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The Mechanics of Neutrophils: Synthetic Modeling of Three Experiments

Biomedical Engineering Department, Boston University, Boston, Massachusetts 02215

Correspondence: Address reprint requests to Micah Dembo, 44 Cummington St., Boston University, Boston, MA 02215. Tel.: 617-353-1671; Fax: 617-353-6766; E-mail: mxd{at}bu.edu.

	ABSTRACT

TOP ABSTRACT INTRODUCTION CONTINUUM MECHANICS AND REACTIVE... SIMULATIONS OF EXPERIMENTS DISCUSSION CONCLUSION APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
Much experimental data exist on the mechanical properties of neutrophils, but so far, they have mostly been approached within the framework of liquid droplet models. This has two main drawbacks: 1), It treats the cytoplasm as a single phase when in reality, it is a composite of cytosol and cytoskeleton; and 2), It does not address the problem of active neutrophil deformation and force generation. To fill these lacunae, we develop here a comprehensive continuum-mechanical paradigm of the neutrophil that includes proper treatment of the membrane, cytosol, and cytoskeleton components. We further introduce two models of active force production: a cytoskeletal swelling force and a polymerization force. Armed with these tools, we present computer simulations of three classic experiments: the passive aspiration of a neutrophil into a micropipette, the active extension of a pseudopod by a neutrophil exposed to a local stimulus, and the crawling of a neutrophil inside a micropipette toward a chemoattractant against a varying counterpressure. Principal results include: 1), Membrane cortical tension is a global property of the neutrophil that is affected by local area-increasing shape changes. We argue that there exists an area dilation viscosity caused by the work of unfurling membrane-storing wrinkles and that this viscosity is responsible for much of the regulation of neutrophil deformation. 2), If there is no swelling force of the cytoskeleton, then it must be endowed with a strong cohesive elasticity to prevent phase separation from the cytosol during vigorous suction into a capillary tube. 3), We find that both swelling and polymerization force models are able to provide a unifying fit to the experimental data for the three experiments. However, force production required in the polymerization model is beyond what is expected from a simple short-range Brownian ratchet model. 4), It appears that, in the crawling of neutrophils or other amoeboid cells inside a micropipette, measurement of velocity versus counterpressure curves could provide a determination of whether cytoskeleton-to-cytoskeleton interactions (such as swelling) or cytoskeleton-to-membrane interactions (such as polymerization force) are predominantly responsible for active protrusion.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION CONTINUUM MECHANICS AND REACTIVE... SIMULATIONS OF EXPERIMENTS DISCUSSION CONCLUSION APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
The mechanical properties of the neutrophil are closely related to its function as the primary foot soldier of the immune system. As such, it has been a favorite subject of experiments aimed at characterizing its ability to change shape and generate forces as dictated by circumstance. This is why, over the last decades, an impressive amount of qualitative and quantitative data have been collected on the mechanical behavior of neutrophils at the macroscopic scale accessible to visible light microscopy. At the same time, investigations at the microscopic level defined by molecular biology techniques have elucidated a number of biochemical pathways. However, despite these efforts, the phenomenology of neutrophil mechanics has lagged, insofar as it has been principally restricted to extensions of liquid droplet models (e.g., see Yeung and Evans, 1989; Drury and Dembo, 2001). This approach has the drawback that it ignores the dual nature of the cytoplasm with its cytosolic and cytoskeletal phases. It also is clearly insufficient to address problems of active motion.

The goal of this article is to begin to bridge the gap between microscopy and biochemistry with a mesoscopic paradigm of neutrophil mechanics applicable to a wide variety of experimental conditions. Within the framework of continuum mechanics, we will propose and discuss simple models, and carry out a numerical analysis of their consequences in the setting of various experiments. Our objective is not to obtain perfect agreement with experimental data, nor is it to provide a detailed connection with biochemical processes. Rather, we are interested in developing synthetic models that can serve as an intuitively accessible but self-consistent context in which the profusion of experimental evidence can be coordinated, and new experiments devised. Depending on taste and personal bias, these models are to be a foundation or a "straw man" that can be further expanded, revised, or contested. Regardless, their principal merit lies in their ability to organize ideas on neutrophil behavior.

In vivo, it is felt that the neutrophil exists in two basic states. In the quiescent or passive state, the neutrophil simply flows with the blood circulation deforming passively with minimal disturbance to its environment. In contrast, the activated state represents a response to inflammatory stimulus: in this incarnation the neutrophil is capable of actively developing forces that lead to adhesion and deformation. Of all the experiments devised to study the two faces of the neutrophil, a few stand out as particularly instructive because they capture some essential aspect of neutrophil mechanics, and because they can be built on to address more complex phenomena:

1. The aspiration of a passive neutrophil into a micropipette provides powerful constraints on mechanical properties on the basis of rate of entry versus time for various aspiration pressures and pipette radii.
   
2. The local stimulation of a neutrophil with fMLP with subsequent pseudopod formation represents the most elementary active shape-changing process that can be conceived.
   
3. The motion of an activated neutrophil in a micropipette toward a chemoattractant against an opposing pressure provides the simplest quantitative measurement of the production of active force.
   

It is worth noting that although it may not be always directly apparent to the reader, almost all the qualitative features and quantitative parameters that figure in our models are deeply interconnected. To keep things tractable, it has been necessary to present experiments sequentially and to order the discussion of the impact of various physical parameters in a succession of individual items (e.g., elasticity, viscosity, swelling, etc.). However, this neat organization obscures the entanglement between all the factors determining neutrophil behavior, which basically makes it impossible to change one thing in a model without changing everything in that model. As a consequence, our models have de facto undergone a stringent portability test by which parameters determined by a particular experiment are validated by other experiments.

