Quantitative Analysis of Forces in Cells...Analysis of MT buckling shapes (15). Open squares show...

Quantitative Analysis of Forces in Cells

• Polymerization forces

• Forces from whole cells and cell aggregates

• Osmotic forces

Anders CarlssonWashington University in St Louis

Basic properties of forces in cells

Measurement methods and magnitudes of particular types of forces:

Fundamental Cell Mechanobiology Fact: Net Forces Are Very Small in Comparison to Individual Force Contributions

Moving/accelerating cell: Fnet=ma=(2000 µm3)(1gm/cm3)(0.01µm/s2)= 2x10-20 N ≃ 0

Fnet = 0 = ∑Fstatic + Fdrag

Fdrag = (6𝜋ηR)v = 6𝜋×(0.01Pa-s)×(20µm)×(0.1µm/s)

= 2x10-13 N ≃ 0

So ∑Fstatic = 0: Force balance

Also, Fgrav = mg = (4𝜋R3/3)(ρcell - ρwater)g

=[4𝜋(20µm)3/3](0.05gm/cm3)(10m2/s) = 2x10-13 N ≃ 0

Force BalanceWe always speak of forces between different entities, like tension or stress:

TT

Tension (or compression)

Shear stress andtensile stress

Pressure is an isotropic compressive stress

Force Balance in cells

(Baum lab Dev Bio 2015)

Basic Units Facts

kT = 4.1 pN-nm

1 Mpa = 1 pN/nm2= 10 atm = 147 psi

Biological Force Measurements Require a Reference Mechanical Scale, Like Temperature or

Stiffness of a Filament/Rod

Persistence length Lp quantifies the stiffness of a biopolymer filament. It is (roughly) the distance it takes a filament to bend 90 degrees by thermal fluctuations

Filament Type

DNAActinMicrotubule

Lp

75 nm10 µm2 mm

(Dogic group Nat Matls 2015)

Thermal fluctuations of two bundled actin filaments

Biological Force Measurements Require a Reference Mechanical Scale, Like Temperature or

Stiffness of a Filament/Rod

Persistence length Lp quantifies the stiffness of a biopolymer filament. It is (roughly) the distance it takes a filament to bend 90 degrees by thermal fluctuations

Filament Type

DNAActinMicrotubuleSpaghetti

Lp

75 nm10 µm2 mm100 light years

(Dogic group Nat Matls 2015)

Thermal fluctuations of two bundled actin filaments

Polymerization Forces - the Brownian Ratchet

8(AEC PRE 2000)

factor of 1/!2. Thus the thermal-ratchet prediction !13" forthis case is v#exp(!Fa/!2kT). It is seen that the correc-tions to the exponential dependence are much smaller for

oblique incidence than for $ i"0. The largest discrepanciesare about 30% in the range Fa/kT"1 to 1.5. The decay rateat the highest force values actually seems to be slower than

that in the exponential curve. We believe that this is caused

by two factors. The first is that, as will be discussed below,

there is a non-negligible probability of monomer addition

even below the critical height for the obstacle. The second is

bending of the filament as a result of the applied forces. For

example, at Fa/kT"2.5, we find that the tip of the fiber atthe end of the growth process %when it is 36 monomers long&is bent about 15° relative to the 45° angle at the base. The

bending is proportional to the square of the fiber length, and

the observed velocities correspond to a weighted average of

the bending between long and short fibers. The filament

bending is expected to cause the growth rate to increase since

the projection of the applied force on the growth axis be-

comes smaller, and also diffusion to the tip becomes less

restricted. %We are plotting the monomer addition rate, notthe rate of growth in the z direction.& We have in fact ob-served that the growth rates become larger for longer fibers.

This may be partly due to such bending effects, and partly

due to the fluctuations of the fiber tip. The latter may be

estimated in terms of the effective elastic modulus of the

fiber tip. As defined by Mogilner and Oster !15", the modu-lus is '"4(kT/l3 sin2 $i , where ( is the persistence length

and l is the length of the fiber. The rms vertical fluctuation of

the fiber tip is then !kT/' . With the persistence length of

FIG. 4. Growth rates %solid circles& vs the total force F. %a& Hard force field, perpendicular incidence. %b& Soft force field, perpendicularincidence. %c& Hard force field, 45° incidence. Rates are given in units of )kTc , where ) is the monomer mobility and c is the concentration.

Force is given in units of a/kT . The solid line corresponds to exponential decay !cf. Eq. %1&". Diamonds in %a& correspond to the mobilityenhanced by a factor of 2 %left& and reduced by a factor of 10 %right&. Dashed curves correspond to the theory of Eq. %12&.

