Measurement of the magnetic moment of single Magnetospirillum gryphiswaldense cells by magnetic tweezers

Magnetospirillum gryphiswaldense is a helix-shaped magnetotactic bacterium that synthesizes iron-oxide nanocrystals, which allow navigation along the geomagnetic field. The bacterium has already been thoroughly investigated at the molecular and cellular levels. However, the fundamental physical property enabling it to perform magnetotaxis, its magnetic moment, remains to be elucidated at the single cell level. We present a method based on magnetic tweezers; in combination with Stokesian dynamics and Boundary Integral Method calculations, this method allows the simultaneous measurement of the magnetic moments of multiple single bacteria. The method is demonstrated by quantifying the distribution of the individual magnetic moments of several hundred cells of M. gryphiswaldense. In contrast to other techniques for measuring the average magnetic moment of bacterial populations, our method accounts for the size and the helical shape of each individual cell. In addition, we determined the distribution of the saturation magnetic moments of the bacteria from electron microscopy data. Our results are in agreement with the known relative magnetization behavior of the bacteria. Our method can be combined with single cell imaging techniques and thus can address novel questions about the functions of components of the molecular magnetosome biosynthesis machinery and their correlation with the resulting magnetic moment.

Materials and Methods

Calculation of the viscous drag coefficients via Stokesian dynamics
The centerline cl of the bacteria's helix with diameter , length in axial direction and helical pitch is given by where u is varied between 0 and to describe the complete helix centerline. Any position H on the bacteria's surface, other than the ends, can be described by since every position has the same distance d to the centerline (except the ends). The unit vector ̂n is the vector normal to the centerline and ̂t ( ) denotes the rotation matrix by an angle [0, 2 ] about an axis in direction of the local unit tangent vector ̂t of the helix centerline. Through variations in the parameters and the bacteria's surface is parameterized. The both ends of the bacteria can be represented, in a similar manner, by fixing = 0 or = and by varying and .
The diameter A is measured directly in the experiment. The quantities and are determined by the experimentally measured arc length s and the measured end-to-end distance ee = � cl � � − cl (0)�.
The surface of the helix is divided and represented by point-like beads with an identical effective radius a. They are located at positions with = 1, … , . A force is required to move the pointlike particle at a position through the fluid because the beads have a drag coefficient = 6 in a solvent of viscosity . The beads generate a flow field by moving through the fluid. The flow ( ) induced by the particle located at at an arbitrary point in space is given by with the Oseen tensor 1 and the tensorial free space Greens function ( ) = 1 �1 + 2 �. The parameters = 1, . .3 and = 1, . . ,3 denote the entries of the matrices and . [3] The flow field ( ), which is caused by particle influences via the fluid, respectively via the Oseen tensor, the motion of all other particles at with = 1, . . , and ≠ . For a helix moving with the velocity , all beads fixed on its surface move with the same speed ̇= ( = 1, … , ). Every moving particle influences via the hydrodynamic interaction all other particles. Therefore, the forces 1 , … , required to move the beads with a given velocity are determined by the coupled linear For one solution with the velocity nearly parallel to the helical axis and with the total force acting on the helical bacterium, = ∑ is parallel to . The friction coefficient trans describes the proportionality between both quantities: trans = . (7) For a helix rotating with a frequency around the axis of through its center = 1 ∑ , the velocity of each bead is given by Eq. 6 can be used to determine the relation between the bead velocities and the required forces. The torque acting on the helix can be expressed in terms of the forces: = ∑ ( − c ) × . For the case in which the rotational axis is nearly perpendicular to the helix axis, the two vectors and are related via the rotational friction: rot = .

Calculation of the viscous drag coefficients by Boundary Integral Method
The Boundary Integral Method (BIM) exploits the fact that the Stokes equation is linear and can therefore be rewritten as an integral equation 2 for the flow velocity u(r) at an arbitrary point r inside the infinite and initially quiescent fluid: where summation over the repeated index j is implied, y is a point on the surface S of the bacterium, η is the fluid viscosity and f is the surface traction.
When the observation point r is moved to the surface, Eq. 9 can be converted to a linear system of equations 2 which in our implementation is solved by GMRES 3,4 . The surface integral in Eq. 9 is computed by discretizing the bacterial surface using flat triangles to interpolate between the surface nodes. To allow a direct comparison, the nodes of the triangles are taken at the same positions as in the Stokesian dynamics calculations (see above).

[4]
To impose a rotation/translation on the bacterium, each node is coupled by a harmonic spring to an auxiliary anchor point. During the simulation, these imaginary anchor points are translated/rotated with a prescribed (angular) velocity. By distributing the force of these springs over the local area surrounding each node, the force is converted into a surface traction f, which is a term in Eq. 9. The solution of the linear system resulting from Eq. 9 then yields the surface velocity u from which the desired drag coefficients rot and trans and can be directly obtained. The translational/rotational velocities have been chosen to be small enough that the relation between force and velocity remains linear.