Mechanisms of Cell Propulsion by Active Stresses

Abstract

The mechanisms by which cytoskeletal flows and cell-substrate interactions interact to generate cell motion are explored using a simplified model of the cytoskeleton as a viscous gel containing active stresses. This model yields explicit general results relating cell speed and traction forces to the distributions of active stress and cell-substrate friction. It is found that 1) the cell velocity is given by a function that quantifies the asymmetry of the active-stress distribution, 2) gradients in cell-substrate friction can induce motion even when the active stresses are symmetrically distributed, 3) the traction-force dipole is enhanced by protrusive stresses near the cell edges or contractile stresses near the center of the cell, and 4) the cell velocity depends biphasically on the cell-substrate adhesion strength if active stress is enhanced by adhesion. Specific experimental tests of the calculated dependences are proposed.

1. Introduction

Directed motion of eukaryotic cells is often achieved by a combination of protrusion at the front of the cell, contraction farther back, and adhesion between the cytoskeleton and the cell's environment. This mechanism encompasses a broad range of observed types of cell motion. For example, in actin-based motility, actin polymerization at the front of the cell causes protrusion, actomyosin-induced contraction occurs farther back, and coupling of the actin cytoskeleton to the cell's environment via transmembrane proteins such as integrins pushes the cell forward. On the other hand, in nematode sperm, protrusion caused by polymerization and bundling of major sperm protein (MSP) at the front of the cell and contraction mediated by MSP depolymerization close to the rear appear to generate motion [1]. Studies of diverse cell types have suggested a continuum of different types of cell motion corresponding to different relative weights of protrusion, contraction, and adhesion [2]. A large body of mathematical modeling work has treated these effects individually and in combination [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. By simulation of detailed models, it has been possible to reproduce migration behaviors which are compatible with experimental observations, including the distribution of internal gel flows and the dependence of cell velocity on adhesion strength. However, there are as yet no clear, explicit relationships giving the velocity and traction forces of a moving cell in terms of the spatial distribution of key driving forces such as protrusion, contraction, and adhesive interactions with the environment. This paper seeks to develop such relationships.

The fact that similar mechanisms are implemented by diverse sets of proteins suggests that a general theory of cell propulsion could encompass many of the known phenomena. I propose such a general theory based on the concept of “active stresses” introduced by Kruse et al [18]. Active stresses lead to flow of cytoskeletal gels, driven by nonequilibrium chemistry, even in the absence of externally applied stresses. They should be distinguished from traction forces. Active stresses are internal to the gel; traction forces act at the interface between the gel and the substrate, and result from the active stresses. The effects of active stresses on cell properties have been explored in a series of papers reviewed in Ref. [19]. They can be either protrusive, as would result from polymerization of cytoskeletal filaments, or contractile, as would result from depolymerization or actomyosin contraction. I develop explicit relationships between the cell velocity and traction forces on one hand, and active stresses and adhesion on the other hand, by solving a simplified mathematical model. This model differs from current models for actin-based motility in that it treats actin polymerization and actomyosin contraction on an equal footing, via protrusive and contractile active stresses, respectively. By treating the effects of active stresses within a general framework, it gives results which are independent of the specific mechanisms creating the stresses. The simplicity of the present model has several advantages. First, it establishes the “bare bones” of cell migration, which furthers basic understanding and aids the development of biomimetic systems for studying cell migration. This makes it possible to correlate a wide body of experimental data within one physically transparent mathematical framework. Second, it allows the ready exploration of a multidimensional parameter space, which may allow the discovery of new propulsion modes. Finally it provides an easily interpretable “prequel” to the more complex simulations, whose results can be difficult to understand.

This paper does not explicitly treat the mechanisms generating active stresses, which may include chemotactic signaling circuits and/or spontaneous cell polarization mechanisms. Rather I focus on the mechanisms by which a given distribution of active stress and friction generates cell motion, and on evaluating the dependence of cell motion and traction forces on key parameters such as the adhesion strength. I focus on two key properties, the cell velocity V_cell and the traction force dipole moment P, defined by

$$P=\int \overrightarrow{r}\cdot {\overrightarrow{f}}_{\mathit{trac}}\left(\overrightarrow{r}\right)\mathit{dA}$$ (1)

where f_trac is the traction force density (the force density exerted by the cell on the substrate), and the integral is performed over the two-dimensional contact area between the cell and the substrate [20]. Since the integrated traction force must vanish or nearly vanish [21], P is the most basic measure of the strength and orientation of the traction forces. P is negative (as in most cells) when traction forces at the front point backwards and those at the rear point forwards. The key results of the paper are the formulas for V_cell and P given in Eqs. (7), (10), and (13) respectively.

