# Performance and application of an open source automated magnetic optical
density meter for analyzing magnetotactic bacteria

Marcel K. Welleweerd1 Tijmen Hageman1,2 Marc Pichel1,2 Dave van As1 Hans
Keizer1 Jordi Hendrix1 Mina M. Micheal1 Islam S.M. Khalil1 Alveena Mir2 Nuriye
Korkmaz2 Robbert Kräwinkel1 Daniel Chevrier3 Damien Faivre3 Alfred Fernandez-
Castane4 Daniel Pfeiffer5 Leon Abelmann1,2 1University of Twente, Enschede,
The Netherlands, 2KIST Europe, Saarbrücken, Germany, 3Aix-Marseille
Universit/’e, CEA, CNRS, BIAM, UMR7265, Saint-Paul lez Durance, France, 4Aston
University, Birmingham, UK, 5University Bayreuth, Bayreuth, Germany
<EMAIL_ADDRESS>
###### Abstract

We present a spectrophotometer (optical density meter) combined with
electromagnets dedicated to the analysis of magnetotactic bacteria. We have
ensured that our system, called MagOD, can be easily reproduced by providing
the source of the 3D prints for the housing, electronic designs, circuit board
layouts, and microcontroller software. We compare the performance of this
novel system to existing adapted commercial spectrophotometers. In addition,
we demonstrate its use by analyzing the absorbance of magnetotactic bacteria
as a function of their orientation with respect to the light path and their
speed of reorientation after the field has been rotated by
$90\text{\,}\mathrm{\SIUnitSymbolDegree}$. We continuously monitored the
development of a culture of magnetotactic bacteria over a period of five days,
and measured the development of their velocity distribution over a period of
one hour. Even though this dedicated spectrophotometer is relatively simple to
construct and cost-effective, a range of magnetic field-dependent parameters
can be extracted from suspensions of magnetotactic bacteria. Therefore, this
instrument will help the magnetotactic research community to understand and
apply this intriguing micro-organism.

###### Contents

  1. I Introduction
    1. I.1 Research question and relevance
    2. I.2 Previous work
    3. I.3 Structure and contents
  2. II Theory
    1. II.1 Angle-dependent scattering $C_{\text{mag}}$
    2. II.2 Dynamic response
    3. II.3 Brownian motion
  3. III Method
    1. III.1 Measurement head
      1. III.1.1 Mechanical
      2. III.1.2 Coil system
      3. III.1.3 Temperature sensor
      4. III.1.4 Light source
      5. III.1.5 Photodiode
    2. III.2 Measurement board
      1. III.2.1 AD converter
      2. III.2.2 Microcontroller
      3. III.2.3 Display
      4. III.2.4 Storage
      5. III.2.5 Current drivers
      6. III.2.6 Power supply
      7. III.2.7 Enclosure
    3. III.3 Cabling
    4. III.4 Software
  4. IV Results
    1. IV.1 Performance
      1. IV.1.1 LED and photodetector
      2. IV.1.2 Magnetic field system
    2. IV.2 Applications
      1. IV.2.1 Transmission as a function of angle (MSR-1)
      2. IV.2.2 Response as a function of field strength and time (MSR-1)
      3. IV.2.3 Comparison of $C_{\text{mag}}$ for different instruments
      4. IV.2.4 Long-term growth monitoring (MSR-1)
      5. IV.2.5 Marathon test: MC-1 velocity measurement
  5. V Discussion
    1. V.1 Possible improvements
      1. V.1.1 Measurement head
      2. V.1.2 Measurement board
      3. V.1.3 Software
    2. V.2 Possible future applications
      1. V.2.1 Flagellar motion
      2. V.2.2 Multi-color optical density
      3. V.2.3 Modulated light intensity
      4. V.2.4 Combined marathon and $C_{\text{mag}}$
      5. V.2.5 Sedimentation
      6. V.2.6 Suspensions of magnetic nanoparticles
  6. VI Conclusion
  7. A $C^{*}_{\text{mag}}$ and $C_{\text{mag}}$ approximations
  8. B Cotangent approximation
  9. C Measurements

##  I Introduction

Magnetotactic bacteria possess a chain of iron-oxide or iron-sulfide
nanocrystals that makes them align with the Earth’s magnetic field [1, 2].
This property allows them to search efficiently for the optimal redox
conditions in stratified water columns [3]. Schüler and colleagues discovered
that the transmission of light through suspensions of magnetotactic bacteria
is influenced by the direction of an externally applied field [4]. This effect
has been successfully applied as a simple method to monitor such processes as
the cultivation of magnetotactic bacteria [5, 6, 7, 8] and to assess their
velocity [9, 10].

