A constitutive law for cross-linked actin networks by homogenization techniques
Inspired by experiments on the actin driven propulsion of micrometer sized beads we develop and study a minimal mechanical model of a two-dimensional network of stiff elastic filaments grown from the surface of a cylinder. Starting out from a discrete model of the network structure and of its microscopic mechanical behavior we derive a macroscopic constitutive law by homogenization techniques. We calculate the axisymmetric equilibrium state and study its linear stability depending on the microscopic mechanical properties. We find that thin networks are linearly stable, whereas thick networks are unstable. The critical thickness for the change in stability depends on the ratio of the microscopic elastic constants. The instability is induced by the increase in the compressive load on the inner network layers as the thickness of the network increases.
The here employed homogenization approach combined with more elaborated microscopic models can serve as a basis to study the evolution of polymerizing actin networks and the mechanism of actin driven motion.
keywords:A-microstructure, B-biological material, B-constitutive behavior, C-stability and bifurcation, homogenization
Msc: 74Q05, 74Q15
Dynamic filamentous polymer networks play an essential role in the mechanics of living cells. However, adequate constitutive equations, which take into account the growth history of the networks and the thereby developping prestresses, are difficult to obtain. One prominent example, which shall serve as a starting point for this paper, is the actin filament network. Actin filaments cross-linked by a special protein complex called Arp2/3 complex111The Arp2/3 complex is a special protein complex which induces the nucleation of new actin filaments at already existing ones, thus triggering the formation of Y-junctions. It is activated by other enzymes (e.g. ActA, Wasp) which are bound to the membrane or the surface of a biomimetic object and therefore Y-junction formation is limited to the space close to the enzyme covered surface. play a key role in cell motility LBP99 () as well as in the motion of cellular organelles TRM00 () or pathogens YTA99 (); CCG95 () inside their respective host cells.
This actin driven motion has been reconstituted under minimal conditions LBP99 () using beads OuT99 (), vesicles UCA03 (), and droplets BCJ04 (). The basic biochemical processes necessary for motion are by now well characterized (for a review see PBM00 () and references therein). Briefly, under the action of several auxiliary proteins, actin polymerizes into a polar filament network, whereby filament elongation and filament linkage are restricted to a zone close to the surface of the propelled object, i.e. the internal interface (The filament network is highly porous and does not hinder the diffusion of monomers to the surface of the object.). Depolymerization takes place far away from the object, i.e. at the external interface. Therefore, new network material is inserted between the already existing network and the object and forces the older network away from the object [see Fig. 1 (a)]. If this type of growth is occurring on curved surfaces, the network experiences large stresses and deformation as growth proceeds.
In experiments with actin-propelled biomimetic objects (e.g. beads) OuT99 (); UCA03 (); BCJ04 () it was shown, that initially the actin networks forms a symmetrical cloud around the object, until a spontaneous symmetry-breaking occurs [see Fig. 1 (b)]. Then the network is growing faster on one side than on the other and finally the object starts to move pushed by an actin comet. Thereby actin is continuously polymerizing in the contact region between the bead and the comet.
Thus the (chemical) out-of-equilibrium process of polymerization/depolymerization (which is nevertheless stress dependent, see FKT07 () and references therein) and the action of enzymes produce a network in a pre-stressed state, which does not relax on the timescale of symmetry-breaking and which depends on the history of growth OuT99 (); NGF00 (); GPP05 (); DSR08 (). Several theoretical studies have addressed this problem either using discrete OuT99 (); MoO03 () or phenomenological continuous models SPJ04 (); LLK05 (); JPK08 () without, however, obtaining an adequate constitutive law for the network. A type of homogenization model has been employed previously to the actin network in the geometry of the lamellipodium OSS08 (). However, this model has a viscoelastic character and stresses build up only transiently due to temporary cross-link formation and not due to static crosslinks formed close to the polymerizing interface of the network. The rigorous derivation of a macroscopic constitutive law of the network is therefore still lacking and is the subject of the present paper which takes up several ideas already outlined in JCP09 ().
One crucial observation is that actin forms a more or less periodic network on the microscopic scale, which is stable on the time scale of symmetry-breaking (i.e. several minutes). The size of each elementary cell, e.g. the distance between two links in the actin network ( several tens nm), is small compared to the total size of the structure (1 µm), which introduces a small parameter into the problem and allows for the upscaling of the microscopic mechanical model to a continuous medium by discrete homogenization methods MoC98 (); ToC98 (); CMR03 ().
In this paper we apply the discrete homogenization method to a simple 2D network structure which has been “grown” from a circle. While we keep the network structure and the microscopic mechanical model simple, we retain crucial biological features. In the Theory section we will define the network topology along with the microscopic mechanical properties and derive a set of continuous equilibrium equations. In the Results section we will solve these equations for the axisymmetric equilibrium state and identify the conditions under which linearly stable solutions exist. Finally, in the Conclusions section we will give some perspectives for this homogenization approach in modeling actin networks.