7086 PRE 62A. E. CARLSSON

Growth velocity is predicted to slow exponentially with opposing force

Membrane

9Kovar and Pollard PNAS 2004

So the growing filament could exert a force of at least 0.8 pN

Green end is attached to “formin”, red end is anchored by inactive myosin

Initial straight length is about 0.7 µm

Measuring Polymerization Force of a Single Actin Filament

Buckling force is 𝜋2kTLp/4L2=0.8 pN

a protein polymer. In the case of MTs, thereis clear experimental evidence that boththeir assembly (4–6) and disassembly (7)can generate force, but limited quantitativedata are available on the actual magnitudeof these forces. In this respect, the study offorce production by the assembly of cy-toskeletal filaments, or by protein aggrega-tion in general, clearly lags behind the studyof force production by motor proteins, forwhich a number of quantitative in vitroassays have been developed (8).

We created an experimental system inwhich growing MTs were made to pushagainst an immobile barrier, and analyzedthe subsequent buckling of the MTs tostudy the forces that were produced; theforce calibration was provided by a mea-surement of the flexural rigidity of the MTs(9). We etched arrays of long channels (30�m wide, 1 �m deep) in glass cover slips(10); the walls of these channels were usedas barriers. Using materials with differentetch rates, we produced walls with an “over-hang” that prevented the MTs from slidingupward along the wall (Fig. 1, A and B).Short stabilized MT seeds, labeled with bio-tin, were attached to the bottom of thestreptavidin-coated channels, and MTs wereallowed to grow from these seeds (Fig. 1A)(11). Because the seeds were randomly posi-tioned in the channels, the MTs approachedthe walls from different angles and distances.We scanned our samples for MTs that weregrowing roughly perpendicular to the wallsand observed them as their growing endsapproached the walls (Fig. 1, C and D) (12).

In many cases, the MT end was caughtunderneath the overhang on the wall, forc-ing the MT to encounter the wall. Afterencountering the wall, most MTs continuedto increase in length, indicating a continu-ing addition of tubulin dimers at the grow-ing MT ends. The virtually incompressible(9) MTs were observed to bend in twodifferent ways to accommodate this con-tinuing increase in length. In some cases,the MT end moved along the side of thewall while the MT bent roughly perpendic-ular to its original direction [these MTswere not followed any further (13)]. In oth-er cases, the MT end, probably hindered bysmall irregularities in the shape of the wall,did not move along the side of the wall; thiscaused the MT to buckle with its end piv-oting around a fixed contact point with thewall (Fig. 1, C and D). The force exerted bythese MTs on the wall was large enough toovercome the critical buckling force (14).

After the initiation of buckling, both themagnitude and the direction of the force fexerted by each MT on the wall (and there-fore by the wall on the MT) were solelydetermined by the elastic restoring force ofthe buckled MT [initially this force should

be roughly equal to the critical bucklingforce (14)]. A considerable component fp ofthis force was directed parallel to the direc-tion of elongation of the MT, thereby op-posing its growth (Fig. 2). Assuming that aMT behaves as a homogeneous elastic rod,the magnitude of the critical buckling forcefc normalized by the flexural rigidity ⇧ ofthe MT is given by fc/⇧ ⇤ A/L2, where L isthe length of the MT. The prefactor Adepends on the quality of the clamp provid-ed by the seed: A ⇥ 20.19 (the maximumvalue) for a perfect clamp that fixes theinitial direction of the MT exactly in thedirection of the contact point with the wall,

A ⇤ ⌅2 (the minimum value) for a seedthat acts as a hinge around which the MT iscompletely free to pivot. Because there wasno reason to assume that either of theseconditions would be perfectly met, we ex-pected buckling forces somewhere betweenthese minimum and maximum values.

To determine the actual force acting oneach buckling MT, we obtained a sequenceof fits to the shape of an elastic rod fromvideo frames spaced 2 s apart (Fig. 2) (15).When no assumptions were made about thequality of the clamp or the magnitude of fc,these fits produced values for f/⇧ , fp/⇧ , andL as a function of time. Fig. 3A shows theparallel component of the normalized forceand the MT length as a function of time forfive different examples, both before andafter reaching the wall. The MT lengthbefore reaching the wall was determined bytracking the end of the growing MT (15)

Fig. 1. In vitro assay to study the force exerted bya single growing MT. (A) Schematic representa-tion of the experiment (shown in perspective froma side view; not to scale). A biotinylated MT seed(black), attached to the streptavidin-coated bot-tom of a channel (indicated by black dots), tem-plates the growth of a freely suspended MT (gray).An overhang was created on the walls of thechannel to prevent the MT ends from sliding up-ward after encountering the wall. (B) Electron mi-croscopy image showing a wall with overhang(scale bar, 1 �m). (C and D) DIC images of twobuckling MTs (top view) (12). The upper panelseach show a MT [arrowhead in top left of (C)] grow-ing from a randomly positioned seed. The lowerpanels are snapshots (separated by 1 min) of eachMT after the growing end has encountered thewall.Becauseof thecontrastproducedby theover-hang on the walls (which vary in size between sam-ples), the last few micrometers of the MTs cannotbe seen. The sharp changes in contrast indicatethe actual locations of the walls. Scale bar, 10 �m.