###  I.1 Research question and relevance

Commonly, the field-dependent transmission of light through a suspension of
magnetotactic bacteria is measured by extending a standard spectrophotometer
with a magnetic add-on. Such spectrophotometers are also known as optical
density meters, and are commonly used in biolabs to determine cell
concentrations.

The modification of existing spectrophotometers with magnetic add-ons has
several disadvantages. (i) These instruments are relatively complex and
expensive, so modifications are usually made to depreciated equipment. (ii)
Most instruments contain magnetic components that disturb the field and there
is generally little space to mount electromagnets, certainly not in three
dimensions. (iii) The various types of spectrophotometers and magnetic field
generators and the variations between laboratories lead to a lack of a
standardized measurement. (iv) More fundamentally, most spectrometers are not
intended for sub-second continuous registration of absorbance over time. They
are operated manually, and often use flash lamps.

In this publication, we present a spectrophotometer that intimately integrates
the optical components with a magnetic field system and is dedicated to
research on magnetotactic bacteria, see Figure 1. Additionally, the design
considers that students at the Master’s or early PhD level should be capable
of constructing such an instrument, both with respect to complexity and cost.
Our main research question addresses how this new magnetic optical density
meter, which we have dubbed MagOD, compares to existing adapted
spectrophotometers, and what novel measurement strategies it enables.

Figure 1: Photograph of an open-source spectrophotometer with magnetic field
option (MagOD). The system consists of a measurement head (left) in which a
cuvette with a suspension of magnetotactic bacteria is inserted. The
measurement board (right) is dedicated to controlling the magnetic field, data
acquisition and communication with the user over a touchscreen and WiFi. The
design of the system is open, including the layout of the electronic circuit
boards (top left), 3D print source files (top right) and control software.

###  I.2 Previous work

The system to be constructed is a spectrophotometer combined with a magnetic
field system. It is therefore useful to compare it with commercial
spectrophotometers. These systems generally use a xenon light source and
monochromator with a wide wavelength range. Table 1 provides an overview of
the specifications of representative commercial systems (Biochrome Ultrospecs
and the Eppendorf Biophotometer used here for comparison), including their
wavelength range $\lambda_{\text{min}}-\lambda_{\text{max}}$, spectral
bandwidth $\Delta_{\lambda}$, maximum absorbance $O_{\text{D}}$ (see Eq. (3))
and approximate price.

The first spectrophotometer modified with a magnetic field module was
presented by Schüler [4]. That device was based on standard optical components
and used a permanent magnet to generate a $70\text{\,}\mathrm{m}\mathrm{T}$
field. Later versions were constructed around commercial optical density
meters such as the ones presented by Lefèvre [9] (based on a Varian Cary 50
UV) and Song [8] (based on a Hitachi U2800). In their case, the magnetic field
is generated by coil systems that can generate adjustable fields up to
$6\text{\,}\mathrm{m}\mathrm{T}$ [9].

Table 1 also lists the parameters of the MagOD system introduced in this
paper. Its optical properties and price range compare well with those of
standard commercial systems, whereas its field range is similar to that of the
adapted systems by Lefèvre and Song.

