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2 Physics of the actin based cytoskeleton

2.1 Actin is a living semiflexible polymer

As illustrated in Figure 5 actin is a living polymer forming double stranded filaments of several 10 µm length which coexist with monomers at a concentration of about 0.1 µM. In appropriate bu er (containing Ca++ or Mg++ and ATP) a random network is formed. The average distance between the filaments (the mesh size ξ) decays with monomer (G-actin) concentration

according to ξ c−A1/2. The actin filaments are typically about 20 µm long and the contour length L is thus large compared to the mesh size. The length of the filaments can be adapted by so called capping molecules which bind to the fast growing end (cf. Ch. 3.1).

Actin is a semiflexible macromolecule. This is demonstrated by pronounced conformational fluctuations which can be directly observed by microscopic observation of fluorescence labelled filaments which are for instance embedded in a network of non-labelled filaments. Another technique for the direct visualization of single filament dynamics will be described below (Fig. 9). The flexibility of the filaments is characterized by the persistence length Lp. It is defined as the contour length, s, over which the

local orientations (characterized by the tangent vector ) are correlated (cf. t

K¨as et al. 1996). The correlation of the orientation of the local tangent to the contour position s = 0 and s decays exponentially (cf. Doi & Edwards Sect. 8.8)

t(0)t(s)		= exp	−Lp		·	(2.1)
				s

Since the manifold of conformations is a consequence of thermally excited bending fluctuations the persistence length can be related to the filament bending elastic modulus, B, by

B = kBT · Lp

(2.2)

B is defined through the bending energy function

		1				∂2u		2
	∆Gela =		B					ds	(2.3)
		2				∂s2

where u is the deflection of the filament at the contour position s and ∂2u/∂s2 is the local curvature. The bending modulus can be measured most directly by Fourier analysis of the bending fluctuations of fluorescent labelled filaments (cf. K¨as et al. 1996) similar to the procedure for the

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Fig. 5. a) Schematic view of actin as living macromolecule. The filaments are double stranded and exist in a dynamic equilibrium with monomers. They exhibit a fast growing end (also called barbed or plus end) where the rate of monomer association is ten fold higher than the dissociation rate (kon/ko 10) while at the opposite end (pointed or minus end) the rate of monomer dissociation is higher (kon/ko 0.3). b) At stationary equilibrium the growth rate at the barbed end equals the dissociation rate at the pointed end (a situation which is called treadmilling), resulting in a random network of filaments coexisting with monomeric actin (G-actin, where “G” stands for “globular”). There exists a threshold concentration for polymerization which is of the order of 0.1 µM. c) Electron micrograph (negative staining) of network of F-actin (of meshsize 1 µm). In addition the crosslinker filamin has been added at a molar ratio actin to filamin of 100:1 which leads to the formation of bundles in the network (cf. Sect. 3.1).

measurement of the membrane bending modulus. The momentary deflec-

tion is
u (s, t) =	uq (t) exp {iqs} ·	(2.4)

q

The total bending energy of a filament of length L is ∆Gela = 1 BL q4u2

2 q q

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and the equipartition theorem (stating that the average energy per mode is 12 kBT ) provides a relationship between uq and B:

Luq2 =	kBT	·	(2.5)
	Bq4

Measurements of uq as function of the wavelength of the mode Λ = π/q allow to measure B rather precisely (cf. K¨as et al. 1996). The bending sti ness of an actin filament is B ≈ 6 × 10−26 J·m and thus the persistence length Lp 15 µm. The persistence length can be modified by a factor of two through binding of actin regulation proteins such as tropomyosin (cf. G¨otter et al. 1996). An interesting (still unsolved) question is wether filaments may also be rendered more flexible by local defects (cf. Piekenbrock & Sackmann 1992). In this context it is interesting to note that the parallel coupling of the single strands is weak and that filaments unwind locally resulting in pronounced local fluctuations of the torsional angle (cf. Bremer et al. 1991). Such fluctuations are expected to cause restricted torsional motions of the filaments.

Further evidence for the role of defects is provided by the finding that binding of the sequestering molecule cofilin to actin filaments results in a shortening of the twist of the actin filament by 75%. The e ect is mediated by the cooperative binding of cofilin between the two strands of a filament. Local binding of cofilins is expected to change the torsional angle of the filament.

