓ(t) of the filament will also change discontinuously due to the jump in σ but the referential length will remain continuous since a finite segment of new material cannot appear in infinitesimal time. Thus from (3.7) and (3.8)

$$\ell (t)=\{\begin{array}{lll}\frac{V_{0}t}{1+{\sigma }_1/E}& \text{for}& t_0<t<t_1,\\ \frac{V_{0}t}{1+{\sigma }_2/E}& \text{for}& t_1<t<t_2,\\ \frac{V_{0}t}{1+{\sigma }_1/E}& \text{for}& t>t_2,\end{array}\dot{\ell }(t)=\{\begin{array}{lll}\frac{V_0}{1+{\sigma }_1/E}& \text{for}& t_0<t<t_1,\\ \frac{V_0}{1+{\sigma }_2/E}& \text{for}& t_1<t<t_2,\\ \frac{V_0}{1+{\sigma }_1/E}& \text{for}& t>t_2.\end{array}$$ 3.9

In figure 3a, we observed that there is effectively a one-parameter family of straight lines on the t, ℓ-plane, the force σA = F being the parameter. During the loading (3.7), the point (t, ℓ(t)) first moves on the straight line corresponding to σA = F₁ in figure 3b. At time t₁, it jumps down to the straight line associated with σA = F₂ and traverses that line for t₁ < t < t₂, and finally at time t₂ it jumps back up to the straight line σA = F₁ where F_i is the force. This figure may be compared with fig. 2 of Brangbour et al. [5].

(c). The model of Brangbour et al. [5].

Brangbour et al. [5] devised a novel method for measuring the force–velocity relation for a system involving a relatively small number of growing actin filaments. They used a suspension containing (actin monomers as well as) magnetic colloidal particles that when subjected to a magnetic field assembled into a linear chain. The surfaces of the colloidal particles were functionalized so that a certain controlled number of actin filaments grew radially from the surface of each particle. The interaction between the filaments on two adjacent particles caused the particles to move apart against the magnetically applied force. In this way, the authors measured the force, the separation between particles and their velocity. Several of their results were described above.

Brangbour et al. [5] arrive at (what is effectively) equation (3.1) using a more accurate version of the following simplified micro-mechanical argument: suppose that actin filaments of stress-free length ℓ_R are attached to a rigid spherical particle of radius R that is in the vicinity of a wall. The wall represents the mid-plane between a pair of adjacent particles. When the distance ℓ from the particle surface to the wall is greater than ℓ_R as in figure 4a, there is no contact between the filaments and the wall and so the filaments carry no force. On the other hand, if the distance to the wall is less than ℓ_R as in figure 4b, some filaments will interact with the wall and carry force. The (compressive) force f in a filament increases as the distance ℓ to the wall decreases. Suppose the force is given by the classical entropic model f = c₁/ℓ for ℓ < ℓ_R, where the parameter c₁ is related to the elastic modulus of a filament. Moreover, as the distance between the particle and wall decreases, the number, N, of filaments interacting with the wall, and therefore carrying force, increases from the value N = 0 at ℓ = ℓ_R. Assume that the number of force carrying filaments is given by the linear relation N = c₂(ℓ_R − ℓ) for ℓ < ℓ_R, where the parameter c₂ is related to the density of filaments on the particle surface. The total force between the particle and wall, fN, when calculated using the two preceding equations, leads to precisely a constitutive relation of the form (3.1) with EA = c₁ c₂. It is worth emphasizing that when the constitutive relation (3.1) is derived in this way, it accounts for both the stress–stretch behaviour of a filament and the changing number of load carrying filaments.

Figure 4.

(a) Case ℓ > ℓ_R: the filaments are not in contact with the wall and so remain stress-free. (b) Case ℓ < ℓ_R: each of N filaments carry a force f. Both N and f increase as ℓ decreases (at fixed ℓ_R). (Online version in colour.)

According to the entropic model c₁ = k_BT where k_B is the Boltzmann constant and T is the absolute temperature, and the linearization of the geometric relation derived by Brangbour et al. [5] gives c₂ = N_GS/(4R) where N_GS is the fixed number of filaments on a colloidal particle. Therefore, one finds

$$EA=c\hspace{0.17em}{k}_{B}T\hspace{0.17em}\frac{N_{GS}}{4R},$$

where c is a factor they introduce to better fit the data. Observe from this that the effective modulus EA can be varied by changing the density of filaments on the particle surface.

Our equation (3.4) is identical to eqn (4)₂ in Brangbour et al. [5] provided we identify their variables X, V₀ t and ck_BTN_GS/(4R) with our ℓ, ℓ_R and E respectively.

4. Growth of a network of actin filaments

We continue to consider the test specimen described in §2 but now with its left end attached to an AFM cantilever and its right end to a (movable) support as depicted schematically in figure 5. The AFM acts like a linear spring whose force–elongation relation is

$$\sigma A=-k_c(y_0-Y_0).$$ 4.1

Here σ(t) is the average (compressive) stress in the specimen, the cross-sectional area of the specimen over which the filaments are distributed is A, Y₀ is the position of the AFM cantilever when it is undeflected, y₀(t) is the position of the left-hand end of the specimen at time t and k_c is the (usual) spring stiffness in units of force/displacement. It is more convenient to set k = k_c/A and write (4.1) as

$$\sigma =-k(y_0-Y_0).$$ 4.2

(a). Constitutive response functions

(i). Stress–stretch–filament density relation

First consider the constitutive function $\mathsf{\Lambda }$ describing the relation between the stress σ, stretch λ and filament density ρ. Suppose that the force f in a filament is related to its stretch by f = E_fA_f(λ⁻¹ − 1) where E_f is its Young’s modulus and A_f its cross-sectional area. This, together with (2.4), gives the constitutive relation for the specimen to be σ = fρ = E(λ⁻¹ − 1) where E = ρE_fA_f.

