INTRODUCTION

Top Abstract Introduction Conclusions Appendix A Appendix B Appendix C Appendix D References

Attempts by theoretical physicists to contribute in some useful way to the study of biology have, so far, been most successful in systems in which all forces and motion can be modeled and mathematized explicitly, or in those governed by equilibrium statistical mechanics, for which equipartition can be invoked. One specific example of such success is the analysis of structural microfilaments, essentially one-dimensional mechanical objects with no moving parts. Despite this unassuming mechanical description, these semiflexible biopolymers are essential for innumerable functions and processes at the molecular and cellular level.

Depending on the bending modulus of the filament in question, experiments investigating the elastic properties of these biopolymers largely rely on either mechanical or statistical techniques. Microtubules, with a persistence length of ~5 mm, are quite amenable to micromanipulation or forcing via hydrostatic drag. Actin and nucleic acids, with persistence lengths near 15 µm and 50 nm, respectively, fall in the realm of statistical mechanics (note that we are here addressing the bending elasticity, not the stretching elasticity, another area of great excitement and successes; Yin et al., 1995; Cluzel et al., 1996; Smith et al., 1996).

An area in which careful analysis has been less prevalent, however, is investigations of dynamics at the single polymer level. Such a theoretical program has only recently been made experimentally relevant through the advent of optical tweezers and the proliferation of similar techniques for precise and controllable micromanipulation. Whereas treatments of the undamped, inertial case have a long history (Harris and Hearst, 1966; Landau and Lifshitz, 1986), and viscously overdamped dynamics have been studied in great detail and with exciting results in bulk for polymer gels (Isambert and Maggs, 1996; MacKintosh and Janmey, 1997), the application of viscous dynamics to single polymers and connections with experiment have not been fully elucidated. We intend this paper to be a complement to the important works done in the inertial and bulk contexts.

Specifically, we here couple elasticity theory and overdamped viscous hydrodynamics (as is appropriate in the biological context) to explore elastohydrodynamics. Although equations with the appropriate units will be sufficient to determine the scales of forces and velocities, if we wish to extract numbers from the experiments it is necessary to perform a thorough analysis. The slenderness of the filaments allows us to simplify greatly the hydrodynamics and arrive at a local partial differential equation of motion. We find that coupling to hydrodynamics allows us to extend the range of mechanical experiments to much smaller bending moduli. For example, whereas measurements of actin's rigidity so far have been via fluctuation analyses invoking equipartition and thus statistical mechanics, we present here an experimental method that does not rely on nonzero temperature. Furthermore, the method allows investigation into questions that have been raised about whether actin can even be treated as a semiflexible polymer, or is in fact scale-sensitive (Käs et al., 1993) or dynamic in its elasticity. Such a purely mechanical treatment obviates the possible complications of statistical treatments like dimensionality (Ott et al., 1993), correlations among sampled images, or self-avoidance.

It is our hope that this new experimental method, as well as the general analytic techniques here outlined, will contribute to the current exciting and active dialogue among physicists and biologists regarding the nature and numbers behind biopolymer rigidity, as well as the effects of associated proteins and varying biochemical environments on elastic moduli. Furthermore, these new methods and analyses should prove useful in the study of other examples of dynamic elastic filaments, such as supercoiled fibers of B. subtilis (Mendelson, 1990). We intend this investigation to be the necessary precursor to such promising extensions.

A useful starting point for developing the dynamics of an elastic filament in a viscous medium will be to consider the simplest time dependencies possible. To that end, recall the classic problems introduced by G. G. Stokes (Stokes, 1851), illustrated in Fig. 1, involving the motion of a viscous fluid bounded by a wall that is either (I) moved impulsively or (II) oscillated. These easily solvable problems capture the essential ideas of viscous diffusion of velocity. The experimental geometry is such that the Navier-Stokes equation for the velocity field u(x, t) is simply the diffusion equation

(1)

where  = µ/ is the kinematic viscosity, and µ and  are the fluid viscosity and density. Subscripts on functions indicate differentiation throughout unless otherwise indicated. The salient features of the solutions are the relationships between length scales, time scales, and material parameters. Specifically, in the impulsive case, the velocity at any point x and time t depends only on the ratio x/(t)^1/2; likewise, in the oscillatory case, deformations decay with a characteristic length that scales as _S() = (/)^1/2.

