Download Torsion stiffness of a protein pair determined by magnetic

Torsion stiffness of a protein pair determined by magnetic particles
X.J.A. Janssena#, J.M. van Noorloosa, A. Jacoba, L.J. van IJzendoorna, A.M. de Jonga, M.W.J. Prinsa,b*
a
b
#
*
Eindhoven University of Technology, Eindhoven, The Netherlands
Philips Research, Eindhoven, The Netherlands
Currently at: Delft University of Technology, Delft, The Netherlands
Correspondence to: [email protected]
Correspondence about the submitted manuscript:
[email protected]
Tel: +31152785394, Fax: +31152781202
Postal Address: Applied Physics, TU Delft
Lorentzweg 1
2628 CJ Delft
The Netherlands
Abstract
We demonstrate the ability to measure torsion stiffness of a protein complex by applying a controlled torque on a
magnetic particle. As a model system we use Protein G bound to an IgG antibody. The protein pair is held
between a magnetic particle and a polystyrene substrate. The angular orientation of the magnetic particle shows
an oscillating behavior upon application of a rotating magnetic field. The amplitude of the oscillation increases
with a decreasing surface coverage of antibodies on the substrate and with an increasing magnitude of the
applied field. For decreasing antibody coverage the torsion spring constant converges to a minimum value of
1.5 · 103 pN·nm/rad that corresponds to a torsion modulus of 4.5 · 10 4 pN·nm2 . This torsion stiffness is an upper
limit for the molecular bond between the particle and the surface that is tentatively assigned to a single Protein
G-IgG protein pair. This assignment is supported by interpreting the measured stiffness with a simple
mechanical model that predicts a two orders of magnitude larger stiffness for the proteinG – IgG complex than
values found for micrometer length dsDNA. This we understand from the structural properties of the molecules,
i.e. DNA is a long and flexible chain-like molecule whereas the antibody-antigen couple is orders of magnitude
smaller and more globular in shape due to the folding of the molecules.
1. Introduction
The advances in single-molecule biophysics research techniques have sparked a strong interest in the
nanomechanical properties of biological molecules. Insights are obtained on the response of biological molecules
to force and torque in direct relation to their function. Research has mostly focused on structural properties of
DNA and the relation with enzyme activity involving gene transcription, replication and chromosomal packaging.
Force extension measurements have revealed structural transitions of DNA and have been used to characterize
binding affinities and binding kinetics for both small molecules and more complex proteins (1,2). The
application of torque to single molecules is achieved by using rotating micropipettes (3), by using magnetic
tweezers (4) and by the optical torque wrench (5). Using these techniques, torque induced structural transitions
of DNA have been found and the uncoiling of DNA by topoisomerase IB has been shown to be torque dependent
(6-12). Proteins are structurally very different and so are their nanomechanical properties. The torsional rigidity
of multi-protein actin fibers has been investigated and shows the presence of discrete twist states which are
related to the rigidity of the actin network in cells (13,14). However, the application of torque to individual
proteins is virtually unexplored. Single-protein measurements are of strong fundamental interest, since such
studies promise to generate insights into energy landscapes and the connection between metastable protein
conformations and protein function. In addition, measurements of the torsional rigidity of individual proteins are
relevant for immunoassay biosensing applications with the aim to reach high selectivity and sensitivity (15,16).
In this paper we demonstrate the ability to measure the torsion stiffness of a biomolecular system with a size of
only a few tens of nanometers, namely a pair of proteins. Due to the small size of proteins, the torsion modulus is
expected to be relatively large. The challenge is to apply directly to the molecules a relatively large but also
accurate and reproducible torque. In this paper, we will demonstrate how the torsion properties of a protein pair
can be measured using magnetic particles in a rotating magnetic field. We will describe the experimental method
and extract torsion stiffness data for a model protein pair consisting of Protein G bound to an IgG antibody.
2. Materials and methods
The experimental arrangement is sketched in figure 1a. We use superparamagnetic particles (Dynal M-270
carboxyl, diameter 2.7 μm) that consist of a composite material of iron oxide nanoparticles in a polystyrene
matrix. In order to be able to visualize the rotation of the particles by optical microscopy, the magnetic particles
are labeled with 250 nm non-magnetic nanoparticles (Figs. 1a and 1b).
A rotating magnetic field with a constant field strength (< 25 mT) is used to apply a torque on the magnetic
particles. We determined the magnitude of the applied torque from the viscous drag at the maximum rotation
frequency of the particle free in solution [the method is described in Ref. (17)]. The Dynal M-270 particles used
in this study show a lineair relationship between the maximum rotation frequency and the applied field strength
in the field regime below 25 mT, which can be explained by the presence of a small remanent moment in the
particles with a magnitude of 1.3  1016 Am2 and a spread of 25%. In previous work (17) the presence of a
remanent moment was attributed to the tail of the iron oxide nanoparticle size distribution inside the
microspheres, for which the magnetic anisotropy exceeds the maximum energy of an iron oxide nanoparticle in
the applied field.
