mechanical amplification is something we experience every day , in the form of gears , pulleys , and levers .
while climbing a hill on a bicycle , for instance , shifting gears increases the force on the wheels while limiting the pressure required on the pedals .
however , energy has to be conserved , and because mechanical work is defined as force  displacement , an increase in force can only be obtained at the expense of displacement .
thus , although shifting gears allows one to develop the additional force needed to go uphill , speed is reduced as each pedal stroke produces a smaller turn of the wheels .
cells have similarly developed microscopic force amplification strategies during evolution . here , we discuss some amplification schemes for one of the major force generators in the cell  actin polymerization .
actin plays a ubiquitous role in cell motility and morphogenesis , spanning many scales of space and time . in fission yeast , for example , a miniature actin machinery only 100 nm across can induce the invagination of an endocytic vesicle in just a few seconds ( picco et al . , 2015
. however , to sever the entire yeast cell , a cytokinetic ring forms with an initial perimeter of 10 m and requires 30 min to drive division ( proctor et al . , 2012 ) .
in other species , considerably larger actin assemblies exist that reach the scale of centimeters , such as in muscle cells .
clearly , actin and its associated factors need to be specifically organized to achieve these different functions ( fig .
1 ) . from a functional point of view , a key problem is to understand how the global architecture of an actin network allows forces that are produced at the molecular scale to be productive for the cell . in this respect , we can distinguish two sorts of components .
active components generate forces from chemical sources of energy and include molecular motors , as well as actin itself , which can push by polymerizing ( kovar and pollard , 2004 ) and possibly pull while depolymerizing .
passive components , such as actin cross - linkers , are essential but can only transmit forces generated by other elements .
networks of actin filaments are essential for many biological processes at the cellular level , and the organization of the filaments in space must be adapted to the task . here , polymerization force ( orange ) of actin filaments ( red ) occurs near the plasma membrane ( blue ) .
linear filopodia bundles with fascin ( black ) can produce high speeds , but represent a weak configuration for force generation .
lamellipodia are thin cellular extensions in which filaments are nearly parallel to the substrate on which the cell is crawling .
the 2d branched network , created by arp2/3 actin - nucleating complexes ( black ) , can produce higher forces at the expense of displacement .
during endocytosis in yeast , actin forms a 3d network at the site of the invagination that appears roughly spherical , but the organization of actin filaments in space is not known .
the coat structure ( yellow ) enables actin to pull the membrane inward and actin polymerizes near the base of the structure , where arp2/3 nucleators are shown in black ( picco et al . , 2015 ) .
the forces developed by an actin meshwork are determined by the organization of its components .
ultimately , these forces must be sufficient to drive biological processes , and thus their scale depends on the physical characteristics of the cell .
for example , in the case of endocytosis in yeast , the turgor pressure pushing the surface of the invagination outward reaches 1,000 pn , which the actin machinery must overcome ( basu et al . , 2014 ) . during cytokinesis
, the actomyosin ring also works against the turgor pressure , which produces high forces on the furrow ( proctor et al . , 2012 ) . for both cases
, these forces have been calculated from measured cellular parameters , particularly the turgor pressure and the dimensions over which the membrane is deformed .
hence , for these processes at least , the two ends of the problem are known : the forces produced by the molecular components make up the input and the force required for the cellular process to occur represents the output . yet the force balance within the system must be considered to understand how the actin machinery harvests the input to produce this output . in this comment ,
we focus on the transmission of forces produced by the polymerization of actin , setting aside turnover and the contribution of molecular motors .
we discuss specifically how the arrangement of the filaments in the system regulates the amount of productive force . in many ways
, the actin machinery behaves analogously to a cyclist : though its power is limited , it can  shift gears  to favor either more displacement ( high gears ) or more force ( low gears ) .
indeed if an actin monomer in solution binds the barbed end of a filament , there is a change of free energy ( gp ) and polymerization will occur if gp < 0 ( fig .
2 a ) . this reaction depends on the concentration ( c ) of monomeric actin and will take place only above a critical concentration ( c * of 0.14 m ; table 1 ; pollard , 1986 ) .
it is associated with gp = kbt ln(c / c * ) , where kb is the boltzmann constant and t is the absolute temperature .
if actin is polymerizing against a load and producing work ( w ) , the change in free energy is gp + w. in this case , polymerization will occur spontaneously if the change is negative , i.e. , gp + w < 0 .
consider an actin filament pushing against a force ( f ) applied parallel to the filament axis ( fig .
