Article
Can Dissipative Properties of Single Molecules Be Extracted from a Force Spectroscopy Experiment? Fabrizio Benedetti,1 Yulia Gazizova,1,3,4 Andrzej J. Kulik,1 Piotr E. Marszalek,2 Dmitry V. Klinov,3 Giovanni Dietler,1 and Sergey K. Sekatskii1,* ´ cole Polytechnique Fe´de´rale de Lausanne, Lausanne, Switzerland; Laboratoire de Physique de la Matie`re Vivante, IPHYS, BSP, E Department of Mechanical Engineering & Materials Science, Duke University, Durham, North Carolina; 3Russian Institute of PhysicalChemical Medicine, Moscow, Russia; and 4Department of Biological and Medical Physics, Moscow Institute of Physics and Technology, Moscow, Russia 1 2
ABSTRACT We performed dynamic force spectroscopy of single dextran and titin I27 molecules using small-amplitude and low-frequency (40–240 Hz) dithering of an atomic force microscope tip excited by a sine wave voltage fed onto the tip-carrying piezo. We show that for such low-frequency dithering experiments, recorded phase information can be unambiguously interpreted within the framework of a transparent theoretical model that starts from a well-known partial differential equation to describe the dithering of an atomic force microscope cantilever and a single molecule attached to its end system, uses an appropriate set of initial and boundary conditions, and does not exploit any implicit suggestions. We conclude that the observed phase (dissipation) signal is due completely to the dissipation related to the dithering of the cantilever itself (i.e., to the change of boundary conditions in the course of stretching). For both cases, only the upper bound of the dissipation of a single molecule has been established as not exceeding 3,107 kg=s. We compare our results with previously reported measurements of the viscoelastic properties of single molecules, and we emphasize that extreme caution must be taken in distinguishing between the dissipation related to the stretched molecule and the dissipation that originates from the viscous damping of the dithered cantilever. We also present the results of an amplitude channel data analysis, which reveal that the typical values of the spring constant of a I27 molecule at the moment of module unfolding are equal to 451:5 mN=m, and the typical values of the spring constant of dextran at the moment of chair-boat transition are equal to 30 50 mN=m.
INTRODUCTION During the last half a century or so, the importance of the elastic and dissipative properties of biomolecules has been widely recognized in the life sciences community, and a number of excellent reviews have been published (see, e.g., (1–3), to cite only a few). Here, we speak not only about such naturally evident aspects as, say, muscle proteins, molecular motors, and the conversion of an external stress into a biochemical signal and its transmission, but also about the much more general influence of the applied force on virtually all characteristics of life, including mechanically induced phase transitions and mechanical activation (or deactivation) of biochemical reactions. To be a bit Submitted February 22, 2016, and accepted for publication August 16, 2016. *Correspondence: [email protected] F. Benedetti’s present address is Center for Integrative Genomics, University of Lausanne, Lausanne, Switzerland. Fabrizio Benedetti and Yulia Gazizova contributed equally to this work. Editor: H. Jane Dyson. http://dx.doi.org/10.1016/j.bpj.2016.08.018
more concrete, in a very recent study (4), we presented evidence suggesting that a force-induced globule-coil transition in laminin-binding protein and its complexes plays an important role in the flavivirus-cell membrane fusion process. Correspondingly, numerous experiments measuring these same mechanical properties have been reported, but although the situation concerning the elastic properties of single molecules appears quite clear (after all, a molecule’s spring constant is just a kind of derivative of the force-extension signal), apparently this is not the case for their dissipative properties. A number of experiments to measure the dissipative properties of single molecules (i.e., their internal friction) have been reported (see, e.g., (5–14)). One of the most straightforward methods employs small-amplitude dithering of an atomic force microscope (AFM) tip and measurements of the phase changes of this oscillatory system in the course of molecule stretching. An alternative, equivalent approach is to analyze the thermal noise (in a broad frequency range) of an AFM signal in the course of molecule stretching.
Ó 2016 Biophysical Society.
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Unfortunately, the unambiguous extraction of singlemolecule dissipation data for such experiments remains difficult. First of all, there persists a fundamental problem related to the adequacy of a simple one-dimensional (1D) damped harmonic oscillator model employed to extract the dissipation constants from experimental data. Indeed, disregarding the manner of excitation (thermal noise, magnetic or acoustic), the oscillatory motion of an AFM cantilever þ stretched molecule system is properly described by a fourth-order partial differential equation together with a complex of suitable initial and boundary conditions that change over the course of stretching (15). Therefore, a transparent procedure that is free from any implicit suggestions to reduce this very complicated system to the desired simple 1D form must be established first. As well formulated by Bippes et al. (8), ‘‘conceptually, the measured dissipation parameter contains the dissipation by the molecule itself as well as the damping induced by the altered boundary condition at the tip,’’ and only a well-justified, clear-cut data treatment procedure can separate these two contributions. Actually, the question is whether the dissipation of the molecule itself can be discerned at all against the background created by altered boundary conditions. This is an especially legitimate concern if one takes into account the very large and often hardly explainable dissipation values reported for single molecules in many of these experiments. All of these considerations make the measurement of the dissipation parameters of a single stretched molecule a truly complicated issue, and the probability of making an error when interpreting the data rather high. In our opinion, it is necessary to start with a somewhat different approach to the problem, looking first for a type of experiment that, although it may be more difficult to implement and may yield data that look less pretty to the eye, would enable a transparent and model-independent interpretation. Here, we present results obtained from low-frequency (40–240 Hz) experimental measurements of the viscoelastic properties of single molecules (dextran and titin I27 protein), which we believe achieve that goal to a large extent. The frequency used in our experiments is much lower than the first resonant frequency of cantilever dithering in air and in liquid, and this gives us the necessary small parameter to enable a Taylor expansion of the quantities in question. We show that the measured phase (dissipation-related) signal comes exactly and exclusively from the change of boundary conditions and has no connection to the single-molecule dissipation itself.
MATERIALS AND MATERIALS A schematic of the experiment is presented in Fig. 1, which in essence shows the original setup we used in a previous study to measure the spring constants of molecular complexes (16,17), slightly modified for the work presented here. In this study, we used a Nanoscope IV Picoforce AFM
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(Bruker, Billerica, MA). An additional sine-modulation voltage from the output of the lock-in amplifier SR850 (Stanford Research Systems, Sunnyvale, CA) was fed at a fixed frequency u to the piezo driver of the cantilever, which is normally used for imaging in tapping mode. This voltage caused a dithering of the cantilever with a peak-to-peak amplitude of 1–2 nm when in deep contact with the sample, and amplitude and phase signals related to this dithering were measured via a lock-in amplifier. A 20 mL droplet of a 5% solution of dextran (500 kDa: 100–1000 monomers; Sigma-Aldrich, St. Louis, MO) in water was deposited onto a glass coverslip, which was cleaned immediately before each experiment in a plasma chamber (PDC-32G; Harrick Plasma, Ithaca, NY) at low intensity with an air pressure of 1 Torr for 5 min. Due to the hydrophilic nature of a glass substrate treated in this manner, a droplet of solution would immediately spread on the surface and dry overnight at room temperature. Next, the coverslip was washed extensively with ultrapure water to remove the layer of dextran that solidified on the surface, and was then ready to be used for experiments. Experiments with dextran were performed in ultrapure water. It is known that dextran molecules are not charged, and water represents a good solvent for force spectroscopy experiments in this case (18,19), similarly to media such as phosphate-buffered saline and ethanol. For the latter, it has been shown that the elastic response of dextran is not influenced by changing buffer conditions (19). Titin I27 polyproteins containing six identical I27 modules (domains) were produced from a plasmid that was derived from the pAFM8 (20) plasmid (a kind gift to P.M. from Jane Clarke, Cambridge University) according to a standard procedure involving transformation into Escherichia coli bacteria. Experiments with these proteins were performed in 10 mM Tris HCl buffer solution, pH 7.3. To prepare a sample, a 10–20 mL droplet of the protein solution with a concentration of ~10 mg/mL was deposited onto a freshly detached template-stripped gold surface (Platypus Technologies, Madison, WI), incubated for 10 min, and then washed out with the Tris buffer. Rectangular silicon nitride-made cantilevers (ORC8-10; Bruker) with a nominal spring constant of 0.05 N/m and geometrical sizes of L ¼ 200 mm, b ¼ 20 mm, and h ¼ 0.8 mm were used. (In some experiments with dextran, triangular AFM cantilevers with similar nominal spring constants were used.) The first resonant frequencies of these cantilevers in air ranged from 12 to 24 kHz. With a nominal value of 18 kHz, this quantity decreased to 1–2 kHz in liquid. In all cases, the exact value of the spring constant was calibrated using the built-in AFM procedure based on the thermal fluctuation method (see (21) and references cited therein). Aside from cleansing, no chemical treatment was applied to the AFM tips. A pulling speed of 200–25 nm/s and a dithering frequency of 40–240 Hz were used. Experimental data were retrieved, stored, and processed using in-house-written LabView (16,17) and Hooke (22) software. Dextran and I27 protein molecules were chosen for these starting experiments because both of these molecules can be regarded as standard (if not calibrating) samples for single-molecule force spectroscopy, and have been widely studied (see, e.g., (8,13,18,19,23–30)). The dissipative properties of both of these molecules have been described elsewhere (8,13). An additional advantage is provided by the characteristic fingerprints of both dextran (a pronounced force plateau occurring at stretching forces of ~800 pN) and I27 (modular nature of the protein, with the characteristic unfolding force of 100–200 pN for a module) molecules. For both of these systems, the observation of these fingerprints helped us to unambiguously distinguish targeted force spectroscopy events against the background.