The Dynal M-270 carboxyl particles are coated with Protein G and labeled with the non-magnetic particles using
the following procedure: (i) Biotin EZ-link (18) is coupled to the M-270 particles by activation of carboxyl
groups using 1-ethyl-3-(3-dimethylaminopropyl) carbodiimide (EDC). (ii) The excess of EDC and unbound
biotin EZ-link is removed. (iii) Protein G is added to let reacting with the remaining activated carboxyl groups in
the presence of N-Hydroxysuccinimide (NHS). (iv) The excess of NHS and unbound Protein G is removed. (v)
Label the M-270 particles with 250 nm streptavidin-coated nanoparticles (19) that bind to the biotin. (vi) Quench
the activated carboxyl groups with ethanolamine. (vii) Wash and store the particles in PBS buffer. A graphical
representation of the functionalization process can be found in Fig S1 in the supporting material. Note that the
activation of the carboxyl groups and the coupling of Protein G is performed in two separate steps, to avoid
cross-linking of carboxyl and amine groups in Protein G by EDC. We verified the presence of Protein G and
biotin on the particles by ELISA tests. Finally the presence of the 250 nm particles is proven using scanning
electron microscopy (SEM) and light microscopy images (Figs. 1b and 1d).
The experiments were performed in a fluidic cell with a diameter of 9 mm and a depth of 0.12 mm, made using a
polystyrene substrate and a Secure-SealTM spacer. Mouse IgG antibodies (8) are physically adsorbed on the
substrate by incubating for 45 minutes with 100 μL of a given concentration of IgG in phosphate buffered saline
(PBS). Thereafter the substrate is rinsed with PBS and blocked with Casein for 45 minutes to reduce nonspecific binding of the particles to polystyrene. After rinsing the sample with PBS, the functionalized particles
are added and the cell is closed with a cover slip. The particles sediment and bind selectively to the substrate due
to the formation of IgG – Protein G bonds. During this incubation, a non-rotating magnetic field of 3 mT is
applied to ensure that the magnetic moments of the particles align parallel with the substrate before binding.
Thereafter the sample is turned upside-down to allow unbound particles to move away from the sensing surface
and sediment to the bottom of the cell. Finally the sample is placed under an optical microscope (Leica
DM6000M) equipped with a water immersion objective (Leica HXC APO L63X0.90 W U-V-I) with a standard
halogen lamp (Xenophot HLX 64625) to illuminate the sample and a high speed camera (Redlake MotionPro
HS-3) recording at 100 Hz and a rotating magnetic field is applied to the particles (Fig. 1c). The homogeneous
rotating magnetic field is created using a quadrupole system with four individually controlled coils and soft-iron
poles connected by a soft-iron yoke. The application of a rotating field causes an angular motion of the particles,
which is analyzed off-line using image analysis software written in MatLab (The Mathworks Inc.).
To find the conditions for particle binding by the ProteinG-IgG system, we determined the fraction of bound
particles (Nbound/Ntotal) for different IgG concentrations during incubation, with Ntotal typically 150 (Fig S3 in the
supporting material). Below 10 nM IgG, less than 20% of the particles bind to the surface which is comparable
to the number of non-specifically bound particles to a casein blocked substrate without IgG. These particles are
only loosely bound to the substrate and detach in the first few seconds of the measurements. This is most
probably caused by the fact that the pole tips are located underneath the substrate, resulting in a small field
gradient directed away from the substrate. This field gradient causes loosely bound particles to be pulled off
substrate and to sediment to the bottom of the flow cell. The particles that remain bound after applying the field,
we assume to be specifically bound and are used for the torsional stiffness measurements. Between 10 nM and
100 nM the bound fraction rises steeply from 20% to 90%, and practically all particles bind at 1 μM. We
attribute the rapid increase in particle binding above 10 nM to the fact that the surface coverage of IgG becomes
high enough so that each particle can form at least one bond. Experimental results will be reported at IgG
concentrations of 50 nM (low number of bonds) and 1 μM (high number of bonds). Since the first particles bind
specifically near 10 nM and we assume that the surface coverage of the IgG increases linear with IgG
concentration during incubation, we can (in first order) estimate that for 50 nM IgG each particle can form at
maximum 5 and at 1 uM IgG at maximum 100 bonds with the substrate.
3. Basic principle of the experiment
We describe a first-order physical model of the experiment to clarify its basic principles. We model the torsion
properties of a protein pair that is captured between a rigid particle and a substrate by a spring that obeys the
angular form of Hooke's law:
  k ( )
(1)
with τ the torque on the spring, k(θ) the angle dependent torsion spring constant and θ the angular rotation of the
spring away from its equilibrium position. The equation of motion of the particle now gives the balance between
the applied magnetic torque (left-hand side) and the sum of the hydrodynamic and spring torsion torque (righthand side):
mB sin(t   )  8 R3C
d
 k ( )
dt
(2)
with m a permanent magnetic moment of the particle that corresponds to the remanent magnetisation of the
particles, B the applied field, ω the field frequency, η the effective viscosity of the fluid and R the radius of the
particle. The hydrodynamic drag on the particle has to be corrected for the close proximity of the substrate. We
simulated a sphere rotating in fluid at various distances from a substrate (Fig S2 in the supporting material). We
found an increase of 22% in the rotational drag when the particle approaches the substrate. In the analysis of our
results in the remaining part of the paper we therefore use a correction factor C of 1.22
We numerically solved this differential equation for a system with an angle-independent torsional spring
constant. We find that the particle shows characteristic movements when a rotating magnetic field is applied (see
Fig 2), depending on the ratio between the magnetic torque (mB) and the torsional spring constant (k). When the
magnetic torque is much smaller than the torsional spring constant (mB/k<<1), the amplitude of the oscillations
is very small. The oscillations become very clear when a magnetic torque is applied that is comparable to the
torsional spring constant. We will now discuss the shape of the oscillations in detail. When at a certain instant
the field is applied at a small angle to the magnetic moment, the particle twists toward the magnetic field and the
molecular spring is loaded. For increasing angular orientation of the field, the torque due to the spring increases
and the angle between the magnetic moment and the field also increases. When the angle between the magnetic
field and the magnetic moment exceeds 90 degrees, the magnetic torque decreases and the particle is rotated in
opposite direction due to the torque of the spring. As a result, the angle between the field and the magnetic
moment increases further and when the angle exceeds 180 degrees, the magnetic torque changes direction and
pulls the particle back even faster (causing the unequal forward versus backward speed of the rotation).