2. Model

2.1. Physical Picture

The conceptual foundation of the model is shown in Fig. 1. A cell contains protrusive active stresses at the front and the contractile active stresses at the rear. Because accepted convention [22] assigns stress the opposite sign from pressure, the stress in the protrusive region (positive pressure) is negative, and that in the contractile region is positive. The protrusive stresses lead to expansion of the cytoskeletal gel. Because this expansion is constrained by the front cell membrane, the gel flows backwards relative to the cell. A similar mechanism for contraction near the rear of the cell, constrained by the rear cell membrane, also causes backward flow there. When the backward gel flow is coupled to the external environment, forward motion of the cell results, much in the same way that backwards motion of the bottom of a paddlewheel propels a riverboat forward. This picture encompasses the actin-myosin mechanism and the MSP mechanisms discussed above. It also has some overlap with currently held theories of apicomplexan motility, in which backward flows of actin relative to the cell body couple to a substrate and thus cause forward motion [23]. However, in this case the flows appear to arise by a different mechanism, in which myosin A inside the cell forms a scaffold fixed relative to the cell, and polarized actin filaments are pushed backwards by their interaction with the myosin. The flows are caused not by protrusive and contractile active stresses, but rather by the fixed myosin scaffold and the polarization of the actin filaments.

Figure 1.

Schematic of forward cell motion resulting from interaction of internal flows with the environment. Protrusion is indicated by “−” sign and contraction by “+” sign, in accordance with accepted convention regarding sign of stress.

Several preceding works have treated viscous flow of the cytoskeletal gel in cells [3, 4, 6, 5, 7, 24, 8, 25, 12, 14]. The current approach is mostly closely related to those of Refs. [24] and [12], which treated gel flow generated by active myosin stresses and actin polymerization. The approaches differ in that I treat actin polymerization as an active stress on an equal footing with myosin contraction, and that the formulas derived for the velocity and traction-force dipole treat spatially varying active stresses.

To define a mathematical model embodying the mechanism of Fig. 1, I make the following assumptions:

• The gel flows may be convergent or divergent. Here divergent means dV/dx > 0 and convergent means dV/dx < 0, where V(x) is the gel flow velocity and x is the direction of motion. Such flow behaviors are clearly demonstrated by speckle-microscopy studies of actin gels [26]. For example, near the leading edge of epithelial cells, the actin retrograde (negative) flow velocity rapidly decreases going into the cell [27], indicating convergent flow. Pronounced variations in flow velocity are also seen in speckle-microscopy studies of keratocytes [28, 29].

• Inhomogeneity in the gel flow leads to internal stresses. A uniform flow field causes no relative motion and requires no stresses for its maintenance. Therefore a natural baseline assumption is that the stress required to maintain the flow is proportional to the gradient of the flow velocity. Such gradients could originate either a) from variations in the height of the cell or b) from processes occurring at constant cell height. In case a) flow naturally slows or accelerates going into the cell if the gel volume is conserved, and the stress arises from the shear viscosity of the gel as it deforms going into the cell. In case b), convergent or divergent flows in steady state must be accompanied by gel disassembly or assembly, to avoid buildup or depletion of the gel. Then the stress required to maintain convergent or divergent flows is related to the dynamics of gel assembly or disassembly. Existing evidence supports case b). In keratocytes, the lamellipodium profile is quite flat for distances up to 10 μm into the cell [30]; in addition, speckle microscopy studies of epithelial cells have revealed correlations between convergent gel flow and depolymerization [31].

• The static elasticity of the actin cytoskeleton is ignored. This assumption is justified by experiments which show that cell-surface-attached beads subjected to constant forces eventually settle down to a constant non-zero velocity of motion [32], which cannot occur with finite zero-frequency elastic moduli. Such a description of the gel stresses is not quantitatively accurate, but I believe that for the slow processes that characterize cell migration, it will correctly identify the crucial effects.

• Propulsion is dominated by the isotropic component of the active stress. This assumption differs from that made in previous work studying lamellipodial flow [24], which emphasized the shear components. I focus on the isotropic component because of its clear relationship to cell propulsion.

• The force exerted by the substrate on the gel is assumed to be proportional to the gel velocity. This assumption has been found to be reasonably accurate by simultaneous measurements of force and velocity in keratocytes [33].

• The gel flows are assumed to be parallel or antiparallel to the direction of motion. The flows, and the profile of the cell, are also assumed to be independent of the position transverse to the direction of motion. These assumptions are made for mathematical simplicity and because the parallel/antiparallel components of the flows should dominate cell motion. The formalism for more general geometries is given in the Supporting Material.

