Atomic Force Microscopy in Cell Biology Episode 1 Part 5 docx

However, to determine elastic properties of cells quantitatively and reproducibly, force curves have to be recorded and analyzed Radmacher, 1995 as a function of position the cell Radmac

Measuring the Elastic Properties of Living Cells by the Atomic Force Microscope

Manfred Radmacher

Drittes Physics Institute Georg-August Universit¨at

37073 G ¨ottingen, Germany

I Introduction

II Principles of Measurement

A Force Curves on Soft Samples

B Range of Analysis III Application to Cells

A Elasticity of the Cytoskeleton

B Are Actin Filaments the Major Contributors to Cellular Mechanics?

C Influence of Mechanical Properties on Resolution in Imaging

D Limitations and Problems

E Tip Shape

F Sample Thickness

G Homogeneity of the Cell

IV Mechanics of Cellular Dynamics

V Summary References

I Introduction

The AFM combines high sensitivity in applying and measuring forces, high precision

in positioning a tip relative to the sample in all three dimensions, and the possibility

to be operated in liquids, especially physiological environments (Drake et al., 1989), and therefore is capable of following biological processes in situ (Fritz et al., 1994; Radmacher, Fritz et al., 1994; Schneider et al., 1997; Dvorak and Nagao, 1998; Rotsch

et al., 1999) One application that makes use of all three features is the investigation of

METHOD IN CELL BIOLOGY, VOL 68

cellular mechanics (Tao et al., 1992; Weisenhorn et al., 1993; Radmacher et al., 1995,

1996; Radmacher, 1997) Technically this was first done by using the force modulation

method (Radmacher, et al., 1992, 1993) However, to determine elastic properties of

cells quantitatively and reproducibly, force curves have to be recorded and analyzed

(Radmacher, 1995) as a function of position the cell (Radmacher, Cleveland et al., 1994; Radmacher et al., 1996) Mechanical properties of many different cell types including glial cells (Henderson et al., 1992), platelets (Radmacher et al., 1996; Walch et al., 2000), cardiocytes (Hofmann et al., 1997), macrophages (Rotsch et al., 1997), endothelial cells (Braet et al., 1996, 1997, 1998; Mathur et al., 2000; Miyazaki and Hayashi, 1999; Sato

et al., 2000), epithelial cells (Hoh and Schoenenberger, 1994; A-Hassan et al., 1998),

fibroblasts (Rotsch et al., 1999; Rotsch and Radmacher, 2000), bladder cells (Lekka et al., 1999), L929 cells (Wu et al., 1998), F9 cells (Goldmann et al., 1998), and osteoblasts (Domke et al., 2000) have been investigated.

It has been postulated that mechanical properties play a major role in cellular proc-esses and can thus serve as indicators for cellular procproc-esses (Elson, 1988) The mech-anical properties of eucaryotic cells are determined mainly by the actin cytoskeleton (Sackmann, 1994a) The protein actin can form double-helical polymer fibers with a periodicity of 3.7 nm A large number of actin-associated proteins control the architecture

of this network (Hartwig and Kwiatkowski, 1991) There are molecules which induce bundling, cross-linking, and anchoring of actin to the cell membrane This network is a very active cellular component which is under constant remodeling; therefore it is not only responsible for the shape of a cell but also plays a major role in dynamic processes such as cell migration (Stossel, 1993) and division (Glotzer, 1997; Robinson and Spudich, 2000)

Only a limited number of techniques are available which probe the mechanical prop-erties of cells Traditionally this question has been tackled with either the help of

mi-cropipettes (Evans, 1989; Discher et al., 1994) or with the so-called cell poker (Felder and Elson, 1990; Petersen et al., 1982; Zahalak et al., 1990), which is conceptually

re-lated to the AFM More recently, several new techniques have emerged, for example, the

scanning acoustic microscope (Hildebrand and Rugar, 1984; L¨uers et al., 1991), optical tweezers (Ashkin and Dziedzic, 1989; Florin et al., 1997), magnetic tweezers (Bausch

et al., 1998; Bausch, 1999), and the atomic force microscope, which will be discussed

here in more detail

In this chapter, I will discuss the principle of the measurement of elastic properties in general, the potential and possibilities when applying it to cells, and potential problems Finally, I will present examples of measurements of the elastic properties of living cells

A good example which proves that the investigation of living cells and hence cel-lular dynamics is possible by AFM can be seen in Fig 1 Here, cardiomyocytes, which

spontaneously pulse even as single cells in culture, were probed (Domke et al., 1999).