Thereafter the angle between the magnetic moment and the field decreases, the particle is pulled back to its
equilibrium position and the cycle is repeated.
When the magnetic torque is much larger than the torsional spring constant (mB/k>>1), the spring rotates over
several revolutions until the torque of the spring equals the magnetic torque. When the field continues to rotate,
the spring can at maximum unwind one turn before the magnetic moment is again overtaken by the field. As a
result the equilibrium position of the oscillations is shifted away from zero. On the molecular level this means
that the molecule is first coiled over several rotations and then rotated back and forth over an angle smaller than
one full rotation.
In the calculation of figure 2, an angle-independent torsional spring constant (k(θ) = k) was used. In reality k(θ)
is likely to be a more complicated function, which may change the detailed shape of the curves and may reveal
detailed molecular properties. However, the dominant oscillatory behavior will remain the same as it arises from
the fact that the field periodically overtakes the magnetic moment.
4. Results and discussion
To determine the torsion properties of the biological bond, we have measured the rotation of particles bound to
surfaces incubated with buffers containing different IgG concentrations. A rotating magnetic field was applied
with a magnitude ranging from 4 mT to 22 mT and a low rotation frequency was selected (0.2 Hz) to minimize
the viscous drag. The angular movement of bound particles appeared to fall into two categories: particles showed
either oscillations of angular orientation (cf. Fig. 2) or were static without any angular motion. The fraction of
oscillating particles decreases with increasing IgG concentration: surfaces incubated with 50 nM IgG yielded
bound particles of which 50% were oscillating, while only a few percent of particles were oscillating on surfaces
incubated with 1 μM IgG. We attribute the decrease of the oscillating fraction with increasing IgG concentration
to the formation of multiple bonds between a particle and IgG molecules at the surface. Consequently, the
effective bond between the particles and the substrate becomes stiffer and the particles are not able to follow the
rotating field.
Figure 3 shows measurements for two oscillatory particles, bound to surfaces prepared with the two different
IgG concentrations. The particles clearly show a repetitive angular motion, following the rotating magnetic field
up to a certain point after which the rotation is reversed. The particles are able to follow the rotating magnetic
field over tens of degrees. This indicates that the applied torque is of the order of the torsional stiffness of the
biological system i.e. for a very weak system the particle would follow the rotating field over a larger angle and
possibly even multiple turns, whereas hardly any rotation would be observed for a very stiff system (see also Fig
2). The artifacts in the rising slopes of the measured saw tooth curves arise from the image analysis which relies
on comparing the images in the movie stream with a reference image. Due to the finite number of pixels in the
image, the cross-correlation has a maximum error when the particle is rotated over 45 degrees with respect to the
reference image. Therefore these artifacts occur each period at the same orientation of the particle. Note that the
error does not propagate throughout the measurement since for each image the absolute angle is determined
rather than a cumulative one. For very low field magnitudes (B < 3 mT) we observe that the particle oscillation
frequency equals the rotation frequency of the field as predicted by the model described above using a permanent
magnetic moment. However, for field amplitudes exceeding 3 mT we observe that the oscillation frequency of
the particle is twice the rotation frequency of the field. We attribute the double frequency to a remagnetization of
the particle when the field opposes the magnetic moment of the particle. A remagnetization is possible when the
applied field exceeds the coercive field of the particle.
The amplitude of the oscillations is measured in more detail as a function of the applied field (Fig. 4a) for the
two different IgG concentrations (50 nM and 1 μM). In each of the two measurements a single particle is studied,
to keep the magnetic moment and hydrodynamic properties constant. For increasing field strength, the magnetic
torque increases and the particle is able to rotate over a larger angle before the torque of the biological system
equals the maximum applied magnetic torque. For increasing IgG concentration we expect that more Protein G IgG bonds are formed between the particle and the surface. As a result, the area over which the particle and the
substrate bind increases in width and therefore the effective bond between the particle and the substrate becomes
stiffer and the maximum angle over which the particle rotates decreases when applying the same magnetic
torque.
As mentioned before, the field frequency of 0.2 Hz was chosen to minimize the influence of the hydrodynamic
drag i.e. the magnetic torque and torque due to the biological binding have to dominate the measured angular
motion of the particle. In order to justify this assumption, we estimated the influence of the hydrodynamic drag.
