SUPPORTING INFORMATION FOR
Comparison of Brownian Dynamic-Based Estimates of Polymer Tension with Direct Force
Measurements
Mark E. Arsenault(a), Prashant K Purohit
(a)
, Yale E. Goldman(b), Henry Shuman(b), Haim H.
Bau(a) ,*
(a)
Department of Mechanical Engineering and Applied Mechanics and
Institute, University of Pennsylvania, Philadelphia, PA, 19104
(b)
Pennsylvania Muscle
*email: bau@seas.upenn.edu
S.1 Optical Trap Calibration
Every bead trapped during our experiments was calibrated in order to convert the
photodiode signals into direct force measurements. The experiments presented in this document
were carried out with aqueous solution as the suspending medium. Similar calibration
experiments (not reported here) were carried out with glycerin. Figure S.1 depicts the power
spectrum G (and the respective fit) of an optically trapped, 1 m-diameter bead undergoing
Brownian motion, sampled at 20 kHz. The power spectrum of the thermal fluctuations of a bead
in a harmonic well is (Svoboda and Block 1994):
G f t  
k BT

 2  fc 2  ft2
,
(S.1)
where ft is the frequency,  =6r is the bead’s hydrodynamic drag coefficient, fc=/(2) is the
trap’s characteristic frequency,  is the trap’s spring constant,  is the solution viscosity, r is the
bead radius, kB is the Boltzmann constant, and T is the absolute temperature. This expression can
be rearranged, yielding
G f t  
4 k B T
1
2
C
  f
1   t
  f c



2



,
(S.2)
where C is the voltage to force conversion factor. The power spectrum in Fig. S.1 is fit by
equation (S.2) (solid line). When ft << fc, G  4kBT/C2. From this low frequency asymptote, we
determine the bead’s voltage to force conversion factor C = 12.2 pN/V. The corner frequency fc =
230 Hz corresponds to spring constant  = 0.014 pN/nm.
As the second calibration technique, we applied hydrodynamic drag to optically-trapped
beads by moving the microscope stage back and forth in a triangle wave form. The triangle wave
displacement profile of the microscope stage exerts a square wave hydrodynamic drag profile on
the bead. Figure S.2 depicts the quadrant-photodiode signal of the same optically-trapped bead
from Fig. S.1 (dashed line) superimposed on the microscope stage’s position (solid line). The
equation of motion of the bead, neglecting the inertial term, is

x
  x  FS t  ,
t
(S.3)
where FS is the force on the bead. Equation (S.3) admits the solution

 t 
FS 
xt  
1 e   ,

 

(S.4)
where FS = u; u = 2A is the stage’s velocity; A is the peak-to-peak amplitude of the stage’s
displacement; and  is the frequency of the triangle wave (s-1). The displacement of the bead out
of the trap is recorded as a signal from the quadrant photodiode. From equation (S.4), the
equilibrium displacement
xeq 
2 A 
C
(S.5)
(measured in V) of a bead attached to a Hookean spring and acted upon by a constant drag force.
By measuring the average steady-state displacement of the bead from its equilibrium positions
(horizontal dotted lines in Fig. S.2) and inserting this value into Eq. (S.5), we estimate the trap’s
voltage to force conversion factor C = 13.7 pN/V. The estimates based on the hydrodynamic
drag method and the power spectrum method agree within 15%.
-5
10
Power
(V22s)s)
PowerSpectrum
Spectrum (V
-6
10
-7
10
-8
10
-9
10
-10
10
0
10
2
10
Frequency (Hz)
(Hz)
Frequency
4
10
Figure S.1: The power spectrum of an optically trapped, 1-m diameter bead undergoing
Brownian motion. The solid line corresponds to equation (S.2) with C = 12.2 pN/V and fc=230
Hz.
0.03
0.01
0
-0.01
-2
0
0.1
0.2
0.3
0.4
Detector Signal
(V)(V)
Detector
Signal
StageDisplacement
Displacement ( m)
Stage
(m)
2
-0.03
0.5
Time (s)
Time
(s)
Figure S.2: The photodiode signal from the same optically-trapped bead (dashed line)
superimposed on the microscope stage’s position (solid line). The stage is oscillating as a
triangle wave ( = 10 Hz). The average steady-state displacement, denoted by horizontal dotted
lines, is compared to the steady-state value expected from equation (S.7). This gives us the
voltage to force conversion factor C = 13.7 pN/V.