Journal of Biochemical and Biophysical Methods, 22 (1991) 41-53 41 Elsevier

JBBM 00852

A torsion pendulum for measurement of the viscoelasticity of biopolymers and its application to actin networks

Paul A. Janmey Hematology Unit, Massachusetts General Hospital, Department of Medicine,

Harvard Medical School, Boston, MA, U.S.A,

(Received 3 April 1990) (Accepted 11 July 1990)

Summaff

This report describes the design, construction, and method of operation of a torsion pendulum which is specifically designed for the measurement of soft and fragile biopolymer gels. The pendulum can be assembled and employed in a standard biological laboratory and provides data that currently require access to specialized equipment usually limited to physics or material science laboratories. This instru- ment measures the shear moduli of viscoelastic materials by applying either steady or oscillating shear forces to a disc-shaped sample and measuring the resulting angular displacement of a pendulum attached to one face of the sample. The device is easily constructed using commercially available materials and no specialized machinery. Shear stresses as low as 0.03 Pa and shear rates as low as 0.00003 s-1 can be measured in steady shear experiments, and dynamic shear moduli from 1 to 2500 Pa measured by oscillatory measurements with sample volumes as low as 0.5 ml. The use of the torsion pendulum is illustrated by measuring the effects of two different actin binding proteins on the viscoelasticity of actin filament networks.

Key words: Torsion pendulum; Actin; Shear modulus; Network; Viscoelasticity; Rheology; Gel; Bio- polymer

Introduction

Much of the structure of biological materials is provided by proteins which polymerize into linear filaments or networks. Part of the biologic function of these polymeric assemblies is to endow given regions of particular cells or tissues with viscoelastic properties [1,2], and the material properties of biopolymer networks

Correspondence address: P.A. Janmey, Hematology Unit, Massachusetts General Hospital, Department of Medicine, Harvard Medical School, Boston, MA 02114, U.S.A.

0165-022X/91/$03.50 © 1991 Elsevier Science Publishers B.V. (Biomedical Division)

42

assembled in vitro can help elucidate the function of these protein in vivo. Because the instrumentation needed for quantitative measurement of viscoelasticity is gener- ally complex and unavailable in biological laboratories, information concerning the rheology of biopolymer networks is limited and often inconsistent. The polymeriza- tion of proteins such as fibrin, collagen, tubulin and actin has been studied in order to relate biochemical changes to resultant changes in the physical properties of the protein assemblies [3]. For some biopolymers, for example fibrin, the relation between macromolecular structure and viscoelasticity is well established. The rheol- ogy of fibrin clots can be interpreted in terms of the concentration, length and stiffness of filaments and the nature of interfilament associations [4-6].

In the case of actin, several different models have been proposed to explain the molecular basis for its viscoelastic properties [7-12]. A major cause of the lack of a coherent model is that measurements of actin viscoelasticity are not in agreement, and none of the current models can fit all the experimental data. The apparent damage to actin gels caused by the rheometers available to measure their viscoelastic properties [12,13] has motivated the design of an apparatus that is delicate enough to measure F-actin and other gels without sample disruption. Actin rheology has previously been studied almost exclusively by one of two methods: capillary viscometry and forced oscillation. The former method measures only viscous prop- erties, and requires high shear rates that disrupt the sample. The latter method requires sophisticated instrumentation that has generally been designed for optimal use with materials having properties different from actin. This paper describes the construction and use of a torsion pendulum, a device that allows two independent rheologic measurements: creep under constant stress and free oscillation either in a strained or unstrained state. The advantages of this device are that it is especially well suited for use with very soft materials such as F-actin solutions and allows for measurements of very slow relaxations, that occur beyond the time scale achievable by forced oscillators. The device is inexpensive, requires no motors, and can be assembled from commercially available components.