The AFM tip was used to monitor the mechanical contraction of a cell at different locations

Fig 1 Pulse mapping of a group of active cardiomyocytes Although these cells were mechanically pulsing,

it was possible to image them In this figure the deflection images of two scans were superimposed The cell margins are sketched in the inset on the top left Several time series of height fluctuations at different locations

on the cells were recorded The presented sequences are scaled identically; their locations on top of the cells are marked with circular spots Pulses on cell 1 (a myocyte) show all the same shape regardless of position.

On cell 2, a fibroblast which is moved passively by the neighboring myocytes, only biphasic pulses are found Reproduced from Domke, J., Dannohl, S., Parak, W J., Muller, O., Aicher, W K., and Radmacher, M (2000) Substrate dependent differences in morphology and elasticity of living osteoblasts investigated by atomic force

microscopy Colloids Surf B, 19, 367–373, with permission from Elsevier Science.

II Principles of Measurement

A Force Curves on Soft Samples

In the AFM, an indentation experiment is done by employing the force curve mode

(Weisenhorn et al., 1989) in which the deflection of the cantilever is plotted as a function

of the z height of the sample On a stiff sample, the deflection is either constant as long

as the tip is not in contact with the sample or proportional to the sample height while the tip is in contact With soft samples, the tip may deform (compress) the sample when the

loading force is increased (Tao et al., 1992; Weisenhorn et al., 1993; Radmacher et al.,

1995) Thus, the movement of the tip will be smaller than the movement of the sample base, the difference being the indentation of the sample In general, this indentation may

be elastic, i.e., reversible; plastic, i.e., irreversible; dynamic, i.e., viscous in nature; or

a combination of the three By tuning the experimental parameters one can determine which of the three contributions is present Viscous effects are proportional to the velocity

of the tip approaching to or retracting from the sample By reducing the scan rate, viscous effects can be minimized and they change sign when the velocity changes its sign So,

by averaging approach and retract data viscous effects are canceled out Since the AFM cantilever is immersed in aqueous buffer, there will always be large viscous damping

by the surrounding fluid (Radmacher et al., 1996) This can be seen in force curves

as an increasing separation of the approach and retract curves while the tip is still off the surface In addition, during contact, there may be viscous forces exerted by the sample itself However, this effect seems to be smaller than the effect of the liquid; thus it will need precise data analysis procedures to separate the contributions from both sample and environment Experimentally this problem could be solved by using

small cantilevers (Sch¨affer et al., 1996, 1997; Walters et al., 1997) as soon as they are

available

The difference between elastic and plastic deformation can be seen by compar-ing approach and retract curves If there are no differences between the two, and if subsequent curves are reproducible, then the deformation caused is reversible and it will be elastic in nature Otherwise the deformation is plastic in nature or damaging However, because of viscous effects this determination has been made at low scan velocities

B Range of Analysis

Typically, force curves are analyzed in a given range of loading forces Thus, deflection values must first be converted into loading force values Since cantilever springs are linear springs for small deflections, Hooke’s law can be applied:

Here kcis the force constant of the cantilever, d is its deflection, and F is the corresponding

loading force exerted by the cantilever In experiments, the deflection is not necessarily zero when the cantilever is free, e.g., because of stresses in the cantilever, which will

Fig 2 Differences in force curves on stiff and soft samples In a force curve (a) the cantilever deflection is plotted as a function of a sample base height On a stiff sample the force curve will show two linear regimes (i) The tip is not touching the sample yet, the deflection is constant (between points 1 and 2 in (a), position 1 in (b)) (ii) The tip is touching the sample and the deflection is proportional to the sample height (between points 2 and 3A) On a soft sample, however, due to deformation of the sample, the deflection will raise slower than the movement of the sample base height (between points 2 and 3B in (a), position 2 and 3 in (b)) In fact the elastic response of the sample will lead to a nonlinear relationship between deflection and sample base height The difference between curve A and curve B is the indentation of the soft sample (An animated version of this graph can be found at http://www.dpi.physik.uni-goettingen.de/ ∼radmacher/animations.html).