In all experiments, the angular velocity of the particles is smaller than 1 rad/s. With the viscosity of water (1 · 10 3
Pa s), this gives a hydrodynamic torque of 70 pNnm. For the field strength ranging from 3 mT up to 20 mT, the
applied magnetic torque increases from 4 · 10 2 pNnm to 3 · 103 pNnm. As a consequence, the contribution of the
hydrodynamic torque decreases from 20% at a field strength of 3 mT down to 2% at a field strength of 20 mT.
Since the torque due to the biological binding is of the same order as the magnetic torque (Fig. 3), it can be
concluded that indeed the magnetic torque and torque due to the biological binding dominate over the
hydrodynamic drag.
When measuring the angular movement of different particles on the same sample, a substantial variation in the
measured maximum angle was found. The amplitude of the oscillations (max) was found to decrease with
increasing IgG concentrations using a rotating field of 20 mT (Fig. 4b). Note that the functionalization of the
particles and the polystyrene substrate are both stochastic processes, which are expected to yield random spatial
distributions of IgG and Protein G on the surfaces. Therefore we attribute the observed variability in the
measured maximum angle at a given IgG concentration to a varying number of bonds between the particles and
the surface as explained before.
From the measured angular orientation in time we can determine the torque k(θ)θ due to the torsional spring by
rewriting the equation motion:
k ( )  mB sin(t   )  8 R3C
d
dt
(3)
Now the saw tooth signal (figure 3) is averaged over 4 periods of the oscillating angular orientation. Thereafter
k(θ)θ is calculated for a number of angular orientations of the particle (θi) using equation 3. The magnetic
moment m, magnetic field strength B and hydrodynamic drag 8R3C in this equation are constants given by the
experimental conditions. The orientation of the field ωt is known from the rotational frequency  of the field and
the rotational speed of the particle dθ/dt is calculated from linearly fitting θ(t) around the angle θi. We performed
this calculation for several angles i and plotted in figure 5 versus the angle i. A linear fit of the data in figure 5
gives a torsion spring constant of (1.5 ± 0.3) · 10 3 pNnm/rad and (4.2 ± 0.6) · 103 pNnm/rad for a substrate
incubated with 50 nM and 1 μM IgG respectively. In a series of experiments with decreasing surface coverage of
antibodies we found the torsion spring constant to converge to its smallest value of 1.5 · 10 3 pNnm/rad, which is
an upper limit for the torsion constant of the molecular bond between the bead and the substrate prepared with
the solution of 50 nM IgG. This molecular bond is tentatively assigned to a single Protein G – IgG protein pair.
The torsion spring constant is not a material property but depends on the length of the spring. One way to
compare systems with different length scales is by calculating the torsion modulus i.e. the torsion spring constant
times the length of the spring. The size of the Protein G molecule depends on its spatial structure and is not
exactly known. For antibodies the size is known and is typically 15 nm x 7 nm x 3.5 nm (20). If we assume a
length of 30 nm (twice the size of the antibody), the torsion spring constant of 1.5 · 103 pNnm/rad translates into
a torsional modulus of 4.5 · 104 pNnm2.
It is interesting to compare the derived protein torsion modulus to the modulus of double stranded DNA (9).
dsDNA moduli have been reported of the order of 4 · 10 2 pNnm2, which is roughly two orders of magnitude
smaller than we find for the Protein G - IgG system. An order of magnitude estimation of the ratio in the torsion
moduli between DNA and a Protein G - IgG complex might be obtained by calculating the torsion modulus of
the biological system in a continuum approximation. The biological system might be considered as a cylindrical
rod with a diameter R and a length L. When the rod is twisted by applying a torque τ, the twist θ is given by (21):

L
JG
(4)
with G the shear modulus of the material and J the polar moment of inertia of a disk ( J  1 / 2R 4 ).
Introduction of the torsion spring constant k = GJ/L gives the angular form of Hooke's law (θ = τ/k). The
torsional modulus (kL) of the cylindrical rod increases with the diameter of the rod to the fourth power:
kL 
GL
1
L  GJ  R 4G
L
2
(5)
Using this relation and assuming equal values for the shear moduli, we can estimate the ratio in torsion modulus
between dsDNA and a protein-protein couple. The radius of a dsDNA strand (1.2 nm (22)) is about three to four
times smaller than that of an antibody. According to equation 5, a radius that is smaller by a factor 3 to 4 gives a
torsion modulus that is smaller by a factor 81 to 256. The ratio between the measured torsion moduli is
approximately 4.5 · 104 / 4 · 102 = 125. Interestingly, the calculated and measured ratios are very similar which
supports our assignment of the molecular bond between the bead and the surface to a single Protein G – IgG pair.
A next step is to perform a more detailed comparison based on molecular models of torsional deformation.