Materials and Equipment

Pendulum The torsion pendulum is based on the design of Plazek et al. [14]. The theory of

operation of torsion pendulums is described by Ferry [15]. The pendulum itself consists of a T-shaped structure suspended from a thin filament, shown schemati- cally in Fig. la. Since dilute polymer gels have low shear moduli, the pendulum can be relatively light weight, and in this case was constructed of Lego (Enfield, CT, U.S.A.) building blocks (set 720 contains all necessary pieces). The use of these blocks allows for easy addition or removal of mass at the end of the torsion arm to alter the moment of inertia. A typical pendulum is assembled from a variety of different Lego blocks to produce a pendulum of linear dimensions and moment of inertia appropriate for the experimental space and the viscoelastic properties of the material. Moments of inertia of individual narrow rectangular blocks are calculated

43

from the relations:

I c = M)~2/12 for blocks centered at the axis of rotation, and

I 0 = M[ ( ~ 2 /3 ) + DX + D 2] for blocks centered off-axis

where M is the mass of block; X, the length of block; D, the distance from axis to innermost edge of block.

A typical single block has X = 1.60 to 8.0 cm and M = 0.83 to 2.8 g. The balance of the pendulum depends on the uniformity of mass and linear dimension of the pieces, and this uniformity is sufficiently good that no discernible imbalance has been detected for pendulums as long as 30 cm in total length. The reproducibility of the mass of 6.35 cm long blocks, for instance, even after repeated use and handling was measured to be 1.347 + 0.016 g (n = 10). A typical pendulum design, with a moment of inertia of 455 g. cm 2 is shown in Fig. lb.

A 25 mm diameter glass disc (for example, microscope cover slip No. 48382-085, VWR, San Francisco CA, U.S.A.) is glued to the bottom of the pendulum. This disc serves as the upper plate of the sample holder. Just above the disc, attached to the center of the torsion arm, is a mirror which is used for optical detection of the rotation of the pendulum. The top of the pendulum is attached to a filament that suspends the pendulum from a fixed surface. A convenient filament is 1 / 4 lb-test nylon monofilament fishing line (0.2 mm diameter). A 15 cm length of such a filament provides approximately 0.4 Pa of shear stress on a sample attached to a 25 mm glass disc when the filament is twisted one revolution. The wire is calibrated as described below. The entire pendulum is enclosed in a transparent plastic box which can be humidified to prevent evaporation from the edges of the sample, and the filament is attached to a rotating disc (also a Lego block, part No. 7159) mounted on the top of the box. The filament can be twisted through any given angle to provide torque during creep experiments. The torsion arm is suspended above a vertical lift platform (Oriel 19018 Stratford CT, U.S.A.) on top of which is rigidly mounted a second glass disc, that serves as the fixed bottom plate of the sample holder. An x - y variable platform can also be mounted on top of the vertical platform for centering of the bottom plate beneath the freely suspended top plate. The sample to be measured, generally unpolymerized, is introduced by a pipette onto the bottom plate which is then lifted until the liquid sample makes contact with the top plate. The gap between the plates is adjusted to give a slightly convex or flat meniscus of the liquid which is held between the two plates by surface tension. With 25 mm glass plates, aqueous samples of volumes between 0.5 and 1.5 ml can be measured.

Detection system The rotation of the torsion pendulum is detected by means of a small beam of

light (an 0.5 mW laser, for example Oriel No. 79255) reflected from a mirror mounted on the axis of rotation of the pendulum. The spot of reflected light is focussed onto the surface of a 50 mm long position-sensitive photodetector

44

(LSC/30D United Detector Technologies, Hawthorne CA, U.S.A.). The signal from the photodetector is amplified and processed to produce an output voltage that is proportional to the linear position of the light on the surface of the photocell (Amplifier 301-DIV from United Detector Technologies). The -10 to +10 V output from the amplifier can either be digitized for data processing by computer [16], or input directly into a paper chart recorder. The latter method is particularly useful for creep measurements and is adequate for most applications.

__ Apply torque for ~ " ~ ~ ~ static measurements

~9~ ~ '~ - i .............