An important quantity characterizing the flexibility of semiflexible filaments is the mean square amplitude of the fluctuations u2 , also called roughness. Integration over all modes defined in equation (2.5) yields the following scaling relation between the roughness and the filament length

u2 =	kBT	L3.	(2.6)
	B

The roughness has important consequences for the behaviour of filaments in networks where the excitation of the conformational degrees of freedom is impeded. Following the strategy of classical polymer physics (cf. Doi & Edwards 1986) the constraints imposed on a single (e.g. fluorescence labelled) test chain by the surrounding network can be accounted for by assuming that the filament is surrounded by a tube of constant diameter ξ which is about equal to the mesh size. As illustrated in Figure 6 the filament exhibits restricted wiggling motions within the tube. It is obvious that the confinement by the tube truncates the long wavelength excitations. According to equation (2.6) the longest wavelength una ected is of the order

Λe ≈ (Lp · ξ2)1/3.

(2.7)

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Fig. 6. Restricted local motion of single test filament embedded in macromolecular network. The constraints imposed by the network are accounted for by a tube of constant diameter. The network thus forms an e ective medium for the test filament determining u2 . Note that due to the constraint a maximum bending excitation wavelength Λe can be defined which characterizes the bending mode with the smallest wave vector not a ected by the wall of the tube.

This follows by noting that u2 ≈ (ξ2) and equation (2.2) and defines a new contour length scale: Le ≈ Λe, called “deflection length” or (Odijk) “entanglement length” which determines the crossover between two types of behaviour of the filaments (cf. Odijk 1983). Over contour lengths L < Le the bending fluctuations are determined by the filament bending energy (as free filaments) while for L > Le (or wave vectors q L−e 1) the bending fluctuations are determined by the fluctuating forces generated by the tube walls. The entanglement length plays a crucial role for the frequency dependence of the viscoelastic impedance of actin networks (cf. Isambert & Maggs 1996; Hinner et al. 1998; Morse 1998).

A special but important feature of semiflexible filaments is that (in contrast to flexible filaments) their elastic behaviour depends on the length, L, of the filament (or of a chain segment considered).

If L Lp the chain behaves as an entropic spring. However, the tension associated with stretching of the filament is not an universal quantity as for flexible polymers but depends sensitively on the bending sti ness. The tension τ associated with an extension δL (in the direction parallel to the long filament axis) is

B2	(2.8)
τ kBT L4 δL

and thus depends strongly on the bending sti ness and the filament length. This relationship follows from the following consideration. The elastic energy of a filament under tension is (similar to the situation of membranes

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forming semiflexible shells)

∆Gela = 2		B	∂s2			+ τ ∂s		ds.	(2.9)
1				∂2u	2		∂u	2

Similar to (1.2), the mean square amplitude of the thermally excited mode of wavelength q is

L uq2 =	kBT	·	(2.10)
	Bq4 + τ q2

(Note that τ has dimension [τ ] = N ).

We consider now the change in length δL2= (L − L◦) and remember

that we can express it as L − L◦ 1/2 ( u)

ds. The total mean square

sum over all modes

u

u

amplitude is obtained by replacing the

q by

an integral L◦/(π)

uq dq (with the limits

a is the

π/L and π/a, where

filament diameter

and L

◦ the contour length). It follows

L

kBT

τ

δL ≈ L◦

√

arctan

·

(2.11)

π

B

Bτ

For small tensions (which are relevant for the present lecture) one can expand δL in terms of τ /B and obtains equation (2.8). It is very important to realize that the extension-versus-force relationship depends critically on the orientation of the force with respect to the average filament axis (cf. Kroy & Frey 1996).

For L ≤ Lp the deformation is no longer temperature dependent and the deformation is highly anisotropic (cf. Frey and Wilhelm for a discussion of the tangential elasticity in terms of the Euler model of rigid rods). For forces perpendicular to the filament axis the deflection Umax is proportional to B−1: Umax f L3/B (cf. Landau & Lifshitz, Sect. 20).

2.2 Actin network as viscoelastic body

A physical property of actin networks of uttermost importance is its viscoelasticity. It controls the dynamics of many cellular processes such as cell locomotion, centripetal endothelial cell contractions (cf. Sect. 5) or cellular shape changes (e.g. under shear flow in blood vessel). From the point of view of biological physics even more important is that measurements of viscoelastic properties yield insights into the structural organization of biomacromolecular networks or into the correlation between macroscopic viscoelasticity and the conformational dynamics of single molecules as will be discussed below. Microrheological techniques provide a powerful tool to study the control of cellular processes by viscoelastic properties of the

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Fig. 7. a) The definition of viscoelastic behaviour of body subjected to sudden shear strain by angle γ. b) Time dependence of elastic shear modulus, G(t), of entangled actin network. Note the composition of G(t) of three regimes: an initial relaxation regime at 0 ≤ t ≤ τe, a plateau regime at τe < t < τd and a terminal regime at t ≥ τd.

cell membrane and cytoskeleton through simultaneous systematic studies of model systems and biomaterials (cf. Bausch et al. 2000; Palmer et al. 1999).