In this section we derive the continuous equilibrium equations governing the homogenized network in an abstract Lagrangian coordinate frame. We start out with the discrete formulation of the network structure then upscale this description to a continuous medium. In a second step we introduce a free energy for the continuous medium and derive the equations of equilibrium using the first variation of the free energy. For completeness we outline in A and B the relation between the equilibrium description based on the stress vectors in the abstract Lagrangian frame as it is introduced in this Theory section and the classical Cauchy stress tensor.
2.1 Description of the discrete network
We consider a planar network around a solid circle of radius . The network consists of stiff elastic filaments linked at their ends by nodes. Its topology is shown in Fig. 2 on the left.
The bars are attached to the cylinder surface at sites evenly located at a distance , i.e. at an angular distance . The network is made of layers of bars in the radial direction. As the network was formed by polymerization of actin monomers at the surface of the cylinder, each radial layer contains the same number of nodes . Each node can be identified by two integers as shown in Fig. 2 on the left. It is assumed that and are very large and of the same order. Consequently, is assumed to be very small and the parameter is of order 1 with respect to .
The elementary cell of such a network consists of one node and three bars and is shown in Fig. 3. Orientation and length of each bar is described by the vector . The tension in the bars is described by a linear law
with . is a spring constant and is the length of the bar at rest. Here we will only consider the situation and . At this stage the choice of a linear law is somewhat arbitrary and constitutes the simplest possible choice. It is consistent with basic principles in mechanics, such as material frame indifference. However, the homogenization procedure is valid for any microscopic law.
So far we have only considered forces associated with the lengths of the filaments and have not included the fact, that we are only interested in networks which keep a certain topology, e.g. the elementary cell has a finite area . To avoid this type of transition, i.e. a collapse of the network into zero area and a complete alignment of the vectors we add the following interaction moment between filaments 1 and 3
where is a constant and denotes the interaction strength. As will be seen later in the derivation of the equilibrium equations in section 2.3, expressions (2) and (3) arise naturally from the first variation of the free energy which contains a potential which diverges for the complete alignment of and .
The upscaling of the network to a continuous medium consists in determining the equivalent stresses from the bar tensions, the equations of equilibrium (or motion) satisfied by these stresses and an equivalent constitutive equation ensuing from the properties of the bars. This can be carried out by using an asymptotic expansion (for an introduction see CaC01 (); Lov44 (); ToC98 ()). Here, as the network structure is simple, a more heuristic presentation can be used. The basic idea of the homogenization process is, that for most of the network motions, the positions of its nodes can be approximated by a continuous deformation function such that the position of the node is with and . The discrete Lagrangian variables become then the set of Lagrangian (curvilinear) coordinates of the equivalent continuous medium with .
In this homogenized network the filament vectors can now be expressed using a simple Taylor expansion up to
where . Since they are associated with a quite simple numbering system for the nodes, the variables and arise naturally as Lagrangian variables of the equivalent continuous medium through the homogenization process. However, they are not the most convenient variables to study axisymmetric equilibrium configurations. Therefore we have introduced the variables and defined by
and which correspond to the radial and angular position of the node in the network [see Fig. 2 to the right]. Setting
with and where . In the following we will employ the simplified notation .
2.3 Equations of equilibrium
The equations of equilibrium can be obtained directly by following the homogenization procedure up to first order in detailed in CMR03 (). However, here we have chosen a simpler approach using only the filament vectors obtained by homogenization and by defining a free elastic energy of the network corresponding to the constitutive functions and the moment defined in (1) and (2). This free energy can be expressed as the functional
where . The first sum in (9) denotes the free energy associated with the compression and elongation of the filaments. The second term is associated with the interaction between the filaments 1 and 3 and serves as a “hard core” potential, which prevents the collapse of the networks, i.e. it prevents the complete alignment of the filaments 1 and 3.
The first variation of with respect to the deformation function reads
where the stress vectors and are given by
Integrating Eq. (10) by parts and taking into account the periodicity in (i.e.the integrals along the boundaries of the network with and cancel each other out) one finds
The last two integrals over the boundary at and in Eq. (13) evaluate to zero since is fixed at the internal boundary, i.e. for , and there are no normal forces acting on the external surface, i.e. for . Here denotes the Cauchy stress tensor (see B) and is the unit outward surface normal vector in the deformed configuration.
The mechanical equilibrium in the absence of volumic forces is therefore given by
with the boundary conditions as mentioned before
where denotes the unit vector in the radial direction in the orthonormal coordinate basis . For a later use we introduce here also the tangential unit vector as .