The filaments in the experiments of Brangbour et al. [5] are relatively short and do not form a network. By contrast, the actin filaments in the experiments of Parekh et al. [9] form a cross-linked network where filament bending becomes important. To account for an analogous phenomenon in foams, Gibson & Ashby proposed the modification $E={\rho }^2{A}_f^2{E}_f$ to the effective Young’s modulus (see [34]). We adopt their model and write the stress–stretch–filament density relation in the form $\lambda =\mathsf{\Lambda }(\sigma ,\rho )$ where

$$\mathsf{\Lambda }(\sigma ,\rho )=\frac{1}{1+\sigma /E},E=E(\rho )={\rho }^2{A}_f^2{E}_f.$$ 4.3

For a three-dimensional constitutive model of actin networks, see for example [35].

Keeping in mind that we are taking compressive stress to be positive, we are concerned with σ ≥ 0 whence λ ≤ 1. Since the total cross-sectional area taken up by the filaments, NA_f, cannot exceed the cross-sectional area A over which the filaments are distributed, it is necessary that

$$0\le \rho \le \frac{1}{A_f}.$$ 4.4

(ii). Kinetic law for filament density

Next consider the kinetic relation $\dot{\rho }=\mathsf{R}(\rho )$ governing the filament density. As the number of filaments increases, the number of monomers in the surrounding monomer pool decreases, and so the rate at which new filaments develop is expected to decrease, i.e. we anticipate $\dot{\rho }$ to be a decreasing function of ρ. Moreover, since the total number of monomers in the system is finite, as well as because of (4.4), ρ cannot increase indefinitely. Therefore for the kinetic law $\dot{\rho }=\mathsf{R}(\rho )$, we take

$${\tau }_{\rho }\hspace{0.17em}\dot{\rho }={\rho }_{\mathrm{\infty }}-\rho ,{\tau }_{\rho }>0,\hspace{0.17em}{\rho }_{\mathrm{\infty }}>0,$$ 4.5

where ${\tau }_{\rho }$ and ρ_∞ are constant parameters. The linear dependence of $\dot{\rho }$ on ρ is similar to that in [11]. Since one can solve (4.5) explicitly, the response predicted by this kinetic relation is

$$\rho (t)={\rho }_{\mathrm{\infty }}+({\rho }_0-{\rho }_{\mathrm{\infty }})\hspace{0.17em}{\text{e}}^{-(t-t_0)/{\tau }_{\rho }},$$ 4.6

where ρ₀ = ρ(t₀) < ρ_∞ is the filament density at some particular instant t₀. Note that ρ(t) increases monotonically and ρ(t) → ρ_∞ as t → ∞.

Since ρ(t) < ρ_∞, this, together with $E={\rho }^2{A}_f^2{E}_f$ and ρ(t) → ρ_∞ as t → ∞, tell us that the effective Young’s modulus obeys

$$E(t)<E_{\mathrm{\infty }}\text{and}E(t)\to E_{\mathrm{\infty }}\text{as}\hspace{0.17em}t\to \mathrm{\infty }\hspace{0.17em}\text{where}\hspace{0.17em}{E}_{\mathrm{\infty }}:={\rho }_{\mathrm{\infty }}^2{A}_f^2{E}_f.$$

Thus, E_∞ is Young’s modulus when the system reaches steady state. It will be convenient to write (4.3)₂ in terms of E_∞ as

$$E=E_{\mathrm{\infty }}\hspace{0.17em}\frac{{\rho }^2}{{\rho }_{\mathrm{\infty }}^2}.$$ 4.7

(iii). Kinetic law for surface growth

We now present two models for the kinetic relation $V=\mathsf{V}(f)=\mathsf{V}(\sigma ,\rho )$ describing surface growth at the left-hand boundary of the specimen where $V=-{\dot{x}}_0$. According to the literature, e.g. [5,9], growth is expected to stall at some critical value f_stall of the filament force. Therefore we take $\mathsf{V}(f)$ to be a monotonically decreasing function of f with $\mathsf{V}(f)\to 0$ as f → f_stall.

We first present the Arrhenius-type kinetic law (1.1) that is exponential in the filament force f with f_stall > 0, V₀ > 0 and ζ ≠ 0 being constant parameters. For reasons that we will explain below, we shall not use this kinetic relation in our calculations but note it here because of its frequent use in this field. Howard [2] gives an expression for the parameter ζ in terms of the temperature T, the Boltzmann constant k_B and the length of a stress-free monomer a:

$$\zeta =\frac{a{f}_{\text{stall}}}{k_{B}T},$$ 4.8

indicating that the parameter ζ should also be positive. Keeping in mind that the filament force f is positive in compression, (1.1) says that growth occurs (V > 0) for compressive forces in the range 0 ≤ f < f_stall and that growth stalls when f → f_stall. The dashed curve in figure 6 shows the variation of V with f according to (1.1) for ζ = 5. When ζ → 0 this curve approaches the straight line V/V₀ = 1 − f/f_stall.

