Supporting text
Used formulas for DNA mechanics and drag coefficients
To describe the force extension behavior of DNA we combined the usually applied formula for
the extensible worm-like chain model (1) with an improved formula for the inextensible worm like-chain model (2) resulting in:
k T
F ( zDNA )  B
p
2

z 
z 
1
1 z 
 + entr   0.5164228 entr   2.737418 entr 

2
 4(1  zentr L0 ) 4  L0 
 L0 
 L0 
4
5
6
3
z 
z 
z 
z 
+16.07497 entr   38.87607 entr  +39.49944 entr   14.17718 entr 
 L0 
 L0 
 L0 
 L0 
7



[1]
,where the entropic part of the DNA extension zentr is given by:
zentr  zDNA  L0 F S .
[2]
L0 denotes the DNA contour length in the absence of force, zDNA the DNA end-to-end distance at
a given force F, S the DNA stretching modulus for which 1000 pN was taken throughout the
analysis, p the DNA persistence length and kBT the Boltzmann constant times temperature.
The relation for the spring constant of the DNA kDNA can be obtained by derivation of Eq. 2 by F,
which provides:
1
kDNA

dzDNA dzentr L0
1
L



 0.
dF
dF
S dF dzentr S
[3]
For the drag coefficients γtrans and γtorque of the microspheres we used the following expressions
(3, 4), which include corrections for the closely located flow cell surface:
 trans 
6R
, and
9R R
57 R 4 R 5
7 R11
R12
1





8 z 2 z 3 100 z 4 5 z 5 200 z11 25 z12
3
 5R3 
γtorque  8πηR3 1 
,
3
 16 z 
[4]
[5]
where R denotes the Radius of the magnetic microsphere, z the distance of the microsphere center
from the surface, and η the viscosity of the surrounding fluid for which 10-3 kg∙s-1∙m-1 was taken
throughout the paper.
-1-
Power spectrum of the coupled rotational and translational fluctuations
The set of linear, coupled Langevin equations describing the coupling between translational and
rotational Brownian fluctuations of the magnetic microsphere (Eq. 2, main text) can be written in
the following matrix form:
  1z  z  F(t ) ,
with
z 
z   rot ,
 z 
 F (t ) 
F(t )   rot ,
 Ftrans (t ) 
k
k
κ   DNA rot
  kDNA
 kDNA 
,
kDNA 
  1
0 
.
μ   rot
1 
0

trans 

To solve this equation set we find a linear transformation z  Aς that decouples these equations,
i.e. resulting in  ς  λς  Φ(t ) . λ is the diagonal matrix with the eigenvalues λ± of the matrix μκ
with λ=A-1μκA, where A is formed by a set of eigenvectors. Φ(t) are the transformed Langevin
forces. Choosing a normalization for A for which A1μT A1  1 , the following expression can
be derived for the power spectrum Sz(f) (5):
S z ( f )  2kBTA(λ 2  (2f ) 2 1) 1 AT
T
For our system the eigenvalues are:
2
k
k
k
1  kDNA  krot kDNA  4kDNAkrot

 
,
  DNA rot  DNA 

2 rot
2 trans 2   rot
 trans 
 trans rot
[6]
and for the power spectrum we obtain:
Sz ( f ) 
4k BT
1  C 2 trans rot
 1
 transC 2
1

 2
2
2
2
  rot   (2f )   (2f )

C
C
 2

2
2
2
   (2f )   (2f )

C
C

 2
2
2
  (2f )   (2f ) 

 rot C 2
1
1


 2  (2f ) 2  trans  2  (2f ) 2 
2
[7]
,with C=(λ+–(kDNA+krot)/γrot)/kDNA.
Low pass filtering due to the finite exposure time of the camera is considered in analogy to the
simple case without rotation fluctuations (see main text) by multiplying the power spectrum with
the squared sinc function. This provides the following expression for the low-pass corrected
power spectrum of the z fluctuation of the magnetic microsphere:
-2-
S zcorr ( f ) 
  rot C 2
 sin 2 (f f cam )
4k BT
1
1