	CONTINUUM MECHANICS AND REACTIVE INTERPENETRATING FLOWS

TOP ABSTRACT INTRODUCTION CONTINUUM MECHANICS AND REACTIVE... SIMULATIONS OF EXPERIMENTS DISCUSSION CONCLUSION APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
Basic concepts
From a mechanical standpoint, neutrophils are made up of three main constituent components: a relatively impermeable membrane (in the sense that the cell maintains a constant volume), an aqueous solvent phase, and a cytoskeletal network phase. It is also clear that the solvent phase is in large part a passive player, as it cannot transmit significant stresses other than through a banal pressure field. Although it is a medium through which all manner of chemical signals are diffusing, by itself the solvent is a bystander that flows through the network as required by volume conservation: swelling in one area of the cell draws water from other parts, while contraction does the opposite.

The principal actors of neutrophil mechanics (and for that matter, many amoeboid cells) are therefore the membrane and the cytoskeleton. The membrane contributes primarily through the surface tension whose meaning and form will be discussed in greater detail (see Membrane Surface Tension) and through boundary conditions imposed by contact surfaces. On the other hand, the mechanical properties of the network are the predominant source of richness of neutrophil behavior and, at the same time, the predominant source of controversy in this area of study. Despite this complexity, a few reasonable general statements can be made. First, the cytoskeleton is able to offer passive resistance to changes in shape; this means that the cytoskeleton is endowed with elastic and/or viscous properties that oppose deformation. Second, the stimulated cytoskeleton is able to produce active forces that result in spontaneous movement, and thus changes of shape. Several classes of forces can be envisioned to be responsible for this: isotropic network-to-network repulsive interactions leading to a natural swelling tendency of the cytoskeleton; directional polymerization forces involving Brownian ratchets; more general network-to-membrane repulsive potentials; directional molecular motors; etc. Fortunately, the general framework of continuum mechanics is sufficiently broad to allow the inclusion of all those alternatives through modification of the momentum equations.

Finally, it is possible to argue that organelles represent a fourth mechanical constituent of the neutrophil. Small organelles such as mitochondria can probably be subsumed into the bulk properties of the cytoplasm through coarse-graining. However, the nucleus occupies 20% of the cell volume (Schmid-Schönbein et al., 1980) and it has been argued by some to be a mechanically important entity (Kan et al., 1999). In the neutrophil, the nucleus is made up of three to four segmented lobules tethered together by flexible neck regions like pearls on a string (e.g., see Sanchez and Wangh, 1999; Campbell et al., 1995). This likely is an adaptation to the demands of passage through narrow capillary beds and diapedesis into tissues, the implication being that the nucleus has evolved to be as mechanically unobtrusive as possible. In the interest of simplicity, we therefore neglect to include a separate compartment for the nucleus in our models. Some weak support for this approach is provided by the fact that we get an adequate fit to the data that we are trying to explain.

Evolution equations
The Reactive Interpenetrating Flow formalism has previously been described in Dembo and Harlow (1986), Dembo (1994a), and He and Dembo (1997). We will therefore limit ourselves to a brief overview. In all that follows, the subscript s denotes solvent-related quantities and n denotes network quantities. When subscripts are dropped, n's should be assumed.

Mass conservation
Let be the volume fraction of a given phase, we then have trivially:

(1)

The incompressibility condition further yields:

(2)

Finally, conservation of network gives:

(3)

where J is simply the net rate of network production by polymerization at a given location.

Momentum conservation
Since only two forces act on the solvent, namely pressure gradients and drag due to relative motion of the network, the solvent momentum equation has a simple form:

(4)

where H is the solvent-network drag coefficient.

In addition to these terms, the network momentum equation must incorporate the forces that have been mentioned above (see Basic Concepts); namely, forces due to swelling, forces due to interaction with the membrane, and forces due to viscoelastic stresses:

(5)

In this equation, ⁿⁿ is the stress (tensor) of the network due to interfilament forces, ^nM is the network-to-membrane interaction term, is the viscosity, and F^el is the elastic force due to deformations.

Constitutive equations
A number of prescriptions are necessary to provide closure to the mass and momentum equations: these are the constitutive equations that establish the link between physical laws and biological behaviors.

Network polymerization
We assume that the net rate of polymerization of network is determined by a logistics type of law:

(6)

This formulation allows one to incorporate the idea that the rate of monomer release and addition should be proportional to the number of filaments. The equilibrium network concentration is given by:

(7)

Here m is a dimensionless concentration of a polymerization messenger produced by the membrane with emissivity _m (cm s^-1), lifetime _m, and diffusion coefficient D_m so that:

(8)

with appropriate von Neumann boundary conditions:

(9)

where n is the unit normal vector to the membrane. In general, we have chosen D_m and _m such that the penetration depth is small; as a result _n = ₀ for most of the interior of the cell. However, when the neutrophil is stimulated, the emissivity rises sharply in certain areas resulting in a localized increase in polymerization.

Network-network interaction
In simulations of the swelling model described below (see Two Models: Network Swelling versus Polymerization Force), we have implemented an isotropic stress term proportional to the network concentration, i.e.:

(10)

where _ij is the usual Kronecker symbol. If is negative, one has a net contractility of the network (e.g., under the influence of myosin motors) while if is positive, one has a repulsive pressure term that causes network swelling.