4. Conclusion
We have used rotational actuation of magnetic particles to measure the torsion stiffness of a small biological
system sandwiched between a particle and a substrate. As a model system we used Protein G on the particles and
an IgG antibody on a polystyrene substrate. The angular orientation of the bound particles shows an oscillating
behavior upon applying a rotating magnetic field. We attribute the decreases of the amplitude with increasing
antibody concentration to the formation of multiple bonds between the particle and the surface. The lowest
torsion spring constant that we found is 1.5 · 10 3 pNnm/rad, which corresponds to an estimated torsion modulus
of 4.5 · 104 pNnm2. We attribute these values to the spring constant and torsional modulus of the molecular bond
between the magnetic particle and the surface. We have shown that the tentative assignment of this bond to a
single Protein G – IgG protein pair is supported by a simple model that predicts the torsional modulus based on a
direct comparison with single molecule deformation experiments of DNA. The modulus found for the Protein G
– IgG pair is at least two orders of magnitude larger than the torsion modulus of double stranded DNA, which we
qualitatively understand from the structural properties of the molecules, i.e. DNA is a long and flexible chainlike molecule whereas proteins are globular in shape due to the folding of the molecules. In conclusion, we have
proven that rotational actuation of magnetic particles can be used to measure the torsion stiffness of a
biomolecular system with a length scale of only a few tens of nanometers. The method opens a new window in
nanomechanical biophysics research, allowing detailed studies of the torsion properties of protein/protein
complexes.
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Figure legends
Figure 1: Overview of the experiment. (a) To scale schematic representation of the model system to measure the
torsional stiffness of a protein pair. The magnetic particles are labeled with 250 nm non-magnetic nanoparticles
in order to be able to record the rotation of the particles with an optical microscope. Inset: Protein G on a
magnetic particle binds selectively to the crystallisable part of an IgG antibody present on the surface of a
polystyrene substrate. The surface of the substrate is blocked with Casein to prevent non-specific binding. (b) M270 magnetic particle labeled with 250 nm non-magnetic polystyrene particles, imaged using a scanning
electron microscope (c) The particle solution is contained in a fluid cell. The cell is made from a polystyrene
substrate (top), a Secure-SealTM spacer (black), and glass cover slip (bottom). The immersion objective of the
microscope is focused onto the particles that are hanging below the polystyrene substrate. The rotational
behavior of the bound particles is studied in a rotating magnetic field, generated by four coils with soft-iron
cores. (d) Microscope image of two magnetic particles. The label nanoparticles are indicated by circles.
Figure 2: Calculated angular orientation (θ) of a particle in time for different ratios between the magnetic
torque (mB) and an angle-independent torsion spring constant k(θ) = k. One rotation of the field is indicated
with T = 2π/ω where ω equals the rotational frequency of the field. The curves are obtained by numerically
solving the equation of motion (Eqn 2). For a very stiff biological system (mB/k <<1) the particle hardly rotates.
For a torsional spring constant comparable to the applied magnetic torque (mB/k = 1), the particle shows an
angular orientation that oscillates around zero. For a very weak torsional spring constant (mB/k >> 1) the
particle follows the rotating field over several revolutions where after the oscillation starts.
Figure 3: Measured angular orientation (θ) of magnetic particles, bound to a substrate coated with 50 nM or 1
μM IgG. The field magnitude was 22 mT. The field rotation frequency was 0.2 Hz. The measured oscillation
frequency of the particles is twice the rotation frequency of the field, which is caused by remagnetization of the
particles.
Figure 4: (a) The amplitude of the oscillation increases with increasing field strength since the magnetic torque
increases with the applied field. (b) For a fixed field strength of 20 mT, the amplitude of the oscillation decreases
with increasing IgG concentration, which we attribute to an increase of the number of bonds between the surface
and the particle. The dashed line is a guide to the eye.
Figure 5: The torque due to the biological system plotted versus the angular orientation of the particle. Linear
fitting of the data gives a constant torsion spring constant of respectively (1.5 ± 0.3) · 10-18 Nm/rad and (4.2 ±
0.6) · 10-18 Nm/rad for a substrate incubated with respectively 50 nM IgG and 1 μM IgG.
Figure 1
a)
b)
Figure 2
Figure 3
c)
d)
Figure 4
Figure 5
Supporting Material
Figure Legends
Figure S1: Schematic representation of the functionalization and labelling of the M-270 particles. A plain M-270
particle terminated with carboxyl groups is functionalized with biotin EZ-linker through EDC activation. This
particle is then functionalized with protein G. This is achieved by simultaneously adding protein G and NHS to
couple the protein G to the remaining free carboxyl groups. The functionalized particle is finally labeled with
streptavadin coated 250 nm particles via the biotin-streptavadin binding.
Figure S2: The correction factor C for the hydrodynamic torque on a sphere rotating in bulk fluid as a function of
the distance between the surface and the bottom of the sphere where the distance is normalized on the sphere’s
radius. The results are obtained from a Comsol Multiphysics FEM simulation.
Figure S3
The fraction of functionalized M-270 particles bound to the IgG coated polystyrene substrate as a function of the
IgG concentration used for the functionalization of the substrate. Each point on the graph entails a measurement
involving about 400 particles. The line is plotted as a guide to the eye.
Figure S1
B io t in E Z - lin k
&
EDC
B io t in E Z - lin k
P r o t e in G
S t r e p t a v id in b e a d s
Figure S2
Figure S3
P r o t e in G
&
NHS
S t r e p t a v id in
beads
Reviewer 1 Comments for the Author...
This is a solid article describing the application of torsional stress to a protein/ligand pair, and the
measurement of the torsional stiffness and modulus of that pair. The approach is novel and interesting,
and generally well-argued. I have a number of concerns, however, that I would like addressed before I
can recommend publication:
1) The authors use a constant bead magnetic moment in their analysis. Since the beads are
superparamagnetic, I would expect the moment to vary as the (constant magnitude) applied field
becomes misaligned with the moment-- i.e., since the component of the field along the moment varies
with the angle (&[omega] t -&[theta]), the induced moment should also vary with angle. Why use a
constant moment?