Table 1: Optical density meters. | $\lambda_{\text{min}}$ | $\lambda_{\text{max}}$ | $\Delta_{\lambda}$ | $O_{\text{D}}$ | $B$ | Price
---|---|---|---|---|---|---
| (nm) | (nm) | (nm) |  | (mT) | (Eu)
Ultrospec 8000 | 190 | 1100 | 0.5 | 8 |  | 12000
Biophotometer D30 | 230 | 600 | 4 | 3 |  | 5000
Ultrospec 10 | 600 | 600 | 40 | 2.3 |  | 1300
Schüler [4], 1995 | 637 | 637 | 18 |  | 70 |
Lefèvre [9], 2009 | 190 | 1100 | 1.5 | 3.3 | 0-6 |
Song [8], 2014 | 190 | 1100 | 1.5 | 6 | 0-4.3 |
MagOD | 465 | 640 | 25 | 2 | 0-5 | 2000

###  I.3 Structure and contents

In this paper, we first discuss a model of the relationship between the
transmission of light and the orientation of magnetotactic bacteria (Section
II). Next to the specifications listed in Table 1, we defined other
specifications that are important for analyzing magnetotactic bacteria and the
open-source nature of the instrument. Our design choices are discussed in
Section III. The results section is divided into two parts. In Section IV.1,
we analyze the performance of our current implementation and compare it with a
commercial optical density meter. Section IV.2 illustrates the possibilities
of our novel system by giving four examples of experiments to extract
information about the magnetic behavior of magnetotactic bacteria. This
instrument is still very much a work in progress, and we invite the
magnetotactic bacteria community to participate in its further development.
For this purpose, we indicate possibilities for improvement and ideas for
additional applications in Section V.

##  II Theory

The standard method to determine the proportion of bacteria with magnetosomes
in a culture is to observe the changes of light transmitted through a
suspension of bacteria under rotation of a magnetic field [4]. The
transmission of light is dependent on the relative orientation of the bacteria
to the light path. For MSR-1, which are long, slender bacteria, transmission
is high when the field is perpendicular to the light path, whereas it is low
when the field is aligned parallel to the light path. This is somewhat
counterintuitive, as MSR-1 have the smallest projected cross section when they
are aligned along the line of view. (Analogously to blinds, MSR-1 let light
pass if the blinds are closed).

It is important to realize that we measure the intensity of light that reaches
the photodetector. The light leaving the source can either be absorbed by the
suspension of bacteria or scattered sideways so that it does not reach the
photodetector. Highly dense suspensions of magnetotactic bacteria are milky
white in appearance. Like milk, it is therefore very likely that magnetotactic
bacteria scatter, rather than absorb, light. MSR-1 are small compared to the
wavelength of the incident light, especially considering the size of their
cross section. Additionally, their refraction index is only slightly higher
than that of the surrounding liquid. These small “optically soft” objects
scatter more light in forward direction if their projected area along the
light path increases [11]. This would explain why the light intensity at the
photodetector drops if the MSR-1 are aligned with the light beam.

For MSR-1, the projected area is roughly proportional to the sine of the angle
between the long axis of the bacteria body and the light path. Owing to
Brownian motion and flagellar movement, the bacteria will not be aligned
perfectly along the field direction but show an angular distribution. The
width of this distribution will decrease with increasing field. In the
following discussion, we develop a simple theory to account for this effect.
As the MagOD meter allows us to adjust the angle and strength of the magnetic
field accurately, we can use it to validate the approximation.

###  II.1 Angle-dependent scattering $C_{\text{mag}}$

We define the angle between the light path and the MSR-1 long axis as
$\alpha$, see Figure 2, and introduce a scattering factor relative to the
intensity of light reaching the photodetector ($I(\alpha)$ with unit
$\mathrm{V}$)

$g(\theta)=\frac{I_{\text{max}}-I(\alpha)}{I_{\text{max}}-I_{\text{min}}}\,.$
(1)

For MSR-1, the photodetector signal $I$ has a maximum when the MSR-1 are
aligned perpendicular to the light beam ($I_{\text{max}}=I(90)=I_{\bot}$), at
which point scattering $g(90)$ is minimal.

Figure 2: Definitions of various angles. In our MagOD system, we set the angle
$\theta$ between the light path and the magnetic field $B$. The bacteria align
with the direction of the field, but can deviate by a small angle $\phi$ in a
cone around the field direction described by $\beta$. As a result, the angle
between the bacteria long axis and the light is $\alpha$. In case of
sufficiently large fields, $\phi=0$, $\alpha=\theta$, and $\beta$ is
irrelevant.