View larger version (14K): [in this window] [in a new window]   FIGURE 1   Geometry of Stokes problems I and II.

We introduce here the analogous two problems in elastohydrodynamics, illustrated in Fig. 2. They involve (I) the deflection of a polymer anchored at one end after the instantaneous introduction of a uniform fluid velocity U, and (II) the steady undulations of a polymer, one end of which is oscillated. Rather than a diffusion equation as in the Stokes problems, the dynamics of small deformations y(x, t) of the filament are governed by a fourth-order partial differential equation of the form

(2)

where  = A/ plays the role of a "hyperdiffusion" coefficient, A is the bending modulus, and  is the drag coefficient. This equation has appeared before in the literature on semiflexible biopolymers (Barkley and Zimm, 1978; Amblard et al., 1996; Gittes et al., 1993), primarily in the context of scaling arguments for relaxation times; our goal here is to provide a complete solution, given arbitrary initial and boundary conditions as dictated by experiment. (Nota bene: In Amblard et al. (1996), Eq. 2 should include a minus sign; as written, the equation is ill-posed.)

View larger version (11K): [in this window] [in a new window]   FIGURE 2   Geometry of elastohydrodynamic problems I and II.

An analysis similar to that presented below of the oscillatory passive elastica was carried out a number of years ago by K. E. Machin (Machin, 1958, 1962), who considered the motion of a driven flagellum. Machin was interested specifically in a semiinfinite active flagellum that was bent with a set of boundary conditions amenable to analysis. Ours will be more malicious, but not subtle.

We first recall some general features of equations of motion for elastica embedded in viscous flow. By illustrating the geometrically exact equation, we hope to make clear how higher order terms will affect the results of linearized analysis. We then apply this dynamic to a number of experimentally relevant scenarios. Inspired by Stokes problems I and II in fluid dynamics (SI and SII), we first solve problems I and II of elastohydrodynamics (EHDI and EHDII), each of whose dynamic mimics its hydrodynamic analog. Problem I requires some mathematical details familiar from elasticity theory to assist our physical intuitions. Specifically, we use a set of basis functions appropriate to the equation of motion and specified boundary conditions. All of the pleasant features found when applying Fourier space to unbounded or periodic systems are found here as well, in what we term -space. Unlike Fourier space, -space respects both the compact support and the boundary conditions of the elastica and thus diagonalizes the equation of motion. We then discuss an experimental realization of problem II and its analysis, which provides a new technique for the measurement of a polymer's bending modulus. Finally, we comment on experiments by a separate group to which the EHDI analysis may be applied.

	ELASTIC FORCES

A bent elastic polymer exerts a restorative force per unit length given by the functional derivative f = /r of a bending energy,

(3)

(Note that we may also include any forces of constraint, such as a Lagrangian tension to enforce inextensibility (Goldstein and Langer, 1995), but such terms will be of higher order in the curvature than we will consider in this investigation.) Here is the curvature, s is the arclength, and A is the bending stiffness constant, with units of energy × length. This may also be expressed as the product EI of Young's modulus E and the moment of inertia I (Love, 1892). For a polymer of persistence length L_p at absolute temperature T, exploring all configurations in D dimensions, we may also derive by equipartition the equivalence A = (D  1)k_BTL_p/2.

Henceforth we consider elastic filaments lying in the plane, the geometry best suited to data acquisition via microscopy. The curvature  may then be expressed exactly as d/ds, where  is the angle between the tangent to the curve and some fixed axis (see Fig. 3), or equivalently as  = y_xx/(1 + y_x²)^3/2. Taking the functional derivative of the energy (Eq. 3), we find the force per unit length, exerted purely in the normal () direction,

(4)

and the boundary conditions  = _s = 0, indicating torquelessness and forcelessness at free ends of elastica (Weinstock, 1974; Landau and Lifshitz, 1986). At hinged or clamped ends different boundary conditions hold, as will be discussed below.