Ideally, the used Dynal M270 beads are superparamagnetic which means that the induced magnetic
moment is always aligned with the applied field. Hence no torque can be applied to truly
superparamagnetic material. However, it is known that these beads can be rotated which means that a
torque can be applied. In previous work, we have shown that besides the induced magnetic moment
(which is parallel with the applied field), the beads also have a small field independent magnetic
moment i.e. permanent moment. This is caused by the tail of the iron oxide nanoparticle size
distribution inside the beads for which the magnetic anisotropy exceeds the magnetic energy of the
iron oxide nanoparticle in the applied field. It is important to realize that the torque applied in our
experiments is calibrated by measuring the viscous drag at the maximum rotation frequency which is
independent of the underlying mechanism. The torque increases linear with the applied field strength
which justifies the assumption of a fixed remanent magnetic moment of the particle. We have clarified
the calibration procedure and linear relationship between torque and applied field (in the relevant
regime of the field strength) in the text.
It is also important to emphasize that the moments are aligned parallel to the substrate by applying a
stationary magnetic field during incubation.This reduces the spread in effective magnetic moment i.e.
the component of the magnetic moment that is parallel with the field and is already mentioned in the
text. During the rotation of the field, the torque on this moment indeed varies with the angle between
the field and the moment. The term sin(t – )
account in our analysis.
2) I would like to see more data supporting their argument that a spring constant of 10^-18 N-m/rad
corresponds to a tether consisting of a single protein pair. The argument relies on the logic that the
lowest measured torsional modulus must result from a single pair. This argument is implicitly
statistical, but no statistics are given. What I mean is this: how can we be sure that the authors have
tried to measure enough beads to create a near-certain probability of getting a single tethering pair?
For example, if they only have data on a few beads, there is a significant probability that they will not
happen to find a bead tethered by a single pair.
I was unable to find any data on how many beads were assayed; the only statement was 'Over multiple
measurements, 1.5 · 10-18 Nm/rad is the smallest torsion spring constant that we measured'. I think it
is critical to actually say how many beads were tested at the 50 nM level. It would be even better if
some analysis was done-- I understand that geometric effects on torsional spring constant preclude
direct counting of the number of tethering pairs for a given bead. But i would think that the authors
can discern (one tethering pair) from (more than one tethering pair); the ratio of those two should be
predictable from a poisson distribution. This analysis would lend a great deal of weight to their
conclusions.
For any given IgG concentration we typically measured several tens of beads. We agree with the
reviewer that our argument of the lowest measured torsional modulus is indeed implicitly statistical
and more measurements should be executed in order to really prove our statement that this torsional
stiffness is due to a single tether. However, the aim of this manuscript is to show the feasibility of
performing torque measurement on biological systems that are significantly shorter than the more
frequently used biopolymers like DNA. Given the low statistics we were not able to measure the
predictable ratio between one and more than one tethering pair. Especially at the low IgG
concentrations, only a few beads bind to the substrate due to the low surface coverage of the
antibodies (Figure S3 of the Supporting Material).
We can (in first order) estimate the number of bonds between the bead and the substrate as follows:
For a IgG concentration of 1 x 10-8 M the beads start to bind specifically to the substrate i.e. the
fraction of bound beads exceeds the background of non-specific binding. Now we assume that: this
increased binding is due to the fact that the beads can form a single tether and that the surface
coverage of the IgG increases linear with IgG concentration during incubation. From these
assumptions we can estimate that for 50 nM IgG, each bead can form at maximum 5 and at 1 uM IgG
100 bonds with the substrate.
We have rephrased parts of our manuscript to point out that we want to show the proof of principle of
our technique and that the lowest found torsional stiffness yields an upper limit for that of a single
protein couple since we can not rule out multiple bonds.
3) The model of the drag on the spinning bead seems simplistic. First, the use of the bulk result for the
drag on a spinning bead can easily be corrected for surface effects-- by my calculation, this could be
something like a 10% increase in the drag on the spinning bead, assuming it spins on-axis. Perhaps
this is negligible at the level of analysis presented. However, it seems possible that the bead might spin
off-axis, for example if a single protein-pair tether forms off of the bead center (see, e.g. ,Klaue and
Seidel, PRL (2009) for discussion of a similar issue in a DNA-tethering experiment). As discussed in
the Klaue paper, such effects can be measured by looking at the bead trajectory under field rotation-an on-center tether would cause the bead to spin in place, while an off-axis center would create a
measurable circular trajectory. The latter would have a different drag coefficient. Has this effect been
controlled here? Can they do controls on the drag torque using a tether that freely swivels?
We agree with the reviewer that surface effects influence the hydrodynamic drag on the particle. We
realized this and we simulated a rotating sphere in fluid at various distances from the surface and
calculated the effective drag on the sphere. The presence of the surface increases the drag on the
particle with 22% as found from FEM simulations in Comsol Multiphysics. We did correct our
measurements but forgot to notice this in the manuscript. We now explicitly state that we perform this
correction and added the results of these simulations to the supporting material.