Next, recall the maximum dissipation kinetic law for growth (3.3)₁ used in §3: V = V₀ for 0 ≤ f < f_stall, V = 0 for f > f_stall. This is shown dotted in figure 6. In this section, we adopt the following regularized (smoothed out) version of this kinetic law:

$$V=\mathsf{V}(f)=V_0[1-{\left(\frac{f}{f_{\text{stall}}}\right)}^m]\text{for}0\le f\le f_{\text{stall}},$$ 4.9

where f_stall, V₀ and m are constant parameters such that

$$V_0>0,f_{\text{stall}}>0,\hspace{0.17em}m>0.$$

For m = 1, this model is linear in the filament force and for m → ∞ it approaches the maximum dissipation kinetic law used in §3. The solid curves in figure 6 show the variation of ${\dot{\ell }}_R$ with f according to (4.9) for some different values of m.

The experimental observations in [9] indicate that the process of growth under spring loading involves an intermediate stage where the elongation-rate is almost independent of the stress, and therefore a plot of $\dot{\ell }$ versus σ involves a more-or-less horizontal segment prior to stall. This is ideally captured by the maximum dissipation kinetic relation (which, as we saw in §3, also did very well in modelling the experiments of Brangbour et al. [5]). Because the approach to stall is gradual and not sudden, a regularized version of that kinetic relation is desirable such as the power-law model (4.9) with a moderate value of m. As can be seen from figure 6, the exponential model (1.1) does not capture this behaviour. The exponential kinetic relation does approach the maximum dissipation kinetic law when ζ → −∞, though negative values of ζ appear not to be reasonable, e.g. Howard’s model (4.8).

Instead of the growth speed parameter V₀ it will sometimes be more convenient to use the time-scale for growth defined by

$${\tau }_R:=\frac{{\ell }_0}{V_0},$$ 4.10

where ℓ₀ is the distance between the AFM cantilever and the other support when the AFM is undeflected. When growth commences, the filaments attached to the AFM do not extend all the way to the other support. Therefore, they initially grow under zero stress. Their length when they first touch the other support is ℓ₀.

The implications of the dissipation inequality on the kinetic law for growth are discussed in electronic supplementary material, section S4.

(iv). Model parameters: parameter values

The constitutive model described in §4a(i)–(iii) involves the following parameters: the stress–stretch–filament density relation (4.3)₁, (4.7) involves the Young’s modulus E_∞ and the maximum filament density ρ_∞. The kinetic law (4.5) for the nucleation of new filaments involves ρ_∞ and the time scale τ_ρ. The kinetic law for surface growth (4.9) and (4.10) involves the time scale τ_R, the stall force f_stall and the exponent m. Since stall occurs when f = f_stall and ρ = ρ_∞ it is convenient to define

$${\sigma }_{\text{stall}}:={\rho }_{\mathrm{\infty }}{f}_{\text{stall}}.$$

In addition, the constitutive relation of the AFM spring involves its stiffness k. It is useful to define an associated stress σ₀ by

$${\sigma }_0:=k{\ell }_0,$$ 4.11

where k is the stiffness of the AFM spring in units of stress/displacement; see (4.2).

While it is natural to work with non-dimensional variables and parameters, we shall not do so here since we want to make quantitative comparisons with the experimental results of Parekh et al. [9].

How we arrived at the following specific values of the various parameters, including the sources of the data, is described in electronic supplementary material, section S2. Here we simply record the values we shall use:

$$\begin{array}{rl}& {\ell }_0=3\hspace{0.17em}\mu \text{m},{\sigma }_{\text{stall}}=0.77\hspace{0.17em}\text{nN}\hspace{0.17em}\mu m^{-2},{\tau }_{\rho }=40\hspace{0.17em}\text{min},\hspace{0.17em}m=5,\\ & {\sigma }_0=0.2362\hspace{0.17em}\text{nN}\hspace{0.17em}\mu m^{-2}(\text{spring used for figs 2 and 3a of [9]}),\\ & {\sigma }_0=0.1575\hspace{0.17em}\text{nN}\hspace{0.17em}\mu m^{-2}(\text{spring used for fig. 3b of [9]}).\end{array}$$ 4.12

The value of ${\tau }_{\rho }$ was chosen arbitrarily, while that of m was chosen to ensure that the kinetic law $V=\mathsf{V}(f)$ was reasonably close to the maximum dissipation kinetic law as depicted in figure 6. This was necessary in order to get qualitative agreement between the theoretical predictions and the experiments. Two different AFM springs were used in [9] leading to the two values of σ₀ = kℓ₀ above. The value of ρ_∞ turns out not to be needed but its value is of the order of 80 filaments per μm², significantly smaller than the maximum filament density 1/A_f = 53 000 μm⁻².

The two main sets of experiments carried out by Parekh et al. [9] pertain to their figs 2 and 3. It can be seen from those figures that the elongation-rate for growth under spring-loading is $\dot{\ell }=72\hspace{0.17em}{\text{nm min}}^{-1}$ in fig. 2 and $\dot{\ell }=129\hspace{0.17em}{\text{nm min}}^{-1}$ in fig. 3. Therefore the conditions under which the two sets of experiments were carried out had to be different, and so it is not unreasonable for the values of certain parameters in the model to also be different. The elongation-rate $\dot{\ell }$ depends sensitively on the growth-speed V₀ which in turn depends on the monomer concentration in the surrounding solvent. We assume that the different values of $\dot{\ell }$ observed is likely due to different monomer concentrations and so take different values for V₀ (equivalently τ_R = ℓ₀/V₀) depending on which experiment we are modelling. Furthermore, it can be readily shown from (4.18) and (4.19) below that the elongation-rate at the initial instant, $\dot{\ell }(0)$, depends sensitively on (the time scale for growth, τ_R, and) Young’s modulus at steady state, E_∞. Therefore, we also take the value of E_∞ to be different in the two analyses. When concerned with the experiments focused on growth under the action of the AFM spring (fig. 2 of [9]), we shall take