• The force required to detach the rear of the cell from the substrate is independent of velocity. This assumption is based on previous analyses [34] which have indicated only a weak dependence of rupture forces on on velocity.

2.2. Mathematical Formulation of Model

The model shown in Fig. 2 embodies these assumptions. The cell has length L in the direction of motion (the x-direction) and width W in the transverse direction. It moves at constant velocity on a flat substrate and its dimensions remain constant. The equations of motion treat the actin gel immediately above the substrate, so that the vertical coordinate does not appear. Because the velocities and cell profile are assumed independent of the coordinate transverse to the direction of motion, I treat a one-dimensional geometry, keeping in mind that it is a slice of the two-dimensional cell. The active stresses σ_act which are two-dimensional (having units of force per length), are related to the three-dimensional stresses by a factor of the cell height. As discussed above, gradients in the gel flow velocity V(x) cause internal stresses. Thus, the internal force density, which is proportional to the gradient of the stress, is proportional to the second derivative of the velocity; the coefficient of proportionality v acts as an effective viscosity. The interaction between the gel and the substrate is described by a friction coefficient per unit area ξ(x) To simplify the mathematics, I assume that σ_act vanishes very near the cell boundaries; this assumption can be relaxed at the expense of making the boundary conditions more complex.

Figure 2.

Side view of mathematical model of cell.

Because the velocities of flow in the cell are very small, the inertia terms, and thus the net force acting on any element of the gel, are also small. Therefore I assume that the force density acting on the gel vanishes everywhere. The contributions to this force density are:

• The force density dσ_act/dx resulting from the active stresses.

• The force density νd²V/dx² resulting from flow gradients.

• The frictional force density −ξ(x)V(x) resulting from interactions between the actin gel and the substrate.

• The force density −F_det (per unit of transverse width) required to detach the rear of the cell from the substrate (where F_det > 0). This force acts at the rear of the cell. However, it may not act directly on the actin gel, but rather on the membrane. The membrane may thus be pulled backwards in such a way that it exerts a force on the actin gel at the front of the cell - in other words the membrane elastically transfers the force from the substrate at the rear of the cell to the actin gel at the front. I thus assume that F_det acts on the gel at x = L/2. The results below for the dependence of cell velocity on F_det are independent of the position where F_detis exerted, so this is not a limiting assumption.

The resulting force balance for the cytoskeletal gel takes the form (see Table 1 for a list of parameters and their units):

$$\nu =\frac{d^{2}V}{{\mathit{dx}}^2}+\frac{d{\sigma }_{\mathit{act}}}{\mathit{dx}}-\xi \left(x\right)V\left(x\right)-F_{\mathit{det}}\delta (x-L∕2)=0$$ (2)

Table 1.

Definitions and units of model quantities and parameters. F denotes force, T denotes time, and L denotes length.

Parameter	Definition	Units
Vcell	Cell velocity	L/T
V(x)	Gel velocity	L/T
σact(x)	Active stress density	F/L
ν	Effective gel viscosity	FT/L
ξ(x)	Cell-substrate friction coefficient	FT/L3
ξ 0	Crossover friction coefficient	FT/L3
P	Traction force dipole moment	FL
Fdet	Cell-rear detachment force density	F/L
α	Fdet/ξ	L2/T

Two boundary conditions apply to Eq. (2):

• The velocities at the front and the back of the cell must be equal for the cell to maintain a steady-state size, so

$$V(L∕2)=V(-L∕2)=V_{\mathit{cell}}.$$ (3)

Note that V_cell is not known a priori but will rather emerge from solution of the equations of motion.

• The total external force exerted by the substrate and the cell membrane on the cytoskeletal gel must vanish, since the gel moves at constant velocity. This zero-force condition has the form

  $${\int }_{-L∕2}^{L∕2}\xi \left(x\right)V\left(x\right)\mathit{dx}+F_{\mathit{det}}=0.$$ (4)

This implies a boundary condition on dV/dx. Integrating Eq. (2) from −L/2 to a point (L/2)_ just inside L/2 yields (recalling that σ_act vanishes just inside ±L/2), and using Eq. (4), gives

$$\begin{array}{cc}\hfill \nu \left[\frac{\mathit{dV}}{\mathit{dx}}{\mid {}_{(L∕2)-}-\frac{\mathit{dV}}{\mathit{dx}}\mid }_{-L∕2}\right]=& {\int }_{-L∕2}^{L∕2}\xi \left(x\right)V\left(x\right)\mathit{dx}\hfill \\ \hfill & =-F_{\mathit{det}}.\hfill \end{array}$$ (5)

Because of Eq. (5), I will treat Eq. (2) on the interval [-L/2, (L/2)_], where only ξ(x)V(x) contributes to the substrate force density. On this interval the simplified equation

$$\nu \frac{d^{2}V}{{\mathit{dx}}^2}=\frac{-d{\sigma }_{\mathit{act}}}{\mathit{dx}}=\xi \left(x\right)V\left(x\right),$$ (6)

holds, with the effects of F_det included via the boundary conditions.