3.1 Axisymmetric state
In the following we will use the simplified notation . Eq. (14) with the boundary conditions (15) was analyzed for axisymmetric solutions of the type . The stress vectors (11) and (12) can then be simplified to
The equilibrium equation (14) becomes
with the boundary conditions
At this stage we point out that the unknown function needs only to be piecewise continuously differentiable. It can have “corner points”, i.e. discontinuities in the first order derivative. The presence of such discontinuous solutions can be inferred from the fact that vanishes for a finite value at the “corner point” . For every continuously differentiable segment of an extremal solution Eq. (18) holds supplemented by the following boundary conditions at the “free corner points” MathHBook ()
The first condition (20) is equivalent to the physical boundary condition, that the normal stresses match at the “cornerpoint”. The second condition (21) minimizes the energy with respect to the position of the “corner point”.
First one might consider the special case . As it is easy to see, solves Eqs. (18) and (19). Under the condition that (recall that denotes the distance at which the nodes are attached at the cylinder surface) one therefore finds the equilibrium solution
At the radial layer number one finds and . At this point the network collapses to zero volume and the filaments , and are oriented in parallel. The addition of further network layers will either lead to a back-folding of the gel into itself (in this case solution (22) holds), or the network layer will rest in this collapsed state, i.e. for . In this case constitutes a “corner point”, where Eqs. (20) and (21) hold.
The back-folding behavior is excluded in a physical situation since it requires the remodeling of the network structure and in the following we will not consider these special cases. The network collapse is unphysical as well since it implies a zero volume of the network layers.
Now we will consider the more physical case . Fig. 4 shows the dependence of the equilibrium network thickness and the associated energy depending on the number of radial network layers for the case and , i.e. all filaments are identical and type 2 filaments are inserted at the internal interface at their equilibrium length.
First, increasing the ratio of the elastic constant decreases the network thickness for a constant number of radial network layers . Second, the continuously differentiable solution branches with cease to exist beyond a critical value . Instead solutions appear which have one or more discontinuities in and which contain solution segments with (not shown).
To exclude the back-folding behavior we have added an additional term in the free energy related to the interaction between and and which prevents the alignment of and . Fig. 4 shows exemplary the network thickness and free energy for a small value of . Solutions exist even for very large networks and the solution behavior for small networks is similar to the behavior with .
Fig. 5 shows exemplary solutions for Eq. (18) for small, intermediate and large networks. For small and intermediate networks, the filament lengths and stresses increase almost linearly with , whereas for large networks the behavior is strongly nonlinear. The network is always under radial compression, whereas it is under tangential compression close to the surface of the obstacle and under tangential extension at the outer surface of the network.
3.2 Linear stability
To study the (mechanical) linear stability of the axisymmetric state we have evaluated the sign of the second variation of with respect to at the prestress axisymmetric state
with the boundary conditions
Details of the stability analysis can be found in C. Briefly, we have used a Fourier mode decomposition of and a direct integration in and finite elements for a numerical integration in the .
Fig. 6 shows the linear stability of depending on the elastic ratio and .
The linear stability of the axisymmetric state depends on the number of radial network layers superimposed over each other, whereby thin networks are stable and thick networks are unstable. Furthermore, networks with higher tangential stresses (large ) destabilize at smaller . The instability occurs independently of the transversal wavenumber and is accompanied by the appearance of a region adjacent to the inner interface where . The appearance of the instability is probably due to the fact, that as the network thickness increases the compressive load acting on type 1 and type 3 filaments in the innermost network layers increases. The elastic stress is released by rotating consecutive network layers against each other. Since the model does not contain any force opposed to a change in orientation of consecutive filaments of type 1 and 3, which introduce an additional length scale related to a “persistence length” of consecutive filaments of the same type, a catastrophic growth of the instability at infinitely small scales occurs (data not shown).
We have introduced a two-dimensional homogenization model to describe the mechanics of stiff elastic filament networks assembled on a cylindrical surface resembling Arp2/3 cross-linked actin networks as they occur in living cells. Our approach allows us to realistically model macroscopic deformations and stresses due to network growth by using only assumptions about the microscopic network properties. The main purpose of the model is to introduce a minimal set of mechanical ingredients as a basis for the more advanced problem of the time-dependent evolution of the network interfaces due to polymerization/depolymerization at the interface with a possible symmetry-breaking and subsequent motion.
The approach is novel in the field of actin driven motion, inasmuch as it allows to calculate equilibrium stresses and the position of the interface between the network and the surrounding solution without any ad hoc assumptions SPJ04 () or restrictions to infinitesimal deformations JPK08 (). Furthermore it will allow a rigorous calculation of the equilibrium state for networks with perturbed interfaces, resulting from spontaneous variations in the polymerization or depolymerization speed at the network interfaces as outlined in JCP09 (). However, in this study we have limited ourselves to fixed interfaces (in the Lagrangian frame) and studied the existence and linear stability of physical and biological relevant axisymmetric equilibrium states.