Fig. 2. Analysis of MT buckling shapes (15). Opensquares show the hand-recorded shapes of theMT shown in Fig. 1D at 12-s intervals (shapes wereanalyzed at 2-s intervals). The dashed line on theleft indicates the position of the seed (xL). Thedashed line on the right indicates the position of thewall (xW) as judged by eye from the images (Fig.1D). The solid lines show fits to the shape of anelastic rod. One (at the top) is shown as an exam-ple. We assumed that the MT was held at its seedand that a force f was applied at the contact pointof the MT with the wall (x0,y0). This contact pointremained fixed in time and was chosen to producethe best combined fit over the entire time sequence(this produces a value of x0 very close to xW). Wefurther assumed that the MT was free to pivotaround the contact point, but we made no as-sumptions about the quality of the clamp providedby the seed. The fits produced the magnitude andthe direction of the force f (normalized by the flex-ural rigidity ⇧ of the MT ) at each time point, as wellas the length of the MT given by the arc lengthbetween x0 and xL. MT growth is opposed by fp,the component of the force that is directed parallelto the axis of the MT. Scale bar, 5 �m.

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(in each case a segment in the time se-quence is missing, during which the end ofthe MT was obscured by the presence of theoverhang on the wall). The growth velocityvaried considerably even at zero force, asreported previously (16, 17). However, allMTs clearly slowed as soon as a force wasapplied. These curves also show that theforces on short buckling MTs tend to begreater than the forces on long bucklingMTs. The total normalized force (not justthe parallel component) as a function ofMT length is shown in Fig. 3B for all MTshapes analyzed. For each MT length, theforces vary over a certain range because ofvariability in the quality of the clamp pro-vided by the seed. The two dotted linesindicate the theoretical limits for fc (dis-cussed above); as expected, we found thatthe restoring forces were between these lim-its, which validated our assumption thatMTs 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 velocityapproaches the velocity of a freely growingMT (�1.2 ⇧m min–1) at low force, anddecreases to �0.2 ⇧m min–1 as more andmore force is applied. This implies that thereduction in growth velocity is controlledby the applied force and is not simplycaused by the proximity of the end of theMT to the glass barrier. The lower x axis inFig. 4 is labeled with values for the normal-

ized force because this is the parameterobtained from our fits. An independentmeasure of � is needed to obtain values forthe absolute force. The flexural rigidity ofpure MTs has been measured using variousmethods; the values reported range over anorder of magnitude, 4 to 40 pN�⇧m2 (6, 19,20). We used an analysis of the thermalfluctuations to measure the rigidity of ourMTs (21) and found values at the upper endof this range: 34 ⇥ 7 pN�⇧m2. This meansthat the largest forces in Fig. 4 are on theorder of 4 pN (the upper x axis is labeledwith absolute values of force derived fromour measurement of the flexural rigidity).

The force-velocity relation in Fig. 4 canbe compared with theoretical predictions.In the absence of force, the growth velocityis given by the difference in the rate ofaddition and removal of subunits, v ⌦ ⌅(⌃c– ), where ⌅ is the added MT length perdimer (⌅ ⌦ ⌅/13 nm for an MT with 13protofilaments), ⌃c is the rate of subunitaddition (the on-rate), c is the tubulin con-centration, and is the rate of subunitremoval (the off-rate). In principle, both ⌃and may be affected by a force that op-poses elongation of the MT ( fp in our case).Thermodynamic arguments (22) show thattheir ratio (which gives the critical tubulinconcentration ccr) must increase with forceaccording to

ccr(fp) ⌦ (fp)⌃(fp)

⌦ (0)⌃(0) exp(fp⌅/kBT) (1)

where kB is the Boltzmann constant and Tis temperature. This leads to

v(fp) ⌦ ⌅{⌃ exp(⇤qfp⌅/kBT)c

⇤ exp[(1 ⇤ q)fp⌅/kBT]} (2)

where q may take any value between 0 and1 (possibly in a force-dependent way). Thestall force fs (the force at which the velocitybecomes equal to zero) is independent of qand is given by

fs ⌦kBT⌅

ln⌃c

(3)