2.3. Solutions of Particular Cases

Here I summarize the solutions of Eq. (6) for some representative cases; details are given in the Supporting Material.

2.3.1. Cell Velocity

Constant Friction

I first obtain a solution for V(x) in the special case of a stress localized at a point, and then write the general solution as a sum of such solutions. The result is

$$V_{\mathit{cell}}=-\frac{1}{\nu }{\int }_{-L∕2}^{L∕2}\frac{{\sigma }_{\mathit{act}}\left(x\right)\text{sinh}\left(\kappa x\right)}{2\text{sinh}(\kappa L∕2)}\mathit{dx}-\frac{F_{\mathit{det}}}{2\nu \kappa \text{tanh}(\kappa L∕2)}.$$ (7)

where

$$\kappa =\sqrt{\xi ∕\nu }.$$ (8)

is a decay rate describing how quickly the flow generated by a point stress source decays away from the stress. A quantity very similar to κ was defined in Ref. [24], but using the shear viscosity rather than the effective viscosity defined here. Thus V_cell contains contributions from different parts of the cell, with those near the cell edges weighted more strongly because of the sinh term. The opposing force F_det slows motion at a rate depending on both ξ and ν.

Weakly Varying Friction

F_det = 0. I take the frictional term in Eq. (6) to have the form

$$\xi \left(x\right)=\stackrel{‒}{\xi }+\Delta \xi \left(x\right),$$ (9)

where $\stackrel{‒}{\xi }$ is constant and $\Delta \xi ⪡\stackrel{‒}{\xi }$ is a small variation. By solving Eq. (6) to first order in Δξ, one obtains the following correction to V_cell:

$$\Delta V_{\mathit{cell}}=-\frac{\stackrel{‒}{\kappa }}{2}{\int }_{-L∕2}^{L∕2}\frac{\text{cosh}\left(\stackrel{‒}{\kappa }x\right)\stackrel{‒}{V}\left(x\right)}{\text{sinh}(\stackrel{‒}{\kappa }L∕2)}\frac{\Delta \xi \left(x\right)}{\stackrel{‒}{\xi }}\mathit{dx},$$ (10)

where $\stackrel{‒}{V}\left(x\right)$ is the value of V(x) in the absence of δξ and $\stackrel{‒}{\kappa }=\sqrt{\stackrel{‒}{\xi }∕\nu }$. Thus a locally increased friction coefficient (δξ(x) > 0) increases the speed of the cell if it occurs in a region of backwards flow ($\stackrel{‒}{V}\left(x\right)<0$).

2.3.2. Traction Force Dipole

I assume constant friction to keep the calculation tractable. The traction force consists of friction terms between the gel and the substrate, and the detachment force F_det. The traction-force dipole moment (1) is

$$P=W{\int }_{-L∕2}^{L∕2}x{f}_{\mathit{trac}}\left(x\right)\mathit{dx}$$ (11)

where

$$f_{\mathit{frac}}=\xi \left(x\right)V\left(x\right)+F_{\mathit{det}}\delta (x+L∕2)$$ (12)

is the force density exerted by the cell on the substrate, and I have assumed that F_det acts on the substrate at the rear of the cell. In the Supporting Material, it is shown that

$$P=-W\left[{\int }_{-L∕2}^{L∕2}\left[1-\frac{(\kappa L∕2)\text{cosh}\kappa x}{\text{sinh}\kappa L∕2}\right]{\sigma }_{\mathit{act}}\left(x\right)\mathit{dx}+L{F}_{\mathit{det}}∕2\right].$$ (13)

This expression generalizes a limiting case, in which the second term in the integral can be ignored, derived for constant active stresses in lamellipodia [24].

3. Biophysical Implications of Model Results

The main results of the model analysis are Eqs. (7), (10), and (13). Here I use them to classify possible mechanisms of cell propulsion, to assess the dependence of V_cell on substrate binding strength, and to relate the traction-force dipole to other measures of actomyosin contractile stress. Because the model assumes that both front and back move at a constant velocity, it does not directly address amoeboid-type motion associated with protrusion cycling and variations in cell length. However, in this case the main factor determining the cell velocity will be the average distribution of active stress as a function of distance from the leading and trailing edges. If one were to identify the active stress in the model with these average stresses, the present results should be applicable to ameboid motion as well.