Dithering at low frequency: the theoretical model and its analysis Our starting point is a well-known equation (15):
~r~S
v2 y vy v4 y gc ¼ EI 4 ; 2 vt vt vx
(1)
Single-Molecule Dissipation
FIGURE 1 Schematic of the experimental setup. Bottom: a typical example of static force (black, lower curve), amplitude (red, middle curve), and phase (blue, upper curve) signals for the case of avidin-BBSA protein pairs, previously obtained at a dithering frequency of a few kilohertz (see (16,17) for details). The phase data, which look quite typical for this type of experiment (cf. (8,10)) have not been presented before because their interpretation was not clear for us. The Nanoscope IV Picoforce AFM was used in our new setup.
which describes the dithering of a homogeneous beam of a constant cross section where we neglect the beam’s structural damping and use a generic coefficient gc that reflects the viscous damping of the cantilever per unit length. In Eq. 1, the coordinate x is directed along the long cantilever axis of symmetry and the coordinate y is perpendicular to the cantilever plane. E is the Young’s modulus of the cantilever material and I is its inertia moment, which for a rectangular cantilever of thickness h and width b is equal to I ¼ bh3 =12. For small viscous damping, in the ~r~S term we have ~ r ¼ rm , which is the density of the beam material, and ~S ¼ S ¼ bh, which is the area of the beam’s cross section. In our case, however, damping is not at all small and the product ~r~S in Eq. 1 cannot be taken as rm S; rather, it is replaced with rS þ ma to take into account the so-called added (hydrodynamic) mass per unit length (see, e.g., chapter 4 of Ref. (15)). This change is very important because the added mass is a few times larger than the rS. Still, we would like to preserve the notation that is very familiar in the field, and therefore we write a generic term, ~r~S ¼ rS þ ma , here. The parameter gc , as well as some others considered here, explicitly depends on the frequency. However, this is not very important for our current purposes, since single-frequency excitation is always in place and no comparison of data obtained at different frequencies is involved. Certainly, Eq. 1 and its connections to the simplest model of a 1D damped oscillator have been discussed many times in the context of AFM experiments (see (15) and references cited therein, as well as
(31–37)), but we were unable to find a case that exactly and fully corresponded to our experiments in the literature. For this reason, we present our own considerations below. For an excitation at a single frequency u, we may introduce a uniquely determined complex parameter k such that
~r~Su2 iugc ¼ EIk 4 :
(2)
Then the spatial part y(x) of the general solution of Eq. 1, yðx; tÞ ¼ expðiutÞyðxÞ, is given by
yðxÞ ¼ a sin kx þ b cos kx þ c sinh kx þ d cosh kx;
(3)
where a, b, c, and d are constants. Our excitation method is an imposed motion at the point where x ¼ 0 (the point where AFM cantilever is rigidly fixed (hinged) on its support) with an amplitude A and frequency u: yð0Þ ¼ Aeiut . The condition ‘‘beam is hinged at x ¼ 0,’’ i.e., dy/dx ¼ 0 for x ¼ 0, immediately gives a ¼ c. The no-moment condition at x ¼ L gives aðsin kL sinh kLÞ b cos kL þ d cosh kL ¼ 0, from which we obtain d ¼ ðaðsin kL þ sinh kLÞ þ b cos kLÞ= cosh kL. From the condition of a given amplitude of the imposed motion, b þ d ¼ A, after simple algebra we get
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b ¼
A cosh kL aðsin kL þ sinh kLÞ cos kL þ cosh kL
(4a)
d ¼
A cos kL þ aðsin kL þ sinh kLÞ : cos kL þ cosh kL
(4b)
Finally, the relation between a and A, i.e., the function a(A), is given by the boundary condition at x ¼ L. In our case, we have the following appliedforce condition (15):
EI
v3 y ¼ ðki þ igi uÞyðLÞ; vx3 L
(5)
where index i describes the spring constant and characteristic damping of a (single) stretching molecule attached to the cantilever tip. Note the different dimensions of the gi ; gc values (which is why we use the symbol gi rather than gi throughout this work). In other words, we have a characteristic equation,
EI ak3 ð cos kL cosh kLÞ þ bðaÞk3 sin kL þ dðaÞk3 sinh kL
A ðsin z cosh z þ cos z sinh zÞz3 2g cos z cosh z : 2 ð1 þ cos z cosh zÞz3 þ gðsin z cosh z sinh z cos zÞ (11)
First, we note that the above formulae have an evident (and, of course, correct) limit in the case of large g outperforming all other terms (this corresponds to a deep contact/strong connection between the AFM tip and the sample): dy=dx j x¼L ¼ A=L. This is a real value that naturally shows no dissipation: one end of the cantilever is fixed while the other end just repeats a sinusoidal displacement of the piezo. Our next aim is to find an approximation to Eq. 11 for values of g and z that are relevant for the experiments in question. We claim that in these cases, the parameter g is not large but also is not excessively small (we have ki < < kc , but the corresponding ratio cannot be very small when, as it actually takes place, an effect of the single-molecule attachment to the cantilever is seen in the course of stretching), so we can suppose that g is on the order of 0.3–0.03 or so. At the same time, z ¼ kL is definitely rather small for the low frequencies exploited in our experiments (keep in mind that kL is on the order of one for the first resonant frequency of cantilever dithering). A Taylor expansion of Eq. 11 performed for small g- and z-values, taking into account Eqs. 4a and 4b, gives
dy=dx j x ¼ L ¼ akðcos z cosh zÞ bk sin z þ dk sinh zy
¼ ðki þ igi uÞðaðsin kL sinh kLÞ þ bðaÞcos kL þ dðaÞcosh kLÞ;
a ¼
(6)
whose solution gives us the spatial part of the function that describes the cantilever dithering, yðxÞ ¼ aðAÞðsin kx sinh kxÞ þ bðAÞcos kxþ dðAÞcosh kx, and also, in particular, the expression
dy dxx ¼ L ¼ aðAÞkðcos kL cosh kLÞ bðAÞk sin kL þ dðAÞk sinh kL;
a ¼
A sin z cosh z þ cos z sinh z 2q cos z cosh z ; 2 1 þ cos z cosh z þ qðsin z cosh z sinh z cos zÞ (8)
where the dimensionless complex parameter q ¼ ðki þ igi uÞ=ðEIk3 Þ, and where we introduce the notation z ¼ kL. This formula, together with Eqs. 4a and 4b, fully describes the solution. Up to now, z and q could be arbitrary. At this stage, we introduce another dimensionless complex parameter:
3 g ¼ qðkLÞ ¼ ðki þ igi uÞL3 ðEIÞ ¼ 3ðki þ iugi Þ kc ;
(9) where
kc ¼ 3EI L3
(10)
is a static spring constant of the cantilever (15), and thus
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(12)
For completeness, here we present the next-order Taylor expansion of the same value obtained using Wolfram Mathematica software:
dy=dx j x ¼ L ¼
(7) which determines the angle of inclination of the end of the cantilever relevant for a standard AFM measurement schema based on light reflection from the cantilever surface. The coefficients a, b, d, and k are complex, so Eq. 7 also gives the relevant phase shift of the signal. A straightforward but slightly cumbersome analysis of Eqs. 2–5 gives
gA Az4 þ : 2L 6L
3Agk 2ð3 þ gÞz
Akz3 ð 2520 189g þ 19g2 Þ 1680ð3 þ gÞ
2
þ O z4 : (13)
Substituting Eqs. 2, 9, and 10, we can rewrite Eq. 12 as
dy=dx j x ¼ L y
A Au ð3gi þ gc LÞ: 3ki þ ~r~Su2 L þ i 2kc L 2kc L (14)
The real part of this expression, which with an appropriate proportionality factor is nothing other than the amplitude signal of the lock-in amplifier, is
Ry
A 2kc L
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 3ki ~r~Su2 L þ u2 ð3gi þ gc LÞ ;
(15)
and the phase signal of the lock-in amplifier is given by
wyarctan u
3gi þ gc L : 3ki þ ~r~Su2 L
(16)
For another pair of standard lock-in output signals, the so-called X and Y outputs corresponding to the presentation of the signal in the form of X cos w þ Y sin w, we obtain (of course, again up to an appropriate proportionality factor)
Single-Molecule Dissipation
X ¼
A r~ SLu2 3ki þ ~ 2kc L
(17)
Au ð3gi þ gc LÞ: 2kc L
(18)
Y ¼
Equations 12–18 constitute the central result of this section.