A similar result is obtained if the growthprocess is pictured as a “Brownian ratchet”(23). In this more mechanistic view, theon-rate depends on the force-dependentprobability that thermal fluctuations (in theposition of the MT end in this case) allowfor a gap between the MT end and thebarrier that is large enough for a dimer toattach to the growing MT end (under op-timal conditions, the size of this gap alongthe direction of MT growth is equal to ⌅,the added length per dimer). If the force isindependent of the size of the gap and thetime required to add a dimer is long relativeto the time required for the MT end todiffuse over a distance ⌅, then

v( fp) ⌦ ⌅[⌃ exp(⇤fp⌅/kBT)c ⇤ ] (4)

(23, 24). This relation assumes that theeffect of force on the off-rate can be neglect-ed. We performed a weighted least-squaresfit of the data in Fig. 4 to both the functionv( fp) ⌦ A – B exp(Cfp/�) (assuming thatonly the off-rate is affected or q⌦ 0) and thefunction v(fp) ⌦ A exp(–Cfp/�) – B (assum-ing that only the on-rate is affected or q ⌦1), where A, B, and C are fitting parame-ters. In the first case, the best fit (�2 ⌦ 1.5)

Fig. 3. MT length and applied force obtained from the analysis of MT buckling shapes such as shownin Fig. 2. (A) For five different MTs, the length L as a function of time (at 2-s intervals) is shown both beforeand after contact with the wall (solid symbols). A segment of time is missing in each case, during whichthe end of the growing MT was obscured by the presence of the overhang on the wall. Open symbolsshow the parallel component of the normalized force, fp/� . The lower left curve corresponds to the MTshown in Figs. 1D and 2. The upper right curve corresponds to the MT shown in Fig. 1C. (B) Totalnormalized force, f/�, as a function of MT length for all MT shapes analyzed (n ⌦ 1316). Each pointcorresponds to one MT shape. The dashed lines indicate the theoretical length dependence of fc in twolimiting cases: fc/� ⌦ 20.19/L2 for a MT with a seed that acts as a perfect clamp (upper curve) and fc/�⌦ ↵2/L2 for a MT with a seed that acts as a perfect hinge (lower curve). In the experiments, the seedsbehaved in an intermediate way and, as expected, the forces obtained from the fits fall between thesetwo limiting curves.

Fig. 4. Average MT growth velocity as a functionof force. Velocity and force were obtained fromcombining data such as shown in Fig. 3A (18). Thelower x axis gives the value of the normalizedforce, fp/�. The upper x axis gives the absolutevalue of the force, based on our measurement ofthe flexural rigidity. The solid line gives the best fitof the data to an exponential decay.

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(Dogterom and Yurke 1997)

Microtubule Polymerization Forces

V decays exponentially, but too rapidly

a protein polymer. In the case of MTs, thereis clear experimental evidence that boththeir assembly (4–6) and disassembly (7)can generate force, but limited quantitativedata are available on the actual magnitudeof these forces. In this respect, the study offorce production by the assembly of cy-toskeletal filaments, or by protein aggrega-tion in general, clearly lags behind the studyof force production by motor proteins, forwhich a number of quantitative in vitroassays have been developed (8).

We created an experimental system inwhich growing MTs were made to pushagainst an immobile barrier, and analyzedthe subsequent buckling of the MTs tostudy the forces that were produced; theforce calibration was provided by a mea-surement of the flexural rigidity of the MTs(9). We etched arrays of long channels (30�m wide, 1 �m deep) in glass cover slips(10); the walls of these channels were usedas barriers. Using materials with differentetch rates, we produced walls with an “over-hang” that prevented the MTs from slidingupward along the wall (Fig. 1, A and B).Short stabilized MT seeds, labeled with bio-tin, were attached to the bottom of thestreptavidin-coated channels, and MTs wereallowed to grow from these seeds (Fig. 1A)(11). Because the seeds were randomly posi-tioned in the channels, the MTs approachedthe walls from different angles and distances.We scanned our samples for MTs that weregrowing roughly perpendicular to the wallsand observed them as their growing endsapproached the walls (Fig. 1, C and D) (12).

In many cases, the MT end was caughtunderneath the overhang on the wall, forc-ing the MT to encounter the wall. Afterencountering the wall, most MTs continuedto increase in length, indicating a continu-ing addition of tubulin dimers at the grow-ing MT ends. The virtually incompressible(9) MTs were observed to bend in twodifferent ways to accommodate this con-tinuing increase in length. In some cases,the MT end moved along the side of thewall while the MT bent roughly perpendic-ular to its original direction [these MTswere not followed any further (13)]. In oth-er cases, the MT end, probably hindered bysmall irregularities in the shape of the wall,did not move along the side of the wall; thiscaused the MT to buckle with its end piv-oting around a fixed contact point with thewall (Fig. 1, C and D). The force exerted bythese MTs on the wall was large enough toovercome the critical buckling force (14).