3.1. Cell Propulsion Mechanisms

Asymmetric active-stress mechanism

Eq. (7) implies that active protrusive stresses near the front of the cell and contractile stresses near the back of the cell enhance cell motion. This occurs because a protrusive stress near the front of the cell generates a diverging flow field which pushes the front membrane of the cell forward; a contractile stress near the rear of the cell causes a converging flow field which pulls the rear membrane forwards. Fig. 3 shows the spatial dependence of the cell velocity induced by a point negative (protrusive) stress as a function of the point of application (essentially the sinh factor multiplying σ_aci in Eq. (7)). For contractile (positive) stress the velocity has the opposite sign. For weak friction, V_cell is seen to depend linearly on the point of application of the stress. This occurs because weak friction implies that κ is small (cf. Eq. (8)); in this limit Eq. (7) reduces to

$$V_{\mathit{cell}}=\frac{1}{\nu L}{\int }_{-L∕2}^{L∕2}x{\sigma }_{\mathit{act}}\left(x\right)\mathit{dx}-\frac{F_{\mathit{det}}}{\xi L}.$$ (14)

Thus the factor multiplying σ_act is proportional to x, and the driving force for cell motion is simply the dipole moment of the active stress. Opposing force slows motion at a rate inversely proportional to ξ. When the friction is stronger, κ in Eq. (7) increases. This occurs because the frictional coupling to the substrate reduces flows more effectively, limiting the distance over which they can propagate in the cell. Therefore the effect of an active stress near the center of the cell cannot propagate to the edge of the cell. This renders the contributions from near the cell edges even more dominant, as seen in Fig. 3.

Figure 3.

Schematic of dependence of cell velocity on position of a unit protrusive (negative) active stress localized at a point inside the cell. Contractile (positive) stress gives velocity of opposite sign.

Fig. 4a illustrates this propulsion mechanism for a hypothetical active-stress distribution in which protrusive stress is concentrated in a narrow region near the front of the cell (to the right), and contractile stress is concentrated farther back. Even though the total amount of contractile stress exceeds the protrusive stress, and the contractile stress is positioned to cause backwards motion of the cell, the protrusive stress outweighs the contractile stress because is it closer to the front of the cell.

Figure 4.

Examples of asymmetric-stress propulsion mechanism (frame a) and asymmetric-friction mechanism (frame b). Dashed line denotes results for Δξ = 0. In a) parameters satisfy L²ξ = 8ν, contractile active stress has magnitude 0.2 (arbitrary units) and extends from 0 to 0.2L, and protrusive active stress has magnitude −1.0 and extends from 0.48L to 0.5L. In b) parameters satisfy L²ξ_min = ν, so L²ξ_max = 8ν, contractile stress has magnitude 1.0 and extends from −0.2L to 0.2L, and ξ switches from ξ_min to ξ_max at 0.3L. Remaining parameter choices affect only vertical scaling.

These predictions are consistent with known or inferred distributions of active stresses in cells. In the actin-myosin and major sperm protein (MSP) systems, polymerization of actin or MSP occur near the front of the cell and contractile activity due to myosin activity or MSP depolymerization occurs in the middle or the rear of the cell. Measured distributions of actin and myosin in cell fragments [35] also show actin at the front and myosin at the rear. The reduced impact of active stresses near the middle of the cell, shown in Fig. 3, might explain the observation that in some cell types myosin inhibition has limited impact on cell velocity [36].

Eq. (7) can be tested directly by localized stimulation of protrusion or contraction. Localized application of calyculin, which enhances myosin-induced contraction, causes keratocytes to move in the opposite direction from the point of application [28], consistent with Eq. (7) and Fig. 3 (with the sign changed to account for contractile stress). Better spatial resolution could be obtained by localized photoactivation of agents acting upstream of actin polymerization, such as the Rho GTPases. By moving the photoactivated spot forwards through the cell, and simultaneously measuring V_cell, one could obtain a plot of V_cell vs. position. Such a plot should reflect the main features of Fig. 3. Photoactivation of actomyosin contraction would have similar effects, but with a sign change.

It is instructive to relate the asymmetric-stress mechanism to a limiting case based on a fixed actin polymerization velocity localized at the front of the cell (x = L/2), independent of ν and ξ. This limit can be treated within the present approach as follows. The polymerization velocity V_p is related to the strain rate: $V_p={\int }_0^{L∕2}(\mathit{dV}∕\mathit{dx})\mathit{dx}$ where it is assumed that V(0) = 0 - the flow velocity at the center of the cell - vanishes because of the substrate friction. If ν increases, then σ_act(x) must increase proportionally to ν in order to maintain a constant strain rate, so the ratio σ_act(x)/ν in Eq. (7) will approach a constant. On the other hand, if the velocity (and thus the protrusive active stress) are localized at x = L/2, the first integral in Eq. (7) becomes independent of κ. There the cell velocity becomes independent of both ν and ξ, as expected for the limit of a fixed actin polymerization velocity.