To introduce the concept of viscoelasticity we make the following Gedankenexperiment (cf. Fig. 7). We confine an actin network between two parallel plates and deflect the top plate tangentially by a distance ∆x corresponding to a sudden shearing of the network by an angle Θc. Observing the e ect of the shear onto a single test filament, we recognize that the test filament within the networks would feel a time dependent stress. In the case of small deformations the elastic response is linear and Hooks law holds for this time dependent elastic stress

σ (t) = G (t) Θc

(2.12)

G (t) is a time dependent elastic modulus called the relaxation modulus. For entangled actin networks we find three time regimes. At short times the networks behave as an elastic body but the elastic constant decreases rapidly with time. After a relaxation time τe (typically 10−1 s), G (t) remains nearly constant over a time regime τe ≤ t ≤ τd (called plateau

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regime). At long times t > τd (called the terminal regime) the entangled network starts to flow like a liquid3.

Viscoelasticity is determined by two sets of viscoelastic parameters. There are two strategies to measure viscoelastic moduli: the creep response experiment and the oscillatory experiment (cf. Doi & Edwards 1983; Tempel et al. 1996). In the first case a sudden strain (or stress pulse) is applied and the stress (or strain) response is measured as a function of time. The response is characterized by a time dependent frictional coefficient ζ(t) (measured in terms of Pa·s) and an elastic modulus G(t) also called relaxation modulus (or an elastic compliance J (t) also called response compliance). G(t) and ζ(t) are interrelated through the response time τ

τ =	ζ(t)	·	(2.14)
	G(t)

In the creep experiments one of the moduli (G(t) or J (t)) and τ are measured. However, in general the relaxation behaviour is determined by a whole spectrum of relaxation times. We will ignore this aspect in the present review and refer the reader to the monograph by Ferry (1980) for a rigorous discussion of this point.

In the oscillatory experiment an oscillatory stress (or strain) is applied and the response is analyzed according to the classical method known from the treatment of damped oscillators. The viscoelasticity is characterized by

a complex impedance according to
σ(ω) = G (ω)	·	γ(ω)	(2.15)
with γ(ω) = γ◦ exp{iωt} and
G (ω) = G (ω) + iG (ω).			(2.16)

The real part, G (ω), (called the “storage modulus”) determines the response in phase with the excitation and is the frequency dependent shear elastic modulus. The imaginary part G (ω) represents the out of phase component and is called “loss modulus” since it characterizes the energy dissipation during one cycle (cf. Eq. (2.19)). G (ω) is related to the shear

3For more general time dependent shear strains one has to consider that the momentaneous stress depends on the pre-history of deformation and equation (2.12) has to be replaced by

σ (t) = G t − t	dα (t )	dt	(2.13)
	dt

where dα/dt is the shear rate.

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viscosity by
	G (ω) = ω	·	η(ω).	(2.17)

Various methods for the high precision measurements of G (ω) of soft actin networks are available. These include (i) torsional rheometry enabling measurements of the elastic impedance of macroscopic networks (cf. M¨uller et al.; Janmey 1995), (ii) magnetic bead microrheometry (cf. Ziemann et al. 1994; Amblard et al. 1996) shown in Figure 8 and (iii) a force free technique suitable for soft materials is based on the Fourier analysis of the random motion of colloidal probes embedded in the networks (cf. Crocker et al. 2000).

As mentionend, the moduli have a simple physical meaning: G is a measure for the energy stored during one cycle of the deformation which is

Wstored =	1	G (ω)γ2	(2.18)
	4
		0

for a half cycle of the oscillatory excitation. G is a measure for the energy dissipated during a half cycle

Wdissip = πG (ω)γ02.

(2.19)

It is often helpful to measure the phase shift between both response functions which is defined as

tan φ =	G (ω)			Wdissip	·	(2.20)
	G (ω)			W
				stored

An impedance spectrum of the entangled network is shown in Figure 8. As expected G (ω) is the mirror image of the relaxation modulus G(t) exhibiting again the three characteristic regimes. It is noteworthy that G has a minimum in the plateau regime of G (ω) which can be used to define the center of the plateau.