Network-membrane interaction
If one assumes, as is the case in our polymerization force model, the existence of forces on the cytoskeleton normal to the membrane (i.e., no shearing stress), the network-to-membrane stress can be written:

(11)

where n is the unit normal vector to the membrane, and where ^nM will depend, among other things, on the distance to the membrane (i.e., far from the membrane one expects ^nM 0). Essentially, this expression embodies a directional swelling (or contractile) stress acting in the network near the membrane. This has the basic effect of applying local pressure to the membrane. In the case of the polymerization force model, ^nM naturally depends on the local polymerization rate as is described next.

Implementation of the polymerization force model
We have used the simplest heuristic approach to a polymerization force term by making it linearly proportional to the local polymerization rate. In our calculations, the magnitude of the stress term is given by:

(12)

where one will recognize J_m as the part of network polymerization that is due only and strictly to the messenger m.

From a thermodynamic point of view, one might prefer a form for this constitutive law involving the net polymerization (J) of the network rather than just the part related to the messenger. However, we have found that this leads to serious difficulties, as when the stress vanishes and the network concentration is frozen at its equilibrium value. Empirically, what happens in these simulations is that after a short transient, as the network concentration builds up, protrusion ceases with

Physically, this may be interpreted by postulating that the stimulated addition of monomers to the network occurs predominantly near the membrane, whereas disassembly happens further back without impediment to the Brownian ratchet or other propulsive mechanism.

Network viscosity and elasticity
Whether the cytoskeleton must be treated as a viscous or elastic phase has been a longstanding matter of debate. Suffice it to say that from a physical standpoint, this depends on the relative magnitude of the strain rate to the cytoskeletal molecular remodeling rate. Both terms are considered in the present work. The viscosity is taken to be linearly dependent on network concentration:

(13)

The incorporation of an elastic stress term in our models involves material history and as such presents considerable technical challenges, especially in the presence of advection. The approach we have used here is a "poor man's" elasticity in the sense that it only accounts for compression and dilation through an isotropic stress term and ignores shear contributions. For this, we introduce the scalar field w > 0 that is evolved as follows:

(14)

where _el is the decay time of elastic memory of the network related to the remodeling rate by the following ansatz:

(15)

As one can see, w is an advected cytoskeletal volumetric coefficient that is <1 if there has been recent dilation, >1 if there has been recent compression, and that decays back to 1 in isovolumetric flow. In a Maxwellian-like model, the elastic contribution to the momentum equation can then be written:

(16)

where is a proportionality constant, the specific network stiffness.

Boundary conditions
Boundary conditions have to be specified for both solvent and network phases. If we assume impermeability of the plasma membrane to both water and cytoskeleton, we have:

(17)

where n is the unit normal vector to the membrane and v_M is the velocity of the membrane.

We further have to consider two possible types of boundary conditions: those that represent motion constrained by a solid surface such as that of a pipette, and those that represent free motion of the membrane. Solid walls impose the trivial constraint:

(18)

and depending on whether the boundary condition is stick or slip, v_n · t will be required to vanish or not. Here t is the membrane tangent vector (well-defined in two dimensions). The solvent boundary condition is always taken to be slip.

For free boundaries, a condition of stress balance across the membrane has to be written by equating the internal stress tensor with external stresses (in the context of this work, these are limited to predetermined external pressures P_ext) while taking into account the contribution of membrane surface tension. Adding Eqs. 4 and 5, and making use of the fact that all forces of interest are written in conservative form, we obtain the following:

(19)

where is the full network stress tensor (including the contributions of ordinary swelling, directional force and elasticity), is the surface tension, and is the mean curvature of the membrane.

Membrane surface tension
One of the main conclusions of the recent work of Drury and Dembo (2001) was that some sort of surface dilation viscosity is necessary to explain the time-dependence of neutrophil entry into a micropipette under aspiration (see Neutrophil Aspiration in a Micropipette). This was implemented by adding a term proportional to _M · v (where _M is the divergence along the plane tangent to the membrane) to the surface tension in the boundary momentum equation (Eq. 19) together with an equation for the evolution of membrane wrinkling. Practically, this meant that the surface tension was taken to increase locally in regions of surface creation.

In contrast, this study takes a simpler approach by considering the surface tension as a global property of the cell, as can be noted from the absence of a divergence term acting on in Eq. 19. In a sense, this is a special case of what was done in Drury and Dembo (2001) as, instead of setting a specific parameter for the diffusion of cortical stress, such diffusion is taken to be infinitely rapid. As a justification of this assumption, note that at the microscopic level, the plasma membrane is essentially inextensible and massless. Furthermore, the fluid mosaic nature of the bilayer allows it to act as a perfect conductor of stress (see Fig. 18). Thus a change in surface tension due to deformation in one region of the cell is instantly communicated to the entire plasma membrane. This global property of membrane tension is supported by the observations of Raucher and Sheetz (1999) in fibroblasts, and the results of Zhelev et al. (1996) that are described in this article's section Pseudopod Formations by fMLP Stimulation.

View larger version (40K): [in this window] [in a new window]   FIGURE 18  Lipid bilayer folds: (A) Flow of bilayer around a solitary transfold staple. (B) Stabilization by membrane-network-membrane interaction.

View larger version (26K): [in this window] [in a new window]   FIGURE 1  Distance of neutrophil entry versus time for three different experimental conditions. Squares and error bars represent data obtained by Drury and Dembo (2001).