Results having some similarities with Eq. (7) has been derived for the portion of the actin gel in the lamellipodium at the leading edge of a cell [24, 37]. Both of these treatments used a force-balance similar to Eq. (2), with σ_act taken positive (contractile) and constant, ξ taken constant, different boundary conditions corresponding to the front and rear of the lamellipodial region, and a varying height. It was found that, as in Eq. (7), the lamellipodium edge velocity depends linearly on the active stress. However, the present treatment, in the absence of external force, would give zero cell velocity for a constant active stress; in Refs. [24, 37], the velocity is nonzero because of the asymmetric boundary conditions and the height variation. Ref. [24] found a linear dependence of velocity on opposing force, as in Eq. (7). However, Ref. [37] found a nonlinear dependence, because of coupling between the external force and the crosslinking properties of the gel.

Asymmetric friction mechanism

Eq. (10) shows that spatial variations in the friction coefficient can cause motion even if the active-stress distribution is symmetric. Motion will occur in the direction opposite to the gel flow in the region of enhanced friction. In the limit of weak friction (small κ), the effect of asymmetric friction on V_cell takes the simple form

$$\Delta V_{\mathit{cell}}=-\frac{1}{L}{\int }_{-L∕2}^{L∕2}\stackrel{‒}{V}\left(x\right)\left[\frac{\Delta \xi \left(x\right)}{\xi }\right]\mathit{dx}.$$ (15)

Thus V_cell is a weighted average involving $\stackrel{‒}{V}\left(x\right)$ and the fractional change in the friction coefficient. When the friction is stronger, the contributions in Eq. (10) are more weighted toward the ends of the cell, and contributions from the middle become relatively insignificant. The asymmetric-friction mechanism may be understood as an additional force acting on the gel: if the gel flow is backwards where the friction is increased, this force points forwards. In the example of Fig. 4b, the cell contains only symmetrically distributed contractile active stresses, which give no motion by themselves. However, they cause inwards gel flow from the cell edges (dashed line in the velocity plot). Because this flow is backwards near the front of the cell where the friction is enhanced, the cell moves forward, at the velocity indicated in the Figure. Here the fractional change in ξ is large, but the numerically obtained results for this case are nevertheless consistent with the predictions of Eq. (10).

The relation between V_cell and the distribution of adhesion strength given by Eq. (10) is consistent with existing data regarding the distribution of adhesion strength in cells. Since the gel at the front of a cell usually flows backwards, enhanced friction near the front of a cell enhances forward motion. Several observations indicate that cells may use such a mechanism to improve the efficiency of motion. For example, combined force and velocity measurements on migrating keratocytes [33] showed a friction coefficient approximately three times larger at the front of the cell than at the back. This work also demonstrated a strong dynamic correlation between the asymmetry of the friction coefficient and the direction of cell motion, as would be expected if friction asymmetry enhances motion. A correlation between asymmetry of cell-substrate interactions and cell motion was also found in a micropatterning study in which cells were found to protrude preferentially in the direction of newly formed adhesions [38]. Finally, measurements of actin speckle flow in keratocytes [28] revealed a reduction in V_cell with increasing slippage at the front of the cell, as Eq. (10) would predict if the frictional coupling at the front were decreased. Motion induced by asymmetric friction is also a possible explanation of the observation that lifting one edge of keratocyte fragments, which should cause a reduction of adhesion at that end, causes motion in the opposite direction [35].

Quantitative tests of Eq. (10) could be accomplished by methods similar to those outlined above for V_cell, including localized photoactivation of key proteins acting upstream of adhesion. Moving the focus of adhesion from front to back should cause a gradual transition from acceleration to deceleration of cell motion as the direction of flow changes from retrograde to anterograde; the induced velocity change should mirror the retrograde flow rate.