Schüler [4] introduced a parameter to characterize the relative proportion of
magnetotactic bacteria by comparing the light reaching the detector for the
magnetic field aligned parallel and perpendicular to the light path
($C_{\text{mag}}$ denotes the “coefficient of magnetically induced
differential light scattering” or the “ratio of scattering intensities” [12]).
Assuming that the scattering intensity can be estimated from the reduction of
light reaching the detector compared to the reference value of a sample
without bacteria ($I_{\text{ref}}$), the original definition is

$C^{*}_{\mathrm{mag}}=\frac{I_{\text{ref}}-I(0)}{I_{\text{ref}}-I(90)}\,.$ (2)

With increasing concentration of bacteria, the total amount of light reaching
the photodetector will decrease. In microbiology, cultures are traditionally
characterized by “optical density”, a parameter that relates the reduction in
light intensity to the reference value on a $10$-base log scale111Analogous to
the Beer–Lambert law. However, it should be noted that the relation between
$O_{\text{D}}$ and cell concentration is only approximate [13].

$O_{\text{D}}(\alpha)=\log\left(\frac{I_{\text{ref}}}{I(\alpha)}\right)=\log\left(I_{\text{ref}}\right)-\log\left(I(\alpha)\right)\,.$
(3)

After the pioneering work of Schüler, researchers started to equip these
optical density meters with magnetic fields [8, 12, 14, 15]. Using these
instruments, it is more convenient to define $C_{\text{mag}}$ as

$C_{\text{mag}}=\frac{O_{\text{D}\parallel}}{O_{\text{D}\bot}}=\frac{\log(I_{\text{ref}})-\log(I(0))}{\log(I_{\text{ref}})-\log(I(90))}\,.$
(4)

Today, the latter definition is commonly used. However, it should be noted
that the values are not identical, not even for $C_{\text{mag}}$ close to
unity (see Appendix A). As $C_{\text{mag}}$ equals unity in the absence of
magnetotactic bacteria, ($C_{\text{mag}}-1$) is often plotted [16, 12, 17, 7,
15].

In addition to the ratio, it is insightful to study the absolute difference
between the absorbances in the parallel and perpendicular directions

$\displaystyle\Delta_{\text{OD}}$
$\displaystyle=O_{\text{D}\parallel}-O_{\text{D}\bot}$
$\displaystyle=\log(I(90))-\log(I(0))\,.$ (5)

This difference is proportional to the absolute amount of magnetotactic
bacteria that rotate in the field.

###  II.2 Dynamic response

When measuring $C_{\text{mag}}$ with adapted photospectrometers, the
$O_{\text{D}}$ values are measured over a long interval, whereas the actual
rotation of the bacteria is not measured. However, our MagOD system can
measure at sub-second intervals and monitor the dynamic behavior of the
bacteria. The response of bacteria to a change in field direction is
determined by the balance between magnetic torque and rotational drag torque
[18, 19, 20]. Alignment of a bacterium to an external magnetic field with
angle $\phi(t)$ (see Figure 2) can be described by a simple differential
equation

$f\frac{\partial\phi(t)}{\partial t}+mB\sin\phi(t)=0\,,$ (6)

where $f$ [$\mathrm{N}\text{\,}\mathrm{m}\text{\,}\mathrm{s}$] represents the
rotational drag coefficient, $m$ [$\mathrm{A}\text{\,}{\mathrm{m}}^{2}$] the
magnetic dipole moment of the bacterium, and $B$ [$\mathrm{T}$] the magnetic
field strength.