View larger version (9K): [in this window] [in a new window]   FIGURE 3   Geometry of an elastic filament. R = local radius of curvature = 1/; t, n = unit tangent and normal, cos  =  · êx; d = diameter of the filament; arclength s varies from 0 to L, the total arclength. Within the approximations of slender-body hydrodynamics, a local anisotropic proportionality is satisfied between an external force per unit length f and the velocity v.

For small deviations from a horizontal line (|y_x|  1),   ê_x,    ê_y, and the linearized force is

(5)

(where ê_y is the unit vector in the y direction), with boundary conditions

(6)

The specification of the filament dynamic is complete upon definition of the hydrodynamic drag, which balances f. We now turn to this problem.

	SLENDER-BODY HYDRODYNAMICS

We consider experiments taking place on cellular biological scales, with typical lengths L in microns, times t in seconds, and a dynamic viscosity µ that of water, in centipoise. The Reynolds number is UL/  L²/t  10⁸/10²  10⁶, so we are safely in the low-Reynolds number or Stokesian regime. In this Aristotelian overdamped limit, forces balance velocities rather than accelerations. For a body whose length is much greater than its width, the well-developed set of calculations known as slender-body hydrodynamics applies (Keller and Rubinow, 1976; Cox, 1970, 1971). If this filamentous polymer has diameter d, length L, and an aspect ratio d/L  1, we have to lowest order in 1/ln(L/d) the simplified, local, anisotropic proportionality between the drag force f_d and the velocity r_t,

(7)

Here f_d(s) is a force per unit length exerted on the filament, and (s) and (s) are unit vectors in the normal and tangential directions at arclength s along the polymer. The products and indicate tensor multiplication, projecting velocities normal and tangential to the curve and relating them via their respective drag coefficients to the applied force. The velocity of the polymer is denoted r_t(s), and u is any background velocity that may be present in the problem; the drag should be a function of the former relative to the latter. The anisotropy, evident when dragging a pencil through molasses, between motions parallel and perpendicular to a slender object's long axis is embodied by the parameter , which depends logarithmically on the aspect ratio, with asymptotic behavior   1/2 as L/d  . For small d/L, the viscous drag coefficient  has the limiting behavior

(8)

where c is a constant of order unity, which depends on the shape of the body (Keller and Rubinow, 1976; Cox, 1970, 1971; Lighthill, 1975; Childress, 1981; Shelley and Ueda, 1996).

We now equate the elastic force per unit length with the drag force (f_d = f) to derive the equation of motion:

(9)

Linearizing the expression for the drag (Eq. 7) for nearly straight polymers and noting that its tangential components are of order y_x², the dynamic reduces to

(10)

Here u  u · ê_y. In the absence of any background flow we recover Eq. 2. This is the simplest linearized expression of elastohydrodynamics: elastic forces, characterized by a fourth spatial derivative, balance viscous drag. It shares many similarities with the diffusion equation (Eq. 1) and may be thought of as "hyperdiffusion" of displacement in analogy with hyperviscosity.

	ELASTOHYDRODYNAMIC PROBLEM I

Now that we have established the equation of motion appropriate to these elastohydrodynamic analogs, we recall the solutions to the fluid dynamics problems SI and SII in hopes of exploiting the analogy as much as possible. In Stokes I (SI), a semiinfinite plane of fluid is driven by a wall that is motionless for time t < 0 and has velocity Uê_y for t > 0. In Stokes II (SII), the wall oscillates as U cos(t)ê_y, and we solve for the behavior after transients have died away.

As illustrated in Fig. 1, velocity gradients are in the x direction in both Stokes problems, and hence are perpendicular to the direction of flow (along the y axis). In the absence of an imposed pressure gradient, the Navier-Stokes equation for the fluid velocity u(x, t) parallel to the wall is simply the linear diffusion equation (Eq. 1), u_t = u_xx.