After the initiation of buckling, both themagnitude and the direction of the force fexerted by each MT on the wall (and there-fore by the wall on the MT) were solelydetermined by the elastic restoring force ofthe buckled MT [initially this force should

be roughly equal to the critical bucklingforce (14)]. A considerable component fp ofthis force was directed parallel to the direc-tion of elongation of the MT, thereby op-posing its growth (Fig. 2). Assuming that aMT behaves as a homogeneous elastic rod,the magnitude of the critical buckling forcefc normalized by the flexural rigidity ⇧ ofthe MT is given by fc/⇧ ⇤ A/L2, where L isthe length of the MT. The prefactor Adepends on the quality of the clamp provid-ed by the seed: A ⇥ 20.19 (the maximumvalue) for a perfect clamp that fixes theinitial direction of the MT exactly in thedirection of the contact point with the wall,

A ⇤ ⌅2 (the minimum value) for a seedthat acts as a hinge around which the MT iscompletely free to pivot. Because there wasno reason to assume that either of theseconditions would be perfectly met, we ex-pected buckling forces somewhere betweenthese minimum and maximum values.

To determine the actual force acting oneach buckling MT, we obtained a sequenceof fits to the shape of an elastic rod fromvideo frames spaced 2 s apart (Fig. 2) (15).When no assumptions were made about thequality of the clamp or the magnitude of fc,these fits produced values for f/⇧ , fp/⇧ , andL as a function of time. Fig. 3A shows theparallel component of the normalized forceand the MT length as a function of time forfive different examples, both before andafter reaching the wall. The MT lengthbefore reaching the wall was determined bytracking the end of the growing MT (15)

Fig. 1. In vitro assay to study the force exerted bya single growing MT. (A) Schematic representa-tion of the experiment (shown in perspective froma side view; not to scale). A biotinylated MT seed(black), attached to the streptavidin-coated bot-tom of a channel (indicated by black dots), tem-plates the growth of a freely suspended MT (gray).An overhang was created on the walls of thechannel to prevent the MT ends from sliding up-ward after encountering the wall. (B) Electron mi-croscopy image showing a wall with overhang(scale bar, 1 �m). (C and D) DIC images of twobuckling MTs (top view) (12). The upper panelseach show a MT [arrowhead in top left of (C)] grow-ing from a randomly positioned seed. The lowerpanels are snapshots (separated by 1 min) of eachMT after the growing end has encountered thewall.Becauseof thecontrastproducedby theover-hang on the walls (which vary in size between sam-ples), the last few micrometers of the MTs cannotbe seen. The sharp changes in contrast indicatethe actual locations of the walls. Scale bar, 10 �m.

Fig. 2. Analysis of MT buckling shapes (15). Opensquares show the hand-recorded shapes of theMT shown in Fig. 1D at 12-s intervals (shapes wereanalyzed at 2-s intervals). The dashed line on theleft indicates the position of the seed (xL). Thedashed line on the right indicates the position of thewall (xW) as judged by eye from the images (Fig.1D). The solid lines show fits to the shape of anelastic rod. One (at the top) is shown as an exam-ple. We assumed that the MT was held at its seedand that a force f was applied at the contact pointof the MT with the wall (x0,y0). This contact pointremained fixed in time and was chosen to producethe best combined fit over the entire time sequence(this produces a value of x0 very close to xW). Wefurther assumed that the MT was free to pivotaround the contact point, but we made no as-sumptions about the quality of the clamp providedby the seed. The fits produced the magnitude andthe direction of the force f (normalized by the flex-ural rigidity ⇧ of the MT ) at each time point, as wellas the length of the MT given by the arc lengthbetween x0 and xL. MT growth is opposed by fp,the component of the force that is directed parallelto the axis of the MT. Scale bar, 5 �m.