RESULTS AND DISCUSSION From the considerations given above, it follows that the Xand Y-output signals of the lock-in amplifier are the most suitable pair for the measurements in question: the former directly gives the value of the stretched molecule’s spring constant (up to some constant factor), and the change of the latter in the course of stretching is related exactly to the single-molecule dissipation. Consequently, for the most part, while performing the measurements we recorded these X and Y outputs. Nevertheless, since another pair of lock-in amplifier outputs, namely, the amplitude and phase signals, are by far more widely used and thus in a sense
are essentially more familiar and convincing for those working in the field, we also recorded these signals in numerous experiments. This is exactly the type of information that enables a direct comparison with the data obtained in other laboratories. Of course, these two pairs of signals (viz. X,Y and amplitude and phase signals) are not independent but are related to each other according to the formulae (15–18). A typical lock-in amplifier does not enable one to record both signal pairs simultaneously, but of course they can be restored (calculated) from each other, as illustrated in Fig. 2 B. An analysis of many thousands of experimental force retraction curves observed for dextran and titin I27 protein shows that the observed dissipation-related signal of the lock-in amplifier was due fully to the cantilever-related dissipation and demonstrated no signs of the dissipation related to a single molecule itself. This result is most easily seen when the X- and Y-output signals of the lock-in amplifier are analyzed. A typical picture of the case of the titin I27 protein is presented in Fig. 2. Although a noisy X-output signal (a high noise level, typically ~20% of the signal, is unavoidable at the low frequency exploited here; for
FIGURE 2 (A–D) Examples of experimental data corresponding to the stretching of a single titin I27 molecule. The static force signal is presented in black, and X and Y channels (volts; for clarity of presentation, the signals are shifted vertically from each other) are shown in green and blue, respectively. (B) For illustration, we also present amplitude and phase curves that were calculated (restored) from X and Y signals using Eqs. 15–18. The pulling speed is equal to 25 nm/s and the dithering frequency is equal to 140 Hz.
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technical reasons, it is impossible to use time constants of the lock-in amplifier greater than 100 ms) still unambiguously reveals that module one-after-one unfolding is essentially a derivative (with the opposite sign) of the static force curve (Eq. 17), no structure exceeding the noise level can be discerned in the Y-output signal. In particular, an extensive quantitative comparison of the statistical distributions of Y signals (histograms of the signal amplitude versus the number of observations were obtained using different resolutions) that are characteristic for bound regions of the force curves (i.e., regions in which the tip and sample surfaces are unambiguously bound by the stretched molecules) and free regions of the same curves did not reveal any statistically significant difference. Similarly, no distinguishable peaks corresponding to the frequency of individual titin domain unfolding of I27 were observed in the Fourier transforms of the Y-channel signal. We drew the same conclusions when we recorded another pair of lock-in amplifier output signals (viz., those related to amplitude and phase). Below, we present and discuss these types of measurements for the case of dextran, which received our special attention. First of all, let us briefly examine the following question: what results can be anticipated for these signals if the contribution from the single-molecule dissipation is negligible? From Eqs. 15 and 16, we get for qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi such a case RyðA=2kc LÞ ð3ki ~ r~ Su2 LÞ2 þ u2 g2c L2 and r~ Su2 LÞÞ. The phase signal is wyarctanðgc Lu=ð3ki þ ~ not at all constant; it changes over the course of stretching
because the molecule spring constant changes (and, we believe, this may create—and actually did create—an illusion of a single-molecule dissipation measurement). If the term ugc L is reasonably smaller than or comparable to ki, we will see a roughly inverse proportionality between these two signals. If this term is much smaller, we will see a saturation of the phase signal, which tends toward a constant zero level when the interaction between the tip and stretched molecule is well present. (The latter circumstance can be easily understood directly from Eq. 12 as corresponding to the situation in which the term proportional to g strongly dominates all others; see the discussion above). Of course, it is not easy to judge unambiguously whether an inverse proportionality exists between two noisy signals; nevertheless, we can say that we observed such a dependence with sufficient reliability in our experiments. (One also should take into account that an absolute value of the measured phase is contaminated with a constant technical phase shift accumulating along the measurement chain lock-in output voltage–lock-in input; each transducer, filter, amplifier, etc. contributes here). A typical example is given in Fig. 3. In Fig. 4, we illustrate the second possibility considered above, viz., that which occurs when the term ugc L is much smaller than ki. It can clearly be seen that the phase signal in the regions of strong interactions between the AFM tip and the sample is the same (and constant) regardless of whether we have a case of deep mechanical contact (0–140 nm in Fig. 4) or a sufficiently strong connection caused by the stretching of
6
5
arbitrary units
4 FIGURE 3 Amplitude (red) and phase (blue) channels of a single stretched dextran molecule obtained with dithering at a frequency of 240 Hz. The pulling speed is equal to 25 nm/s. In green we present an inverse of the properly shifted phase signal, which is hardly distinguishable from the amplitude signal. For clarity of presentation, the graphs are shifted vertically from each other.
3
2
1
0 550
600
650
700 x, nm
750
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800
850
Single-Molecule Dissipation 15
Volts, rad, nN
10
FIGURE 4 Example of experimental data corresponding to the stretching of a single dextran molecule. The static force signal is presented in black, and amplitude and phase signals (volts) are shown in red and blue, respectively. The pulling speed is equal to 25 nm/s and the dithering frequency is equal to 240 Hz. In this figure, a chair-boat transition of dextran that occurs at a force of ~600 nN (215 nm) is barely discerned in the static force signal but is very well seen in the amplitude-related signal. The same transition is also shown in Fig. 3 (~660 nm).
5
0
100
150
200
250
nm
dextran (190–220 nm). For these particular types of force retraction curves, an inverse proportionality between the amplitude and phase signals was seen only in narrow intermediate regions between the cases of strong interactions and free cantilever dithering (the latter takes place in the regions of 160–190 nm and >230 nm in Fig. 4; certainly we have the same constant phase signal in these regions). We now present the results obtained from analyzing the X (or amplitude) signal of the lock-in amplifier. Interpretation of these data does not pose any problem, enabling us to directly obtain the value of the spring constant of the stretching complex ki. The corresponding procedure is especially easy for the X signal: according to Eq. 17, this signal, apart from some (small) constant factor, is directly proportional to ki. The necessary calibration is also easily done by looking into the deep-contact part of the force curves. For this part, the signal is determined simply by the known amplitude of the cantilever dithering A (one cantilever end is rigidly fixed while the other end repeats the motion of the piezo driver). Such a procedure reveals a gradual increase, without any peculiarities, of the titin I27 spring constant from zero to a value of ki ¼ 451:5 mN=m at the moment of complete module unfolding (the local maximum of the signal; the precision of each individual measurement is estimated as 50:3 mN=m). This value corresponds well to, but is more precisely measured than, the data obtained by direct differentiation of the static force signal that allows for the simultaneous action of two springs. In the framework of a simple 1D model,
which is sufficient for our current purposes where phase data are not involved, the springs of the stretched complex and the cantilever work in parallel. Good expositions and explanations of this model can be found, for example, in (10,38). We can directly compare our results with those of Khatri et al. (13), who reported a value of ki Li y1 2 nN (a spring constant normalized on the stretched module length Li) for the same protein. Correspondingly, taking the value of Li to be ~20–30 nm, we obtain a much larger value, not smaller than ~30 mN/m, for the spring constant of the I27 protein. In our opinion, this value is inconsistent with the derivative of a static force signal. Note also that a persistence length of the protein as small as 0.25 nm (we believe this is too small; cf. the data given in (39)) was invoked by Khatri et al. (13) to explain their observation. In the case of dextran, for which the amplitude-andphase pair of signals has mainly been measured, a more complicated procedure was needed to extract the spring constant from the experimental data, due to the nonlinear character of Eq. 15 and the nonnegligible contribution of the ðugc LÞ2 term occurring under the radical sign there. Here, we will not discuss this procedure in full; that will be done elsewhere. Briefly, we proceeded as follows: for parameters occurring in a real experiment (Eq. 15) in a linearized form (Taylor expansion), R ¼ a0 ki þ c0 , where a0 , c0 are appropriate constants, can reasonably be used. This means that the spring constant of interest is a linear function of an amplitude signal, but now we do not see an easy way to obtain the necessary coefficients directly
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from experimental data. To do so, i.e., to calibrate the sensitivity of the measurements, we numerically determine the appropriate coefficients (constants) as the values that minimize the (square of the) difference between an experimental force curve Fexp(x) and integral R x of the experimental phase signal R(x): Fcalc ðxÞ ¼ 0 ðaRðx0 Þ þ cÞdx0 (the initial point of integration corresponds to the beginning of the stretching event, and the final point corresponds to the end of such an event). The thusdetermined constants are then used to extract the value of the spring constant of a dextran molecule from experimental amplitude signal curves, ki ¼ aR þ c. In typical cases, this procedure gives the local maximum values ki ¼ 30 50 mN=m at the moment of the chair-boat transition of dextran (in certain cases, values as high as ki ¼ 100 120 mN=m have been measured), and roughly two times greater values (up to 0.25 N/m in certain cases) at the moment of full stretching of a dextran molecule (upon its detachment from the surface). These results correspond well with previously reported data for the spring constant of dextran (see especially (8)). Estimation of the upper bound of single-molecule dissipation The main results of this work are the nonobservation of a single-molecule dissipation signal and an explanation for the possibility of an apparent (illusory) observation of such a signal. Of course, this also naturally raises the question as to what upper limit of the corresponding value can be derived from the reported nonobservation. This question is not easy to answer. Certainly, it is clear that despite a rather noisy observed signal, in our experiments we were able to measure the dissipation related to the stretching molecule equal to at least 10–15%, and in any case to 20% of own cantilever dissipation, gc L. However, it is not immediately clear what the value of gc L is in our experiments. The first possibility is to ascribe to cantilever dissipation the value of the dissipation constant of the cantilever in water obtained from an analysis of the thermal noise power spectrum density (PSD) function. Various studies have examined this issue (see, e.g., (31–37)), and actually this parameter is routinely obtained using the built-in procedure (based on the approximation of an experimental PSD by a single Lorentzian) of the Picoforce AFM software (21). In doing so (in some cases, we also independently recorded the thermal noise PSD using an SR 760 FFT frequency analyzer (Stanford Research Systems), which led to essentially the same results), we regularly obtained for our cantilevers a value of approximately gc =Ly5,104 kg=ðm,sÞ; hence, gc Ly107 kg=s. Correspondingly, by pursuing this approach, we found that for both single dextran and titin I27 molecules, the dissipation related to molecule unfolding does not exceed the value of 2,108 kg=s.