A convenient method of solving SI with the associated boundary condition is to postulate a scaling solution inspired by dimensional analysis: u(x, t) = UF(), with   x/(t)^1/2. The scaling function F then obeys a nonautonomous ordinary differential equation 1/2F = F, the solution of which is F = erfc (/), where erfc is the complementary error function. Rewriting Eq. 1 in this form illustrates the scaling behavior alluded to after Eq. 1.

Armed with some understanding of SI, we now turn to problem I of elastohydrodynamics (EHDI). In problem I, we consider an elastic filamentous polymer that is anchored at the origin. For t < 0 it lies along the line segment {y = 0; 0 < x < L}. We then may consider forcing the filament by moving one end relative to the fluid (moving the anchor) or moving the fluid relative to the polymer (moving, for example, the coverslip). We will first attempt to do this in a way as analogous to SI as possible.

The strict analog of SI involves a polymer of infinite extent, obviating the problem of boundary conditions at the "right" end. Although this scenario is of limited value in comparing to experiments on actin or microtubules, where thermal fluctuations dominate on scales longer than the persistence length, it is useful both in illustration of how Eqs. 1 and 2 differ, and in application to more rigid biofilaments, e.g., filaments of B. subtilis, whose persistence length is ~10 m (Pederson and Goldstein, unpublished data).

Defining   x/(t)^1/4, the scaling ansatz y = utF() transforms Eq. 2 into F  1/4F = F. Demanding that F()  0, we find that the slope y_x = utF/(t)^1/4 grows in time without bound, thus failing to meet the criterion on which the linearization of Eq. 9 was predicated: |y_x|  1. We therefore turn instead to the case of the finite elastica, clamped at one end and free at the other, and subject to impulsive hydrodynamic drag. Our analysis is applicable to experiments in which either the coverslip is moved or, as a special case, in which the elastica is allowed to relax from some initial condition in the absence of flow.

To make the mathematics as transparent as possible, we first nondimensionalize the equation of motion (Eq. 10). Distances in x are rescaled by the total length L, time by the elastohydrodynamic time scale L⁴/A, and distances in y by the (constant) velocity u times this time scale:

(11)

The governing equation, y_t  u = y_xxxx, then becomes

(12)

The homogeneous equation is g = g. Well versed in the litany of Fourier transforms, we first left-multiply by an as yet arbitrary function _k() (where k indicates a parameter rather than a derivative) and integrate over the domain of ,

(13)

Integration by parts of the fourth-order derivative introduces eight separate surface terms. The boundary conditions implied by the functional derivative dictate the vanishing of the second and third derivatives at the free end (x = L). Requiring g to satisfy these conditions eliminates two of the eight terms.

The left end of the polymer is clamped at the origin, so y(x = 0) = y_x(x = 0) = 0. Demanding this behavior of g eliminates two additional surface terms. We now choose _k to satisfy the same boundary conditions as y, g, and h: _k(0) = _k(0) = ²_k(1) = ³_k(1) = 0. This annihilates the remaining four surface terms. Finally, we choose _k to obey

(14)

Defining g_{k  0}¹ d_kg, the equation of motion becomes g_k = k⁴g_k, the solution to which is

(15)

If we wish to describe the dynamics in such terms, we must construct the _k, which necessitates that we identify the allowed values of k.

A moment's thought reveals that the _k cannot simply be constructed out of the familiar sin's and cos's of Fourier space, which are incompatible with boundary conditions in which successive derivatives vanish. A countably infinite family of such _k can, however, be constructed by including hyperbolic trigonometric functions as well in the basis of the function space. The general solution of Eq. 14 is (Landau and Lifshitz, 1986)

(16)

The expression has four unknowns, as a solution to a fourth-order problem must. Inserting the four boundary conditions leads to a solvability condition for k:

(17)

This transcendental equation has an infinite number of solutions. For large values of k, as 1/cosh k  0, the solutions approach the solutions of the Fourier-like solvability condition cos k = 0, i.e., k_n+1  /2 + n. The first few solutions are

(18)

The first three normalized eigenfunctions are shown in Fig. 4.