< s,MPLE

12RI

~ ) Apply momentary ! ~ displacement for dynamic

measurements

B

45

M a t e r i a l s

Actin [17], gelsolin [18], and f i lamin (ABP) [19] were purif ied from skeletal muscle, blood plasma, and smooth muscle, respectively, using publ ished methods. A 20 Poise (2 Pa . s) viscosity s tandard was obta ined from the Dow Chemical Co. and provided by Dr. Ken S. Zaner. Polyacrylamide was a gift from Niels Henr ik Borch, Roskilde University.

Methodology

D y n a m i c m e a s u r e m e n t s

If the torsion arm is given a momen ta ry displacement and then released, the arm will oscillate if the material between the plates is sufficiently elastic. The frequency of the resulting free oscillations and the rate at which they are damped are funct ions of the elastic and viscous properties of the material, the shape and d imens ion of the sample, and the momen t of inert ia of the torsion arm. They are independen t of the ampli tude of the initial displacement of the arm. This latter observat ion allows for conf i rmat ion that free oscillations are in fact observed, and that no damage to the

sample occurs as a result of the displacement. The measurement also requires that the restoring force provided by the suspending wire is negligible. This assumpt ion can be checked by measuring the frequency of oscillation in the absence of a sample, a value used to calibrate the f i lament for static measurements . The storage

Fig. 1. Schematic diagram of torsion pendulum (A) and photograph of the assembled instrument (B). A pendulum appropriate for gels having a shear modulus on the order of 10 Pa has a length of 16 cm and mass of 28.4 g, with a moment of inertia, •=455 g.cm 2, suspended from a 15 cm long nylon monofilament of diameter 0.2 mm. The frequency of free oscillations of a 1 ml sample using this pendulum would be approximately 1 I-Iz, and a 360 degree displacement of the suspending filament would produce a 4% strain, corresponding to an angular displacement of 0.4 degrees at the edge of the sample and a linear displacement, x = 0.44 cm on the photodetector placed L = 34 cm away from the axis of rotation.

Summary of the equations needed for calculating viscoelastic parameters

Static measurements Dynamic measurements

G (t) = stress/strain = o/3' o(stress) = 2"r/~rR 3 ,r(torque) = Ka a = angular displacement of wire x = spring constant of wire y(strain) = Or/h 0 = angular displacement of sample

x / 2 L x = displacement of light on detector L = distance from detector to mirror

G' ( 6o ) = ( ~o2I/b )(1 + AZ/4~r 2) G " ( ~ ) = (~o2I/b)(,a /~r )

~0 = angular frequency of oscillation A = logarithmic decay of amplitude I = moment of inertia of pendulum arm b = ~rR4/2h

46

and loss shear moduli (G'(¢0) and G"(~0), respectively) are determined from the relations:

G'(~o) = ~2I/b(1 + AZ/47r 2) and

c " ( ,o ) = ,o21/b( )

where ~0 is the angular frequency of oscillation (-- 2Try; where v is the frequency in Hz); b = ~rRa/2h is a form factor, with R and h the sample radius and height, respectively; and A is the logarithmic decrement, defined as A = In(A/A,+ l) where A, is the amplitude of the nth oscillation. Since the sample is usually introduced as a liquid, the form factor, b, is more conveniently calculated from the relation b = qr2R6/2V, where V is the sample volume. The moment of inertia, I, is calculated from the weights and dimensions of the blocks used to construct the pendulum as described above. If R, V, and I are given in units of cm, ml, and g . c m 2, respectively, the quantities calculated for G' and G " will be in units of d y n / c m 2 (= 0.1 Pa or 0.1 N/m2).