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However, such a not-fully-justified use of a simple 1D damping oscillator model to describe the cantilever motion is exactly what we want to avoid, so we cannot rely on this approach. Of course, in our experiments, whenever free cantilever dithering in a liquid takes place, we can directly measure the parameter wyarctanðgc L=~r~SuLÞ. However, the problem is that we do not know, and have no straightforward calibration tools to determine, the value of the product ~r~S that should be taken to estimate gc L from the corresponding measurements. Thus, we need to rely on theoretical estimations here. Details of these estimations are given in Supporting Materials and Methods in the Supporting Material. Here we would like to note that, again, the low frequencies used in our experiment constitute an advantage, making the corresponding values rather general and not too dependent on the individual peculiarities of each concrete experiment. As an estimation, we obtain the value of gc Ly1:6,106 kg=s, and this enables us to state the following rather conservative conclusion: our experiments show that for both single dextran and titin I27 molecules, the dissipation related to molecule unfolding does not exceed the value of 3,107 kg=s. CONCLUSIONS In all of our low-frequency, single-molecule force spectroscopy experiments with dextran and titin I27 molecules, we were unable to record the dissipation related to the stretched molecule itself, regardless of whether we used a phase-output signal of the lock-in amplifier or its X-/Y-output signals. The observed dissipation signal was fully due to the dissipation of the AFM cantilever itself. The experimental data are consistent with the elaborated theoretical model, which uses a quite general partial differential equation that describes the beam dithering in liquids, as well as an appropriate set of initial and boundary conditions as a starting point, and does not exploit any implicit suggestions to treat the data. Note that the value gi < 3,107 kg=s presented here as an upper limit of single-molecule dissipation is roughly the same as that given as a typical (or better to say ‘‘maximal observed’’) experimentally measured value for similar single molecules. Of course, this is not surprising given the discussion concerning the problems with the phase signal interpretation presented above, and the circumstance that the cantilever dissipation gc estimated from a 1D model of thermal noise PSD is approximately an order of magnitude smaller than the corresponding value estimated from direct phase measurements. The theoretical analysis of single-polymer-molecule dissipation using the hydrodynamic Rouse-with-internal-friction model of a single protein molecule (40–43) predicts a few orders of magnitude smaller values for this dissipation. (For example, a seven orders of magnitude difference between measured and theoretical values for I27 protein molecules was discussed in
Single-Molecule Dissipation
Ref. (13).) For this reason, other theoretical models, such as the model of diffusion in a rough potential proposed by Zwanzig (44), are already being discussed as a possible explanation for such a discrepancy. We believe that the results presented here clearly make this and similar discussions somewhat premature. As is clear from the above considerations, the difficulties of interpreting single-molecule dissipation data become even more prominent in experiments involving higher frequencies close to the (first) resonance of the cantilever dithering (in the terminology of this work, when the values of the parameter kL are comparable or larger than unity) or a broad frequency range (the thermal excitation case). The analysis would become much more difficult for such a case, but we expect that the main conclusions would remain the same, and it is barely possible to distinguish the part of the signal that is due to the dissipation of the stretched molecule itself. It also should be noted that a higher frequency implies a higher Reynolds number (see Supporting Materials and Methods), which means a smaller contribution from targeted viscous effects and a larger contribution from kinematic (momentum) forces, which are irrelevant for the experiments in question.
4. Zaitsev, B. N., F. Benedetti, ., V. B. Loktev. 2014. Force-induced globule-coil transition in laminin binding protein and its role for viral-cell membrane fusion. J. Mol. Recognit. 27:727–738. 5. Meiners, J. C., and S. R. Quake. 2000. Femtonewton force spectroscopy of single extended DNA molecules. Phys. Rev. Lett. 84:5014– 5017. 6. Humphris, A. D. L., J. Tamayo, and M. J. Miles. 2000. Active quality factor control in liquids for force spectroscopy. Langmuir. 16:7891– 7894. 7. Humphris, A. D. L., M. Antognozzi, ., M. J. Miles. 2002. Transverse dynamic force spectroscopy: a novel approach to determine the complex stiffness of a single molecule. Langmuir. 18:1729–1733. 8. Bippes, C. A., A. D. L. Humphris, ., H. Janovjak. 2006. Direct measurement of single-molecule visco-elasticity in atomic force microscope force-extension experiments. Eur. Biophys. J. 35:287–292. 9. Kawakami, M., K. Byrne, ., D. A. Smith. 2004. Viscoelastic properties of single polysaccharide molecules determined by analysis of thermally driven oscillations of an atomic force microscope cantilever. Langmuir. 20:9299–9303. 10. Janovjak, H., D. J. Mu¨ller, and A. D. L. Humphris. 2005. Molecular force modulation spectroscopy revealing the dynamic response of single bacteriorhodopsins. Biophys. J. 88:1423–1431. 11. Kawakami, M., K. Byrne, ., D. A. Smith. 2005. Viscoelastic measurements of single molecules on a millisecond time scale by magnetically driven oscillation of an atomic force microscope cantilever. Langmuir. 21:4765–4772. 12. Kawakami, M., K. Byrne, ., D. A. Smith. 2006. Viscoelastic properties of single poly(ethylene glycol) molecules. ChemPhysChem. 7:1710–1716.
SUPPORTING MATERIAL
13. Khatri, B. S., K. Byrne, ., T. C. McLeish. 2008. Internal friction of single polypeptide chains at high stretch. Faraday Discuss. 139:35– 51, discussion 105–128, 419–420.
Supporting Materials and Methods is available at http://www.biophysj.org/ biophysj/supplemental/S0006-3495(16)30706-8.
14. Kageshima, M. 2012. Magnetically-modulated AFM for analysis of soft matter systems. Curr. Pharm. Biotechnol. 13:2575–2588.
AUTHOR CONTRIBUTIONS A.J.K., P.E.M., G.D., and S.K.S. designed the research. F.B. and Yu.G. performed the research. P.E.M., A.J.K., and D.V.K. selected the molecules to be studied and provided the necessary reagents. S.K.S., with the help of Yu.G., F.B., and A.J.K., elaborated the theoretical model. S.K.S., F.B., Yu.G., A.J.K., D.V.K., and G.D. interpreted the results. F.B., Yu.G., and G.D. processed and analyzed the data. All authors contributed to writing the manuscript and approved the final version.
ACKNOWLEDGMENTS The authors thank Ge´rard Gremaud for useful discussions and Carine Ben Adiba for preparation of the I27 proteins. This work was supported by the Swiss National Science Foundation (grant No. 200021-150161) and the National Science Foundation (MCB 1517245 to P.M.).
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15. Magrab, E. B. 2012. Vibration of Elastic Systems With Applications to MEMS and NEMS. Springer, Berlin. 16. Chtcheglova, L. A., G. T. Shubeita, ., G. Dietler. 2004. Force spectroscopy with a small dithering of AFM tip: a method of direct and continuous measurement of the spring constant of single molecules and molecular complexes. Biophys. J. 86:1177–1184. 17. Favre, M., L. A. Chtcheglova, ., G. Dietler. 2007. Force-clamp spectroscopy with a small dithering of AFM tip, and its application to explore the energy landscape of single avidin-biotin complex. Ultramicroscopy. 107:882–886. 18. Marszalek, P. E., H. Li, ., J. M. Fernandez. 2002. Chair-boat transitions in single polysaccharide molecules observed with force-ramp AFM. Proc. Natl. Acad. Sci. USA. 99:4278–4283. 19. Lee, G., W. Nowak, ., P. E. Marszalek. 2004. Molecular dynamics simulations of forced conformational transitions in 1,6-linked polysaccharides. Biophys. J. 87:1456–1465. 20. Steward, A., J. L. Toca-Herrera, and J. Clarke. 2002. Versatile cloning system for construction of multimeric proteins for use in atomic force microscopy. Protein Sci. 11:2179–2183. 21. Bruker. 2009. NanoScope V Controller Manual NanoScope Software Version 8. 22. Sandal, M., F. Benedetti, ., B. Samorı`. 2009. Hooke: an open software platform for force spectroscopy. Bioinformatics. 25:1428–1430. 23. Rief, M., F. Oesterhelt, ., H. E. Gaub. 1997. Single molecule force spectroscopy on polysaccharides by atomic force microscopy. Science. 275:1295–1297. 24. Marszalek, P. E., A. F. Oberhauser, ., J. M. Fernandez. 1998. Polysaccharide elasticity governed by chair-boat transitions of the glucopyranose ring. Nature. 396:661–664. 25. Marszalek, P. E., H. Li, and J. M. Fernandez. 2001. Fingerprinting polysaccharides with single-molecule atomic force microscopy. Nat. Biotechnol. 19:258–262.