(20)

where A is the total surface area of the cell, A₀ is the area of the sphere of equivalent volume, and is a relaxation time. The "slack" coefficient S [0,1] is determined by the amount of slack available in the membrane before unfurling begins in earnest when the area becomes greater than a threshold value A_s. We have used the following ansatz:

(21)

In our work, A_s = (1 + s)A₀ where s, the threshold coefficient for surface viscosity, is 5%.

Of note is the absence of an elastic term in Eq. 20 for the surface tension (i.e., depends linearly on dA/dt rather than A). Such a term was taken into account by Drury and Dembo (2001) but found to have only a small importance in the pipette aspiration problem as long as the pipette is not too narrow. While there is no doubt that it is present and important at larger deformations than those considered here (e.g., see Needham and Hochmuth, 1992), we have chosen to neglect the elastic term to avoid adding yet another parameter to our models.

Two models: network swelling versus polymerization force
One of the central issues of cell motility revolves around the origin of forces that are produced by the cytoskeleton according to the needs of the cell, especially in cases when there is no clear evidence that molecular motors are involved. It is important to recognize that how this issue is approached depends somewhat on whether one subscribes to a predominantly fluid versus elastic physical picture for the cytoskeleton.

In the fluid picture, the cytoskeleton is viewed as a highly dynamic structure that is continually recycled at a turnover timescale that is shorter than the prevalent strain rate. Arguments in favor of this are provided by in vitro studies such as those of Kuhlman et al. (1994) or Wachsstock et al. (1994) that show the turnover of a common actin crosslinker, -actinin, to occur on timescales of less than 1 s. In addition, in vivo studies of neutrophils (Cassimeris et al., 1990; Cano et al., 1991) and other amoeboid cells (e.g., Sund and Axelrod, 2000) have shown fast actin subunit cycling in and out of the polymerized state over the course of cell motion. The cytoskeleton is then conceived as a disorganized structure with isotropic properties, most notably when it comes to force generation.

In the elastic picture, the cytoskeleton is viewed as a more permanent, organized scaffolding that allows for the directional production of force. An example where this is clearly the case is given by the ultrastructure of skeletal muscle cells. The picture provided by electron-microscopy of neutrophils (Ryder et al., 1984) or other amoeboid cells is less clear, however; on the one hand, the very existence of a connected network of filaments gives credence to the idea of a degree of rigidity, but on the other hand, the apparent spatial disorganization of those structures argues for amorphous properties.

Those two pictures in turn lead to different ideas about the yet undetermined process of force generation in the absence of molecular motors. Since there is no preferred cytoskeletal direction in the cytoskeleton-as-a-fluid picture, the cytoskeleton is endowed with an isotropic equation of state that is devoid of memory terms. More concretely, since we know that actin polymers carry a large negative charge (isoelectric point 5.4; see also Xian et al., 1999) that will lead to interfilament repulsive forces, and also that thermal agitation of the network tends to lead to the least constrained configuration possible, it is reasonable to posit the existence of a swelling stress that tends to expand the cytoskeleton in regions where it is overdense. This has been used in the prior modeling studies of Dembo (1989) and He and Dembo (1997).

Within the "cytoskeleton-as-scaffolding" picture, the polymerization force model has gained increasing visibility over the last few years. The basic idea is that the free energy released by the addition of monomers to a filament is transduced to generate a pressure against a membrane that sterically interferes with the reaction, as observed, for instance, in the sickling of erythrocytes. Originally formulated by Hill and Kirschner (1982), the concept was revived in the form of a rectified Brownian ratchet mechanism (Peskin et al., 1993) to explain amoeboid cell motion (Mogilner and Oster, 1996). Other forms of network-membrane interactions are possible, but our implementation of a membrane-cytoskeleton repulsion term that is dependent on polymerization rate should address those too (see Brownian Ratchets). From a phenomenological point of view, the main strength of the polymerization force model is that it allows for the directed application of force in cellular activities, while its principal weakness is that it lacks a clear answer to the question of how this level of directionality is maintained.

Because there is, as yet, no definitive evidence that indicates which approach is correct, we have chosen to present both in this study. It should be pointed out, however, that there are yet alternative views that are not considered here—for instance, the recent suggestion of prestressed network in the model for motility of Listeria of Gerbal et al. (2000), or other models of cellular motility involving molecular motors.

Choice of parameters
From a computational point of view, the equations for both models are exactly the same with the difference between models stemming from a different choice of parameters intervening in the constitutive laws. In the case of the swelling model, the specific swelling stress is nonzero while the polymerization force strength and the specific network stiffness are both set to zero. Conversely, in the elastic/polymerization force model (sometimes equivalently labeled in this article as the polymerization-force model), and are nonzero while vanishes (see Table 1).

As indicated in Table 1, the parameters can be subdivided into those that affect cytoskeletal kinetics and those that deal with stresses. We discuss the former first.

Kinetics of network polymerization
From a numerical standpoint, the baseline network density is almost arbitrary in the sense that coefficients such as specific network viscosity can easily be rescaled to provide the same momentum equation for different network concentrations. As it is, the cytoskeletal volume fraction in the passive neutrophil was taken to be ₀ = 0.1% (see estimates from Watts and Howard, 1993).

The network turnover and decay time was picked to be 20 s; such a timescale has been found to be appropriate, as argued in the section Two Models: Network Swelling versus Polymerization Force. A change by a factor of 2 either way does not change our results markedly. An upper limit is, however, provided by the time is takes for the neutrophil to extend a pseudopod (less than a minute). A lower limit is provided by the fact that the cytoskeleton appears to have a persistence timescale that is at least greater than a few seconds.