3.2. Variation of V_cell with substrate binding strength

Manipulation of cell-substrate adhesion by several different methods [39, 40] has shown that V_cell first grows with increasing adhesion, reaches a maximum, and then drops. To address this biphasic behavior, I assume that ξ is proportional to the substrate binding strength. Because the binding strength varied in Refs. [39] and [40] refers to detachment forces rather than frictional sliding forces, this assumption needs to be tested. Such tests could be performed, for example, by combined measurements of gel flow and substrate forces, as in Ref. [33]. I also assume that F_det in Eq. (7) is proportional to ξ: F_det = αξ, where α is a constant. Finally, I assume that propulsion is dominated by protrusion near the front of the cell. Then Eq. (7) gives a monotonic drop of V_cell with increasing ξ, rather than biphasic behavior. This indicates that a crucial ingredient is missing from the model. Several explanations have been proposed for the biphasic behavior [3, 11, 15]. Recent experiments in epithelial cells [41], showing that increased substrate adhesion accelerates retrograde flow, shed light on this issue. If σ_act were independent of adhesion or dropped with increasing adhesion, adhesion would instead slow retrograde flow. Therefore I assume that substrate adhesion enhances σ_act. This assumption is consistent with the signaling of adhesions to the actin cytoskeleton [42], and also with the autocatalytic branching theory in which actin polymerization is enhanced by opposing force [43]. To model the feedback of adhesion onto active-stress generation, I take σ_act to have the simple mathematical form

$$\begin{array}{cc}{\sigma }_{\mathit{act}}\left(x\right)=-{\sigma }_{\mathit{max}}\xi ∕(\xi +{\xi }_0)\hfill & L∕2-\Delta L<x<L∕2\hfill \\ {\sigma }_{\mathit{act}}\left(x\right)=0\hfill & x\le L∕2-\Delta L\hfill \end{array}$$ (16)

where σ_max is the magnitude of the saturation value of σ_act, ξ₀ is the crossover value of ξ, and ΔL is the width of the region where the active stress is exerted. This corresponds to propulsion dominated by protrusion near the front of the cell, enhanced by ξ.

Evaluation of Eq. (7) using Eq. (16), over a broad range of model parameters, reveals that in some parameter ranges V_cell is always zero. But for sufficiently large V_maxαξ₀, V_cell has a biphasic behavior as a function of ξ. The possible velocity behaviors are summarized in the solid and dashed curves in Fig. 5. I have fitted the solid curve to the data points of Ref. [40]. A reasonably close fit to the data is obtained, indicating that positive feedback of adhesion onto active-stress generation is a possible origin of biphasic adhesion-velocity curves.

Figure 5.

Cell velocity as function of ξ. Solid and dashed curves exemplify behaviors generated by Eq. (7), with active stress determined by Eq. (16), using ΔL = 0.2L. V_max and ξ_max are values of V_cell and ξ at maximum velocity. Parameter values are such that L²ξ₀ = ν. For solid line, σ_∞/αξ₀ = 2.95, for dashed line σ_∞/αξ₀ = 5.5. Remaining parameter values affect only vertical and horizontal scaling of plot. Solid squares are data taken from Fig. 4a of Ref. [40].

Other possible mechanisms causing a biphasic Vcell-adhesion curve include drag terms that are independent of ξ but proportional to V. Such terms could come from the viscous drag of the medium (if it is much more viscous than water), the drag resisting motion of the cell membrane over the substrate, or the drag resulting from internal deformation of the cell if it rolls over the substrate. I have implemented such terms by taking ξ₀ = 0 (no positive feedback of adhesion onto active stress), and adding a second opposing-force term in Eq. (7) proportional to V_cell. The resulting equation is straightforwardly solved. In this case, the ξ − V_cell curve always emerges linearly from the origin, and the descending branch is broader than the ascending branch. This yields curves which are broader than the experimental data shown in Fig. 5.

3.3. Factors favoring negative traction-force dipole

Traction forces usually point backwards at the front of the cell and forwards at the back, so P < 0. Eq. (13) shows that protrusive (negative) stresses near the cell edges, and contractile stresses near the center of the cell, produce negative values of P. This dependence can be interpreted as follows. A focus of contractile stress near the center of the cell will produce a converging flow field, which exerts a backwards force on the substrate ahead of the point of application, and a forwards force behind it. This leads to P < 0. On the other hand, a center of protrusive stress near the front of the cell will produce mainly retrograde flow, which generates backwards forces near the front of the cell and thus again leads to P < 0. But a center of contractile stress very near the cell rear would produce mainly backwards flow, which would lead to forwards traction forces and thus to P > 0. Note that V_cell and P are determined by very different aspects of σ_act(x): V_cell is determined by components of σ_act that are antisymmetric with respect to the center of the cell, while P is determined by symmetric components.

These factors favoring negative P are consistent with known properties of most moving cells, in which protrusive stresses from polymerization are near the front cell edge, and contractile stresses are closer to the center of the cell. Quantitative tests of Eq. (13) can be obtained by methods parallel to those proposed above for Eqs. (7), if the contractile active stress can be spatially controlled by photoactivation.