View larger version (21K): [in this window] [in a new window]   FIGURE 4   The first three eigenfunctions for EHD problem I. The dotted line indicates the normalized third-order polynomial describing an elastica bent by a point force at the right end. Note the surprising overlap with k1, as exploited in the text (The Simple EHDI Experiment).

Note that had we chosen other boundary conditions, a different solvability condition and eigenfamily would have resulted (cf. Appendix B). For example, in the case of the elastica with free ends, we employ an expansion of y with the basis functions of Eq. B2.

With appropriate boundary conditions, the operator ⁴ can be proved to be self-adjoint, and thus the eigenfunctions constitute a complete basis in function space onto which we may project initial data and relate to later-time solutions via Eq. 15 in the standard Green's function way:

(19)

where the Green's function is

(20)

This is the exact solution of the linearized homogeneous equation. Note that the compact support and the boundary conditions break translation invariance, reflected in the fact that  cannot be expressed as (  '; ).

We note from the solution (Eq. 20) that each mode g_k decays independently and exponentially with time. This is to be compared with diffusive problems, in which each mode decays exponentially in time, except for the zero (average) mode, which is constant. In this experiment, the boundary conditions are incompatible with the existence of a zero mode. The system "hyperdiffuses" to homogeneity.

Because g(, ) decays to zero, we have h(, )  () as   , where

(21)

Returning to the clamped polymer in the presence of some background flow, we project the definitional statement h(, ) = g(, ) + () onto the _k():

(22)

which implies the initial condition g_k(0) = h_k(0)  _k. Recalling the simple time dependence of the modes g_k from Eq. 15, we see

(23)

The dynamic thus mimics that of a capacitor, charging up with the final shape state and draining of the initial shape state, each mode governed independently by decay rate k⁴. In the experiment considered, the initial condition is a flat polymer: h(,  = 0) = 0. Because is the solution to = 1, with boundary conditions (0) = (0) = (1) = (1) = 0, we find upon integrating by parts that _k = k⁴_k, where _{k  0}¹ d_k, _k   d_k, and thus

(24)

Evaluating the first few integrals, we find for _k, _k1  6 × 10², _k2  9 × 10⁴, _k3  7 × 10⁵, _k4  1 × 10⁵. ... Each mode with k > k₁ decays exponentially faster in time than the lowest mode, which thus dominates as   , so

(25)

Our picture of the impulsive dynamic of elastica in viscous flow is thus as follows: we project onto a special function space in which the long-time solution and the difference between initial data and the long-time solution exponentially charge and decay, respectively, each mode behaving independently. We are left with only the long-time solution as the asymptotic limit   .

	ELASTOHYDRODYNAMIC PROBLEM II

In Stokes II, the driving force is exerted by a wall oscillating with velocity u = Uê_y cos (t), or position y = y₀ cos (t). To solve the steady-state limit of SII, we postulate u(x, t) = U(e^itG()), where  = x(/)^1/2 and (z) indicates the real part of z. Inserting into Eq. 1, we then see that G satisfies

(26)

for which the solution vanishing as    is G = e. We then find that u(x, t) = Ue^/ cos(t  /), or, in a form useful for comparison to the elastohydrodynamic case,

(27)

where C = cos(/4) and S = sin(/4). This solution describes right-moving waves of velocity _S/C, decaying as x   with decay length _S/S.

We now consider a polymer held by an optical trap that moves with position y(x = 0) = y₀ cos (t). Because we have shown in the previous section that all modes satisfying the homogeneous equation of motion with homogeneous boundary conditions decay exponentially, we must only find a solution in the presence of inhomogeneity (here, the driving) to find the long-time limit of the dynamic.

To verify the validity of our analysis as well as the plausibility of EHDII as a method for measuring biopolymer rigidity, we conducted the experiment (Riveline et al., 1997) and analyzed image data as described below. A scaling relation predicted by the analysis was confirmed, and a new method for measurement of the persistence length of actin was demonstrated.

The experimental setup is shown in Fig. 5: F-actin is bound to a latex bead, which is optically trapped. As the position of the bead oscillates sinusoidally in time, the filament wiggles back and forth, propagating waves of displacement down its length. The motion relative to the fluid is opposed by the fluid viscosity, and the "wiggles" are opposed by the elasticity of the polymer.

