## Abstract

Forces generated by protein polymerization are important for various forms of cellular motility. Assembling microtubules, for instance, are believed to exert pushing forces on chromosomes during mitosis. The force that a single microtubule can generate was measured by attaching microtubules to a substrate at one end and causing them to push against a microfabricated rigid barrier at the other end. The subsequent buckling of the microtubules was analyzed to determine both the force on each microtubule end and the growth velocity. The growth velocity decreased from 1.2 micrometers per minute at zero force to 0.2 micrometer per minute at forces of 3 to 4 piconewtons. The force-velocity relation fits well to a decaying exponential, in agreement with theoretical models, but the rate of decay is faster than predicted.

It has long been speculated that the assembly and disassembly of cytoskeletal filaments, such as microtubules (MTs) and actin, can generate forces that are important for various forms of cellular motility. Examples include the motions of chromosomes during mitosis that depend on both the assembly and disassembly of MTs (1, 2), actin-dependent motility such as cell crawling and the propulsion of *Listeria* through a host cell (3), and possibly the MT-dependent transport of intracellular membranes (4). To understand the role of force production by protein polymerization in vivo, it is important to determine the maximum forces that can be generated and the effect of an opposing force on the assembly dynamics of a protein polymer. In the case of MTs, there is clear experimental evidence that both their assembly (4-6) and disassembly (7) can generate force, but limited quantitative data are available on the actual magnitude of these forces. In this respect, the study of force production by the assembly of cytoskeletal filaments, or by protein aggregation in general, clearly lags behind the study of force production by motor proteins, for which a number of quantitative in vitro assays have been developed (8).

We created an experimental system in which growing MTs were made to push against an immobile barrier, and analyzed the subsequent buckling of the MTs to study the forces that were produced; the force calibration was provided by a measurement of the flexural rigidity of the MTs (9). We etched arrays of long channels (30 μm wide, 1 μm deep) in glass cover slips (10); the walls of these channels were used as barriers. Using materials with different etch rates, we produced walls with an “overhang” that prevented the MTs from sliding upward along the wall (Fig.1, A and B). Short stabilized MT seeds, labeled with biotin, were attached to the bottom of the streptavidin-coated channels, and MTs were allowed to grow from these seeds (Fig. 1A) (11). Because the seeds were randomly positioned in the channels, the MTs approached the walls from different angles and distances. We scanned our samples for MTs that were growing roughly perpendicular to the walls and observed them as their growing ends approached the walls (Fig. 1, C and D) (12).

In many cases, the MT end was caught underneath the overhang on the wall, forcing the MT to encounter the wall. After encountering the wall, most MTs continued to increase in length, indicating a continuing addition of tubulin dimers at the growing MT ends. The virtually incompressible (9) MTs were observed to bend in two different ways to accommodate this continuing increase in length. In some cases, the MT end moved along the side of the wall while the MT bent roughly perpendicular to its original direction [these MTs were not followed any further (13)]. In other cases, the MT end, probably hindered by small irregularities in the shape of the wall, did not move along the side of the wall; this caused the MT to buckle with its end pivoting around a fixed contact point with the wall (Fig. 1, C and D). The force exerted by these MTs on the wall was large enough to overcome the critical buckling force (14).

After the initiation of buckling, both the magnitude and the direction of the force *f* exerted by each MT on the wall (and therefore by the wall on the MT) were solely determined by the elastic restoring force of the buckled MT [initially this force should be roughly equal to the critical buckling force (14)]. A considerable component *f*
_{p} of this force was directed parallel to the direction of elongation of the MT, thereby opposing its growth (Fig. 2). Assuming that a MT behaves as a homogeneous elastic rod, the magnitude of the critical buckling force*f*
_{c} normalized by the flexural rigidity κ of the MT is given by *f*
_{c}/κ =*A*/*L*
^{2}, where *L* is the length of the MT. The prefactor *A* depends on the quality of the clamp provided by the seed: *A* ≈ 20.19 (the maximum value) for a perfect clamp that fixes the initial direction of the MT exactly in the direction of the contact point with the wall,*A* = π^{2} (the minimum value) for a seed that acts as a hinge around which the MT is completely free to pivot. Because there was no reason to assume that either of these conditions would be perfectly met, we expected buckling forces somewhere between these minimum and maximum values.

To determine the actual force acting on each buckling MT, we obtained a sequence of fits to the shape of an elastic rod from video frames spaced 2 s apart (Fig. 2) (15). When no assumptions were made about the quality of the clamp or the magnitude of*f*
_{c}, these fits produced values for*f*/κ , *f*
_{p}/κ , and *L* as a function of time. Fig. 3A shows the parallel component of the normalized force and the MT length as a function of time for five different examples, both before and after reaching the wall. The MT length before reaching the wall was determined by tracking the end of the growing MT (15) (in each case a segment in the time sequence is missing, during which the end of the MT was obscured by the presence of the overhang on the wall). The growth velocity varied considerably even at zero force, as reported previously (16, 17). However, all MTs clearly slowed as soon as a force was applied. These curves also show that the forces on short buckling MTs tend to be greater than the forces on long buckling MTs. The total normalized force (not just the parallel component) as a function of MT length is shown in Fig. 3B for all MT shapes analyzed. For each MT length, the forces vary over a certain range because of variability in the quality of the clamp provided by the seed. The two dotted lines indicate the theoretical limits for*f*
_{c} (discussed above); as expected, we found that the restoring forces were between these limits, which validated our assumption that MTs behave as homogeneous elastic rods.