Static measurements For shear compliance measurements, a constant stress is applied by twisting the

wire suspending the top plate of the sample, and the resulting displacement, quantified as shear strain, is measured as a function of time. The shear compliance is defined as;

J ( t ) = v ( t ) / o

The strain T is defined as:

V(t) =~(O)r /h

where a(0) is the angular displacement of the upper plate relative to the fixed lower plate, and h is the height of the sample. For parallel plate geometry, the strain varies as a function of the distance r from the axis of rotation. In the limit of small strains, where the shear modulus is independent of strain, this feature does not complicate the analysis, but for non-linear viscoelasticity, the measured shear modulus represents an average value, though one which is heavily weighted toward the maximum strain at r = R, the sample radius. The radial dependence of the strain can be eliminated by using cone and plate sample geometry.

For small angular displacements, a is calculated from the expression:

a = tan a = 1 / 2 ( x / L )

where x is the linear distance that the light moves across the photocell, and L is the distance of the photocell from the mirror.

The stress o is equal to:

2KO/,n'R 3

47

0.07 • , - i - , - , • , • , •

0.06

~ , 0.05

co 0.04

0.03

a::

co 0.02

0.01

0,00 i i a I I I • ' • i . 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0

A n g u l a r displacement of wire ( d e g r e e )

Fig. 2. Shear rate of a 2.0 Pa.s viscosity standard at various shear stresses produced by angular displacement of suspending filament.

where ~ is the wire constant, determined experimentally; 0 is the angular displace- ment of the wire; and R is the radius of the sample. The wire constant, K, is obtained by measuring the period of oscillation of a torsion arm of known moment of inertia in the absence of a sample. The wire constant is then given by:

~: = ~Oo2I

where ~o 0 is the angular frequency of rotation in the absence of a sample ( = 2, t /T, where T is the period), and I is the moment of inertia.

Viscosity of Newtonian liquids The simplest rheologic measurement is of the viscosity of materials which exhibit

a constant rate of shear flow under constant stress. Fig. 2 shows the shear rate of a 2.0 Pa. s viscosity standard at different torques provided by varying the angular displacement of the suspending filament. The linear relation between angular displacement and shear rate confirms that the shear stress imposed by the pendulum is proportional to the angular displacement of the filament. Since viscosity is defined as the ratio of shear stress to strain rate, the slope of plots such as this allows an alternate means to calibrate the filament if the viscosity of the fluid is known.

Creep under constant shear stress For viscoelastic materials, such as actin networks, imposition of constant stress

does not lead to a constant rate of strain, but to more complex creep curves. Fig. 3a shows the time course of deformation (strain) when different shear forces are applied to samples of actin in which the average filament is fixed at 540 nm by a 1 : 200 molar ratio of gelsolin [20]. Under small stresses, the actin sample exhibits a rapid initial strain followed by a non-linear change in strain with respect to time and then a very slow linear increase at longer times. When the stress is removed by

48

z

r r

15 H . . . . , . . . . , . . . .

a remove stress

0 5 0 100

T I M E (s)

In

150

i i° Q oO~ . .

-1.5 -1.0 ~0.5 log (Stress in Pa)

4

0

1 , i i I

- 5 . 0 - 4 . 0 - 3 . 0 - 2 . 0 log (Shear rate in l / s )

Fig. 3. Creep curves of actin filament solutions under various shear stresses. Actin (2 mg/ml = 48 jaM in 0.2 M CaCI 2, 0.5 mM ATP, 0.5 mM 2-mercaptoethanol, and 2 mM Tris, pH 7.0) was polymerized by addition of 2 mM MgC12 and 150 mM KC1 in the presence of 0.24 I~M gelsolin. Two hours after polymerization, various steady shear stresses, indicated in the figure, were imposed and the resulting strain measured. Measurements were made sequentially starting from the smallest stress (0.039 Pa) and

ending at 0.5 Pa, at which point the sample ruptured.

rotating the suspending filament to its original position, part of the strain is recovered. As the stress is increased, the amount of unrecovered strain also increases until above a critical stress between 0.33 and 0.5 Pa the sample flows very rapidly and without limit in the amount of strain, features characteristic of sample rupture. Fig. 3b shows that the viscosity calculated from the linear range of the plot of strain vs time is approximately independent of stress below the point at which filament rupture occurs. Fig. 3c shows the viscosity vs shear rate, which demonstrates that below the critical rupture stress the viscosity is approximately independent of shear rate, but that above the critical stress, viscosity decreases strongly with shear rate, consistent with previous" studies that show viscosity inversely proportional to shear rate for F-actin solutions measured at higher shear rates [12,21].