As mentioned in the section Constitutive Equations, network polymerization above and beyond the baseline equilibrium level is taken to be driven by a diffusible chemical messenger m emitted by the membrane. This is of course not intended literally, but rather as a catch-all for a complicated collection of biochemical intermediates such as Arp2/3 and others (e.g., see Weiner et al., 1999; Machesky et al., 1997). Of note is that a similar approach was recently adopted by Rappel et al. (2002) in a sophisticated model of Dictyostelium polarization. The messenger is characterized by a lifetime _m and a diffusion coefficient D_m; those parameters can be rescaled by arbitrary factors with little change. However, when combined, they yield a penetration depth from the membrane that has critical physical importance, inasmuch as it defines the range of stimulation of polymerization.

In the case of the polymerization force model, only polymerization immediately next to the membrane contributes to protrusive forces. It is therefore reasonable to expect that the effective range of active polymerization from the area of stimulation should be small; for our calculations that distance is 0.3 µm. A shorter distance would not be resolvable by the mesh, whereas from a purely utilitarian perspective, a larger distance would essentially waste the polymerization away from the membrane.

In the case of the swelling model, polymerization plays a role by increasing the network density in an entire compartment of the cell and thus generating a swelling stress. Thus strict localization is not needed except for the creation of thin structures. In the case of pseudopod formation, this corresponds to 1–2 µm so that we have set a diffusion distance 0.5 µm.

Baseline membrane emissivity of the polymerizing messenger in the unstimulated neutrophil was taken to be small such that there is little excess cytoskeleton over the baseline network level ₀. Activation of the neutrophil causes an increase in network polymerization; in our calculations, this is mediated by an increase in emissivity by 2 orders of magnitude at localized patches of the membrane. Specific values are discussed below with the presentation of our results for each of the experiments.

Stress parameters
To set our stress parameters, we have made liberal use of the constraints obtained by Drury and Dembo (2001) in their experiments on passive neutrophil aspiration. A difference, however, is that our work does not include shear thinning. While there is little doubt that shear thinning does occur and is important in the dynamics of the neutrophil (e.g., see Tsai et al., 1993), introducing it enlarges the parameter space to such an extent that we thought it preferable to ignore it for clarity of exposition. Instead, we have restricted ourselves to the modeling of experiments that all have comparable shear rates.

Cytoplasmic viscosity in the passive neutrophil has previously been estimated by Drury and Dembo (2001) to be 1000–6000 poise (100–600 P_a s) which is also consistent with other experimental data (e.g., see Evans and Yeung, 1989; Hochmuth et al., 1993). In our models where only the network phase contributes to the cytoplasmic viscosity, the baseline viscosity ₀₀ was found be fairly tightly constrained in the range 3000–6000 poise. One should note that the assumption of linear dependence of viscosity on network density may not be correct; it is not unreasonable to think that there might be increased cross-linking with densities leading to gelation. This was not explored in the calculations presented here.

The main difference between our two models lies with the mode of active force generation. The swelling model includes a baseline swelling stress of 3000 dyn cm^-2 (3 x 10^-3 atm or 300 P_a) for a network volume fraction of ₀ = 10^-3. From a purely thermodynamical point of view, this is within plausible limits since this corresponds to a energy density of 6 k_BT per monomer incorporated in the network (see Appendix, Another Look at the Swelling Force). A formal justification of this value however would necessitate a detailed thermodynamic model of the cytoskeleton and the ambient solvent. In the present work, this magnitude is set by the constraints from the pseudopod experiments (see Pseudopod Formation by fMLP Stimulation), rather than fundamental principles.

The polymerization force model includes a network-membrane repulsive force that was determined empirically by the following constraints: the force should be as small as allowed without requiring an unduly high polymerization rate to produce active movement. The connection of the polymerization force strength with Brownian ratchet models is discussed in detail in Brownian Ratchets in the Appendix.

As discussed, the swelling force model views the cytoskeleton as an isotropic fluid and therefore, the elastic force term is set to zero. On the other hand, elasticity is crucial to the polymerization force model, since in the absence of a swelling stress it is required to prevent separation of the cytoskeletal and aqueous phases in circumstances such as aspiration of a neutrophil (see Elastic Force versus Swelling).

The solvent-network drag coefficient, H, is of order 1.6 x 10¹¹ poise cm^-2 as per the estimates of Dembo and Harlow (1986). It is worth noting that compared to the other terms in the network momentum equation, its contribution is small, and that solutions are therefore not sensitive to its precise value.

The static surface tension of the cortical membrane has been measured by numerous experimentalists and has been found to be of order 2.5 - 3.5 x 10^-2 dyn cm^-1 or 2.5 - 3.5 x 10^-2 mN m^-1 (Evans and Yeung, 1989; Zhelev et al., 1996). The surface tension viscosity, ₀, which expresses the increase in cortical tension under conditions of area dilation (see Membrane Surface Tension), was found by Drury and Dembo (2001) to be of order 100 poise cm (or 0.1 N m^-1 s) under a somewhat different model than the one used in this article. Using kinematic information from neutrophil aspiration, we have found the optimum value to be 75 poise cm for both the polymerization and swelling force models (see The Effect of Membrane Dilation Viscosity). Finally, it is apparent from experimental data that a small amount of membrane is immediately available for deformation as "slack" without inducing important dilation viscosity effects (see Membrane Surface Tension, Eqs. 20 and 21). This fractional amount s is found to be 5% as described in Neutrophil Aspiration in a Micropipette and Pseudopod Formation by fMLP Stimulation.