To determine $C_{\text{mag}}$, we rotate the field by
$90\text{\,}\mathrm{\SIUnitSymbolDegree}$ very quickly. Therefore, we can
initially assume the bacterium to be orthogonal to the magnetic field
$\phi(0)=\nicefrac{{\pi}}{{2}}$. Solving the differential equation then yields

$\displaystyle\phi(t)$
$\displaystyle=2\cot^{-1}\exp{\left(\frac{mB}{f}t\right)}$
$\displaystyle\approx\frac{\pi}{2}\exp{\left(-0.85\frac{mB}{f}t\right)}\,.$
(7)

This approximation is better than
$0.065\text{\,}\mathrm{r}\mathrm{a}\mathrm{d}$ (see Appendix B). The angle
$\phi$ can be estimated indirectly from the measured scattering as described
by Eq. (1) if we assume that the bacteria remain in the plane of rotation
($\beta=0$). The settling time of this transition period is characterized by
the time constant $\tau=f/mB$. As in Pichel _et al._ [18], we scale the
response time to the magnetic field and introduce a general rotational
velocity parameter $\gamma$ ($\mathrm{rad}\mathrm{/}\mathrm{T}\mathrm{s}$)

$\gamma=\frac{m}{\pi f}=\frac{1}{\pi\tau B}\,.$ (8)

###  II.3 Brownian motion

When we remove the magnetic field, magnetotactic bacteria will quickly
reorient in a random orientation distribution via Brownian motion, and
possibly via flagellar motion. For the same reason, the bacteria will not
align perfectly along the magnetic field. The alignment will become better at
higher fields, so we may expect $C_{\text{mag}}$ to be field-dependent. Let us
first consider the effect of Brownian motion.

The probability distribution of finding magnetotactic bacteria tilted at an
angle of $\phi_{0}$ from the magnetic field direction $b(\phi_{0})$ is
determined by the ratio of magnetic ($-mB\cos(\phi)$) and thermal energy
($kT$) according to the Boltzmann distribution, see e.g. the textbook by
Kittel, Chapter 12 [21]. We should take into account that energy states for a
specific value of $\phi$ exist in a full revolution around the field axis
($\beta=0..2\pi$). Therefore

$\displaystyle b(\phi_{0})=$
$\displaystyle\frac{\int_{0}^{2\pi}e^{a\cos\phi_{0}}\sin(\phi_{0})d\beta}{\int_{0}^{\pi}\int_{0}^{2\pi}e^{a\cos\phi}d\beta
d\phi}$ $\displaystyle=$
$\displaystyle\frac{a}{2\sinh(a)}\sin(\phi_{0})e^{a\cos(\phi_{0})}\,,$

where $a=mB/kT$, with $k$ ($\mathrm{J}\text{\,}{\mathrm{K}}^{-1}$) being the
Boltzmann constant and $T$ ($\mathrm{K}$) the temperature.

To achieve a first-order approximation, we assume that the scattering factor
is proportional to the projection of the bacteria shape onto the light
direction. Defining $\alpha$ as the angle between the bacteria long axis and
the light path, the scattering factor in Eq. (1) becomes

$\displaystyle g(\alpha)=1-\lvert\sin(\alpha)\rvert\,.$

The angle $\alpha$ is the combined result of the angle between the light and
the field direction $\theta$ and the angle between the bacteria and the field
$\phi$. One can show that the relation between $\alpha$ and these three angles
is

$\displaystyle\cos(\alpha)=-\sin(\theta)\sin(\phi)\cos(\beta)+\cos(\theta)\cos(\phi)\,,$

resulting in the following expression for the scattering factor:

$\displaystyle g(\theta,\phi,\beta)=1-\sqrt{1-\cos(\alpha)^{2}}\,.$

The average scattering factor can be obtained by double numerical integration,
first over all values of $\beta$ and then over the distribution of $\phi$

$\displaystyle<g(\theta)>=\int_{0}^{\pi}g(\theta,\phi)b(\phi)d\phi\,.$

The numerical integration was performed in Python, the source code of which is
available as Supplementary Material (angular.py). Figure 3 shows the resulting
average scattering factor as a function of the applied field angle for varying
energy product $mB$. At an energy $mB$ well above
$40\text{\,}\mathrm{k}\mathrm{T}$, the angular dependence approaches a
$1-\sin(\theta)$ relationship.