$$E_{\mathrm{\infty }}=3.7\hspace{0.17em}\text{nN}\hspace{0.17em}\mu {\text{m}}^{-2},{\tau }_R=34\hspace{0.17em}\text{min},$$ 4.13

whereas when modelling the experiments focused on stress jumps (fig. 3 of [9]), we take

$$E_{\mathrm{\infty }}=0.7\hspace{0.17em}\text{nN}\hspace{0.17em}\mu {\text{m}}^{-2},{\tau }_R=10\hspace{0.17em}\text{min}\hspace{0.17em}\text{(fig. 3 a)},{\tau }_R=13\hspace{0.17em}\text{min}\hspace{0.17em}\text{(fig. 3 b)}.$$ 4.14

As can be seen from the electronic supplementary material, the values in (4.13) and (4.14) are within the range of experimentally measured data.

(b). Growth under the action of the atomic force microscope spring

First consider the problem where the specimen grows under spring-loading. Our main interest is in calculating the force, specimen length and elongation-rate as functions of time, and then looking at a plot of elongation-rate versus force.

In these calculations, the position y₁ of the support on the right-hand side is held fixed and the velocity ${\dot{y}}_0$ of the AFM cantilever is measured. Thus, the unstressed length of the specimen at the initial instant is ℓ₀ : = y₁ − Y₀ which represents the distance between the support and AFM cantilever when the AFM is not deflected. It then follows because ℓ = y₁ − y₀ that ℓ₀ − ℓ = y₀ − Y₀, and so the spring loading equation σ = −k(y₀ − Y₀) can be written as σ = k(ℓ − ℓ₀):

$$\ell ={\ell }_0+\frac{\sigma }{k}.$$

The system of four equations to be solved to find ℓ(t), ℓ_R(t), σ(t), ρ(t) are

$$\ell =\mathsf{\Lambda }(\sigma ,\rho ){\ell }_R,{\dot{\ell }}_R=\mathsf{V}(\sigma ,\rho ),\dot{\rho }=\mathsf{R}(\rho ),\ell ={\ell }_0+\frac{\sigma }{k},$$ 4.15

having used ${\dot{\ell }}_R=-{\dot{x}}_0=V$. From (4.15)₄,

$$\dot{\sigma }=k\dot{\ell }.$$ 4.16

Differentiating (4.15)₁ with respect to time and using (4.16) leads to

$$\dot{\sigma }/k=\frac{{\mathsf{\Lambda }}_{\rho }{\ell }_R\hspace{0.17em}\dot{\rho }+\mathsf{\Lambda }{\dot{\ell }}_R}{1-k{\mathsf{\Lambda }}_{\sigma }{\ell }_R},$$ 4.17

where we have set ${\mathsf{\Lambda }}_{\sigma }=\mathrm{\partial }\mathsf{\Lambda }/\mathrm{\partial }\sigma$ and ${\mathsf{\Lambda }}_{\rho }=\mathrm{\partial }\mathsf{\Lambda }/\mathrm{\partial }\rho$. However from (4.15)₁ and (4.15)₄

$${\ell }_R=\frac{\ell }{\mathsf{\Lambda }}=\frac{k{\ell }_0+\sigma }{k\mathsf{\Lambda }}=\frac{{\sigma }_0+\sigma }{k\mathsf{\Lambda }}.$$

Using this to eliminate ℓ_R from (4.17) yields

$$\frac{\dot{\sigma }}{{\sigma }_0}=\frac{\mathsf{\Lambda }{\dot{\ell }}_R/{\ell }_0+(1+\sigma /{\sigma }_0)\dot{\rho }{\mathsf{\Lambda }}_{\rho }/\mathsf{\Lambda }}{1-(\sigma +{\sigma }_0){\mathsf{\Lambda }}_{\sigma }/\mathsf{\Lambda }}.$$ 4.18

Equation (4.15)₃ can be solved for ρ(t). When this expression for ρ(t), together with $\mathsf{\Lambda }=\mathsf{\Lambda }(\sigma ,\rho )$ and ${\dot{\ell }}_R=\mathsf{V}(\sigma ,\rho )$, are substituted into (4.18), the resulting equation has the form $\dot{\sigma }=F(\sigma ,t)$. This can be solved for σ(t). Thereafter, one can calculate the associated value of $\dot{\ell }(t)$ from

$$\frac{\dot{\ell }}{{\ell }_0}\stackrel{4.16}{=}\frac{\dot{\sigma }}{k{\ell }_0}\stackrel{4.11}{=}\frac{\dot{\sigma }}{{\sigma }_0},$$ 4.19

and thus one can construct a parametric plot of $(\sigma (t),\dot{\ell }(t))$ on the $\sigma ,\dot{\ell }$-plane with time t as the parameter. The resulting figure can be compared with figs 2B and 2C of Parekh et al. [9].

(i). Simple ρ-independent model

Before presenting the results of the preceding analysis, it is illuminating to first consider a simpler model in which the number of filaments is fixed and does not evolve. Then we drop ρ from the model⁷ so that equations (4.18) and (4.19) governing ℓ(t), σ(t) specialize to

$$\dot{\ell }=\frac{\mathsf{\Lambda }(\sigma )\mathsf{V}(\sigma )}{1-(\sigma +{\sigma }_0){\mathsf{\Lambda }}^{\prime }(\sigma )/\mathsf{\Lambda }(\sigma )},\frac{\dot{\sigma }}{{\sigma }_0}=\frac{\dot{\ell }}{{\ell }_0},$$ 4.20

where we have set V = V(σ). Equation (4.20)₁ gives $\dot{\ell }$ as a function of stress σ, which can be plotted on the $\sigma ,\dot{\ell }$-plane. Moreover, integrating (4.20) from σ = 0 at t = t₀ to σ = σ_stall at t = t_stall gives the time t_stall at which growth stalls.