Furthermore, we agree with the reviewer that off-axis spinning of the bead will further increase the
hydrodynamic drag. However in our system, the tethers are so short (few nanometers) that off-axis
binding is limited to a very small region around the contact point between the bead and the substrate.
Furthermore, we incubated the particles while applying a magnetic field, causing the moments to align
parallel to the substrate. Configurations as mentioned in the work of Klaue et al. where the tether can
bind to the equator of the bead are therefore highly unlikely and off-axis spinning of the beads was not
observed during our measurements.
4) The data trajectories (Fig. 3) do not show the simple constant-slope rising edge predicted by the
model; instead, there is a visible deviation that occurs consistently at the same point in the cycle. The
deviation is a distinct downturn in the 1 uM data, and a decrease in slope in the 50 nM data. I would
think such a feature might indicate torque-induced rupture of a protein pair (or an antibody ripping off
a surface), but no explanation is given. Can the authors comment on this?
The deviations are artifacts caused by the analysis routine. The angle of the bead is determined from a
cross-correlation of the image with a reference image. Because of the finite number of pixels in the
image, there is a maximum error in the cross-correlation when the two images are 45 degrees rotated
with respect to each other. Note that this is not necessary to happen at an absolute angle 
degrees but depends on when the reference image is taken. This explains why this feature is visible at
~45 degree for the 1 uM measurement and at ~25 degrees for the 50 nM measurement. During the
experiments a constant-slope rising edge was observed when looking at the videostream. We do not
attribute these features in the slopes to bond breaking of any kind.
5) It appears that the line fit to the 50 nM data in Fig. 5 does not go through the origin, while the
torque should be zero at zero angle. Is this a mistake, or have I misinterpreted this plot?
The reviewer did not misinterpret the plot in figure 5. The torque should be zero at zero angle and the
linear fit to the 50 nM data in figure 5 indeed does not go through the origin. We have refitted the data
correctly i.e. fit through zero.
6) Throughout, the authors use SI units. I think this paper would be easier to follow (and, particularly,
to compare their data to other single-molecule manipulation papers) if they quoted torques in pN nm
units, as is relatively standard in the field.
We changed quoted torques in pNnm units.
7) I think the discussion on the number of protein pairs that tether a bead would be clarified if Fig. 1
had a scale drawing of the geometry, so that the huge difference between the bead size and protein size
was clear.
Figure 1a is changed such that the picture is a scale drawing with a close-up of the biological system
in the desired configuration i.e. only one tether.
Reviewer 2 Comments for the Author...
The authors who have developed recently a method to measure and control the torque acting on single
magnetic beads apply this method here to measure the torsional stiffness of a complex of protein G
and IgG antibody. The find the its torsional modulus to be much stiffer than that of DNA.
Overall this is a nice study, which also includes some small, but nice part of physics about the
dynamics of this rotationally driven system. I generally support publication but the authors should
address a couple of significant issues I see.
1. While results and theory part of the paper are appropriate in length and provide sufficient detail, the
introduction and the discussion are extremely short. The authors should spend more time in explaining
why torsional measurements on proteins and other biological macromolecules are currently of
significant interest and what (biological) relevance their findings have for the field.
Also they should acknowledge better other achievements in the field. They should cite previous work
on the torsional properties of actin filaments. While they mention previous measurments in which the
torsional properties of DNA were studied, they almost exclusively cite work from a single group in the
Netherlands. I recommend to include also other studies from this rather broad field, e.g.
Bryant et al. Nature 2003
Maffeo et al. Phys Rev Lett 2010
Mosconi et al. Phys Rev Lett 2009
Celedon Nano Lett 2009
We have extended the introduction referring to single molecule techniques that apply force and torque
to biological molecules. Advances in DNA research are properly addressed and the relevance of
applying torque to proteins and measuring the torsional rigidity of proteins have been clarified.
2. The authors use Dynal M-270 beads and take as far as I can see a remanent magnetic moment from
M-280 beads. Have the authors verified that this parameter is the same for both types of bead? If not,
this would be a severe caveat! I am not entirely convinced about the existence of a remanence for
these particles. A different study using M-280 beads provided an alternative explanation for the
magnetic anisotropy of such particles (Klaue et al. Phys Rev. Lett. 2009). This work should be
mentioned and the issue be thoroughly discussed.
We have used the M270 beads and measured their permanent magnetic moment the same way as we
did with the M280 beads in our previous paper (17). The permanent magnetic moment of this batch of
M270 beads is found to be a factor 10 smaller than that of the M280 beads. Furthermore, the spread in
permanent moment of the M270 beads is smaller (25%) than the spread of the M280 beads (spread of
a factor 10). To avoid any confusion, we have rephrased the section of the manuscript to make it clear
that we did measure the properties of the M270 beads.
Regarding the work of Klaue et al.: for an induced magnetization that has a preferred orientation inside
the bead due to any anisotropy, the applied torque should scale with the square of the field as long as
the magnetization is not saturated. In our measurements we applied fields up to 25 mT and for these
low fields we see a linear relation between the applied field and the torque applied to the bead. For us
this indicates that, for these low field strengths, the torque is due to a permanent magnetic moment.