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(in each case a segment in the time se-quence is missing, during which the end ofthe MT was obscured by the presence of theoverhang on the wall). The growth velocityvaried considerably even at zero force, asreported previously (16, 17). However, allMTs clearly slowed as soon as a force wasapplied. These curves also show that theforces on short buckling MTs tend to begreater than the forces on long bucklingMTs. The total normalized force (not justthe parallel component) as a function ofMT length is shown in Fig. 3B for all MTshapes analyzed. For each MT length, theforces vary over a certain range because ofvariability in the quality of the clamp pro-vided by the seed. The two dotted linesindicate the theoretical limits for fc (dis-cussed above); as expected, we found thatthe restoring forces were between these lim-its, which validated our assumption thatMTs 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 velocityapproaches the velocity of a freely growingMT (�1.2 ⇧m min–1) at low force, anddecreases to �0.2 ⇧m min–1 as more andmore force is applied. This implies that thereduction in growth velocity is controlledby the applied force and is not simplycaused by the proximity of the end of theMT to the glass barrier. The lower x axis inFig. 4 is labeled with values for the normal-

ized force because this is the parameterobtained from our fits. An independentmeasure of � is needed to obtain values forthe absolute force. The flexural rigidity ofpure MTs has been measured using variousmethods; the values reported range over anorder of magnitude, 4 to 40 pN�⇧m2 (6, 19,20). We used an analysis of the thermalfluctuations to measure the rigidity of ourMTs (21) and found values at the upper endof this range: 34 ⇥ 7 pN�⇧m2. This meansthat the largest forces in Fig. 4 are on theorder of 4 pN (the upper x axis is labeledwith absolute values of force derived fromour measurement of the flexural rigidity).

The force-velocity relation in Fig. 4 canbe compared with theoretical predictions.In the absence of force, the growth velocityis given by the difference in the rate ofaddition and removal of subunits, v ⌦ ⌅(⌃c– ), where ⌅ is the added MT length perdimer (⌅ ⌦ ⌅/13 nm for an MT with 13protofilaments), ⌃c is the rate of subunitaddition (the on-rate), c is the tubulin con-centration, and is the rate of subunitremoval (the off-rate). In principle, both ⌃and may be affected by a force that op-poses elongation of the MT ( fp in our case).Thermodynamic arguments (22) show thattheir ratio (which gives the critical tubulinconcentration ccr) must increase with forceaccording to

ccr(fp) ⌦ (fp)⌃(fp)

⌦ (0)⌃(0) exp(fp⌅/kBT) (1)

where kB is the Boltzmann constant and Tis temperature. This leads to

v(fp) ⌦ ⌅{⌃ exp(⇤qfp⌅/kBT)c

⇤ exp[(1 ⇤ q)fp⌅/kBT]} (2)

where q may take any value between 0 and1 (possibly in a force-dependent way). Thestall force fs (the force at which the velocitybecomes equal to zero) is independent of qand is given by

fs ⌦kBT⌅

ln⌃c

(3)

A similar result is obtained if the growthprocess is pictured as a “Brownian ratchet”(23). In this more mechanistic view, theon-rate depends on the force-dependentprobability that thermal fluctuations (in theposition of the MT end in this case) allowfor a gap between the MT end and thebarrier that is large enough for a dimer toattach to the growing MT end (under op-timal conditions, the size of this gap alongthe direction of MT growth is equal to ⌅,the added length per dimer). If the force isindependent of the size of the gap and thetime required to add a dimer is long relativeto the time required for the MT end todiffuse over a distance ⌅, then

v( fp) ⌦ ⌅[⌃ exp(⇤fp⌅/kBT)c ⇤ ] (4)

(23, 24). This relation assumes that theeffect of force on the off-rate can be neglect-ed. We performed a weighted least-squaresfit of the data in Fig. 4 to both the functionv( fp) ⌦ A – B exp(Cfp/�) (assuming thatonly the off-rate is affected or q⌦ 0) and thefunction v(fp) ⌦ A exp(–Cfp/�) – B (assum-ing that only the on-rate is affected or q ⌦1), where A, B, and C are fitting parame-ters. In the first case, the best fit (�2 ⌦ 1.5)

Fig. 3. MT length and applied force obtained from the analysis of MT buckling shapes such as shownin Fig. 2. (A) For five different MTs, the length L as a function of time (at 2-s intervals) is shown both beforeand after contact with the wall (solid symbols). A segment of time is missing in each case, during whichthe end of the growing MT was obscured by the presence of the overhang on the wall. Open symbolsshow the parallel component of the normalized force, fp/� . The lower left curve corresponds to the MTshown in Figs. 1D and 2. The upper right curve corresponds to the MT shown in Fig. 1C. (B) Totalnormalized force, f/�, as a function of MT length for all MT shapes analyzed (n ⌦ 1316). Each pointcorresponds to one MT shape. The dashed lines indicate the theoretical length dependence of fc in twolimiting cases: fc/� ⌦ 20.19/L2 for a MT with a seed that acts as a perfect clamp (upper curve) and fc/�⌦ ↵2/L2 for a MT with a seed that acts as a perfect hinge (lower curve). In the experiments, the seedsbehaved in an intermediate way and, as expected, the forces obtained from the fits fall between thesetwo limiting curves.