1  C 2 trans rot   2  (2f ) 2  trans  2  (2f ) 2  (f f cam ) 2
Numeric integration of the spectrum from 0 to +∞ provides the RMS amplitude
[8]
z2
corr
coupl
for the
coupled translational and rotational fluctuations corrected for the low pass filtering by the camera
(see Fig. 2.), with fcam being the reciprocal value of the exposure time.
Magnetic force and torsional stiffness of the microsphere from known material properties
The force Fi acting on a superparamagnetic nanoparticle i with magnetic moment mi in a
magnetic field B can be written as (6):
Fi  (m i  )B .
The force on the nanoparticle varies in time due to thermal rotational fluctuations of the dipole
moment. For the average force on the dipole the following expression can be derived:
Fi  mi (cos  )B
with  being the angle between dipole moment and magnetic field.
Using Boltzman statistics an analytical expression for the average cos can be derived (6) and
the upper equations turns into:
mB
Fi  mi L i B ,
 kBT 
in which L is the Langevin function, kB the Boltzman constant and T the temperature. For the
whole superparamagnetic microsphere comprising many small particles the magnetic force is
obtained from the sum of the individual forces:

 m B 
Fmag   ∑mi L i  ∇B  mB ∇B .
[9]
 k BT  
 i
Thus by knowing the field-dependent apparent magnetic moment m(B), one can calculate the
force acting on the microsphere from the gradient of the magnetic field. For this, we measured
the magnetic field B along the y-direction using a small hall probe (CY-P3A, Chen Yang
Technologies GmbH & Co. KG) at various distances along the z-direction. For the apparent
magnetic moment we used magnetization curves from literature (7). To simplify the force
calculations, both the magnetic field decay and the magnetization curve were approximated by
analytical functions. We find a good agreement of the calculated forces with the average
measured forces (SI Fig. 3a). This serves as an independent control for the measured magnetic
-3-
field and the used magnetization curves (7), which will be important for calculating the torsional
spring constnt (see main text Fig 4c).
Theoretical prediction of ktorque in dependence of the anisotropy constant C
In order to calculate the torsional spring constant acting on a single superparamagnetic
nanoparticle Eq. 3 (main text) can be applied. For a particle with fixed orientation  of its
anisotropy axis to the magnetic field the angle  between magnetic moment and anisotropy axis
(SIFig. 3c, main text) can be calculated by equating to zero the derivative of Eq. 3 with respect to
. Knowing , the torque on the microsphere for a given orientation  is obtained from the
derivative of Eq. 3 (main text) with respect to  and the torsional spring constant from the second
derivative with respect to . Since no analytical solution for  can be obtained, the torsional
spring constant was calculated numerical. In order to obtain an effective torsional spring constant
for the whole magnetic microsphere the individual spring constants have to be added assuming a
certain angular distribution of the anisotropy axis of the nanoparticles. SI Fig. 3b shows the
development of the torsional spring constant under the applied magnetic field for different values
of the anisotropy constant C assuming that the anisotropy axes of all nanoparticles are perfectly
aligned. As expected (see main text), the torsional spring constant saturates at high magnetic
fields for all considered values of C. A similar saturation is also observed for more randomized
distributions of the nanoparticles.
-4-
References
(1)
Wang MD, Yin H, Landick R, Gelles J, Block SM (1997) Stretching DNA with optical
tweezers. Biophys J 72:1335–1346.
(2)
Bouchiat C, et al. (1999) Estimating the persistence length of a worm-like chain molecule
from force-extension measurements. Biophys J 76:409–413.
(3)
Schäffer E, Nørrelykke SF, Howard J (2007) Surface forces and drag coefficients of
microspheres near a plane surface measured with optical tweezers. Langmuir 23:3654–3665.