Free oscillations Materials that have a dynamic storage modulus (G ' ) much greater than their loss

modulus (G" ) undergo free oscillations in the torsion pendulum. A well char- acterized gel such as polyacrylamide serves to confirm the damped oscillatory motion of the pendulum and may also be used to calibrate the moment of inertia of the pendulum if the shear modulus of the gel is already known. Fig. 4 shows the excellent fit of the function a- e -bt • cos(ct) to points taken from the tracing of the motions of the pendulum during oscillation of a polyacrylamide gel. The value of G ' derived from the frequency (b) was 2500 Pa, in agreement with the value 3100 Pa measured for the same material using a Rheometrics RFS85000 instrument [22].

To illustrate the response of purely viscous fluids and of two types of actin sample in dynamic measurements using the torsion pendulum, Fig. 5 shows the time

49

• t . . . . i . . . . i

1 . 5

1 . 0

0 . 5

" ~ 0 . 0

-0.5

-1.0

- 1 . 5 . . . . . . . . . j , ,

0 1 2

T i m e ( s )

Fig. 4. Fit of the three parameter function a . e -b t . cos (c t ) to points taken from the experimentally-de- rived tracing during free oscillation of a polyacrylamide gel. The composition of the gel was 6.0%

acrylamide, 0.2% bis(acrylamide); 0.05% ammon i um persulfate; 0.002% T E M E D in water.

8 . . . . , . . . . , . . . . i . . . . 7 6 ~ A

o • • , , 0 5 10 15 20

0.3 . . . . , . . . . , . . . . , . . . . o2~ B

v O l ~ k S O

-01 U N

-0.2 ~ . . 200 :1 .

0 3 I . . . . i . . . . i . . . . , . . . .

0 5 10 1 5 2 o

o.3 . . . j . . . . , . . . . , . . . .

~ -01 -o2 ~ 6 " - . 2oo:1 :!.s - 0 3 i - - . . . . , . . . . i . . . . , . . , ,

0 5 1 0 1 5 2 0

T i m e ( s e c )

Fig. 5. Strain following momentary displacement of the pendulum arm of 87% glycerol (A) actin-gelsolin filaments in the absence (B) and presence (C) of the crosslinking protein ABP at the molar ratios shown.

Experimental conditions as described in the legend to Fig. 3.

50

. . . . , . . . . i . . . . , . . . . i . - • 4.0

3 0

• =C 2.0

. . . . 1 0 . 0

5 10 15 20

Oscillation number

Fig, 6. Logari thmic decay of ampl i tudes during free osc i l lat ions of act in-gelsol in f i lament samples in the absence (B) and presence (C) of ABP. D a t a taken from Fig. 5B and C .

course of strain following a momentary displacement of samples composed of 87% glycerol (Fig. 5A), or actin/gelsolin filaments of number average length 540 nm in the absence (Fig. 5B) or presence (Fig. 5C) of the crosslinking protein ABP.

The displacement of the glycerol-containing sample produces a slow increase in strain followed by a very slow return to the equilibrium position, characteristic of the response of a purely viscous fluid. In contrast, the samples containing actin filaments display typical viscoelastic responses: oscillations characteristic of elastic materials and a damping of the oscillation amplitude due to viscous dissipation of the stored elastic energy. The greater frequency of oscillation in the ABP-containing sample reflects the contribution of crosslinks to the elasticity of actin filament networks composed of relatively short filaments.

. . . . , . . . . , . . . . , . . . . , . . . . , . . . .