Numerical implementation
The simulations presented in this article were obtained by solving the model equations through a Galerkin finite element method using a mesh of quadrilaterals as described in Dembo (1994a), He and Dembo (1997), and Drury and Dembo (1999). Boundary conditions are as specified in the next section, Simulations of Experiments, for individual experiments.

Briefly, the calculation is advanced over a time-step t determined by the Courant condition or other fast timescale of the dynamics. We evolve over t by means of five sequential operations:

1. We advect the mesh boundary according to the network flow and then reposition mesh nodes for optimal resolution while preserving mesh topology, boundaries, and interfacial surfaces (Knupp and Steinberg, 1994).
   
2. We advect mass from the old mesh positions to the new mesh using a general Eulerian-Lagrangian scheme with upwind interpolation (Rash and Williamson, 1990).
   
3. We conduct diffusive mass transport and simultaneously carry out any chemical reactions. This is done according to a backward Euler (implicit) scheme coupled with a Galerkin finite element treatment of spatial derivatives and boundary conditions.
   
4. We use constitutive laws to compute necessary quantities such as viscosities and surface tensions.
   
5. Finally, the momentum equations and the incompressibility condition together with the applicable boundary conditions are discretized using the Galerkin approach and the resulting system is solved for the pressure, network velocity, and solvent velocity on the advected mesh using an Uzawa style iteration (Temam, 1979). Of note is that since we enforce a global surface tension that depends on the rate of change of area dA/dt (see Membrane Surface Tension), it was necessary to add a procedure that iterates between the velocity solution v that depends on , and dA/dt that depends on v, until all three are self-consistent.
   

The cylindrical symmetry of the cases under consideration in this article allows the use of a two-dimensional mesh—some of the figures presented here simply correspond to recovery of the third dimension by rotation of the two-dimensional solution. Numerical convergence was confirmed by checking that the results were not sensitive to variations of the tolerance of the different iterations performed by the code as well as to variations of the spatial resolution.

Calculations were conducted using 64-bit arithmetic on a Linux PC workstation. The code was compiled with the Absoft Fortran 90 compiler. Post-processing was performed with a variety of publicly available software packages (SuperMongo, DISLIN, GMV, and ANA) as well as with customized code.

	SIMULATIONS OF EXPERIMENTS

TOP ABSTRACT INTRODUCTION CONTINUUM MECHANICS AND REACTIVE... SIMULATIONS OF EXPERIMENTS DISCUSSION CONCLUSION APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
Neutrophil aspiration in a micropipette
This experiment has already been modeled by Drury and Dembo (2001) with a single-phase, viscous flow approach. Major differences in this study encompass the inclusion of two-phase flow, a different treatment of the cortical tension, the absence of shear thinning, and the addition of cytoskeletal swelling and elastic effects.

Experimental findings
Evidence related to aspiration experiments has already been thoroughly reviewed by Drury and Dembo (2001); we will therefore limit ourselves to a brief summary. Regardless of the details of the aspirating pressure or the pipette radius, neutrophil aspiration into a pipette takes place in three stages (see Fig. 1): TYPE="1">

After 10% of the cell volume is aspirated and the surface area has increased by 5% from the initial spherical configuration, entry slows down to a nearly steady rate of entry that lasts for most of the aspiration time.

After 60% of the cell volume is aspirated and the surface area has reached 95% of its final value in the terminal ‘sausage’ configuration of the cell, the rate of entry accelerates markedly once again.

1. Throughout aspiration, the unaspirated portion of the neutrophil that remains outside the micropipette retains a shape that is close to semispherical, and does not show much flattening.
   
2. There does not appear to be marked phase separation between the cytoskeleton and the cytosol.
   

Baseline simulations
Initial conditions consisted of a spherical model neutrophil of radius R_c = 4.25 µm that was numerically relaxed for several virtual minutes in the absence of external forces to ensure chemical and dynamical steady state. As in Drury and Dembo (2001), the neutrophil was then considered to be in critical contact at the edge of a pipette of radius 2.2 or 3.2 µm and a negative pressure was applied to the portion of the free boundary within the pipette as given by Table 2. Boundary conditions at the pipette walls were assumed to be slip.

View larger version (80K): [in this window] [in a new window]   FIGURE 2  Neutrophil aspiration 55 s after beginning of entry (2.2-µm radius pipette, aspiration pressure of 1430 dyn cm-2). Left corresponds to the polymerization/elastic force model and right corresponds to the swelling force model. Computational meshes are overlaid. Color represents the volume fraction of network.

If a small region A (the tip of the aspirated part of the neutrophil) is depleted of network without the network in region B (the main cell body of much larger volume than A) being significantly compressed, we will have after the equation for elastic stress (Eq. 16):

In region since _n 0 (network depletion) even though w 1 (network dilation),

In region since w 1 (no deformation) even though _{n 0}.

Consider for instance Fig. 3, which depicts simulations where the specific network stiffness and the specific network swelling were halved with respect to the baseline parameter. Runaway phase separation is evident in the elastic force model, whereas it remains negligible in the swelling force model. In addition to not having been observed, it should be noted that, should phase separation occur, it would work strongly against the final acceleration during the aspiration, as a solid plug of network would be slower to aspirate at the end of the process.

View larger version (39K): [in this window] [in a new window]   FIGURE 3  Neutrophil aspiration 8 s after beginning of entry for decreased specific network stiffness (left) and decreased specific network swelling 0 = 3 x 106 (right) (2.2-µm radius pipette, aspiration pressure of 1430 dyn cm-2). Phase separation is apparent in the elastic/polymerization force model (left).