We find that in a triangular network where only changes in the length of the edges contribute to the free energy the existence of sensible solutions is limited by the number of radial network layers superimposed over each other. As one increases beyond a critical number the network starts to fold back into the negative radial direction, since the extension of the microscopic filaments in the tangential direction is unfavorable as one moves away from the cylinder surface. The inclusion of an additional term in the free energy related to the interaction between two selected filaments prevented this back-folding.
Furthermore, the linear stability of the axisymmetric solution depends on the microscopic elastic properties. In general, thin networks are stable and thick networks are unstable, whereby a higher elastic constant for filaments oriented in the tangential direction favors the occurrence of the instability for thinner networks. The instability is independent of the transversal wavenumber of the harmonic perturbations. A closer inspection of the instability reveals that it occurs first at the cylinder surface and is reminiscent of a buckling instability of type 1 and 3 filament, i.e. a counterrotation of consecutive radial network layers. However, there is no mechanism in this simple model to stabilize this instability. A more advanced approach would have to include a more detailed microscopic mechanical model with nonlocal contributions (i.e. in the spirit of second gradient model), which is beyond the scope of this paper and will be the subject of further investigation.
Appendix A Derivation of the Cauchy stress tensor in the network
The first to define a stress tensor in a discrete medium in the Lagrangian frame were Cauchy and Poisson whose derivation can be found in Lov44 (); CaC01 (). We present here the basic ideas in two dimensions to facilitate the comprehension of the Theory section.
For the derivation of the stress one considers a periodic structure of nodes and their connections. Forces act only along the directions of the node connections. The periodic structure is divided into elementary cells which are numbered by . The connections are identified by the triplet , where identifies the connection within the elementary cell.
The stress tensor is defined over a force acting on a arbitrary facet S with unit normal placed in the periodic medium as shown in Fig. 7. The force is obtained by summing up the forces acting along the node connections which are cut by the facet. Thereby one assumes that the length of the facet is sufficiently small that the forces exerted by identical connections are constant over the facet. Therefore the force is given by
Here and denote the tension and number of node connections of type . The term accounts for the orientation of the connections. There are different ways to determine , we give here only the result derived in detail in CaC01 (). The basic idea is, that the facet is cut by the vectors of one type into segments and therefore . As can be seen in Fig. 7 for the example of the type 1 filament, the height of the elementary cell is and consequently its area is given by . Therefore the number of links of type cut by the facet S can be expressed as
After some rearrangements and using the identity
Appendix B Relation between the stress vectors and the Cauchy stress tensor
In the present section we demonstrate the relation between the classical equilibrium equation known from continuum mechanics using the Cauchy stress tensor in the deformed configuration and the equilibrium formulation we have used throughout this paper defined on the Lagrangian reference configuration. Note, that we will use the Einstein notation, i.e. summation over repeated indices, and that we will use the abbreviation which denotes a partial derivation with respect to the coordinate in the Lagrangian reference configuration.
We start out by introducing some equalities. Let be the deformation mapping of the network with respect to its Lagrangian configuration and let be the position in the deformed configuration of the network point of the Lagrangian reference configuration. Let be the difference of positions of two neighbouring points and , then
where . Consequently one finds
with the property
Let now be a virtual velocity field which, due to the one to one mapping from to , can be considered either as a function of or as a function of . is assumed to be differentable and let be its differential with respect to and be its partial derivative with respect to , then
In the absence of body forces the virtual power formulation of the equilibrium equation (14) reads
where the virtual velocity field vanishes on the boundaries and of and is with respect to .
The change of variables in the intergral of the previous equation yields
where denotes the area element of the body in the deformed configuration with . The previous equation also reads
where denotes the usual twice contracted product between the second order tensors and .
Comparing with the classical virtual power formulation of the equilibrium in the Eulerian variable which reads
we can see that the expression of the Cauchy stres tensor in terms of the ’s reads
Appendix C Linear stability analysis
Since the boundary conditions are periodic in one can decompose into a Fourier series
where denotes the wavenumber and , , , denote perturbation amplitudes which depend only on . Using ansatz (LABEL:fsd) one can perform the integration of (23) in directly whereby different wavenumbers decouple. One finds then
where takes the form
with and for and , respectively. The factors are evaluated at the axisymmetric solution and are given by
The integral (44) was calculated numerically by using finite elements, whereby the functions , , , and were approximated by P-elements and the functions were approximated by P elements. This results in the quadratic form
where indicates a vector containing function values evaluated at the discretization points and is a symmetric matrix.
For the axisymmetric state to be linearly stable , i.e. needs to be a positive definite matrix. We have tested this condition for wavenumbers . But it turned out that an instability appears independently of when the factor changes sign at . The corresponding phase diagram is shown in Fig. 6.
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