Nevertheless, our measurements do not exclude that at higher fields strengths (several hundred mT as
in the work of Klaue et al.) another mechanism (e.g. anisotropy of field induced magnetization)
contributes to the torque. For the scope of this manuscript (showing the feasibility of measuring the
torsional stiffness of small molecules/couples) the torque generating mechanism is less of importance
as long as the torque to the particle can be measured. It is important to realize that the torque applied in
our experiments is calibrated by measuring the viscous drag at the maximum rotation frequency which
is independent of the underlying mechanism of torque generation.
3. A few details are missing in the the Methods section:
How do the non-magnetic beads bind to the large beads? What is the modification and the vendor of
these beads?
We added the vendor and type of the beads.
Why did the authors coat the magnetic beads with biotin before Protein G coupling. Why was then
streptavidin added?
In order to clarify the functionalization steps in our experiments we added a graphical representation
of the scheme in the supplementary data.
At what frame rate were the images of the rotating beads acquired?
The frames were recorded at a rate of 100Hz, this has been added in the section Materials and Methods.
What light source was used for sample illumination?
For the illumination the standard halogen lamp of the Leica microscope was used, the type number is
now specified in the section Materials and Methods.
What happens below 20 nM IgG? Is the observed binding due to non-specific interactions? What
happens if the torsional measurements are carried out with these particles?
Below 20 nM IgG hardly any beads (<10%) bind to the substrate. We attribute the beads that bind
below 20 nM IgG to non-specific binding since this fraction is comparable to the number of beads that
bind to the substrate in absence of IgG. The fraction of beads binding as a function of IgG
concentration is added to the supplementary data. Torsional measurements on these beads have not
been performed in detail since the measured curves (angle in time) show non reproducible behavior i.e.
no saw tooth curve is measured. Furthermore almost all of these non-specific bound beads detach from
the surface during the first few seconds of the measurement. This is most probably caused by the small
gradient that applies a force on the particles that is directed away from the substrate. These findings
have been added to the section Materials and Methods.
4. What is the equation used for the estimation of the hydrodynamic drag? Is it simply 8*pi*eta*R^3?
In such close proximity to a surface the largely increased hydrodynamic drag should be appropriately
considered.
The simple 8*pi*eta*R^3 formula for the hydrodynamic drag is corrected with a factor 1.22 for the
proximity of the substrate. The presence of the surface increases the drag on the particle with 22% as
found from FEM simulations in Comsol Multiphysics. We simulated a rotating sphere at various
distances from the surface and calculated the effective drag on the sphere. We did correct our
measurements but forgot to notice this in the manuscript. We now explicitly state that we perform this
correction and added the results of these simulations to the supporting material.
5. How was the conversion of “the data in figure 3" done to obtain the plots in figure 5? What were the
fixed parameters that entered the conversion and where were they taken from? Given the many data
points of the curves in Fig. 3 one should expect also more data points in figure 5. Also mostly data is
presented for only two selective measurements at 1 uM and 50 nM. Figure 5 would be a good spot to
show also torque plots for the other measurements.
The conversion of the data is done using equation 3: with t - ) the angle between the field and the
bead at a given time. This angle is calculated using the measured angle of the bead in time and the
frequency of the field. The angular speed of the bead d/dt is fitted from the data presented in figure 3
on a limited number of positions in time after averaging over 4 periods of the saw tooth.
6. The most severe problem I see for the paper: How can the authors make sure that they really
measure the torque on a single IgG protein complex? If they simply take the lowest observed value
then I am skeptical whether this lowest value could simply be due to non-specific bead attachment. Is
there a way to estimate the average number of bonds for a given IgG concentration based on the
titration the authors did? What would be the average number of bonds at 50 nM? Also, why do the
authors say that they obtain a lower limit for the torsional spring constant? Given that they could have
multiple bonds I would say this is an upper limit.
We do not attribute the lowest measured torque to non specific binding, since beads that are non
specifically bound (for substrates below 20 nM and in absence of IgG) show non reproducible
behavior over several rotations of the field i.e. no sawtooth-like curve could be measured. Furthermore
almost all of these non-specific bound beads detach from the surface during the first few seconds of
the measurement. This is most probably caused by the small gradient that applies a force on the
particles that is directed away from the substrate. The beads that remain attached to the surface
consistently show the same saw tooth behavior. This behavior is not observed in the absence of IgG.
We realize that these indications do not give solid proof, but given the low fraction of non-specifically
bound bead in absence of IgG it is highly unlikely that all the beads we measured are non-specifically
bound.
We have added to our manuscript the fact that we observe consistently the same saw tooth behavior
for many beads and that the measured torsion constant converges to lowest observed value for a series
of experiments with decreasing surface coverage of IgG. We describe in the text that this is the upper
limit of the torsion constant of the molecular bond between the surface and the bead. In a second step
we tentatively assign this bond to that of a single protein pair and use the model as corroborative
evidence to support this assignment. This way of presenting the data does not contain any unjustified
conclusions and still acknowledges the strong arguments we have to identify the measured torsion
constant as that of a single protein pair.
7. The torsional modulus of DNA (also from the cited source) should be 4*10^-28 Nm^2 rather than
3*10^-28 Nm^2
We changed the value in the manuscript.
8. What exactly is theta max in Fig. 4?
Theta max in figure 4 is the maximum angle over which the bead is rotated i.e. the top of the saw tooth.
9. Spelling:
rotaton should become rotation on page 3 (1st paragraph)
proteomic molecules should become proteins/protein complexes
Both spelling errors are corrected.