Magnetic Tweezers: Micromanipulation and Force Measurement at the Molecular Level

1. 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]

2. 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.

3. 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.

4. 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

5. 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

6. 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 • 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.

7. 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.

8. 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.

9. 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.

10. 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. For a given image of time δt we have the forcees: , so , and leads to Fig.9 Mean square fluctuations vs. kz and α. . Note: equipartition theorem only 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

11. 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 Imufig.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.

12. 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:

13. 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.

14. 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.

15. 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.