View larger version (19K): [in this window] [in a new window]   FIGURE 5   Experimental setup.

The elastic constant A has units of energy × length, and the viscous force per unit length per unit velocity has the dimensions of a viscosity or action density µ:

(28)

Thus the natural length obtained from A, , and the frequency of oscillation  is

(29)

Nota bene that () is not a mere rescaling of the persistence length.

With a previously published persistence length for actin of L_p  15 µm (Ott et al., 1993), a viscosity µ = 0.01 cp, k_BT  4 × 10¹⁴ erg at T = 300 K, and measuring  in units of s¹, we obtain

(30)

Thus for frequencies on the order of 1 Hz, we obtain length scales on the order of microns, somewhat below the persistence length. This range of frequencies seems quite advantageous for experiment.

This elastohydrodynamic length () is precisely the length found upon nondimensionalizing the equation of motion (Eq. 2). By analogy to SII, we define the dimensionless coordinate  = x/()^1/4 = x/() = x(A/)^1/4 and rewrite the solution as

(31)

and Eq. 2 as

(32)

The solutions of Eq. 32 are of the form h() = ce, where  may be any one of the four distinct (complex) numbers such that ⁴ = i. These are _j = i^j exp(i/8), j = 1 ... 4. The general solution is the sum of these four solutions,

(33)

where z₀  e^i/8  0.92  0.38i. The unpleasant (but certainly not subtle) remainder of the problem is to solve for the four c_j's, given some four boundary conditions. At the left (x = 0) end, we enforce the position and the condition of torquelessness (as appropriate for an optical trap): y_xx(0) = 0. The right end must satisfy the free end boundary conditions (Eq. 6). The c_j derived from these conditions are functions of a rescaled polymer length   L/() and may properly be written as c_j().

Semiinfinite polymer

The exact solution for h() is presented in Appendix C; it simplifies greatly, however, for extreme values of L/(). For this reason we include a discussion of the polymer of infinite extent. In this limit, the two coefficients c_j for which _j has a nonnegative real part must be zero, allowing only decaying solutions as x  .

The solution consistent with the two left-end boundary conditions is

(34)

where  = cos(/8) and  = sin(/8). Compare with the solution to SII (Eq. 27). The semiinfinite solution (Eq. 34) is shown at the bottom of Fig. 6 for t = n2/6, n = 1 ... 6. In the hydrodynamic case, the solution of Eq. 27 describes exponentially decaying right-moving traveling waves of transverse velocity. In the elastohydrodynamic case, the higher order derivative allows more complicated behavior: right- and left-moving waves of displacement, with different decay rates and velocities. In this case, the right-movers have a slower decay (because   0.38 < 0.92  ), and might be expected in some sense to dominate over the left-movers. This mechanism will be elaborated on below (under Propulsive Force).

Finite polymer

In the limit of a short or stiff polymer,   1, we rewrite  = ,  = x/L  (0, 1) and expand, yielding

(35)

Equivalently, we may derive this polynomial by truncating a series expansion for h in  and enforcing the equation of motion (Eq. 32) and the boundary conditions (Eq. 6). Using Eq. 35, all four boundary conditions are satisfied exactly, whereas Eq. 32 is solved to order (⁴).

The exact solution is shown in Fig. 6 for  = 1, 2, 4, and  and t = n2/6, n = 1 ... 6. Note the existence of a pivot point at x = 2L/3 as   0. This behavior is described by the (⁰) term in Eq. 35: as   0, the polymer acts as a rigid rod. As a consequence, it is impossible to tell if a movie of such a polymer is being played forward or backward. Indeed, this is a filamentous version of the famous "one-armed swimmer" or "scallop" example, illustrating the lack of net propulsion for rigid objects executing time-reversible motions in low Reynolds number flow (Purcell, 1977; Childress, 1981).

View larger version (25K): [in this window] [in a new window]   FIGURE 6   Solutions to EHD problem II for filaments of various rescaled lengths .