In Fig. 4 the average growth velocity 〈*v*〉 is plotted as a function of force (the force-velocity curve) for all data combined (18). This plot shows that the growth velocity approaches the velocity of a freely growing MT (∼1.2 μm min^{–}
^{1}) at low force, and decreases to ∼0.2 μm min^{–}
^{1} as more and more force is applied. This implies that the reduction in growth velocity is controlled by the applied force and is not simply caused by the proximity of the end of the MT to the glass barrier. The lower*x* axis in Fig. 4 is labeled with values for the normalized force because this is the parameter obtained from our fits. An independent measure of κ is needed to obtain values for the absolute force. The flexural rigidity of pure MTs has been measured using various methods; the values reported range over an order of magnitude, 4 to 40 pN·μm^{2} (6, 19, 20). We used an analysis of the thermal fluctuations to measure the rigidity of our MTs (21) and found values at the upper end of this range: 34 ± 7 pN·μm^{2}. This means that the largest forces in Fig. 4 are on the order of 4 pN (the upper *x* axis is labeled with absolute values of force derived from our measurement of the flexural rigidity).

The force-velocity relation in Fig. 4 can be compared with theoretical predictions. In the absence of force, the growth velocity is given by the difference in the rate of addition and removal of subunits,*v* = δ(α*c* – β), where δ is the added MT length per dimer (δ = δ/13 nm for an MT with 13 protofilaments), α*c* is the rate of subunit addition (the on-rate),*c* is the tubulin concentration, and β is the rate of subunit removal (the off-rate). In principle, both α and β may be affected by a force that opposes elongation of the MT (*f*
_{p} in our case). Thermodynamic arguments (22) show that their ratio (which gives the critical tubulin concentration *c*
_{cr}) must increase with force according to (1)where *k*
_{B} is the Boltzmann constant and*T* is temperature. This leads to
(2)where *q* may take any value between 0 and 1 (possibly in a force-dependent way). The stall force*f*
_{s} (the force at which the velocity becomes equal to zero) is independent of *q* and is given by(3)A similar result is obtained if the growth process is pictured as a “Brownian ratchet” (23). In this more mechanistic view, the on-rate depends on the force-dependent probability that thermal fluctuations (in the position of the MT end in this case) allow for a gap between the MT end and the barrier that is large enough for a dimer to attach to the growing MT end (under optimal conditions, the size of this gap along the direction of MT growth is equal to δ, the added length per dimer). If the force is independent of the size of the gap and the time required to add a dimer is long relative to the time required for the MT end to diffuse over a distance δ, then (4)(23, 24). This relation assumes that the effect of force on the off-rate can be neglected. We performed a weighted least-squares fit of the data in Fig. 4 to both the function *v*(*f*
_{p}) =*A* – *B*exp(*Cf*
_{p}/κ) (assuming that only the off-rate is affected or *q =* 0) and the function*v*(*f*
_{p}) = *A*exp(–*Cf*
_{p}/κ) – *B* (assuming that only the on-rate is affected or *q =* 1), where*A*, *B*, and *C* are fitting parameters. In the first case, the best fit (χ^{2} = 1.5) produced extremely large values for the parameters *A* and*B*, and a value for *C* nearly equal to zero (corresponding to almost a straight line). Experimental results show, however, that *B* is very small in the absence of force (25). Fixing the maximum value of *B* at 0.5 μm min^{–}
^{1} produced a fit that was much worse (χ^{2} = 2.5), and smaller values of *B* produced fits that were even worse. Consequently, it is unlikely that the only effect of force is an increase in the off-rate. In the second case, a more reasonable result (indicated by the solid line in Fig. 4) was obtained: χ^{2} = 0.43 with *A* = 1.13 ± 0.11 μm min^{–}
^{1},*B* = –0.08 ± 0.12 μm min^{–}
^{1}, and *C* = 18 ± 4 μm^{2}. This indicates the possibility that the only effect of force is a decrease in the on-rate (26). Although*B* is expected to be small, its true value should be greater than zero. Because of the uncertainty in *B*, it is impossible to extract a good estimate of *f*
_{s} from this fit.

The value predicted for the parameter *C* is equal to κδ */k*
_{B}
*T*, which, given our measured value for κ, corresponds to 5 ± 1 μm^{2}. This is smaller than the value obtained from the fit for *q =*1, 18 ± 4 μm^{2}, which implies that the growth velocity decreases faster with force than would be expected from theoretical arguments [this discrepancy becomes even larger if we assume a smaller value for *q* (26)]. The theoretical rate corresponds to an optimum situation in which the free energy available from the assembly of all 13 protofilaments is converted into mechanical work. Despite the relatively large experimental error bars, our data indicate that this is not the case under our conditions. It may be that, if the end of a growing MT is not blunt but pointed, only a few protofilaments are supporting the load. If this is pictured as a ratchet, gaps closer to the size of a full dimer may be required to squeeze in the next subunit, which would increase the predicted value for *C*. Also, growth may occur through the closure of a sheet of protofilaments (17), which could make the gap size needed for this process even larger than the size of a dimer.

We have presented a quantitative method for studying the force that can be produced by a single growing MT in interaction with a nonspecific glass barrier. Considering that under these conditions less force is produced than is theoretically possible, a logical next step would be to study whether the interaction of the growing MT end with a specific attachment site modifies this result. In principle it should be possible to coat the walls [or simply a pattern of lines (27)] with isolated chromosomes (7) or kinetochore constructs (28) and repeat the same experiment. This system can also be used to study the effect of force on the catastrophe frequency of MTs (the probability of switching from the growing to the shrinking state). In our experiments, growth often persisted after the initiation of buckling, which implies that an opposing force does not markedly increase the catastrophe frequency. Quantitatively verifying this possibility would require the observation of many catastrophe events both before and during the application of force.