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Benedetti et al. 26. Higgins, M. J., J. E. Sader, and S. P. Jarvis. 2006. Frequency modulation atomic force microscopy reveals individual intermediates associated with each unfolded I27 titin domain. Biophys. J. 90:640–647. 27. Rief, M., M. Gautel, ., H. E. Gaub. 1997. Reversible unfolding of individual titin immunoglobulin domains by AFM. Science. 276:1109– 1112. 28. Marszalek, P. E., H. Lu, ., J. M. Fernandez. 1999. Mechanical unfolding intermediates in titin modules. Nature. 402:100–103. 29. Carrion-Vazquez, M., A. F. Oberhauser, ., J. M. Fernandez. 1999. Mechanical and chemical unfolding of a single protein: a comparison. Proc. Natl. Acad. Sci. USA. 96:3694–3699. 30. Fowler, S. B., R. B. Best, ., J. Clarke. 2002. Mechanical unfolding of a titin Ig domain: structure of unfolding intermediate revealed by combining AFM, molecular dynamics simulations, NMR and protein engineering. J. Mol. Biol. 322:841–849. 31. Salapaka, M. V., H. S. Bergh, ., E. McFarland. 1997. Multi-mode noise analysis of cantilevers for scanning probe microscopy. J. Appl. Phys. 81:2480–2487. 32. Rabe, U., K. Janser, and W. Arnold. 1996. Vibrations of free and surface-coupled AFM cantilevers: theory and experiment. Rev. Sci. Instrum. 67:3281–3293.
35. Chon, J. W. M., P. Mulvaney, and J. E. Sader. 2000. Experimental validation of theoretical models for the frequency response of atomic force microscope cantilever beams immersed in fluids. J. Appl. Phys. 87:3978–3988. 36. Green, C. P., and J. E. Sader. 2005. Frequency response of cantilever beams immersed in viscous fluids near a solid surface with applications to the atomic force microscope. J. Appl. Phys. 98:114913. 37. Paul, M. R., and M. C. Cross. 2004. Stochastic dynamics of nanoscale mechanical oscillators immersed in a viscous fluid. Phys. Rev. Lett. 92:235501. 38. Burnham, N. A., G. Gremaud, ., F. Oulevey. 1996. Materials’ properties measurements: choosing the optimal scanning probe microscope configuration. J. Vac. Sci. Technol. B. 14:1308–1312. 39. Cheng, S., M. Cetinkaya, and F. Gra¨ter. 2010. How sequence determines elasticity of disordered proteins. Biophys. J. 99:3863–3869. 40. Rouse, P. 1953. A theory of linear viscoelastic properties of dilute solutions of coiling polymers. J. Chem. Phys. 21:1272–1280. 41. Doi, M. 1995. Introduction to Polymer Physics. Oxford University Press, Oxford, UK. 42. Poirier, M. G., and J. F. Marko. 2002. Effect of internal friction on biofilament dynamics. Phys. Rev. Lett. 88:228103.
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Can Dissipative Properties of Single Molecules Be Extracted from a Force Spectroscopy Experiment? Fabrizio Benedetti,1 Yulia Gazizova,1,3,4 Andrzej J. Kulik,1 Piotr E. Marszalek,2 Dmitry V. Klinov,3 Giovanni Dietler,1 and Sergey K. Sekatskii1,* ´ cole Polytechnique Fe´de´rale de Lausanne, Lausanne, Switzerland; Laboratoire de Physique de la Matie`re Vivante, IPHYS, BSP, E Department of Mechanical Engineering & Materials Science, Duke University, Durham, North Carolina; 3Russian Institute of PhysicalChemical Medicine, Moscow, Russia; and 4Department of Biological and Medical Physics, Moscow Institute of Physics and Technology, Moscow, Russia 1 2
ABSTRACT We performed dynamic force spectroscopy of single dextran and titin I27 molecules using small-amplitude and low-frequency (40–240 Hz) dithering of an atomic force microscope tip excited by a sine wave voltage fed onto the tip-carrying piezo. We show that for such low-frequency dithering experiments, recorded phase information can be unambiguously interpreted within the framework of a transparent theoretical model that starts from a well-known partial differential equation to describe the dithering of an atomic force microscope cantilever and a single molecule attached to its end system, uses an appropriate set of initial and boundary conditions, and does not exploit any implicit suggestions. We conclude that the observed phase (dissipation) signal is due completely to the dissipation related to the dithering of the cantilever itself (i.e., to the change of boundary conditions in the course of stretching). For both cases, only the upper bound of the dissipation of a single molecule has been established as not exceeding 3,107 kg=s. We compare our results with previously reported measurements of the viscoelastic properties of single molecules, and we emphasize that extreme caution must be taken in distinguishing between the dissipation related to the stretched molecule and the dissipation that originates from the viscous damping of the dithered cantilever. We also present the results of an amplitude channel data analysis, which reveal that the typical values of the spring constant of a I27 molecule at the moment of module unfolding are equal to 451:5 mN=m, and the typical values of the spring constant of dextran at the moment of chair-boat transition are equal to 30 50 mN=m.
INTRODUCTION During the last half a century or so, the importance of the elastic and dissipative properties of biomolecules has been widely recognized in the life sciences community, and a number of excellent reviews have been published (see, e.g., (1–3), to cite only a few). Here, we speak not only about such naturally evident aspects as, say, muscle proteins, molecular motors, and the conversion of an external stress into a biochemical signal and its transmission, but also about the much more general influence of the applied force on virtually all characteristics of life, including mechanically induced phase transitions and mechanical activation (or deactivation) of biochemical reactions. To be a bit Submitted February 22, 2016, and accepted for publication August 16, 2016. *Correspondence: [email protected] F. Benedetti’s present address is Center for Integrative Genomics, University of Lausanne, Lausanne, Switzerland. Fabrizio Benedetti and Yulia Gazizova contributed equally to this work. Editor: H. Jane Dyson. http://dx.doi.org/10.1016/j.bpj.2016.08.018
more concrete, in a very recent study (4), we presented evidence suggesting that a force-induced globule-coil transition in laminin-binding protein and its complexes plays an important role in the flavivirus-cell membrane fusion process. Correspondingly, numerous experiments measuring these same mechanical properties have been reported, but although the situation concerning the elastic properties of single molecules appears quite clear (after all, a molecule’s spring constant is just a kind of derivative of the force-extension signal), apparently this is not the case for their dissipative properties. A number of experiments to measure the dissipative properties of single molecules (i.e., their internal friction) have been reported (see, e.g., (5–14)). One of the most straightforward methods employs small-amplitude dithering of an atomic force microscope (AFM) tip and measurements of the phase changes of this oscillatory system in the course of molecule stretching. An alternative, equivalent approach is to analyze the thermal noise (in a broad frequency range) of an AFM signal in the course of molecule stretching.
Ó 2016 Biophysical Society.
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Unfortunately, the unambiguous extraction of singlemolecule dissipation data for such experiments remains difficult. First of all, there persists a fundamental problem related to the adequacy of a simple one-dimensional (1D) damped harmonic oscillator model employed to extract the dissipation constants from experimental data. Indeed, disregarding the manner of excitation (thermal noise, magnetic or acoustic), the oscillatory motion of an AFM cantilever þ stretched molecule system is properly described by a fourth-order partial differential equation together with a complex of suitable initial and boundary conditions that change over the course of stretching (15). Therefore, a transparent procedure that is free from any implicit suggestions to reduce this very complicated system to the desired simple 1D form must be established first. As well formulated by Bippes et al. (8), ‘‘conceptually, the measured dissipation parameter contains the dissipation by the molecule itself as well as the damping induced by the altered boundary condition at the tip,’’ and only a well-justified, clear-cut data treatment procedure can separate these two contributions. Actually, the question is whether the dissipation of the molecule itself can be discerned at all against the background created by altered boundary conditions. This is an especially legitimate concern if one takes into account the very large and often hardly explainable dissipation values reported for single molecules in many of these experiments. All of these considerations make the measurement of the dissipation parameters of a single stretched molecule a truly complicated issue, and the probability of making an error when interpreting the data rather high. In our opinion, it is necessary to start with a somewhat different approach to the problem, looking first for a type of experiment that, although it may be more difficult to implement and may yield data that look less pretty to the eye, would enable a transparent and model-independent interpretation. Here, we present results obtained from low-frequency (40–240 Hz) experimental measurements of the viscoelastic properties of single molecules (dextran and titin I27 protein), which we believe achieve that goal to a large extent. The frequency used in our experiments is much lower than the first resonant frequency of cantilever dithering in air and in liquid, and this gives us the necessary small parameter to enable a Taylor expansion of the quantities in question. We show that the measured phase (dissipation-related) signal comes exactly and exclusively from the change of boundary conditions and has no connection to the single-molecule dissipation itself.