Assuming a dipole moment of
$0.25\text{\,}\mathrm{fA}\text{\,}{\mathrm{m}}^{2}$ as reported in our earlier
work [18], $mB$=$40\text{\,}\mathrm{k}\mathrm{T}$ corresponds to a field of
about $0.7\text{\,}\mathrm{m}\mathrm{T}$. Therefore fields on the order of a
few mT may be sufficient to obtain the maximum value of $C_{\text{mag}}$.

When the field is removed, the scattering factor is $g_{0}=$0.2146$$. In this
case, the intensity at the detector is $I_{0}=g_{0}I(0)+(1-g_{0})I(90)$, which
we can relate to the average $O_{\text{D}}$ of the suspension

$\displaystyle O_{\text{D}}=$
$\displaystyle\log(\frac{I_{\text{ref}}}{I_{0}})$ $\displaystyle=$
$\displaystyle-\log\left(g_{0}10^{-O_{\text{D}\parallel}}+(1-g_{0})10^{-O_{\text{D}\bot}}\right)\,.$

In the above, we ignored the disturbing force caused by the flagella.
Flagellar motion is complex, making the disturbing force difficult to
calculate. However, we know that, in natural conditions, magnetotactic
bacteria can use the Earth’s magnetic field of about
$50\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{T}$ to navigate. In this low
field, $mB$ is only $3\text{\,}\mathrm{k}\mathrm{T}$. If the stochastic energy
provided by the flagella is much greater than this value, the bacteria would
not be able to follow the field. This suggests that, for fields on the order
of mT, flagellar motion can be ignored.

Figure 3: Calculation of the average scattering factor as a function of the
angle of the field with respect to the light incidence for varying values of
the product of the magnetic moment of the magnetosome chain $m$ and the
applied field $B$, expressed in units of $kT$ at room temperature. When all
bacteria are perfectly aligned ($mB/kT=\infty$), the average scattering factor
is inversely correlated to the projection cross section of the bacteria on the
light path ($g=1-\sin(\theta)$). At lower fields, the loss of alignment
reduces the angular dependence, which disappears for $mB<kT$.

##  III Method

Figure 4: Diagram of the system: Measurement head: LED, refdiode, diode,
amplifier stages, coils, cuvette, bacteria. Measurement board: outlet to 12
supply, $12\text{\,}\mathrm{V}$ to $5\text{\,}\mathrm{V}$ analog and digital,
microcontroller, AD converter, LED driver, motor shields, current sense, HDMI
cable, coil cable, SD card, WiFi. User interface: resistive touchscreen.
Measurement board and user interface reside in the same housing.

Our MagOD system is an alternative to the modified commercial optical density
meters currently used in magnetotactic bacteria research. It should therefore
use compatible cuvettes and have comparable specifications. The preferred
wavelength at which absorbance is measured is approximately
$600\text{\,}\mathrm{n}\mathrm{m}$, and the maximum absorbance is
approximately $1.4\text{\,}$ [9]. Intensity variations due to the change in
direction of the magnetic field can be as high as $200\text{\,}\mathrm{\char
37\relax}$, but values as low as $2\text{\,}\mathrm{\char 37\relax}$ have been
reported [8]. Although fields up to $70\text{\,}\mathrm{m}\mathrm{T}$ are
applied [4], there are indications that saturation occurs as low as
$2\text{\,}\mathrm{m}\mathrm{T}$ [8]. Our design therefore should have a
wavelength of approximately $600\text{\,}\mathrm{n}\mathrm{m}$, an absorbance
range of at least $1.5\text{\,}$, intensity resolution better than
$1\text{\,}\mathrm{\char 37\relax}$, and a magnetic field above
$2\text{\,}\mathrm{m}\mathrm{T}$.

The MagOD system has two main components, see Figure 4. The cuvette filled
with the sample to be investigated is inserted into the _measurement head_ ,
which holds the light source and photodetector circuit boards, the three coil
sets and additional sensors (such as temperature). The measurement head is
connected to the _measurement board_ , which holds the analog-to-digital (AD)
converters, the drivers for the magnetic field generation, and the light
source. A microcontroller is mounted on the measurement board, which is
connected over the board to the AD converters, the data storage card, and a
touchscreen.