We now specialize (4.20) to the particular choices (4.3), (4.9),

$$\mathsf{\Lambda }(\sigma )=\frac{1}{1+\sigma /E},\mathsf{V}(\sigma )=V_0[1-{\left(\frac{\sigma }{{\sigma }_{\text{stall}}}\right)}^m],$$ 4.21

with the various parameters having the values given in (4.12) and (4.13). Figure 7a shows the plot of $\dot{\ell }$ versus σ according to this model. Observe that the elongation-rate starts at $\dot{\ell }=83\hspace{0.17em}{\text{nm min}}^{-1}$ and eventually stalls when the force reaches the value σ_stall A = 293 nN. The time taken for the stress to reach 99% of σ_stall is 235 min. This figure may be compared with figs 2B and 2C of Parekh et al. [9] where the elongation-rate starts at about 72 nm min⁻¹ and growth stalls at a force of about 300 nN in a little over 200 min.

Figure 7.

(a) Elongation-rate $\dot{\ell }$ versus stress σ according to the simple model (4.20) and (4.21). (b) Elongation-rate $\dot{\ell }$ versus stress σ according to the general model (4.18) and (4.19).

(ii). General ρ-dependent model

Now consider the more detailed model (4.18) and (4.19), specialized to the constitutive descriptions (4.3), (4.5) and (4.9). The parameters have the same values used in the preceding subsection together with ${\tau }_{\rho }$ taking the value given in (4.12)₃. The initial condition for ρ is chosen (arbitrarily) to be ρ(t₀) = 0.5ρ_∞. Figure 7b shows a parametric plot of $(\sigma (t),\dot{\ell }(t))$ on the stress–elongation-rate plane with time being the parameter. The elongation-rate starts at the value 70 nm min⁻¹ and rises to a maximum value of 75 nm min⁻¹. The elongation-rate remains at about 70 nm min⁻¹ during a more-or-less load-independent intermediate stage after which $\dot{\ell }$ begins to decrease more rapidly as the system approaches stall. The stress increases monotonically throughout this calculation until it reaches⁸ 99% of the stall force in 200 min, the stall force being 293 nN. As noted above, the corresponding experimental values from [9] are an initial elongation-rate of about 72 nm min⁻¹, growth stalling at a force of about 300 nN in a little over 200 min.

(c). Stress jumps

We now turn to two calculations motivated by figs 3a and 3b of Parekh et al. [9] where the force on the specimen is suddenly decreased while it is growing. Our primary interest is in calculating the resulting jump in the elongation-rate $\dot{\ell }$ which, the experiments indicate, undergoes a striking sudden increase.

It is worth noting at the outset that the kinetic law $\dot{\rho }=\mathsf{R}(\rho )$ is independent of stress and so is unaffected by the details of how the stress varies. Thus, we take ρ(t) and $\dot{\rho }(t)$ to vary continuously throughout. Consequently the effective Young’s modulus, $E={\rho }^2{E}_{\mathrm{\infty }}/{\rho }_{\mathrm{\infty }}^2$, is also a continuous function of time. Moreover, keeping in mind that the referential length of the specimen ℓ_R(t) changes only due to growth, and assuming that a finite segment of new material cannot appear in an infinitesimal interval of time, we require ℓ_R(t) to be a continuous function of time. On the other hand the specimen length ℓ(t) and elongation-rate $\dot{\ell }(t)$ will be discontinuous when the stress is discontinuous.

In the first calculation, the specimen grows under spring loading conditions during an initial period t₀ < t < t₂; see figure 9a. At time t₂, the force is suddenly decreased to a smaller value and clamped at that value from then on. (The specimen is not under spring-loading conditions for t > t₂.)

Figure 9.

(a) Stress and (b) elongation-rate versus time; compare with fig. 3a of [9]. The specimen grows under spring loading for t₀ < t < t₂ and with the stress fixed for t > t₂.

In the second calculation (see figure 10a) the specimen grows with the force clamped at a fixed value during an initial stage t₀ < t < t₁. At the instant t = t₁ the force clamp is released and the specimen is allowed to grow under spring loading conditions for a period t₁ < t < t₂. At the instant t = t₂, the force is suddenly decreased back to the value it had during the original constant force stage and clamped at that value for t > t₂.

Figure 10.

(a) Stress and (b) elongation-rate versus time; compare with fig. 3b of [9]. The specimen grows with the stress fixed at the same value for both t₀ < t < t₁ and t > t₂, and under spring loading for intermediate times t₁ < t < t₂.

In both of these calculations, the load levels are always in the ‘load-independent’ range of figure 7b.