MATERIALS AND MATERIALS A schematic of the experiment is presented in Fig. 1, which in essence shows the original setup we used in a previous study to measure the spring constants of molecular complexes (16,17), slightly modified for the work presented here. In this study, we used a Nanoscope IV Picoforce AFM
1164 Biophysical Journal 111, 1163–1172, September 20, 2016
(Bruker, Billerica, MA). An additional sine-modulation voltage from the output of the lock-in amplifier SR850 (Stanford Research Systems, Sunnyvale, CA) was fed at a fixed frequency u to the piezo driver of the cantilever, which is normally used for imaging in tapping mode. This voltage caused a dithering of the cantilever with a peak-to-peak amplitude of 1–2 nm when in deep contact with the sample, and amplitude and phase signals related to this dithering were measured via a lock-in amplifier. A 20 mL droplet of a 5% solution of dextran (500 kDa: 100–1000 monomers; Sigma-Aldrich, St. Louis, MO) in water was deposited onto a glass coverslip, which was cleaned immediately before each experiment in a plasma chamber (PDC-32G; Harrick Plasma, Ithaca, NY) at low intensity with an air pressure of 1 Torr for 5 min. Due to the hydrophilic nature of a glass substrate treated in this manner, a droplet of solution would immediately spread on the surface and dry overnight at room temperature. Next, the coverslip was washed extensively with ultrapure water to remove the layer of dextran that solidified on the surface, and was then ready to be used for experiments. Experiments with dextran were performed in ultrapure water. It is known that dextran molecules are not charged, and water represents a good solvent for force spectroscopy experiments in this case (18,19), similarly to media such as phosphate-buffered saline and ethanol. For the latter, it has been shown that the elastic response of dextran is not influenced by changing buffer conditions (19). Titin I27 polyproteins containing six identical I27 modules (domains) were produced from a plasmid that was derived from the pAFM8 (20) plasmid (a kind gift to P.M. from Jane Clarke, Cambridge University) according to a standard procedure involving transformation into Escherichia coli bacteria. Experiments with these proteins were performed in 10 mM Tris HCl buffer solution, pH 7.3. To prepare a sample, a 10–20 mL droplet of the protein solution with a concentration of ~10 mg/mL was deposited onto a freshly detached template-stripped gold surface (Platypus Technologies, Madison, WI), incubated for 10 min, and then washed out with the Tris buffer. Rectangular silicon nitride-made cantilevers (ORC8-10; Bruker) with a nominal spring constant of 0.05 N/m and geometrical sizes of L ¼ 200 mm, b ¼ 20 mm, and h ¼ 0.8 mm were used. (In some experiments with dextran, triangular AFM cantilevers with similar nominal spring constants were used.) The first resonant frequencies of these cantilevers in air ranged from 12 to 24 kHz. With a nominal value of 18 kHz, this quantity decreased to 1–2 kHz in liquid. In all cases, the exact value of the spring constant was calibrated using the built-in AFM procedure based on the thermal fluctuation method (see (21) and references cited therein). Aside from cleansing, no chemical treatment was applied to the AFM tips. A pulling speed of 200–25 nm/s and a dithering frequency of 40–240 Hz were used. Experimental data were retrieved, stored, and processed using in-house-written LabView (16,17) and Hooke (22) software. Dextran and I27 protein molecules were chosen for these starting experiments because both of these molecules can be regarded as standard (if not calibrating) samples for single-molecule force spectroscopy, and have been widely studied (see, e.g., (8,13,18,19,23–30)). The dissipative properties of both of these molecules have been described elsewhere (8,13). An additional advantage is provided by the characteristic fingerprints of both dextran (a pronounced force plateau occurring at stretching forces of ~800 pN) and I27 (modular nature of the protein, with the characteristic unfolding force of 100–200 pN for a module) molecules. For both of these systems, the observation of these fingerprints helped us to unambiguously distinguish targeted force spectroscopy events against the background.
Dithering at low frequency: the theoretical model and its analysis Our starting point is a well-known equation (15):
~r~S
v2 y vy v4 y gc ¼ EI 4 ; 2 vt vt vx
(1)
Single-Molecule Dissipation
FIGURE 1 Schematic of the experimental setup. Bottom: a typical example of static force (black, lower curve), amplitude (red, middle curve), and phase (blue, upper curve) signals for the case of avidin-BBSA protein pairs, previously obtained at a dithering frequency of a few kilohertz (see (16,17) for details). The phase data, which look quite typical for this type of experiment (cf. (8,10)) have not been presented before because their interpretation was not clear for us. The Nanoscope IV Picoforce AFM was used in our new setup.
which describes the dithering of a homogeneous beam of a constant cross section where we neglect the beam’s structural damping and use a generic coefficient gc that reflects the viscous damping of the cantilever per unit length. In Eq. 1, the coordinate x is directed along the long cantilever axis of symmetry and the coordinate y is perpendicular to the cantilever plane. E is the Young’s modulus of the cantilever material and I is its inertia moment, which for a rectangular cantilever of thickness h and width b is equal to I ¼ bh3 =12. For small viscous damping, in the ~r~S term we have ~ r ¼ rm , which is the density of the beam material, and ~S ¼ S ¼ bh, which is the area of the beam’s cross section. In our case, however, damping is not at all small and the product ~r~S in Eq. 1 cannot be taken as rm S; rather, it is replaced with rS þ ma to take into account the so-called added (hydrodynamic) mass per unit length (see, e.g., chapter 4 of Ref. (15)). This change is very important because the added mass is a few times larger than the rS. Still, we would like to preserve the notation that is very familiar in the field, and therefore we write a generic term, ~r~S ¼ rS þ ma , here. The parameter gc , as well as some others considered here, explicitly depends on the frequency. However, this is not very important for our current purposes, since single-frequency excitation is always in place and no comparison of data obtained at different frequencies is involved. Certainly, Eq. 1 and its connections to the simplest model of a 1D damped oscillator have been discussed many times in the context of AFM experiments (see (15) and references cited therein, as well as
(31–37)), but we were unable to find a case that exactly and fully corresponded to our experiments in the literature. For this reason, we present our own considerations below. For an excitation at a single frequency u, we may introduce a uniquely determined complex parameter k such that
~r~Su2 iugc ¼ EIk 4 :
(2)
Then the spatial part y(x) of the general solution of Eq. 1, yðx; tÞ ¼ expðiutÞyðxÞ, is given by
yðxÞ ¼ a sin kx þ b cos kx þ c sinh kx þ d cosh kx;
(3)
where a, b, c, and d are constants. Our excitation method is an imposed motion at the point where x ¼ 0 (the point where AFM cantilever is rigidly fixed (hinged) on its support) with an amplitude A and frequency u: yð0Þ ¼ Aeiut . The condition ‘‘beam is hinged at x ¼ 0,’’ i.e., dy/dx ¼ 0 for x ¼ 0, immediately gives a ¼ c. The no-moment condition at x ¼ L gives aðsin kL sinh kLÞ b cos kL þ d cosh kL ¼ 0, from which we obtain d ¼ ðaðsin kL þ sinh kLÞ þ b cos kLÞ= cosh kL. From the condition of a given amplitude of the imposed motion, b þ d ¼ A, after simple algebra we get
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b ¼
A cosh kL aðsin kL þ sinh kLÞ cos kL þ cosh kL
(4a)
d ¼
A cos kL þ aðsin kL þ sinh kLÞ : cos kL þ cosh kL
(4b)
Finally, the relation between a and A, i.e., the function a(A), is given by the boundary condition at x ¼ L. In our case, we have the following appliedforce condition (15):
EI
v3 y ¼ ðki þ igi uÞyðLÞ; vx3 L
(5)
where index i describes the spring constant and characteristic damping of a (single) stretching molecule attached to the cantilever tip. Note the different dimensions of the gi ; gc values (which is why we use the symbol gi rather than gi throughout this work). In other words, we have a characteristic equation,
EI ak3 ð cos kL cosh kLÞ þ bðaÞk3 sin kL þ dðaÞk3 sinh kL
A ðsin z cosh z þ cos z sinh zÞz3 2g cos z cosh z : 2 ð1 þ cos z cosh zÞz3 þ gðsin z cosh z sinh z cos zÞ (11)
First, we note that the above formulae have an evident (and, of course, correct) limit in the case of large g outperforming all other terms (this corresponds to a deep contact/strong connection between the AFM tip and the sample): dy=dx j x¼L ¼ A=L. This is a real value that naturally shows no dissipation: one end of the cantilever is fixed while the other end just repeats a sinusoidal displacement of the piezo. Our next aim is to find an approximation to Eq. 11 for values of g and z that are relevant for the experiments in question. We claim that in these cases, the parameter g is not large but also is not excessively small (we have ki < < kc , but the corresponding ratio cannot be very small when, as it actually takes place, an effect of the single-molecule attachment to the cantilever is seen in the course of stretching), so we can suppose that g is on the order of 0.3–0.03 or so. At the same time, z ¼ kL is definitely rather small for the low frequencies exploited in our experiments (keep in mind that kL is on the order of one for the first resonant frequency of cantilever dithering). A Taylor expansion of Eq. 11 performed for small g- and z-values, taking into account Eqs. 4a and 4b, gives
dy=dx j x ¼ L ¼ akðcos z cosh zÞ bk sin z þ dk sinh zy
¼ ðki þ igi uÞðaðsin kL sinh kLÞ þ bðaÞcos kL þ dðaÞcosh kLÞ;
a ¼
(6)
whose solution gives us the spatial part of the function that describes the cantilever dithering, yðxÞ ¼ aðAÞðsin kx sinh kxÞ þ bðAÞcos kxþ dðAÞcosh kx, and also, in particular, the expression
dy dxx ¼ L ¼ aðAÞkðcos kL cosh kLÞ bðAÞk sin kL þ dðAÞk sinh kL;
a ¼
A sin z cosh z þ cos z sinh z 2q cos z cosh z ; 2 1 þ cos z cosh z þ qðsin z cosh z sinh z cos zÞ (8)
where the dimensionless complex parameter q ¼ ðki þ igi uÞ=ðEIk3 Þ, and where we introduce the notation z ¼ kL. This formula, together with Eqs. 4a and 4b, fully describes the solution. Up to now, z and q could be arbitrary. At this stage, we introduce another dimensionless complex parameter:
3 g ¼ qðkLÞ ¼ ðki þ igi uÞL3 ðEIÞ ¼ 3ðki þ iugi Þ kc ;
(9) where
kc ¼ 3EI L3
(10)
is a static spring constant of the cantilever (15), and thus
1166 Biophysical Journal 111, 1163–1172, September 20, 2016
(12)
For completeness, here we present the next-order Taylor expansion of the same value obtained using Wolfram Mathematica software:
dy=dx j x ¼ L ¼
(7) which determines the angle of inclination of the end of the cantilever relevant for a standard AFM measurement schema based on light reflection from the cantilever surface. The coefficients a, b, d, and k are complex, so Eq. 7 also gives the relevant phase shift of the signal. A straightforward but slightly cumbersome analysis of Eqs. 2–5 gives
gA Az4 þ : 2L 6L
3Agk 2ð3 þ gÞz
Akz3 ð 2520 189g þ 19g2 Þ 1680ð3 þ gÞ
2
þ O z4 : (13)
Substituting Eqs. 2, 9, and 10, we can rewrite Eq. 12 as
dy=dx j x ¼ L y
A Au ð3gi þ gc LÞ: 3ki þ ~r~Su2 L þ i 2kc L 2kc L (14)
The real part of this expression, which with an appropriate proportionality factor is nothing other than the amplitude signal of the lock-in amplifier, is
Ry
A 2kc L
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 3ki ~r~Su2 L þ u2 ð3gi þ gc LÞ ;
(15)
and the phase signal of the lock-in amplifier is given by
wyarctan u
3gi þ gc L : 3ki þ ~r~Su2 L
(16)
For another pair of standard lock-in output signals, the so-called X and Y outputs corresponding to the presentation of the signal in the form of X cos w þ Y sin w, we obtain (of course, again up to an appropriate proportionality factor)
Single-Molecule Dissipation
X ¼
A r~ SLu2 3ki þ ~ 2kc L
(17)
Au ð3gi þ gc LÞ: 2kc L
(18)
Y ¼
Equations 12–18 constitute the central result of this section.