###  III.1 Measurement head

We designed the measurement head to be as compact as possible to keep the
volume and power consumption low. The dimensions of the standardized cuvette
($12.5\text{\times}12.5\text{\times}45\text{\,}\mathrm{m}\mathrm{m}$)
determine the size of the coil system, which essentially sets the outer
dimensions of the measurement head. The circuit boards for the light source
and sensors are embedded inside the coil system, with sensors located as close
to the cuvette as possible.

####  III.1.1 Mechanical

As the measurement head carries all components, it is a complex structure that
must be modified regularly to accommodate changes in component dimensions and
added functionality. Therefore we decided to fabricate the structure by 3D
printing so that modifications can be easily implemented. Printing in metal is
still prohibitively expensive, so the measurement head itself cannot act as
electromagnetic shielding. Instead, shields will have to be implemented on the
circuit boards. However, it is possible to 3D-print with black nylon to shield
the photodetector from external light and to allow the parts to be easily
disinfected with a $70\text{\,}\mathrm{\char 37\relax}$ ethanol/water
solution.

The measurement head consists of more than a dozen parts. The design is
parameterized using the open-source OpenSCAD language, so that dimensions can
be easily changed. The source files are available at Thingiverse.com.

####  III.1.2 Coil system

We can choose between permanent magnets or electromagnets with or without
cores to apply a magnetic field. As the field to be applied is relatively low,
electromagnets without cores provide the simplest solution. The field is
directly proportional to the current, and there is no hysteresis, so no
additional magnetic field sensors are required. The disadvantage of not having
a core is that the maximum field is limited to a few mT. Higher fields can
only be applied for short periods of time, limited by coil heating.

The magnetic field is generated by three orthogonal sets of two coils located
on either side of the sample. The dimensions of the coils are more or less
defined by the cuvette height, but we can choose the wire diameter to optimize
the number of windings $N$. The field in the coil is proportional to the
product of the current $I$ and $N$. The resistance $R$ of the coil scales
approximately with $N^{2}$ for fixed coil dimensions. Therefore, the power
dissipated in the coils ($I^{2}R$) is relatively independent of the number of
windings for a given field strength. The inductance of the coil $L$ scales
with $N^{2}$, so the cutoff frequency (proportional to $R/L$) is also fairly
independent of the coil wire diameter. The choice of wire diameter is
therefore determined mainly by the availability of power supplies,
specifications of H-bridges and current ratings on connectors. Table 2 shows
the specifications of two commercially available coils (Jantzen Audio 000-1235
and 000-0996) that are both suitable for our application. The number of turns
was estimated from the coil resistance (using literature values for wire
resistance) and the coil inductance [22]. Our MagOD system incorporates the
coil with the higher number of windings ($996$) to benefit from the
substantially lower currents, but at the expense of a slightly higher cutoff
frequency and higher power consumption.

Table 2: Examples of coil specifications. Jantzen Audio coil no. | 1235 | 0996 |
---|---|---|---
Wire gauge | 18 | 22 | AWG
Wire diameter | 1.0 | 0.64 | mm
Resistance | 21 | 53 | $\text{\,}\mathrm{m}\mathrm{\SIUnitSymbolOhm}\mathrm{/}\mathrm{m}$
Inner diameter | 42 | 42 | mm
Outer diameter | 57 | 53 | mm
Height | 21 | 21 | mm
Inductance | 0.94 | 2.9 | mH
Resistance | 0.5 | 2.1 | $\text{\,}\mathrm{\SIUnitSymbolOhm}$
Cutoff frequency | 85 | 115 | Hz
Windings222estimated from resistance and inductance | $80(3)\text{\,}$ | $140(7)\text{\,}$ |
Current for $1\text{\,}\mathrm{m}\mathrm{T}$ | 0.9333estimated from number of windings | 0.5444from Figure 9 | A
Voltage for $1\text{\,}\mathrm{m}\mathrm{T}$ | 0.44 | 1.1 | V
Power | 0.4 | 0.5 | W