In all processes, whether the stress is held constant or the specimen is spring-loaded, we have

$$\ell =\mathsf{\Lambda }{\ell }_R,{\dot{\ell }}_R=\mathsf{V}(\sigma ,\rho ),\dot{\rho }=\mathsf{R}(\rho ).$$ 4.22

Differentiating (4.22)₁ with respect to time, and then using (4.22)₁ and (4.22)₂ to eliminate ℓ_R and ${\dot{\ell }}_R$ from the result, leads to

$$\dot{\ell }=\mathsf{\Lambda }\mathsf{V}+\ell \hspace{0.17em}\frac{{\mathsf{\Lambda }}_{\sigma }}{\mathsf{\Lambda }}\hspace{0.17em}\dot{\sigma }+\ell \hspace{0.17em}\frac{{\mathsf{\Lambda }}_{\rho }}{\mathsf{\Lambda }}\dot{\rho }.$$

In processes where $\sigma (t)=\mathrm{constant}$, this reduces to

$$\text{Constant stress: }\dot{\ell }=\mathsf{\Lambda }\mathsf{V}+\ell \frac{{\mathsf{\Lambda }}_{\rho }}{\mathsf{\Lambda }}\dot{\rho },$$ 4.23

whereas when the specimen is spring-loaded, so that σ = k(ℓ − ℓ₀), it yields

$$\text{Spring loading: }\dot{\ell }=\frac{\mathsf{\Lambda }\mathsf{V}+\dot{\rho }\hspace{0.17em}{\ell }_0(1+\sigma /{\sigma }_0)\hspace{0.17em}{\mathsf{\Lambda }}_{\rho }/\mathsf{\Lambda }}{1-(\sigma +{\sigma }_0){\mathsf{\Lambda }}_{\sigma }/\mathsf{\Lambda }},$$ 4.24

where σ₀ = kℓ₀ as before.

Consider the instant t₂ at which the stress changes discontinuously. For any time-dependent function g(t) that suffers a finite jump discontinuity at time t₂ we write

$$g^{+}=g(t_2^{+})\text{and}{g}^{-}=g(t_2^{-}),$$

and if g(t) is continuous at t₂ we simply write g = g(t₂). In both of Parekh et al.’s experiments, the specimen is spring-loaded just before the stress jump and has the force clamped just after. Thus (4.24) holds at $t_2^{-}$, while (4.23) holds at $t_2^{+}$. The elongation-rates just before and just after the stress jump are therefore

$${\dot{\ell }}^{-}=\frac{{\mathsf{\Lambda }}^{-}{\mathsf{V}}^{-}+\dot{\rho }\hspace{0.17em}{\ell }_0(1+{\sigma }^{-}/{\sigma }_0){\mathsf{\Lambda }}_{\rho }^{-}/{\mathsf{\Lambda }}^{-}}{1-({\sigma }^{-}+{\sigma }_0){\mathsf{\Lambda }}_{\sigma }^{-}/{\mathsf{\Lambda }}^{-}},{\dot{\ell }}^{+}={\mathsf{\Lambda }}^{+}{\mathsf{V}}^{+}+\dot{\rho }{\ell }^{+}\frac{{\mathsf{\Lambda }}_{\rho }^{+}}{{\mathsf{\Lambda }}^{+}},$$ 4.25

where we have written ${\mathsf{\Lambda }}^{\pm }=\mathsf{\Lambda }({\sigma }^{\pm },\rho )$ and ${\mathsf{V}}^{\pm }=\mathsf{V}({\sigma }^{\pm },\rho )$ having used the fact that ρ(t) varies continuously.

Consider an instant at which the stress has the values σ^±. In order to calculate ${\dot{\ell }}^{+}$ using (4.25)₂ we need the value of ℓ⁺. While we can use σ⁻ = k (ℓ⁻ − ℓ₀) to calculate ℓ⁻, we cannot use σ⁺ = k (ℓ⁺ − ℓ₀) to calculate ℓ⁺ since the specimen is not spring-loaded at time t⁺. Instead, we calculate ℓ⁺ using $\ell /{\ell }_R=\mathsf{\Lambda }$, i.e.

$${\ell }^{+}=\frac{\mathsf{\Lambda }({\sigma }^{+},\rho )}{\mathsf{\Lambda }({\sigma }^{-},\rho )}\hspace{0.17em}{\ell }^{-}.$$ 4.26

In writing (4.26), we have used the fact noted at the beginning of this section that the filament density ρ(t) and referential length ℓ_R(t) vary continuously.

(i). Simple ρ-independent model

In order to get a sense for how the elongation-rate suffers a sudden increase when the stress is suddenly decreased, consider again the special case where the constitutive functions $\mathsf{\Lambda }$ and $\mathsf{V}$ are both independent of the filament density ρ. Then equations (4.23) and (4.24) take the forms

$$\text{Constant stress: }\dot{\ell }=\mathsf{\Lambda }(\sigma )\mathsf{V}(\sigma )$$ 4.27

and

$$\text{Spring loading: }\dot{\ell }=\frac{\mathsf{\Lambda }(\sigma )\mathsf{V}(\sigma )}{1-(\sigma +{\sigma }_0){\mathsf{\Lambda }}^{\prime }(\sigma )/\mathsf{\Lambda }(\sigma )}.$$ 4.28

These equations tell us how the elongation-rate $\dot{\ell }$ varies as a function of stress σ for the two types of loading.

Subject to mild assumptions on $\mathsf{\Lambda }$ and $\mathsf{V}$, the curve on the $\sigma ,\dot{\ell }$-plane defined by (4.28) lies below the one described by (4.27). For example, figure 8 shows these curves for the particular choice (4.21) with the parameters having the values given in (4.12) and (4.14).

Figure 8.

Elongation-rate $\dot{\ell }$ versus stress σ for loading at constant stress (equation (4.27), upper curve) and spring-loading (equation (4.28), lower curve). Suppose that the spring-loaded specimen evolves from A to B, at which point the stress is suddenly decreased back to the value σ(A) and clamped at that value. Accordingly the system must jump from B to C and remain there from then on. Thus during the jump, the stress decreases discontinuously from σ(B) to σ(C) ( = σ(A)) while the elongation-rate increases discontinuously from $\dot{\ell }(B)$ to $\dot{\ell }(C)$. (Online version in colour.)