deform it even without an external load Therefore the offset d0must be subtracted from all deflection values This offset can be easily obtained by calculating the mean deflection

in the off-surface region (between points 1 and 2 in Fig 2a) In principle, it would be fine to pick just one deflection value from this region, but due to noise in the data it is better to take an average Thus, Eq [1] transforms to

The indentationδ is given by the difference between the sample base height z and the

deflection of the cantilever d:

Here again, the offsets must be considered, so I can rewrite Eq [3] as

δ = (z − z0)− (d − d0) [4a]

δ = z − z0− d + d0, [4b]

where d0 is the zero deflection as above and z0 is the z position at the point of

con-tact For quantifying the elastic properties of a sample, a range of data from the force curve to be analyzed and an appropriate model for analysis must be chosen The most simple model comes from continuum mechanics and is based on the work of Heinrich Hertz (Hertz, 1882), which was extended by Sneddon (Sneddon, 1965) For an intro-duction in continuum mechanics one may visit the work of Fung (Fung, 1993), Johnson (Johnson, 1994), or Treloar (Treloar, 1975) The Hertzian model describes the elastic indentation of an infinitely extended sample (effectively filling out a half-space) by

an indenter of simple shape Two shapes often used in AFM are conical or parabolic indenters:

Fcone= π2 · E

(1− ν2)· δ2· tan (α) [5]

Fparaboloid= 4

(1− ν2)· δ3/2·√R [6]

Here, Fcone is the force needed to indent an elastic sample with a conical indenter,

whereas Fparaboloidis the force needed to indent the sample with a parabolic indenter For

a small indentation, a spherical indenter will follow the same force deflection relation

as the parabolic indenter The indentation is denoted byδ, whereas E is the elastic or

Young’s modulus,ν is the Poisson ratio, α is the half-opening angle of the cone, and R is

either radius of curvature of the parabolic indenter or the radius of the spherical indenter, respectively In the case of cells, the sample is virtually incompressible and thereforeν

can be chosen to be 0.5 (Treloar, 1975)

I can now combine Eqs [4b] and [5] in the case of a conical indenter to obtain

kc· (d − d0)= 2

π ·

E

1− ν2 ∗ tan (α) · (z − z0− (d − d0))2. [7] This mathematical function describes the force curve as measured on a soft sample

I can rearrange Eq [7] to obtain

z − z0= (d − d0)+



kc· (d − d0)(1− ν2) (2/π) · E tan (α) , [8]

the mathematically more convenient form

Most of the quantities in Eq [8] either are known or can be measured

experimen-tally The force constant kcand the half-opening angleα can be either obtained from

the manufacturer’s data sheet or determined before or after the experiment A stan-dard method for calibrating the force constant of soft cantilevers is the thermal noise method introduced by Butt and Jaschke (1995) The Poisson ratio was set to 0.5, as

discussed earlier The zero deflection d0 can also be obtained easily from the data

as described previously The deflection d and the sample base height z are the quantities measured in the force curve, leaving only two unknown variables: the elastic modulus E and the contact point z0 The elastic modulus E is the quantity of interest, but the proce-dure to obtain the point of contact z0needs to be discussed also

In a stiff sample (see Fig 2a, trace A, Fig 3a) the contact point separates two linear regimes in the force curve with different slopes Thus, the contact point can be obtained easily by either fitting a line to each of the two regimes or calculating the slope and looking for a discontinuity in it The deflection rises slowly and smoothly at the contact point (Fig 2a, trace B, Fig 3a) for a soft sample There is no jump in slope but only a

Fig 3 Experimental force curves on a stiff substrate and a soft sample (cell) (a) In the stiff sample, the contact point can easily be obtained either from the data or by checking the slope (first derivative) for a discontinuity (b) In the soft sample, the data and the slope are continuous The second derivative should show

a jump, but this discontinuity is no longer detectable due to noise in the data (c).