40 ~ AC:geI:ABP 200:1:1.5

3O

v

~ 5 2 O

0 Ac:gel 200:1

10

0 5 10 15 20 25 3O

TIME (min)

Fig. 7. Increase of shear modulus during polymerization of actin in the presence of two different concentrations of gelsolin with and without the crosslinking protein ABP. Experimental conditions as

described in the legend to Fig. 3.

51

The viscous loss of the two actin-containing samples, quantified as tan ~ = G"/G', is calculated from plots of logarithm of amplitude vs oscillation number, as shown in Fig. 6. The good fit of a line to the data confirms that a single dissipative process, characteristic of the material, causes the damping of oscillations seen in Fig. 6. The slope of the line fit to the data of Fig. 6 and the frequency of oscillation, obtained from the number of oscillations observed in a given period, are sufficient to calculate G' and G", once the sample volume and the moment of inertia of the pendulum are known.

Values of G' obtained from such data are conveniently measured as a function of time during the polymerization of the sample within the apparatus. Fig. 7 shows a typical experiment in which a constant amount of actin polymerized in the presence of two different concentrations of the filament shortening protein gelsolin with and without the crosslinking protein ABP.

Discussion

A simple torsion pendulum can be constructed from commercially available materials at a cost of approximately $2000. The advantages of this device are its sensitivity and the ability to perform several different measurements on the same sample. These features are useful in the study of actin networks and other fragile biopolymers because such materials form gel-like structures at very low protein concentrations: as low as 0.05% volume fraction. These dilute gels have nearly infinite viscosity at very low shear rates, but such shear rates are impractical to measure using conventional viscometers, and the imposition of larger shear stresses to produce larger shear strains fragments the filaments and leads to greatly lowered viscosity [12]. Filament fragmentation also lessens the elasticity of these networks. The abifity to measure the shear modulus by two independent means: from stress/strain curves and by free oscillation, provides for more accurate determina- tion of the elastic properties of gels and avoidance of damage to the samples caused by the measurements themselves. This instrument has been useful in determining the zero shear rate limit of viscosity of F-actin solutions of known average length to determine that these solutions behave as expected for freely diffusing rigid-rodlike macromolecules [12] and that unusual viscoelastic properties at higher shear strains [21] are due to the inability of extremely long filaments to diffuse in response to the imposed stress rather than to intrinsic crosslinks between these filaments. The extreme simpficity of the pendulum design and the use of glass microscope slides as the sample holders facilitate the adaptation of this device for measurements in which particular sizes of the pendulum are required and also allow for visual observation or fight-scattering measurements of the sample during its deformation.

Simplified description of the method and its applications

This paper describes the design and use of a torsion pendulum that can measure the viscoelastic properties of small quantities of soft gels and viscous solutions of biopolymers. The advantage of the

52

instrument design is that it is constructed entirely from inexpensive commercially available materials and can apply very small stresses which permit theologic measurements of biopolymer gels under conditions that do not damage the structure of these dilute and fragile materials. The accuracy of measurements made by this torsion pendulum is illustrated by measurement of both a viscosity standard and a highly elastic gel that was also measured by a Rheometrics rheometer, a much more sophisticated instrument that operates on a different principle. The applicability of the torsion pendulum to biological samples is illustrated by measurement of the effects of crosslinking and filament severing proteins on the theology of actin filament networks.

Acknowledgements

I a m gra te fu l to J o h n D. F e r r y for the i n t r o d u c t i o n to p o l y m e r phys ics a n d the

to r s ion p e n d u l u m , and to T h o m a s P. Stossel for e n c o u r a g i n g the s tudy of ac t in

rheo logy . I a lso t h a n k J e n n i f e r L a m b , S~ren H v i d t , and N ie l s H e n r i k B o t c h fo r he lp

wi th e x p e r i m e n t s and J o h n H. H a r t w i g and T a I to for the gif t of A B P . Th i s w o r k

was s u p p o r t e d by g ran t s f r o m the U n i t e d Sta tes Pub l i c H e a l t h Se rv ice ( A R 3 8 9 1 0 )

a n d N A T O .

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