The effect of membrane dilation viscosity
As argued by Drury and Dembo (2001), the shape of the aspiration curve argues toward a strong component of surface dissipation, inasmuch as simple viscous droplet models cannot account for the three stages of neutrophil aspiration. Dilation viscosity resisting the gradual unfurling of the cortical membrane, as the shape of the neutrophil changes from a ball into a sausage, provides a natural explanation for these features.

If one posits that the membrane is responsible for most of the resistance to aspiration, the initial jump of the neutrophil at the beginning of aspiration leads one to postulate the existence of a certain amount of slack that allows a small (5%) surface increase before dilation viscosity kicks in. This is illustrated in Fig. 4, which shows the difference between early entry curves with and without slack. Furthermore, pertaining to pseudopod growth, we shall see in Pseudopod Formation by fMLP Stimulation that there is experimental evidence supporting this finding.

View larger version (19K): [in this window] [in a new window]   FIGURE 4  Detail of the initial jump phase of neutrophil entry versus time for various parameter choices.

(22)

where A_pip is the area of the pipette. The energy dissipation through unfurling of the neutrophil membrane can be written:

(23)

where A_cell is the total neutrophil surface area and is the effective surface tension including dilation viscosity effects. Fig. 6 provides the relevant quantities for one aspiration calculation in which one finds: and Thus, in this case, the major part (80%) of energy dissipation in resistance to aspiration is contributed by membrane dilation with the balance principally due to cytoskeletal viscosity and compression.

View larger version (18K): [in this window] [in a new window]   FIGURE 5  Detail of the steady phase of neutrophil entry for low surface tension viscosity (upper pair of curves), baseline surface tension viscosity (middle pair of curves), and high surface tension viscosity (lower pair of curves). Note that 1 poise = 0.1 Pa s.

View larger version (14K): [in this window] [in a new window]   FIGURE 6  Surface area, surface creation rate, surface tension, and entry velocity as a function of time for the swelling force model (2.2-µm radius pipette, aspiration pressure of 1430 dyn cm-2). Note that the final few seconds of the aspiration are subject to numerical errors that make the precise values of derivative quantities such as surface area creation rate and surface tension uncertain due to resolution problems near the entrance of the pipette.

View larger version (14K): [in this window] [in a new window]   FIGURE 7  Comparison of numerical simulations of a model with baseline parameters and a model with increased network viscosity (33%) and decreased surface dilation viscosity (67%) (2.2-µm radius pipette, aspiration pressure of 1430 dyn cm-2).

View larger version (77K): [in this window] [in a new window]   FIGURE 8  Comparison of numerical simulations of a model with baseline parameters (top) and a model with increased network viscosity (33%) and decreased surface dilation viscosity (67%, bottom) (2.2-µm radius pipette, aspiration pressure of 1430 dyn cm-2). Color indicates network concentration, and arrows indicate network flow velocity field.

We conclude that, for the conditions encountered in this article, 3000 poise (300 P_a s) is an upper limit to cytoplasmic viscosity in the inactivated neutrophil (higher and lower viscosities are possible at lower and higher shear rates due to shear thinning). As to a lower limit of cytoplasmic viscosity, it is provided by aspiration data in 3.2-µm radius pipettes. Because these larger pipettes require less deformation for entry, cytoplasmic viscosity plays a more important role in resisting aspiration (as already noted by Yeung and Evans, 1989). Viscosities significantly less than 3000 poise lead to unacceptably short aspiration times (data not shown).

Pseudopod formation by fMLP stimulation
In a series of technically challenging experiments, Zhelev et al. (1996) have characterized aspects of the response of neutrophils to the chemoattractant fMLP. Using a pipette with gentle suction to maintain a neutrophil in place, they exposed a local region of the antipodal side of the cell to minute quantities of fMLP delivered by another micropipette (Fig. 9). They then observed the growth of a pseudopod extending toward the source of fMLP and were also able to simultaneously measure the cortical tension with the holding pipette by a law of Laplace method (Evans and Yeung, 1989). This study provides a remarkable probe of a cellular shape-changing process that is not dominated by surface boundary constraints.

View larger version (76K): [in this window] [in a new window]   FIGURE 9  Microphotographs of a neutrophil being held by a pipette and extending a pseudopod toward the right where another micropipette is dispensing fMLP (left; from Zhelev et al., 1996; courtesy R. Hochmuth, copyright, Wiley-Liss). Computed time series of shapes for a polymerization force model (center) and a swelling force model (right).

1. Noticeable growth of a pseudopod extending toward the fMLP source begins approximately 20–30 s after beginning of the exposure. It proceeds at a velocity 0.1 µm s^-1 for several tens of seconds before slowing down and/or stagnating. After a period of varying length (sometimes several minutes), the pseudopod retracts into the cell at a velocity somewhat less than the original extension speed.
   
2. Cortical tension initially remains at the baseline value of 0.025 dyn cm^-1 (= 0.025 mN m^-1), and only begins to increase after the pseudopod is several µm in length. It then rises rapidly sixfold. Subsequently it returns to near baseline as soon as the pseudopod stagnation or retraction phase has been reached, well before the pseudopod has been resorbed.
   
3. The foremost 2 µm of the pseudopod is devoid of granules, probably indicating a region of high F-actin cytoskeletal density.
   
4. Initially, the morphology of the cell remains relatively spherical with a straight pseudopod extruding; during stagnation or retraction, the pseudopod thickens and the cell appears to become somewhat ovoid (prolate).
   

