Transcript PowerPoint-præsentation

Magnetic Tweezers:
Micromanipulation and Force Measurement at the
Molecular Level
Charlie Gosse and Vincent Croquette, Feb. 2002
The advantage of MT:
-No photodamage to material.
-Rotary motions perpendicular to the optical axis.
-Dynamometer properties:
trap stiffness ca. 10-7 N/m, 3D speed of 10 μm/s , rotfreq. 10 Hz.
-Calibration against viscous drag or use of Brownian motions.
- Use magnetic field gradients and feedback loops.
Experiments:
Stretching DNA between a magnetic particle and a glass surface.
Find: vertical force [50fN; 20pN], horizontal force [5pN]
Setup
Fig.1
-PC determines xyz positions at
video rate.
-Feedback loop adjusts the current
in each magnets to cancel differens
between disired and obsserved pos.
-Amplifire suply the individual
Magnets.
Fig. 2
Hexagonal vertical magnets. The soft
at the top close the field Lines, thus
increasing the magnitude of the field.
Plexiglass makes the constr. Stiffer.
Mumetal is ferromagnetic with low
remanent magnetization (infrozen).
Aviod hysteresis in current change by
adding an exp. decaying oscilating current.
Fig. 3
Bead movement
Torque N= μ x B; align the magnetic
moment of the bead with the B field
Fmag=grad(μ · B); pull the bead
upward
Equilibrium Fmag=Fg and set |Iz| = I0.
Fig. 4 verticaly from the top
4.A; vertical move in z direction:
+Iz on 0,1,2 and, –Iz on 3,4,5
The bead moves toward the magnets
having the strongest B field, becouse of
the fieldgradient.
Fx,y ≈ Ix,y if Ix,y are small < 0,03A
Digital feedback loop: are used to lock the bead in a given position.
Iv,y ≈ Iz so Iu = -IzCu, Cu= [Pu ·u + Ku∑(u)]
Cu: normalized correction signal
Pu: proportional coefficient of the feedback loop
Ku: integral coefficient
u: error signel between observed and set position
∑(u): sum over previous error signals
Using the ”≈” correction implies Fu= kuu, ku is the stiffness.
Know Fu=AuIu = -IzAuPuu with Ku∑(u) = 0
For small force: Iz= I0(1-½zPz) and Fz= mg – AzI02Pzz
Gives the vertical stiffness kz= -dF/dz = AuI02Pz
Experiment: Passive tweezers
force measurements along z.
Use config. Fig. 4.A and stretching DNA
fastened to the surface.
δx: the Brownian transverse fluctuations.
l: extension of the DNA.
Fmag= kbT/<δx2> from the equipartition
theorem.
Fig. 6
A) Shows F(l) and it fit well til the WCM
with a persistence length of 50 nm.
B) Shows F(Iz). Note the Fz ≈ Iz2, Fz< 1pN
i.e. unsaturated magnetic materials.
Modulation of the force direction
for Fz > 1pN
Fig.7
Position of the bead-DNA chain.
+ moderate modulation.
o extreme modulation.
-- indicates the shift between normal
and altered config.
Se Table 1. for current settings.
Note:Pulling angle reaches +/- 70o at
high stretching force.
Active tweezers
Simple model with an instantaneous proportional feedback, (ku, Γηr).
The bead is locked 10μm above surface in a pot. well of low ku ≤ 10-7:
Eq. of motion:
, overdamped.
deviation, mass, radius, viscosity, viscous drag coeff = 6π.
FL : stochastic Langevin force, |FL(f)|2= 4kbTΓηr in Fourier space.
Thus |u(f)|2=
,
cutoff frequency of the
bead-molecule system.
Fig. 8.
f>>fc for small Pu the velocity fluctuations
, D diffusion coeff.
So can obtain Γ.
Find ku
use the equipartition theorem: kbT/2 = ku<u2>/2
In Fourier space:
, only for small Pu and
12,5s-1 = fs/2>>fc>>fL= ΔT-1
Fig. 9. (dashed line)
ku is given with an accuracy = √(1/fcΔT), ΔT recording time
If we want to have an accuracy ≤10% then ΔT ≥ 264s.
Digital feedback loop delay. At high
feedback parameters, u is described wirh
the recurrent relation:
α feedback correction,
Δ = 2 csmera and computer delay,
Ln displacement due to Langevin force.
Fig.9
For a given image of time δt we have
the forcees:
, so
Mean square fluctuations vs. kz and
α. .
Note: equipartition theorem only
, and leads to
valid for low feedback parameters.
Multiplied with C(f) to correct for the camera light
integration function. For high Pu it is fitted in fig. 8
to obtain L02, α, Δ and then plotted in fig. 9.
Additional integral correction Ku∑(u) will filter out the bed fluctuatoins
and leads to
Calibration against the viscous drag
The bead is locked in (x,y,z) 10μm above surface in a pot. Well.
Moving the bead in deriction u while limit the
current Iu to a maxumum Imu fig.10 A), Imu then
defines the the max. bead velocity Vu B).
Vu is plottet against Imu fig.11.
Fig.11. show Vu= BuIu.
By Stokes’s Fu= ΓηrVu and Fu=AuIu we have Au=Bu Γηr.
Calibration using the Brownian fluctuations.
The bead is locked in (x,y,z) 10μm above surface in af pot. Well.
Asymptotic model:
At low frequency f<<fc we have
Fu2=FL2 so in
x or y: Au2Iu2 = 4kbT Γηr
z: 4Az2I02Iz2 = 4kbT Γηr Γηr
Remember for f>>fc: Vu2= 4kbT/ 4kbT
to obtain:
with Au=Bu Γηr:
and in z:
Fitted model: By fitting the bead fluctuations to eq.18 we find L02 and α
and thus:
Comparison of the three calibration methods.
Note: Γz is a bit larger than Γx,y because of the shape of the cell.
See Γx,y is linear dep. of r but Γz
is hyperbolic to r.
Micromanipulation
When the bead is locked it remains under control as long as it is in the
field of view of the camera.
Using the current config. Fig.4.A, rotation is obtained by circular
permutation of the currents applied to the magnets with an shift in
angles of 60o. Rotational speed and position can be measured.
Conclusions
- Good monitoring and determination of bead position in 3D.
- Generate force and measure of force in 3D.
- Can be calibrated against viscous drag or by analyzing the
Brownian fluctuations of the trapped object.
- Rotational speed, position and force can be mesaured.