When σ = 0 we have $\mathsf{\Lambda }(0)=1,{\mathsf{\Lambda }}^{\prime }(0)=-1/E$ and $\mathsf{V}(0)=V_0$, and therefore from (4.27) and (4.28)

$$\dot{\ell }=\{\begin{array}{ll}{V}_0& \text{(constant force), }\\ \frac{V_0}{1+{\sigma }_0/E}& \text{(spring loading), }\end{array}\text{at}\hspace{0.17em}\sigma =0.$$

These are the values of $\dot{\ell }$ at which the curves in figure 8 intersect the vertical axis. Therefore, the separation between the two curves (at least at σ = 0) increases as σ₀/E increases. Therefore in order to increase the separation at σ = 0 we should decrease E. In our quantitative calculations, we have therefore taken the smallest value of E from the range of possible values determined experimentally. The value of σ₀ is determined by the stiffness of the AFM spring.

Now consider the response of a spring-loaded specimen starting from point A in figure 8 where the stress is σ(A). The point $(\sigma (t),\dot{\ell }(t))$ evolves along the lower curve starting from A and moving to the right (towards stall). Suppose that when it reaches point B the stress is suddenly decreased back to the value σ(A) and clamped at that value. Then $(\sigma (t),\dot{\ell }(t))$ jumps from point B to point C (and remains there for subsequent time). Therefore, as the stress decreases suddenly from σ(B) to σ(C) ( = σ(A)) the elongation-rate increases discontinuously from the value $\dot{\ell }(B)$ to $\dot{\ell }(C)$.

In our first⁹ quantitative calculation, we took the values of the force just before and after the jump from fig. 3a of [9] and determined the elongation-rates from

$${\dot{\ell }}^{-}=\frac{\mathsf{\Lambda }({\sigma }^{-})\mathsf{V}({\sigma }^{-})}{1-({\sigma }^{-}+{\sigma }_0){\mathsf{\Lambda }}^{\prime }({\sigma }^{-})/\mathsf{\Lambda }({\sigma }^{-})},{\dot{\ell }}^{+}=\mathsf{\Lambda }({\sigma }^{+})\mathsf{V}({\sigma }^{+}),$$ 4.29

which follow from (4.27) and (4.28). The functions $\mathsf{\Lambda }$ and $\mathsf{V}$ are given by (4.21) with the parameters having the values in (4.12) and (4.14). Equation (4.29) then led to

$${\dot{\ell }}^{-}=120\hspace{0.17em}{\text{nm min}}^{-1}\text{and}{\dot{\ell }}^{+}=204\hspace{0.17em}{\text{nm min}}^{-1}.$$

The corresponding experimentally measured elongation-rates were ${\dot{\ell }}^{-}=129\hspace{0.17em}{\text{nm min}}^{-1}$ and ${\dot{\ell }}^{+}=275\hspace{0.17em}{\text{nm min}}^{-1}$.

In our second numerical calculation, the values of the force just before and after the jump were taken from fig. 3b of [9]. The elongation-rates were calculated as above and this led to

$${\dot{\ell }}^{-}=162\hspace{0.17em}{\text{nm min}}^{-1}\text{and}{\dot{\ell }}^{+}=239\hspace{0.17em}{\text{nm min}}^{-1},$$

whereas the experimentally measured values were ${\dot{\ell }}^{-}=129\hspace{0.17em}{\text{nm min}}^{-1}$ and ${\dot{\ell }}^{+}=270\hspace{0.17em}{\text{nm min}}^{-1}$.

(ii). General ρ-dependent model

When the filament density ρ is taken into account, a simple graphical description based on (4.23) and (4.24) is no longer possible since the right-hand sides of those equations now depend on both σ and t (through ρ(t)). Instead we solve the relevant initial-value problem based on (4.23) and (4.24) to calculate the response of the specimen and in particular to determine the elongation-rates just before and after the stress jump.

In the first calculation, we solve the differential equation (4.24) for t₀ < t < t₂ using suitable initial conditions at t₀. The conditions at time $t_2^{+}$ are then determined using (4.26) and this information is used as initial conditions to solve (4.23) for t > t₂. The details of this calculation can be found in the electronic supplementary material, section S3. Figure 9 shows plots of σ(t) and $\dot{\ell }(t)$ versus time as predicted by our model which may be compared with fig. 3a of [9]. In particular, we find $\sigma (t_2^{-})=0.373\hspace{0.17em}\text{nN}\hspace{0.17em}\mu {\text{m}}^{-2}$, $\dot{\ell }(t_2^{-})=112\hspace{0.17em}{\text{nm min}}^{-1}$ and $\dot{\ell }(t_2^{+})=212\hspace{0.17em}{\text{nm min}}^{-1}$, the corresponding experimentally determined values being $\sigma (t_2^{-})=0.381\hspace{0.17em}\text{nN}\hspace{0.17em}\mu {\text{m}}^{-2}$, $\dot{\ell }(t_2^{-})=129\hspace{0.17em}{\text{nm min}}^{-1}$ and $\dot{\ell }(t_2^{+})=275\hspace{0.17em}{\text{nm min}}^{-1}$.