2 b ) . because the addition of one actin monomer produces a displacement (  = 2.75 nm ; table 1 ; holmes et al . , 1990 ) , the mechanical work is w = f  . forces that are antagonistic to elongation can impede actin assembly ( peskin et al . , 1993 )
. the critical force under which the filament would cease to elongate is called the polymerization force ( fa ) . using a physiological concentration ( c of 40 m ; wu and pollard , 2005 ) ,
the polymerization force is thermodynamically limited to kbt ln(c / c*)/ = 9 pn ( hill , 1981 ) . within such limits ,
a first study used optical traps on bare filaments , giving a force of 1 pn ( footer et al . , 2007 ) .
by monitoring the buckling of filaments capped with formins , a second study found the force to be 1.3 pn ( kovar and pollard , 2004 ) . in both cases ,
the concentration of actin was an order of magnitude lower than in vivo , and the measured forces were in fact close to the theoretical maximum under the experimental conditions .
we further assume that an actin filament is able to elongate as long as the parallel component of the antagonistic force at its barbed end remains lower than fa , irrespective of the perpendicular components ( fig .
we discuss various examples of force amplification in which the network develops forces that exceed fa per filament , without breaking the thermodynamic requirement for actin polymerization ( gp + w < 0 ) .
( a ) during polymerization , the addition of one actin monomer ( orange ) corresponds to an elongation (  ) at the barbed end of an actin filament ( red ) and is associated with a change of free energy ( gp = kb t ln(c / c * ) ) .
( b ) the work required to push a load over a distance ( h ) with a force ( f ) is f  h , and thus assembly remains favorable as long as gp + f  h < 0 . in the case where polymerization occurs straight against a load ( h =  ) ,
the maximal force ( fa ) is fa = kb t ln(c / c*)/ ( hill , 1981 ) .
( c ) if the filament encounters the load with an angle (  ) , then h =  sin and the maximal force is consequently increased : f = fa / sin. ( d ) in the branched network of a lamellipod , actin grows against the leading membrane at an angle (  = 54 ) . in the absence of friction ,
the force between the polymerizing tip ( orange ) of the actin and the membrane ( blue ) is perpendicular to the membrane .
it can then reach a maximum magnitude of fa / sin. the sum of the forces produced by the two filaments is then 2.5 fa .
the device illustrated here is composed of a growing actin filament with a  leg  on its side . by elongating
, the filament will induce rotation around the pivot point , where the leg is contacting the membrane .
high forces can be exerted on a load supported at the branch point , as a result of the amplification achieved by the lever arm and contact angle .
( f ) the highest forces are generated if a filament polymerizes parallel to the surface . in the illustrated configuration , elongation of the filament
the maximal force is calculated as in e , except that anchoring has to be assumed at the pivot point to balance forces horizontally .
the device can sustain high forces applied on the top of the dome because the upward movement is small compared with the elongation of the filament .
1 ) , which are thin tubular actin - rich cytoplasmic projections extending forward and orthogonally to the leading edge of motile cells . extending a filopod should require a force ( f ) > 10 pn ( mogilner and rubinstein , 2005 ) to overcome membrane tension and rigidity . in a filopod
if the load is distributed over all barbed ends , then each end sustains a fraction of the total force ( f / n ) .
extension will then be possible only if the polymerization force is larger than the fraction of force experienced by each filament ( f / n < fa ) and thus requires sufficient barbed ends to distribute the force
i.e. , it is the highest gear of the actin machinery . assembling more filaments can increase the force , but because the molecular forces are always equal to the productive force , there is no mechanical amplification .
if each filament can produce the same amount of force parallel to its axis , the push on the membrane can be higher as a result of the contact angle ( usually  = 54 ) at which actin filaments encounter the membrane ( fig .
parallel to the axis of a filament corresponds to a proportional force perpendicular to the membrane ( fa / sin ) .
the total pushing force ( f ) on the membrane , then , is the sum of such perpendicular forces applied by n filaments ( f = n
 fa / sin ) . because sin(54 ) < 1 , the productive force is increased .
this occurs at the detriment of displacement achieved by each actin monomer , which is also proportional to the contact angle (   sin ) .
importantly , the contact angle is not solely determined by the branching angle imposed by arp2/3 , the primary nucleating complex for branched actin filaments , because the branched network can adopt different orientations with respect to the leading edge ( weichsel and schwarz , 2010 ) .
