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Supplementary methods
The torque exerted by the magnetic field upon the superparamagnetic particles (Dynabeads, Dynal
Biotech, Oslo, Norway) that constitute the filament has been variously attributed to dipole-dipole
interactions between the beads1 and to anisotropic magnetic susceptibility2. Both effects can be
incorporated into a single model. We define a preferred magnetisation direction upon a bead, pointing
in the direction of the unit vector s. Then under an external field B an otherwise isolated bead will
acquire a dipole moment m 
4 a 3
χB where scalar susceptibilities   ,   describe the
3 0
magnetisability of the beads parallel to and orthogonal to s: χ   ss    (1  ss) . During the assembly
phase we expect all of the beads to align to minimise the energy of the filament, so that all of the
individual preferred magnetisation vectors are directed along the backbone of the filament. Since the
dipole field decays rapidly with distance, it can be assumed that each bead feels only the external field,
and that of its two nearest neighbours. Letting s be the unit tangent to the filament, and n the unit
normal, if an external field Be=Bss+Bnn is applied to the filament, then the tangential and normal
components of the magnetic moment upon the filament are given by:
ms 
4 a3   Bs
4 a3   Bn
, mn 
30 (1    / 6)
30 (1    /12)
(1.1)
Through a combination of anisotropy and cooperative magnetic effects, the dipole moment is not in
general aligned with the external field, which therefore exerts a torque resultant
1
 a2            / 4 
  b  (m  Be ) 

 (Be  n)(Be  s)
2a
30  (1    / 6)(1    /12) 
(1.2)
per unit length of filament, in which we have defined a unit binormal vector b  s  n . For a
sinusoidally varying transverse field Be=Bxx+Bysin(wt)y, the torque varies with time and filament
orientation as:

2 a 2 Bx2            / 4 

 S ( , t; b0 )
30  (1    / 6)(1    /12) 
(1.3)
where θ is the angle that the filament makes with the x-axis, bo=By/Bx, and the angle and time variation
of the torque is contained in a single dimensionless function:
1
S ( , t ; b0 )  b0 sin  t cos 2  (1  b02 sin 2  t ) sin 2
2
(1.4)
The filament is taken to be inextensible, and parametrised by the arc-length s along its backbone, with
the s=0 corresponding to the tethered end, and s=L to the free end. We consider only deformation of
the filament in the plane of the crossed magnetic fields. A resultant tension Λ(s,t) and normal force
N(s,t) act across each cross-section of the filament. Slender-body theory for low-Reynolds number
flow3 is used to approximate the viscous drag upon every segment of filament, so that the velocity
v=ss+nn can be related to the stress resultants by:
  vs 


N
s
s
,   vn 
N


s
s
(1.5)
where   ,   are the parallel and perpendicular drag coefficients, which will be determined below.
From torque and force balance upon an infinitesimal element of filament we arrive at equations of
inextensibility and motion4:


 2    
 
 2   2
 2       (  1)
  2  2  M n S ( , t; b0 ) 
s
s s
s  s
 s 


2
2
2
2

  
     
   

 2   (  1)
  
 2   2  M n S ( , t; b0 ) 



t s
s s   s  s   s

2
(1.6)
(1.7)
The equations have been non-dimensionalised by scaling all lengths by L, times by ω, and stresses by
Kb/L2, where Kb is the bending modulus of the filament. The system of equations can be solved
numerically by approximating the spatial derivatives by finite differences, and using an implicit
integration scheme (the MATLAB routine ode15s) to perform the time evolution4. A complete
description of the physics is provided by four dimensionless variables: the ratio of transverse to
uniform fields b0, drag anisotropy factor     /   , and a dimensionless frequency     L4 / K b
representing the ratio of the elastoviscous relaxation time, in which in the absence of a magnetic field,
curvature is eliminated from an untethered filament, to the period of the applied field. It has previously
been shown to be useful to define a quantity Sp=1/4; the sperm number5. If the filament were not
internally driven, but rather deformed in response to a point force or torque of the same characteristic
frequency, applied at one end, then the length of filament mobilised would scale, in dimensional terms,
like L/Sp.
The relative importance of magnetic to elastic stresses is encoded into the magnetoelastic number
2 (aBx L)2            / 4 
Mn 


30 Kb  (1    / 6)(1    /12) 
(1.8)
For a given magnetic field strength, the magnetoelastic number can be measured directly by the hairpin
technique described in the methods section, but unless the anisotropy in susceptibility is known, it is
not in general possible to back the raw value of the bending stiffness (which is needed for accurate
determination of Ω) out from this. We assume that the anisotropy effect is weak, using the values
published by Dynal Biotech:       1 .
The filament swims near the floor of the capillary tube and therefore experiences greatly enhanced
viscous drag. Far from any rigid boundary it is known that the drag coefficients for a chain of spheres
are almost indistinguishable to those of prolate spheroid of the same length and aspect ratio6. No
analytic expression exists for the drag coefficients when the filament-to-floor distance, h, is comparable
to the filament length, but it has been shown that a far-field expression, formally only valid in the limit
h<<L, gives acceptable accuracy down to the order of separation observed in experiment:
where  =or 
 =or 
 = or  
3  L
1  = or 
32 h
are the corresponding far field drag coefficients, taking the spheroid values7:
  
(1.9)
2
4
,   
log( L / a)  1/ 2
log( L / a)  1/ 2
(1.10)
Since the value of the rotational drag is not known directly (but is for modelling purposes estimated
from
the
filament-wall
separation),
a
rescaled
sperm
number
must
be
introduced;
S p  (  / 4 )1/ 4 S *p , and the dimensionless quantity S *p is plotted as the horizontal ordinate in Fig. 4.
1.
Biswal, S. L., Gast, A. P., Rotational dynamics of semiflexible paramagnetic particle chains,
Physical Review E 69, 041406 (2004)
2.
Strick, T. R., Allemand, J.-F., Bensimon, D., Bensimon, A., Croquette, V., The elasticity of a
single supercoiled DNA molecule, Science 271, 1835-1837 (1996)
3.
Batchelor, G. K., Slender-body theory for particle of arbitrary cross-section in Stokes flow,
Journal of Fluid Mechanics 44, 419-440 (1970)
4.
Roper, M. L., Dreyfus, R., Baudry, J., Fermigier, M., Bibette, J., Stone, H. A., On the dynamics
of magnetically driven elastic filaments, Journal of Fluid Mechanics (submitted) (2005).
5.
Lowe, C. P., Dynamics of filaments: modelling the dynamics of driven microfilaments.
Philosophical Transactions of the Royal Society of London Series B-Biological Sciences 358,
1543-1550 (2003)
6.
Zahn, K., Lenke, R., and Maret, G., Friction coefficient of rod-like chains of spheres at very
low Reynolds numbers. Part I. Experiment, Journal de Physique II 4, 555-560 (1994) and
Meunier, A., Friction coefficient of rod-like chains of spheres at very low Reynolds numbers.
Part II. Numerical simulation, Journal de Physique II 4, 561-564 (1994)
7.
Cox, R. G., The motion of long slender bodies in a viscous fluid. Part I. General Theory.
Journal of Fluid Mechanics 44, 791-810 (1970)