RESULTS AND DISCUSSION From the considerations given above, it follows that the Xand Y-output signals of the lock-in amplifier are the most suitable pair for the measurements in question: the former directly gives the value of the stretched molecule’s spring constant (up to some constant factor), and the change of the latter in the course of stretching is related exactly to the single-molecule dissipation. Consequently, for the most part, while performing the measurements we recorded these X and Y outputs. Nevertheless, since another pair of lock-in amplifier outputs, namely, the amplitude and phase signals, are by far more widely used and thus in a sense
are essentially more familiar and convincing for those working in the field, we also recorded these signals in numerous experiments. This is exactly the type of information that enables a direct comparison with the data obtained in other laboratories. Of course, these two pairs of signals (viz. X,Y and amplitude and phase signals) are not independent but are related to each other according to the formulae (15–18). A typical lock-in amplifier does not enable one to record both signal pairs simultaneously, but of course they can be restored (calculated) from each other, as illustrated in Fig. 2 B. An analysis of many thousands of experimental force retraction curves observed for dextran and titin I27 protein shows that the observed dissipation-related signal of the lock-in amplifier was due fully to the cantilever-related dissipation and demonstrated no signs of the dissipation related to a single molecule itself. This result is most easily seen when the X- and Y-output signals of the lock-in amplifier are analyzed. A typical picture of the case of the titin I27 protein is presented in Fig. 2. Although a noisy X-output signal (a high noise level, typically ~20% of the signal, is unavoidable at the low frequency exploited here; for
FIGURE 2 (A–D) Examples of experimental data corresponding to the stretching of a single titin I27 molecule. The static force signal is presented in black, and X and Y channels (volts; for clarity of presentation, the signals are shifted vertically from each other) are shown in green and blue, respectively. (B) For illustration, we also present amplitude and phase curves that were calculated (restored) from X and Y signals using Eqs. 15–18. The pulling speed is equal to 25 nm/s and the dithering frequency is equal to 140 Hz.
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technical reasons, it is impossible to use time constants of the lock-in amplifier greater than 100 ms) still unambiguously reveals that module one-after-one unfolding is essentially a derivative (with the opposite sign) of the static force curve (Eq. 17), no structure exceeding the noise level can be discerned in the Y-output signal. In particular, an extensive quantitative comparison of the statistical distributions of Y signals (histograms of the signal amplitude versus the number of observations were obtained using different resolutions) that are characteristic for bound regions of the force curves (i.e., regions in which the tip and sample surfaces are unambiguously bound by the stretched molecules) and free regions of the same curves did not reveal any statistically significant difference. Similarly, no distinguishable peaks corresponding to the frequency of individual titin domain unfolding of I27 were observed in the Fourier transforms of the Y-channel signal. We drew the same conclusions when we recorded another pair of lock-in amplifier output signals (viz., those related to amplitude and phase). Below, we present and discuss these types of measurements for the case of dextran, which received our special attention. First of all, let us briefly examine the following question: what results can be anticipated for these signals if the contribution from the single-molecule dissipation is negligible? From Eqs. 15 and 16, we get for qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi such a case RyðA=2kc LÞ ð3ki ~ r~ Su2 LÞ2 þ u2 g2c L2 and r~ Su2 LÞÞ. The phase signal is wyarctanðgc Lu=ð3ki þ ~ not at all constant; it changes over the course of stretching
because the molecule spring constant changes (and, we believe, this may create—and actually did create—an illusion of a single-molecule dissipation measurement). If the term ugc L is reasonably smaller than or comparable to ki, we will see a roughly inverse proportionality between these two signals. If this term is much smaller, we will see a saturation of the phase signal, which tends toward a constant zero level when the interaction between the tip and stretched molecule is well present. (The latter circumstance can be easily understood directly from Eq. 12 as corresponding to the situation in which the term proportional to g strongly dominates all others; see the discussion above). Of course, it is not easy to judge unambiguously whether an inverse proportionality exists between two noisy signals; nevertheless, we can say that we observed such a dependence with sufficient reliability in our experiments. (One also should take into account that an absolute value of the measured phase is contaminated with a constant technical phase shift accumulating along the measurement chain lock-in output voltage–lock-in input; each transducer, filter, amplifier, etc. contributes here). A typical example is given in Fig. 3. In Fig. 4, we illustrate the second possibility considered above, viz., that which occurs when the term ugc L is much smaller than ki. It can clearly be seen that the phase signal in the regions of strong interactions between the AFM tip and the sample is the same (and constant) regardless of whether we have a case of deep mechanical contact (0–140 nm in Fig. 4) or a sufficiently strong connection caused by the stretching of
6
5
arbitrary units
4 FIGURE 3 Amplitude (red) and phase (blue) channels of a single stretched dextran molecule obtained with dithering at a frequency of 240 Hz. The pulling speed is equal to 25 nm/s. In green we present an inverse of the properly shifted phase signal, which is hardly distinguishable from the amplitude signal. For clarity of presentation, the graphs are shifted vertically from each other.
3
2
1
0 550
600
650
700 x, nm
750
1168 Biophysical Journal 111, 1163–1172, September 20, 2016
800
850
Single-Molecule Dissipation 15
Volts, rad, nN
10
FIGURE 4 Example of experimental data corresponding to the stretching of a single dextran molecule. The static force signal is presented in black, and amplitude and phase signals (volts) are shown in red and blue, respectively. The pulling speed is equal to 25 nm/s and the dithering frequency is equal to 240 Hz. In this figure, a chair-boat transition of dextran that occurs at a force of ~600 nN (215 nm) is barely discerned in the static force signal but is very well seen in the amplitude-related signal. The same transition is also shown in Fig. 3 (~660 nm).