thus , this quasi-2d system works like a gearbox , where the coefficient ( sin ) can vary , allowing a lamellipod to generate nanonewton scale forces ( prass et al . , 2006 )
2 e , in which two asymmetrically branched filaments engage the membrane , but only the long branch polymerizes whereas the short branch provides support by transmitting force between the membrane and the filament network . upon polymerization
, the whole construction rotates around a pivot point at the base of the supporting branch , and the contact angle of the polymerizing filament becomes shallower in comparison to the symmetrically polymerizing configuration .
strikingly , this configuration can develop more force than the symmetric case , as an additional amplification ( x + y)/x is associated with the lever arms ( compare fig .
this illustrates that the network force is not solely proportional to the number of polymerizing barbed ends .
the geometry of the system , particularly the angle at which the filaments contact the membrane , and the lever arms can further affect and amplify the total forces generated by the network .
to interpret in vitro experiments in which actin polymerizes around beads ( achard et al .
, 2010 ; dmoulin et al . , 2014 ) , it has been suggested that resistance from a load could cause actin to polymerize parallel to the surface . in this simple configuration
, a filament is confined between a base and a load , which is pushed upward as the filament grows ( fig . 2 f ) .
the upward displacement of the load is determined by the thickness of the actin filament (  ) and by the lever arms x and y , relative to the pivot point .
2 e , but the new device offers better performance ; whereas the long filament in fig .
2 e can bend all the more as it elongates , this configuration works well even with flexible filaments . in the geometry suggested by fig .
2 f , the load is lifted by the filament thickness once the filament has polymerized over the entire base . in a more realistic 3d network , the relationship between polymerization and displacement will not be as simple , because the arrangement of filaments in 3d networks is intricate .
nevertheless , the mechanical concepts remain valid and , in particular , polymerization parallel to a surface could lead to strong orthogonal forces . in yeast endocytosis , actin polymerizes at the bottom of the network in a configuration resembling the wedge ( picco et al . , 2015 ) .
this may perhaps resolve the apparent mismatch between the number of polymerizing filaments and the force resulting from pressure ( basu et al . ,
the force generated by the network depends critically on the network architecture , as this determines the constraints under which filaments grow ( carlsson and bayly , 2014 ) . in general , the force that can be exerted on a load will also depend on the mechanics of the entire structure .
network elasticity allows the polymerization force to be stored as stress , whereas stress relaxation by disassembly and turnover will decrease the force the network can exert ( zhu and mogilner , 2012 ) .
in 1d structures , such as filopodia , force balance forbids mechanical amplification ; however , in 2d structures , the contact angle between the barbed end and the membrane provides a mechanism for tradeoff between force and displacement , and thus allows for force amplification .
configurations in which filaments grow parallel to the membrane , and thus act like wedges , produce the highest forces .
of course , energy conservation dictates that displacement is reduced as force is increased , such that there is a 
a key parameter of our considerations is the force that a polymerizing actin filament can support ( fa ) .
energetic consideration provides an upper bound of 9 pn , but so far direct measurements have yielded lower values , around 1 pn .
thermal fluctuations provide a scale to which this can be compared . at a given temperature ( t ) ,
the characteristic energy associated with thermal fluctuations is kbt , where kb is the boltzmann constant ; at room temperature , the associated force ( kbt/ ) corresponds to 1.5 pn . hence ,
if fa is truly 1 pn , it would imply that actin polymerization is hardly more efficient than thermal fluctuations .
it is to be hoped that future experimental studies , possibly closer to in vivo conditions , will reveal higher forces , as it would be truly astonishing if actin used only 10% of the available energy . in conclusion
, the architecture of a network determines the productive force , often in a nonintuitive manner .
hence , once a system has been well characterized experimentally , mechanical theory should be used to balance the forces within the network .
when this can not be done , energetic considerations , in which the mechanical work of the forces are summed and compared , are informative .
a thorough analysis of force transduction in the system makes it possible to predict the most efficient architecture for performing a given task ( ward et al . , 2015 ) , which is of outstanding value when comparing different modus operandi across species .
 the actin cytoskeleton drives many essential processes in vivo , using molecular motors and actin assembly as force generators . 
 we discuss here the propagation of forces caused by actin polymerization , highlighting simple configurations where the force developed by the network can exceed the sum of the polymerization forces from all filaments . 
Introduction
The force generated by actin polymerization
The high gear: actin pushing forward
Intermediate gears: actin pushing with an angle
The low gear: actin like a wedge
Conclusion