Fig. 4. Average MT growth velocity as a functionof force. Velocity and force were obtained fromcombining data such as shown in Fig. 3A (18). Thelower x axis gives the value of the normalizedforce, fp/�. The upper x axis gives the absolutevalue of the force, based on our measurement ofthe flexural rigidity. The solid line gives the best fitof the data to an exponential decay.

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m

(Dogterom and Yurke 1997)

Microtubule Polymerization

V decays exponentially, but too rapidly

11

Polymerization Forces of Small Number of Actin Filaments

Growing from a Bundle

(Footer et al 2007)

Measured forces of 1-2 pN are smaller than expected from a bundle of filaments

Bead is held in “keyhole” trap of known stiffnessForce is obtained from deflection of bead

Growing actin filament bundle

Actin “Comet Tail” Forces Measured by Micromanipulation

But Actin Gel is Strongly Attached to Obstacle

•  Tail growing on bead attached to cantilever was pulled off with a suction pipette

•  Attachment force is 1pN/filament or more

(Marcy et al 2004)

(Marcy et al 2004)

1.0

1.5

0.5

V/V0

-2 0 2 4F (nN)

Bead with actin nucleator on surface

Growing actin “comet tail”

Growth velocity drops gradually with opposing forceGrowth is accelerated by pulling force

V0 = growth velocity at zero force

13

Measuring Forces Generated by Whole Cells

Scale bar: 5 microns

Prass et al, JCB 2006

Front view of cantilever

Side view

Since stiffness of cantilever is known, force can be obtained from measured displacement

Cantilever approach

1) Initial contact2) Deflection of

lamellum3) Contact with

nuclear mound4) Maximum force5) Release

Cell Exerting a Force on a Barrier

(Prass et al 2006)

Quantifying Motion and Force

• Initial peak at 25s, 2.2 nN - lamellipodium (leading edge)• Then lamellipodium sneaks around cantilever• Later contact is with mound of the cell body• Initial contact area is about 1 µm2

Measurements of Force Distribution on Substrate (Traction Force)

If the elastic properties of the gel are known, the bead displacement response to a given distribution of forces can be calculated

This relationship is inverted to calculate the force distribution given from the bead displacements

http://bam.lab.mcgill.ca/project_pages/TFM_Silicone.html

http://bam.lab.mcgill.ca/project_pages/TFM_Silicone.html

Traction Forces of Cardiac Myocytes

Cells pull inward on substrate, in time with spontaneous contractions

Strength of contraction increases with stiffness of substrate(Hoffmann group, Biology Open 2013)

Scale bar = 10 microns

Stresses and Forces in Layers

0 5 10 15 20 25 30-1

-0.5

0

0.5

1

0 5 10 15 20 25 30-5-4.8-4.6-4.4-4.2-4-3.8-3.6-3.4-3.2-3-2.8-2.6-2.4-2.2-2-1.8-1.6-1.4-1.2-1-0.8-0.6-0.4-0.200.20.40.60.811.21.41.6

1.822.22.42.62.833.23.4

3.63.844.24.44.64.85

0 5 10 15 20 25 30-1.5-1.4-1.3-1.2-1.1-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.100.10.20.30.40.50.60.70.80.911.1

1.21.31.41.5

Force f

Stress σ

Layer of cells or molecules with alternating displacement

Stress is positive when cells/molecules pull on each other (tension)Force is largest where stress is changing most rapidly

f ∝ dσ/dx

SERIES ON MECHANOBIOLOGYSERIES ON MECHANOBIOLOGYR E V I E W

a

e

q r s t

i

m n o p

k lj

f g h

b c d

TensionPressure

15s5 s1s0s

R1 R2

P1 P2–2.5 0.0 0.2Normal stress (nN µm–2 ) 0.

050.

200.

35FR

ET in

dex

VinTS

R1R2

P2

P1

–200 0 200Traction (Pa)

–100 0 400Epithelial stress (Pa)