5
0
100
150
200
250
nm
dextran (190–220 nm). For these particular types of force retraction curves, an inverse proportionality between the amplitude and phase signals was seen only in narrow intermediate regions between the cases of strong interactions and free cantilever dithering (the latter takes place in the regions of 160–190 nm and >230 nm in Fig. 4; certainly we have the same constant phase signal in these regions). We now present the results obtained from analyzing the X (or amplitude) signal of the lock-in amplifier. Interpretation of these data does not pose any problem, enabling us to directly obtain the value of the spring constant of the stretching complex ki. The corresponding procedure is especially easy for the X signal: according to Eq. 17, this signal, apart from some (small) constant factor, is directly proportional to ki. The necessary calibration is also easily done by looking into the deep-contact part of the force curves. For this part, the signal is determined simply by the known amplitude of the cantilever dithering A (one cantilever end is rigidly fixed while the other end repeats the motion of the piezo driver). Such a procedure reveals a gradual increase, without any peculiarities, of the titin I27 spring constant from zero to a value of ki ¼ 451:5 mN=m at the moment of complete module unfolding (the local maximum of the signal; the precision of each individual measurement is estimated as 50:3 mN=m). This value corresponds well to, but is more precisely measured than, the data obtained by direct differentiation of the static force signal that allows for the simultaneous action of two springs. In the framework of a simple 1D model,
which is sufficient for our current purposes where phase data are not involved, the springs of the stretched complex and the cantilever work in parallel. Good expositions and explanations of this model can be found, for example, in (10,38). We can directly compare our results with those of Khatri et al. (13), who reported a value of ki Li y1 2 nN (a spring constant normalized on the stretched module length Li) for the same protein. Correspondingly, taking the value of Li to be ~20–30 nm, we obtain a much larger value, not smaller than ~30 mN/m, for the spring constant of the I27 protein. In our opinion, this value is inconsistent with the derivative of a static force signal. Note also that a persistence length of the protein as small as 0.25 nm (we believe this is too small; cf. the data given in (39)) was invoked by Khatri et al. (13) to explain their observation. In the case of dextran, for which the amplitude-andphase pair of signals has mainly been measured, a more complicated procedure was needed to extract the spring constant from the experimental data, due to the nonlinear character of Eq. 15 and the nonnegligible contribution of the ðugc LÞ2 term occurring under the radical sign there. Here, we will not discuss this procedure in full; that will be done elsewhere. Briefly, we proceeded as follows: for parameters occurring in a real experiment (Eq. 15) in a linearized form (Taylor expansion), R ¼ a0 ki þ c0 , where a0 , c0 are appropriate constants, can reasonably be used. This means that the spring constant of interest is a linear function of an amplitude signal, but now we do not see an easy way to obtain the necessary coefficients directly
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from experimental data. To do so, i.e., to calibrate the sensitivity of the measurements, we numerically determine the appropriate coefficients (constants) as the values that minimize the (square of the) difference between an experimental force curve Fexp(x) and integral R x of the experimental phase signal R(x): Fcalc ðxÞ ¼ 0 ðaRðx0 Þ þ cÞdx0 (the initial point of integration corresponds to the beginning of the stretching event, and the final point corresponds to the end of such an event). The thusdetermined constants are then used to extract the value of the spring constant of a dextran molecule from experimental amplitude signal curves, ki ¼ aR þ c. In typical cases, this procedure gives the local maximum values ki ¼ 30 50 mN=m at the moment of the chair-boat transition of dextran (in certain cases, values as high as ki ¼ 100 120 mN=m have been measured), and roughly two times greater values (up to 0.25 N/m in certain cases) at the moment of full stretching of a dextran molecule (upon its detachment from the surface). These results correspond well with previously reported data for the spring constant of dextran (see especially (8)). Estimation of the upper bound of single-molecule dissipation The main results of this work are the nonobservation of a single-molecule dissipation signal and an explanation for the possibility of an apparent (illusory) observation of such a signal. Of course, this also naturally raises the question as to what upper limit of the corresponding value can be derived from the reported nonobservation. This question is not easy to answer. Certainly, it is clear that despite a rather noisy observed signal, in our experiments we were able to measure the dissipation related to the stretching molecule equal to at least 10–15%, and in any case to 20% of own cantilever dissipation, gc L. However, it is not immediately clear what the value of gc L is in our experiments. The first possibility is to ascribe to cantilever dissipation the value of the dissipation constant of the cantilever in water obtained from an analysis of the thermal noise power spectrum density (PSD) function. Various studies have examined this issue (see, e.g., (31–37)), and actually this parameter is routinely obtained using the built-in procedure (based on the approximation of an experimental PSD by a single Lorentzian) of the Picoforce AFM software (21). In doing so (in some cases, we also independently recorded the thermal noise PSD using an SR 760 FFT frequency analyzer (Stanford Research Systems), which led to essentially the same results), we regularly obtained for our cantilevers a value of approximately gc =Ly5,104 kg=ðm,sÞ; hence, gc Ly107 kg=s. Correspondingly, by pursuing this approach, we found that for both single dextran and titin I27 molecules, the dissipation related to molecule unfolding does not exceed the value of 2,108 kg=s.
1170 Biophysical Journal 111, 1163–1172, September 20, 2016
However, such a not-fully-justified use of a simple 1D damping oscillator model to describe the cantilever motion is exactly what we want to avoid, so we cannot rely on this approach. Of course, in our experiments, whenever free cantilever dithering in a liquid takes place, we can directly measure the parameter wyarctanðgc L=~r~SuLÞ. However, the problem is that we do not know, and have no straightforward calibration tools to determine, the value of the product ~r~S that should be taken to estimate gc L from the corresponding measurements. Thus, we need to rely on theoretical estimations here. Details of these estimations are given in Supporting Materials and Methods in the Supporting Material. Here we would like to note that, again, the low frequencies used in our experiment constitute an advantage, making the corresponding values rather general and not too dependent on the individual peculiarities of each concrete experiment. As an estimation, we obtain the value of gc Ly1:6,106 kg=s, and this enables us to state the following rather conservative conclusion: our experiments show that for both single dextran and titin I27 molecules, the dissipation related to molecule unfolding does not exceed the value of 3,107 kg=s. CONCLUSIONS In all of our low-frequency, single-molecule force spectroscopy experiments with dextran and titin I27 molecules, we were unable to record the dissipation related to the stretched molecule itself, regardless of whether we used a phase-output signal of the lock-in amplifier or its X-/Y-output signals. The observed dissipation signal was fully due to the dissipation of the AFM cantilever itself. The experimental data are consistent with the elaborated theoretical model, which uses a quite general partial differential equation that describes the beam dithering in liquids, as well as an appropriate set of initial and boundary conditions as a starting point, and does not exploit any implicit suggestions to treat the data. Note that the value gi < 3,107 kg=s presented here as an upper limit of single-molecule dissipation is roughly the same as that given as a typical (or better to say ‘‘maximal observed’’) experimentally measured value for similar single molecules. Of course, this is not surprising given the discussion concerning the problems with the phase signal interpretation presented above, and the circumstance that the cantilever dissipation gc estimated from a 1D model of thermal noise PSD is approximately an order of magnitude smaller than the corresponding value estimated from direct phase measurements. The theoretical analysis of single-polymer-molecule dissipation using the hydrodynamic Rouse-with-internal-friction model of a single protein molecule (40–43) predicts a few orders of magnitude smaller values for this dissipation. (For example, a seven orders of magnitude difference between measured and theoretical values for I27 protein molecules was discussed in
Single-Molecule Dissipation
Ref. (13).) For this reason, other theoretical models, such as the model of diffusion in a rough potential proposed by Zwanzig (44), are already being discussed as a possible explanation for such a discrepancy. We believe that the results presented here clearly make this and similar discussions somewhat premature. As is clear from the above considerations, the difficulties of interpreting single-molecule dissipation data become even more prominent in experiments involving higher frequencies close to the (first) resonance of the cantilever dithering (in the terminology of this work, when the values of the parameter kL are comparable or larger than unity) or a broad frequency range (the thermal excitation case). The analysis would become much more difficult for such a case, but we expect that the main conclusions would remain the same, and it is barely possible to distinguish the part of the signal that is due to the dissipation of the stretched molecule itself. It also should be noted that a higher frequency implies a higher Reynolds number (see Supporting Materials and Methods), which means a smaller contribution from targeted viscous effects and a larger contribution from kinematic (momentum) forces, which are irrelevant for the experiments in question.
4. Zaitsev, B. N., F. Benedetti, ., V. B. Loktev. 2014. Force-induced globule-coil transition in laminin binding protein and its role for viral-cell membrane fusion. J. Mol. Recognit. 27:727–738. 5. Meiners, J. C., and S. R. Quake. 2000. Femtonewton force spectroscopy of single extended DNA molecules. Phys. Rev. Lett. 84:5014– 5017. 6. Humphris, A. D. L., J. Tamayo, and M. J. Miles. 2000. Active quality factor control in liquids for force spectroscopy. Langmuir. 16:7891– 7894. 7. Humphris, A. D. L., M. Antognozzi, ., M. J. Miles. 2002. Transverse dynamic force spectroscopy: a novel approach to determine the complex stiffness of a single molecule. Langmuir. 18:1729–1733. 8. Bippes, C. A., A. D. L. Humphris, ., H. Janovjak. 2006. Direct measurement of single-molecule visco-elasticity in atomic force microscope force-extension experiments. Eur. Biophys. J. 35:287–292. 9. Kawakami, M., K. Byrne, ., D. A. Smith. 2004. Viscoelastic properties of single polysaccharide molecules determined by analysis of thermally driven oscillations of an atomic force microscope cantilever. Langmuir. 20:9299–9303. 10. Janovjak, H., D. J. Mu¨ller, and A. D. L. Humphris. 2005. Molecular force modulation spectroscopy revealing the dynamic response of single bacteriorhodopsins. Biophys. J. 88:1423–1431. 11. Kawakami, M., K. Byrne, ., D. A. Smith. 2005. Viscoelastic measurements of single molecules on a millisecond time scale by magnetically driven oscillation of an atomic force microscope cantilever. Langmuir. 21:4765–4772. 12. Kawakami, M., K. Byrne, ., D. A. Smith. 2006. Viscoelastic properties of single poly(ethylene glycol) molecules. ChemPhysChem. 7:1710–1716.
SUPPORTING MATERIAL
13. Khatri, B. S., K. Byrne, ., T. C. McLeish. 2008. Internal friction of single polypeptide chains at high stretch. Faraday Discuss. 139:35– 51, discussion 105–128, 419–420.
Supporting Materials and Methods is available at http://www.biophysj.org/ biophysj/supplemental/S0006-3495(16)30706-8.
14. Kageshima, M. 2012. Magnetically-modulated AFM for analysis of soft matter systems. Curr. Pharm. Biotechnol. 13:2575–2588.
AUTHOR CONTRIBUTIONS A.J.K., P.E.M., G.D., and S.K.S. designed the research. F.B. and Yu.G. performed the research. P.E.M., A.J.K., and D.V.K. selected the molecules to be studied and provided the necessary reagents. S.K.S., with the help of Yu.G., F.B., and A.J.K., elaborated the theoretical model. S.K.S., F.B., Yu.G., A.J.K., D.V.K., and G.D. interpreted the results. F.B., Yu.G., and G.D. processed and analyzed the data. All authors contributed to writing the manuscript and approved the final version.
ACKNOWLEDGMENTS The authors thank Ge´rard Gremaud for useful discussions and Carine Ben Adiba for preparation of the I27 proteins. This work was supported by the Swiss National Science Foundation (grant No. 200021-150161) and the National Science Foundation (MCB 1517245 to P.M.).
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