In the second calculation, we start by solving (4.23) for t₀ < t < t₁ using suitable initial conditions at t₀. The conditions at time $t_1^{+}$ are then deduced from continuity and the results are used as initial conditions to solve (4.24) for t₁ < t < t₂. The conditions at time $t_2^{+}$ are then calculated using (4.26). Finally we solve (4.23) for t > t₂ using the information from $t_2^{+}$ as initial conditions. The details of these calculations can be found in the electronic supplementary material, section S3. Figure 10 shows plots of σ(t) and $\dot{\ell }(t)$ versus time as predicted by our model which may be compared with fig. 3b of [9]. In particular we find $\sigma (t_2^{-})=0.216\hspace{0.17em}\text{nN}\hspace{0.17em}\mu {\text{m}}^{-2}$, $\dot{\ell }(t_2^{-})=119\hspace{0.17em}{\text{nm min}}^{-1}$ and $\dot{\ell }(t_2^{+})=187\hspace{0.17em}{\text{nm min}}^{-1}$, the corresponding experimentally determined values being $\sigma (t_2^{-})=0.218\hspace{0.17em}\text{nN}\hspace{0.17em}\mu {\text{m}}^{-2}$, $\dot{\ell }(t_2^{-})=129\hspace{0.17em}{\text{nm min}}^{-1}$ and $\dot{\ell }(t_2^{+})=270\hspace{0.17em}{\text{nm min}}^{-1}$.

In this second calculation, the stress is held fixed at the same value 0.218 nN μm⁻² for both t₀ < t < t₁ and t > t₂. Therefore on both these time intervals, the corresponding elongation-rate is given by (4.23) with σ = 0.218 nN μm⁻². However, the right-hand side of (4.23) also involves the filament density ρ(t) and therefore though σ has the same constant value, the elongation-rate evolves as a function of time due to the evolution of ρ(t). In the specific calculation above, we find (in particular) that ρ(t₁) = 0.838ρ_∞ and ρ(t₂) = 0.861ρ_∞ at t₁ = 73 min and t₂ = 79 min. This small difference in the filament densities leads to a small difference in the corresponding elongation-rates, viz. $\dot{\ell }(t_1^{-})=186\hspace{0.17em}{\text{nm min}}^{-1}$ and $\dot{\ell }(t_2^{+})=187\hspace{0.17em}{\text{nm min}}^{-1}$. This is in contrast to the large difference, $\dot{\ell }(t_1^{-})=170\hspace{0.17em}{\text{nm min}}^{-1}$ and $\dot{\ell }(t_2^{+})=270\hspace{0.17em}{\text{nm min}}^{-1}$, observed in the experiments. In order to capture this, we will need to modify the kinetic relation $\dot{\rho }=\mathsf{R}(\rho )$ and possibly ${\dot{\ell }}_R=\mathsf{V}(\sigma ,\rho )$ and $\lambda =\mathsf{\Lambda }(\sigma ,\rho )$ as well.

Finally, it is worth noting the qualitative similarity between figure 10b and fig. 3b from our earlier calculation related to the experiments of Brangbour et al. [5]. In both cases, the force on the specimen is held constant at the same value for t₀ < t < t₁ and t > t₂. In the case of fig. 3b, the force was fixed at a smaller value during the intermediate interval t₁ < t < t₂, whereas the specimen grew under spring loading in figure 10. In fig. 3b, the curve (straight-line) pertaining to t > t₂ is the continuation of the curve pertaining to t₀ < t < t₁. Because of the dependency on the filament density ρ(t), this is not true of the corresponding curves in figure 10.

5. Conclusion

In this paper, we have shown that a continuum model of surface growth and filament nucleation can quantitatively capture the evolution of growing dendritic actin networks. Such growing networks provide the propulsive force for a variety of processes in live cells. A surface growth model is appropriate because polymerization in dendritic actin occurs in a narrow zone next to the load surface, not all over the network. A distinguishing feature of our model is that it describes the actin network as a growing continuum subject to the balance laws of continuum thermomechanics together with constraints imposed by the dissipation inequality. The microscopic details of polymerization of individual filaments are distilled into a kinetic law for the propagation speed of the growing surface. This continuum model is applied to two different experiments. The first is the set of experiments of Brangbour et al. [5] in which the density of filaments does not change and the filaments are not cross-linked. Our model is in remarkable agreement with experiment on the evolution of filament lengths, forces (or stresses), etc., using a maximally dissipative kinetic law. The second is the set of experiments of Parekh et al. [9] in which the density of filaments changes and the filaments are cross-linked. Once again, a smoothed version of the maximally dissipative kinetic law is able to describe the evolution of variables in this experiment. Our kinetic law relates growth rate (in the reference configuration) to an (average) force per filament to facilitate comparison with well-known ‘force–velocity’ relations for single growing filaments. We find that the kinetic law that best describes the two experiments is different from the exponential force–velocity relation that is often used in the context of single filaments. This is hardly surprising since the narrow zone in which polymerization of dendritic actin occurs is a network of interacting filaments which are not parallel and are acted upon by different forces. This kinetic law is one of many that satisfy the dissipation inequality and there could yet be others that perform better; we use it here because of its simplicity and its ability to capture the convex approach to stall that is distinct from the well-established exponential force–velocity relation. The continuum framework proposed here may be applicable to growing networks of other biological and non-biological filaments such as those of collagen, microtubules, carbon nanotubes, etc.

Data accessibility

All data used have been taken from the references cited in the main paper and the electronic supplementary material.

Authors' contributions

The two authors contributed equally to all aspects of this work. Both authors drafted and revised the paper and gave final approval for publication and agree to be held accountable for the work performed therein.

Competing interests

We declare we have no competing interest.

Funding

P.K.P. acknowledges generous support from the Department of Mechanical Engineering at MIT and NIH grant no. R01 HL 135254.