View larger version (22K): [in this window] [in a new window]   FIGURE 10  Distance of pseudopod extension (dashed lines) and cortical tension (solid lines) versus time for the stimulation of a neutrophil with fMLP. Bottom panel is the data from Zhelev et al. (1996), second panel shows result from the polymerization force model, and third panel shows results from the swelling model. Top panel shows the time course of the excitatory polymerizing messenger signal (normalized). Note that 1 dyn cm-1 = 1 mN m-1.

View larger version (77K): [in this window] [in a new window]   FIGURE 11  Calculation 60 s after fMLP stimulation. Left corresponds to the polymerization/elastic force model and right corresponds to the swelling force model. Computational meshes are overlaid. Color represents the volume fraction of network. The solid bar at the base is 4-µm long.

The main results are shown in Figs. 9–12. Note first that the gross experimental findings are largely recovered; namely, that the pseudopod dimensions and morphology correspond to what is observed, and that we have indeed a frontal plug of dense cytoskeleton at the leading edge of the pseudopod.

View larger version (71K): [in this window] [in a new window]   FIGURE 12  Detail of the pseudopod network velocity field at 60 s for the polymerization force model (left) and the swelling force model (right). Color indicates network concentration. The stimulatory part of the membrane is highlighted. Note the disjoining flow in the polymerization force model (left) versus the explosionlike flow in the swelling force model (right).

The central role of the polymerization signal
Since force production in our two models relies on the creation of network at precise locations in the cell, it should be obvious that how this occurs will be key to the characteristics of pseudopod extension. As discussed in Kinetics of Network Polymerization, the models include a polymerization messenger that is produced at the membrane and diffuses inside the cell with a finite lifetime. This is, of course, not to be taken literally; i.e., there undoubtedly is a complex pathway involved, but this approach has the benefit of simplicity and of encompassing some of the basic realities of such signaling: external stimuli are sensed at the external face of the membrane and converted to cytoplasmic signals through enzymatic activity at the internal face of the membrane. Ultimately, this all boils down to three issues: the spatial extent of the signal, the temporal course of the signal, and the intensity of the signal.

In our calculations, the way by which network creation is locally induced is through increased emission of the polymerization messenger at a defined patch of membrane. For the polymerization force model, the region of emitting membrane was taken to be a cap of curvilinear radius 0.75 µm from the symmetry axis while, on the other hand, for the swelling force model this was taken to be a cap of radius 0.5 µm. The main constraint in setting those dimensions was provided by the girth of the pseudopod. In the case of the network swelling model, the application of force is more diffuse, and this is why a smaller area of activation is required.

The time-dependence of the polymerization was set to approach the temporal behavior shown in Fig. 10 (bottom). As can be seen, the cortical tension becomes maximum after 90 s of stimulation and returns to near baseline 30 s later. From a qualitative point of view, this indicates that the driving force of pseudopod extension goes from zero to a maximum and back to zero in the same timeframe, and that presumably, the polymerization signal does the same. As the simplest possible guess, we have chosen to assume a linear variation of the messenger emission as shown in Fig. 10. The free parameter of the maximum value of the messenger emission was then adjusted to give an approximately correct pseudopod maximum length.

Impact of the cytoskeletal viscosity
In the absence of an unmovable external element to counteract protrusive force, network viscosity plays a key role in enabling the extension of a pseudopod. The basic idea is that as outward force somehow develops (through polymerization, swelling, or otherwise), bracing is provided by the viscosity of the network that prevents inward expansion and forces outward protrusion. However, for the purpose of a quantitative determination of the viscosity, it is preferable to focus on the subsequent recovery phase for which the confounding factors of putative swelling, elastic, and polymerization forces play less of a role.

The key assumption that is made here is that pseudopod retraction is a mostly passive process for which the principal determinants are the cortical tension (which is measured by Zhelev et al., 1996) and the network viscosity. This is defensible in light of the fact that the timescale for retraction of the pseudopod is of the same order of magnitude as the time for recovery of a passive, elongated neutrophil back to a spherical shape after pipette aspiration and expulsion (Tran-Son-Tay et al., 1991). Note that active depolymerization is not necessary for pseudopod retraction. In both models, the interruption of polymerization leads to a rapid decay of protrusive force, and even without allowing depolymerization, retraction proceeds according to viscosity and surface tension. That is not to say that active depolymerization does not occur, but the details of the biochemical kinetics cannot be constrained with the data at hand.

Fig. 13 shows the time course of pseudopod retraction for both models with varying specific viscosity ₀ = 3 x 10⁶ - 6 x 10⁶ - 1.2 x 10⁷ poise (effective viscosity is ₀ = 10^-3 x these values). Clearly, ₀ = 6 x 10⁶ gives the best fit to the data. This value is double what was deduced from the aspiration experiments (see The Effect of Cytoskeletal Viscosity) and may possibly be interpreted as the consequence of increased network cross-linking due to neutrophil activation. This increased viscosity was also necessary in our modeling of the activated neutrophil crawling in a micropipette (see Active Motion of a Neutrophil Inside a Micropipette). Finally, it is notable that Bathe et al. (2002) have recently also found that fMLP stimulation apparently increases the internal viscosity of neutrophils.

View larger version (16K): [in this window] [in a new window]   FIGURE 13  Retraction of the pseudopod versus time for varying viscosities (1.2 x 104 - 6 x 103 - 3 x 103 poise). Note that 1 poise = 0.1 Pa s.