Fig 4 Example of the fit procedure From the experimental data (hair-crosses) the zero deflection d0 can

be obtained by averaging some part of the force curve in the flat off-surface region The range of analysis is

defined by its lower and upper limits, d1and d2 The Hertzian model is fitted to the data within this range and

results in values for the Young’s modulus and the contact point z0 As can be checked visually the fit follows

the data very closely and yields a reasonable contact point at z0 = 610 nm (The fitted Young’s modulus is

E= 5100 Pa.) continuous change in slope (Fig 3b) Although the second derivative should then show

a discontinuity at the contact point for soft samples, this is often buried in the noise (Fig 3c) Therefore, it is very difficult or even impossible to determine the contact point

in force curves of soft samples in this simple way from experimental data

Another simple, straightforward, reliable, and robust method to obtain the contact

point was established in previous work (Rotsch et al., 1999) Briefly, the elastic modulus

is obtained by fitting Eq [8] within a given range to the experimental data In addition

to using E as a fit parameter the contact point z0can also be included as a fit parameter

In fact, this yields a very good value for the contact point, and in our experience this procedure is more stable, reliable, and robust than any other method in use Figure 4 shows the procedure in more detail A range of analysis is chosen by defining an upper

and a lower limit of deflection values (d1 and d2), which correspond to a range of

loading forces F1 and F2, given by F1= kc∗ d1 and F2= kc∗ d2 This also defines

a range of analysis in terms of z height, given by z1 and z2 By employing a Monte

Carlo fit, optimized values, which fit the data best, for E and z0are obtained Although

z0 will always be outside of the range of analysis, it turned out to be a very reliable quantity There are two major advantages in this procedure (i) Because the contact point

is obtained by fitting a range of data, noise will average out (ii) Because in the range

of analysis the tip is in contact, noise will be smaller compared to data recorded off the surface

III Application to Cells

The elastic response of cells in indentation experiments could stem from several cel-lular compartments Since the tip approaches from the extracelcel-lular medium, it will first

encounter the glycocalix, then the membrane, and then either the intracellular organelles

or the cytoskeleton The glycocalix and the membrane, in the case of eucaryotic cells, turn out to be very soft and can be neglected in AFM experiments First, I want to estimate the elastic response of these components

The glycocalix is a soft polymer brush, whose elastic properties can be estimated in the framework of a worm-like chain model In this model the elastic response of a polymer

molecule with contour length L is determined by its persistence length lp The persistence length is the length at which the orientation of the molecules becomes uncorrelated

due to thermal bending Recently the relation between force Fwlcand extension x for a

single polymer chain was derived (Marko and Siggia, 1995);

Fwlc=kT

lp ∗



x

L + 1 4(1− x

L)2 −1 4



For an extension of 50% of the contour length and a persistence length of 3 ˚A, which

is a reasonable value for a polysaccharide chain (Rief et al., 1997), a force of about 10

pN is obtained This is about the sensitivity of a state-of-the-art AFM using the softest cantilevers available Since the AFM tip may be in contact with several polymer chains

at the same time, it is conceivable that the elasticity of the glycocalix could be detectable However, since the glycocalix is supported by the cell membrane, the membrane’s elastic resilience must be evaluated first

The lipid membrane is much softer than AFM cantilevers as can be seen by the following argument The AFM cantilever is a thermodynamic system with one degree

of freedom, which will fluctuate in position in thermodynamic equilibrium The average energy of this fluctuation will be given by

1/2 kbT = Eavg= 1/2 kc∗ <x2>, [10]

the equipartition theorem, where kbis the Boltzmann constant, T is the absolute temper-ature, kcis the force constant of the cantilever, and<x2> is the time average of the

mean-square displacement Typical values for the displacement will be several ˚Angstrøms in the case of very soft cantilevers (10 mN/m) In fact Eq [10] is the basis for the method mentioned previously for calibrating force constants of AFM cantilevers For softer can-tilevers the fluctuations will be larger Therefore, this method works best with ultrasoft cantilevers Lipid membranes will also show thermal fluctuations in which the restoring force stems from the bending modulus of the lipid bilayer membrane (Helfrich, 1973)

In cellular membranes (like the membrane of erythrocytes), the fluctuations can be on the order of 100 nm, therefore they are detectable in the optical microscope (Sackmann,

1994b; Svoboda et al., 1992; Zeman et al., 1990) Equation [10] demonstrates that lipid

bilayers are several orders of magnitude softer than AFM cantilevers A similar result was obtained from a more thorough theoretical estimation of the response of cellular membranes to the indentation by AFM cantilevers (Boulbitch, 1998) Thus, the elastic response of the membrane is not detectable Consequently the elastic response of the glycocalix was not observed, since its supporting structure, the cell membrane, was too soft This is only true in the case of eucaryotic cells with soft cell membranes For