–0.0

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Figure 1 Force quantification techniques for cell biology. (a–d) 2D traction microscopy and monolayer stress microscopy. Phase contrast microscopy of migrating MDCK cells (a). Superposition of microbead images in the substrate underlying cells during migration (pseudocoloured in green in the inset) and after cell removal via trypsin (pseudocoloured in red in the inset) (b). Traction forces along the x-axis of the cells shown in a (c). Inter- and intracellular stresses (xx-component of the epithelial stress tensor) (d). Scale bar, 50 μm. (e,f) Micropillars. Scanning electron micrographs of a micropillar array without cells (e) and with an adhered smooth muscle cell (f). Scale bar, 10 μm. (g,h) Cantilevers. Images of a C2.7 cell adhered to a rigid substrate at the bottom and to a flexible plate (cantilever) at the top. After the cell establishes initial contact (g) it adheres to both substrates and exerts contractile force (h), deflecting the cantilever. (i,j) Inserts. Confocal section through an aggregate of GFP-positive tooth mesenchymal cells (green) containing fluorocarbon droplets (red) coated externally with ligands for integrin receptors (i). 3D reconstruction of fluorocarbon droplets showing values of the anisotropic stresses mapped on the droplet surface (j). Scale bar, 20 μm. (k,l) Molecular sensors. Colour map images displaying the FRET index in a transfected vinculin tension sensor (VinTS) and localized to cell–ECM adhesions (k), with corresponding inset images (l). Low index indicates high force. Scale bar, 20 μm. (m–p) Laser ablation. Incision of a stress fibre in living cells via a laser nanoscissor. A laser nanoscissor severs a single stress fibre bundle in an endothelial cell expressing EYFP-actin (arrowhead indicates the position of the laser spot; scale bar, 10 μm) (m). Ends of the severed stress fibre (inset in p) splay apart over a period of 15 seconds (n–p). (q–t) Force inference. Cells from the amnioserosa and adjacent lateral ectoderm (upper left corner) of a Bownes stage 13 Drosophila embryo (q). Water shedding is used to segment the image to obtain cell boundaries (r). Circular fitting is used to determine edge curvatures and edge tangent angles at the triple junctions (s). The force-inference equation sets are solved and edge tensions and inner cell pressures are computed in relative units (t). Figures adapted with permission from: a–d, ref. 90, Elsevier; e,f, ref. 44, PNAS; g,h, ref. 151, PNAS; i,j, ref. 60, Nature America Inc.; k,l, ref. 66, Macmillan Publishers Ltd; m–p, ref. 95, Elsevier; q–t, ref. 152, Elsevier.

2 NATURE CELL BIOLOGY ADVANCE ONLINE PUBLICATION


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Migrating kidney cells on elastic substrate containing fluorescent beads

Displacements of beads are measured with (green) and without (red) forces from cells


(Trepat lab, Nat Cell Biol 2017)

Measured Stresses and Forces in Cell Layers

x-direction traction force density obtained from bead displacements

Compressive (dark blue) and tensile (red/green/blue) stress corresponding to measured force density

Force Measurement Using Micropillars


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Micropillar array

Forces from smooth muscle cell bend pillars inward

(C. S. Chen lab, PNAS 2003)

http://ej.iop.org/images/0034-4885/75/11/116601/Full/rpp301965f09_online.jpg

http://ej.iop.org/images/0034-4885/75/11/116601/Full/rpp301965f09_online.jpg

Molecular-Level Adhesion Forces

22(Dunn group, Nano Letters 2015)

“Donor” can emit green fluorescence

But by Förster resonance energy transfer (FRET), energy is instead transferred to the “acceptor” if is close to the donor

For low force, FRET occurs and the green fluorescence is not seen

For high force, FRET is prevented and green fluorescence is seen

Molecular Force Sensors

Donor

Acceptor

The protein paxillin is concentrated in regions of high force

Stress Measurement Using Ablation


Arrow indicates target of laser on a stress fiber

Ends of fiber retract immediately after laser pulse; initial rate is set by viscosity of medium


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(Ingber lab, Biophys. J. 2006)

Ends of stress fiber continue to move apart; final displacement is determined by elastic properties of medium


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Measuring Osmotic Pressure Difference in Walled Cells

Osmotic pressure difference (turgor pressure) is 𝚷= kT(Cin - Cout)

Chemical measurement of 𝚷 : increase external ion

concentration by an amount ΔCout=(Cin - Cout) so that 𝚷 vanishes. Then collapse of membrane away from cell wall can be observed.

Cell wall

Cout

Cin

(Klipp lab, Eur. Biophys. J. 2010)

Cusp in volume variation occurs when membrane leaves cell wall

𝚷= kTCP=0

𝚷= 0.5 MPa

Budding Yeast

Measuring Osmotic Pressure Difference in Walled Cells

(Klipp lab, Biophys. J. 2016)

Physics analysis shows that

𝚷=(spring constant k)/𝜋(cell radius) = 0.2 Mpa

Nanoindentation

Cell wall

(Boudaoud lab, JRSI 2011)

Budding Yeast

Conclusions

• We can only measure a limited range of forces in cells

• New technologies are pushing the field forward

• Measurement of stress inside cells remains a hard problem

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Quantitative Analysis of Forces in Cells...Analysis of MT buckling shapes (15). Open squares show...

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