3.4. Derivation of magnitude of σ_act from measured traction forces

Evaluation of σ_act from measured traction-force distributions is complicated by our incomplete knowledge of F_det. However, recent experiments have shown that abrogating the motor activity of myosin II reduces the traction forces greatly [44, 45, 46]. Therefore I assume that the traction forces in myosin-active cells are caused by the myosin contribution σ_myo to σ_act, and modify Eq. (13) as follows:

$$P=-W{\int }_{-L∕2}^{L∕2}\left[1-\frac{(\kappa L∕2)\text{cosh}\kappa x}{\text{sinh}\kappa L∕2}\right]{\sigma }_{\mathit{myo}}\left(x\right)\mathit{dx}.$$ (17)

Since P < 0 and σ_myo > 0 (because the myosin stress is contractile), the second term in the integral can only reduce the magnitude of P, so

$$\mid P\mid <W{\int }_{-L∕2}^{L∕2}\mid {\sigma }_{\mathit{myo}}\left(x\right)\mid \mathit{dx}=A\mid {\stackrel{‒}{\sigma }}_{\mathit{myo}}\mid =V\mid {\stackrel{‒}{\sigma }}_{\mathit{myo}}^{3d}\mid$$ (18)

where A and V are the area and volume of the cell, the overbars denote spatial averaging, and ${\sigma }_{\mathit{myo}}^{3d}$ the three-dimensional active myosin stress. Thus |P| cannot exceed the volume-integrated active stress. Equality holds in Eq. (18) when the second term inside the integral in Eq. (17) can be ignored; this will be valid when kL is large and σ_myo(x) is mainly concentrated away from the edges of the cell. In this case the estimate of Ref. [24] for lamellipodia is reproduced.

In Fig. 6a of Ref. [47] for Dictyostelium during early spreading, one finds |f_trac| ≃ 40 Pa and a cell area of about 700 μm². Dividing the measured traction forces into forwards and backwards components, and taking these components to act at points ±8μm from the center of the cell (a distance estimated from the Figure), one finds that P = −40Pa × 700μm² × 8μm ≃ −2 × 10⁻¹³ J. Furthermore, Ref. [48] found a cell volume of about 600μm³. Thus a lower bound on ${\stackrel{‒}{\sigma }}_{\mathit{myo}}^{3d}$ is obtained:

$$\mid {\stackrel{‒}{\sigma }}_{\mathit{myo}}^{3d}\mid >\mid P\mid ∕V\simeq 2\times {10}^{-13}J∕600\mu m^3\simeq 300\mathit{Pa}$$ (19)

Is this bound consistent with known properties of myosin force generation? There are no direct measurements of the magnitude of ${\sigma }_{\mathit{myo}}^{3d}$ However, measurements of cytokinesis in Dictyostelium [48] provide an estimate of myosin contraction forces. The force required for inward motion of a contractile ring of volume 20μm³ and diameter about 3 μm was estimated to be 7 nN. If the contractile ring is a torus, its cross-sectional area is about 20μm³/3πμm ≃ 2μm². Then ${\sigma }_{\mathit{myo}}^{3d}\simeq 7nN∕2\mu m^2=3500\mathit{Pa}$ ≃ 7nN/2μm² = 3500Pa. It is reasonable to assume that the active-stress generation capacity is proportional to the myosin concentration c_myo. The bulk myosin concentration was about six times smaller than that in the contractile ring, suggesting a bulk active-stress generation capacity of about 3500Pa/6 ≃ 600Pa. This is consistent with the present lower bound of 300Pa.

4. Conclusion

This work has treated a simple mathematical model of cell propulsion incorporating active stresses and adhesion in a minimal fashion. It has shown that the cell velocity and a measure of the traction forces are given by very simple formulas based on the spatial distributions of active stress and the adhesion strength. The spatial distributions of protrusive/contractile active stress and adhesion observed in real cells are consistent with those which, according to the model, provide efficient propulsion and negative traction-force dipoles. It is hoped that the fundamental understanding gained from this model analysis can be helpful in the design of biomimetic systems for moving cells. Biomimetic systems for Listeria, in which active stresses due to actin polymerization are generated externally (outside the moving object) have shed great insight not only into Listeria motility, but also on force generation by actin polymerization at the leading edge of cells. A biomimetic system in which actin polymerization occurs internally, and contractile forces are included, would be a much closer analog to cell migration. The results of this paper suggest that such a system could be obtained if a spatial gradient of active stresses in a gel, combined with a frictional coupling of the gel to the external medium, can be created. The efficiency of such a system would be enhanced if the frictional couplings could be focused at the front of the cell.