Propulsive force

Problem II and its associated experiment are sufficiently reminiscent of flagellar hydrodynamics to motivate a calculation of the propulsive force F generated in the x direction by the wiggling. This can be done by integrating f, the force exerted by the polymer on the fluid, along the length of the filament. We then contract this instantaneous total force with ê_x and average over one period. This force is equal to and opposite the propulsive force exerted by the fluid on the polymer.

Noting that the force per unit length in Eq. 4 is a total derivative,

(36)

and recalling the boundary conditions imposed on  and _s, we have

(37)

This is geometrically exact. We now wish to calculate the time average over one period. Within the linearized solution, _s sin   y_xxxy_x. Recalling the expression for y in Eq. 31, we obtain

(38)

where () is the characteristic length and  is a scaling function conveniently normalized (see below).

The exact solution to EHDII given in Appendix C can be used to calculate the function  for all values of the polymer length. The asymptotic behavior as    is

(39)

When the length is short compared to the characteristic length, the polymer flexes very little, so

(40)

As Fig. 7 illustrates, the short-length approximation (Eq. 40) shows good agreement with the exact solution for   3, as does the large- approximation (Eq. 39) for   3. The approach to the asymptotic limit is oscillatory, with a maximum near  4, the value at which {h} acquires its first root, and a local minimum near   6, the value at which {h} acquires its second root. The unexpected local maximum indicates that there is an optimal combination of A, , and a finite L.

View larger version (16K): [in this window] [in a new window]   FIGURE 7   Scaling function  for propulsive force. The large  expansion is plotted for  > 2, and the small- solution is plotted for  < 3.5.

Inserting typical numbers from the experiment (in cgs), we find that F()  2 × 10⁹ dynes = 3 × 10² pN. For a trap stiffness of ~0.02 pN/nm, this would induce a displacement of 1.5 nm, at the lower limit of experimental observation. The production and measurement of propulsive force by an artificial flagellum were attempted by G. I. Taylor (Taylor, 1952), using a glycerine-filled tub to mimic the low Reynolds numbers found in vivo. Taylor struggled to drive the flagellum without inducing unwanted torque or disturbing the flow, a difficulty obviated by the use of optical traps.

Returning to the asymptotic expressions for h derived in the previous two sections, we observe a pleasant accordance with the qualitative features of Fig. 7. In the semiinfinite case, we noted the presence of right- and left-movers, with right-moving waves of displacement exhibiting slower decay. Such a dominance accounts for the nonzero propulsive force in the    limit, where a net propulsion to the left survives. In the   0 case, we recovered a shape that approaches a pivoting rigid rod, not unlike a one-armed swimmer. As we expect from life at low Reynolds number (Purcell, 1977; Childress, 1981), such a motion, invariant under t  t, can produce no net propulsion.

As a further illustration of the relationship between low-Reynolds-number swimming and cyclic motions, we observe that the lowest-order expression for the time-averaged force is equal to

(41)

or, noting that ₀^L dx y(x, t) is simply the area (t) under the curve y(x, t), and that the slope at the left is to first order simply the tangent angle ₀,

(42)

This result can be interpreted quite simply: the propulsive force results from pushing aside some volume (or in two dimensions, an area) of fluid, projected in the direction of propulsion ê_x an amount proportional to ₀. Note that had we been interested in the propulsion in the transverse (ê_y) direction, the ₀ would not appear, leaving the absence of net forcing: F   d = 0, as we would expect.

The net force, then, is the area enclosed by a trajectory in   ₀ space during some cyclic motion. This representation is independent of the particular motion exhibited, although we have here considered simple periodic motion, for which the trajectory is always an ellipse. As   0, the elliptical trajectory thins to a straight line, encloses no area, and thus produces no force.

This representation makes clear that in an inertialess world, net motion is principally geometric in origin rather than dynamic (Shapere and Wilczek, 1987). In a manner analogous to the importance of path rather than kinetics in generating net work in a Carnot diagram, we see that we can remove time entirely from the expression and consider instead a path in a low-dimensional projection of t