It is also important and helpful to realize that the loss modulus G (ω) exhibits maxima in the frequency regimes associated with relaxation processes. Examples are the stress relaxation by selfdi usion of the chains within the confinement tube leading to the low frequency band of G (ω) or the relaxation of the filament bending modes leading to the sharp rise of G (ω) at ω > 1 rad/s in Figure 8c.

The viscoelasticity of the network is characterized by two elastic moduli corresponding to the two frequency regimes. The major parameter characterizing the elasticity at low frequencies ( 10−1 − 1 Hz) is the value of G (ω) in the plateau regime II (also called rubber plateau regime since it is also a characteristic feature of the viscoelastic behaviour of rubber).

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Fig. 8. a) Magnetic bead microrheometry: a powerful tool to study local viscoelastic parameters of soft macromolecular networks. Magnetic beads (called magnetic tweezers) are embedded in the network a) and the bead deflection induced by an oscillatory magnetic field or a force pulse is analyzed by fast image processing. By analyzing the bead deflection with ultramicroscopy deflection amplitudes of 5 nm may be observed and the time resolution is 10−2 s. Forces from femto-Newton to nano-Newton may be applied. The maximum stress achieved with iron oxide beads of 4 µm diameter is about 500 Pa. By embedding also non-magnetic beads b) the strain field induced by local forces can be determined simultaneously. c) Viscoelastic moduli G (ω) and G (ω) for entangled actin network of mesh size ξ 1 µm (cA = 300 µg/ml). Note that G (ω) is shifted in the vertical direction by an order of magnitude to facilitate better distinction between and that G and G cross at ω ≈ 2 rad/s.

The plateau shear modulus G◦ is related to the mesh size ξ and the chain length through the following power law

G◦ kBT Lp−1/5ξ−14/5 kBT Lp−1/5cA7/5

(2.21)

where we made use of the relationship ξ c−A1/2 (cf. Hinner et al. 1998). The high frequency regime (ν > 10 Hz) is determined by the tension of

the filaments. The theoretical prediction for this relaxation modulus in this regime is (cf. Morse 2001; Gittes & MacKintosh 1998)

G (t) ρ Lp		kBT t			3/4
					(2.22)
	kBT L		ζLp3

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where ρ is the polymer density (number of filaments of length L per unit area) and ζ is the frictional coe cient of the filament in the tube. A measure for the cross-over frequency between the two regimes is the reciprocal relaxation time of the mode of wavelength Λe

τe	ζΛ4	(2.23)
	Be ·

The power laws have been verified experimentally (cf. Isambert & Maggs 1996; Hinner et al. 1998) for a large range of concentrations. This scaling law is very helpful (i) to compare the data measured for di erent concentrations, (ii) to check the purity of actin preparations with respect to crosslinkers or (iii) to estimate the elastic modulus of the actin cortex of cells if the mesh size is known.

2.3Correlation between macroscopic viscoelasticity and molecular motional processes

The three regimes can be related to distinct molecular processes by studying the conformational dynamics and di usion of simple test filaments labelled with fluorescent markers (cf. K¨as et al. 1996; Amblard et al. 1996) or colloidal probes (cf. Dichtl & Sackmann 1999; cf. also Fig. 9). In the latter case, the local motion of the test filament is analyzed by measuring the mean square displacements of the colloidal bead as a function of time in the direction parallel |U (t) − U (0)|2 and perpendicular |U (t) − U (0)|2 to the long axis of the filament (which defines also the local axis of the tube). By using confocal ultramicroscopy the position of the bead (in the image plane) can be determined with an accuracy of ±5 nm.

The mean square displacements exhibit a short time regime t < 0.5 s where the local segments move isotropically and exhibit a time dependence

|U (t) − U (0)|2 = |U (t) − U (0)|2 t0.75	2		(2.24)
and a long time behaviour where |U (t) − U (0)|			saturates

( |U (t) − U (0)|2 = U 2 0) while the parallel component increases linearly with time |U (t) − U (0)|2 t. The short time motional behaviour for both directions is determined by the local wiggling motion of a semiflexible filament which is confined within a tube and therefore subjected to a line tension (cf. Gittes & MacKintosh 1998; Amblard et al. 1996).

The di erent long time behaviour for the two directions yields the following important information:

1.From the distribution P U 2 0 of the saturation values U 0 of the transverse bead motion one can determine the local interaction potential V (U ) characterizing the confinement of the filament by