(4)
Goldman AJ, Cox RG, Brenner H (1967) Slow viscous motion of a sphere parallel to a
plane wall - I Motion through a quiescent fluid. Chem Eng Sci 22:637–651.
(5)
Moffitt JR, Chemla YR, Izhaky D, Bustamante C (2006) Differential detection of dual
traps improves the spatial resolution of optical tweezers. Proc Natl Acad Sci U S A 103:9006–
9011.
(6)
Bryant HC, et al. (2007) Magnetic needles and superparamagnetic cells. Phys Med Biol
52:4009–4025.
(7)
Fonnum G, Johansson C, Molteberg A, Morup S, Aksnes E (2005) Characterisation of
Dynabeads (R) by magnetization measurements and Mossbauer spectroscopy. J Magn Magn
Mater 293:41–47.
(8)
Howard J (2001) Mechanics of Motor Proteins and the Cytoskeleton (Sinauer Associates,
Sunderland, MA).
-5-
SI Fig. 1. Accuracy of the position detection. (a) Two immobilized 3.2 µm polystyrene
microspheres (normally used as reference) measured simultaneously. The z position axis
coorespondes to the difference of the z coordinates of both microspheres. The grey line is the
measured signal obtained with a sampling frequency of 120 Hz. The RMS amplitude is 0.25 nm.
The red line is the 1 Hz filtered data. (b) A single microsphere measured while stepping the
objective in 1 nm steps along the z coordinate. The grey line is the measured data recorded at 120
Hz. The red line is the 3 Hz filtered signal.
-6-
SI Fig. 2. Alternative representation of the model shown in Fig. 4a , describing the idealized
system with basic mechanical elements. The drag is modeled by a dashpot where force develops
linear with velocity. The stiffness is expected to be constant in the range of movement and can be
modeled by spring where force develops linear with the displacement. The mass can be
neglected, because the system is overdamped (8).
-7-
SI Fig. 3. Influence of coupling between rotational and translational fluctuations on the noise
along the z coordinate. (a) Small off-center attachment. Calculated noise for a 2µm long DNA
molecule, attached to a 2.8 µm magnetic microsphere with an off-center attachment of 0.2µm and
a torsional stiffness of 100 pN∙µm∙rad-1. The sampling frequency was 60 Hz. RMS amplitudes as
function of force were calculated for translational fluctuations only (black line, calculated
according to Eq. 1, main text), for uncoupled translational and rotational fluctuations (green line,
calculated according to Eq. 1, main text, for both types of fluctuations) and for coupled
translational and rotational fluctuations (red line, calculated according to SI Eq. 8). (b) Large offcenter attachment. Calculated noise as function of force with the same parameters as in a except
the off-center attachment for which 1.2 µm was taken.
-8-
SI Fig. 4. Calculating force and torsional spring constant for a single magnetic microsphere. (a)
Force and magnetic field as function of the magnet distance from the flow cell. The magnetic
field (green line) was measured using a hall probe. Forces were either directly measured from
thermal fluctuations of the microsphere (grey dots with blue circles representing the mean value
for a given magnet position) or calculated according to SI Eq. 9. The field-dependent
magnetization of a single microsphere (inset) was inferred from bulk magnetization curves (7).
(b) Torsional spring constant ktorque as function of the magnetic field calculated for different
values of the magnetic anisotropy constant C with C = 7000, 4000, 2000 or 1000 J m-3 (red lines
from top to bottom). Calculation were performed assuming perfect parallel alignment of the
anisotropy axis of all nanoparticles of the microsphere. The black line is the expected ktorque for a
microsphere with fixed orientation of the magntic dipole moment, i.e. with C being infinite. (c)
Sketch representing an anisotropic iron oxide nanoparticle with magnetic field (blue arrow) and
magnetic moment (green arrow) and corresponding angles. The red line shows the anisotropy
axis resulting from shape- and crystalline anisotropy.
-9-