####  III.1.3 Temperature sensor

Electromagnets—especially those without cores—produce heat as a byproduct of
the magnetic field. In the absence of active cooling, the temperature of the
sample under investigation can rise quickly. This is especially problematic
when one is working with micro-organisms. Therefore, it is important to
monitor the temperature of the cuvette. The best option would be to insert a
temperature sensor into the cuvette, but this method is cumbersome and risks
exposing the sample to the outside air. The temperature of the coils can be
estimated from their resistance, but that would overestimate the temperature
of the cuvette. Therefore, we chose to mount a simple negative temperature
coefficient (NTC) sensor in the housing as close as possible to the cuvette.

####  III.1.4 Light source

Ideally, the absorption pattern of a specimen is measured over a broad range
of wavelengths. Most optical density meters use a wide-spectrum xenon flash
lamp combined with a monochromator. However, this is not only a rather power-
hungry, bulky solution ($>$$10\text{\,}\mathrm{W}$,
$20\text{\,}\mathrm{m}\mathrm{m}$), it is also overkill for observing
magnetotactic bacteria. Instead, we chose an RGB LED as source. LEDs are
simple to control, can be mounted close to the cuvette, operated in continuous
mode and easily adjusted in intensity using pulse width modulation (PWM).
However, the wavelength cannot be chosen continuously, whereas the wavelength
spectrum is determined by the LED type. Moreover, the wavelength bandwidth per
color is rather large ($25\text{\,}\mathrm{n}\mathrm{m}$ compared to
$5\text{\,}\mathrm{n}\mathrm{m}$ for monochromators). Finally, the light
intensity of a LED is low compared to that of xenon lights or lasers. Based on
the manufacturer’s data, the LED power in our current implementation is
approximately $0.2\text{\,}$, $0.1\text{\,}$ and
$0.7\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{W}\mathrm{/}\mathrm{m}\mathrm{m}^{2}$
for $645\text{\,}$ (red), $520\text{\,}$ (green) and
$460\text{\,}\mathrm{n}\mathrm{m}$ (blue) light, respectively. This is
sufficient for most suspensions of magnetotactic bacteria.

The LED has a non-diffuse housing such that the light output in the direction
of the sample is optimal. The LEDs can easily be exchanged, for instance for a
yellow or UV LED, because they are mounted on a separate board.

The LED is mounted in common anode configuration such that (i) it can be
driven by NPN MOSFETs and (ii) the supply difference between the LED
($5\text{\,}\mathrm{V}$) and the microcontroller ($3.3\text{\,}\mathrm{V}$) is
inconsequential. The frequency of the PWM signal is well above the cutoff
frequency of the photodetector amplifier. Since the brightness of LEDs
decreases with time, we monitor the LED intensity. For this purpose a
photodiode is placed in close vicinity before the light enters the cuvette.

####  III.1.5 Photodiode

The light passing the cuvette can be detected with photomultiplier tubes,
avalanche photodiodes and silicon photodiodes [23]. Photomultipliers are
highly sensitive, but are also quite bulky, require high voltages and perform
less well at long wavelengths. Avalanche photodiodes are also very sensitive,
but suffer from nonlinearity, noise, a high temperature dependence, and
require high voltages to operate. As the transmission of light through most
magnetotactic bacteria suspensions is high and we work at low acquisition
frequencies, the sensitivity of silicon photodiodes is sufficient and allows
us to take advantage of its small form factor, linearity and ease of
operation. We used the more light-sensitive large-area photodiodes to boost
sensitivity. The diode is operated in photovoltaic mode. In this mode, the
bias voltage is zero, so the dark current, which is highly temperature-
sensitive, is minimized.

In our MagOD implementation, the photodiode current is amplified by a two-
stage operational amplifier (opamp) circuit because the signal is too weak for
a single-stage amplifier. The first stage is a current-to-voltage converter. A
low-noise JFET opamp is applied because this type of opamp has a low input
current offset, which reduces DC errors and noise at the output. The first
stage has the largest amplification in order to minimize the amplification of
noise. The amplifier circuit is located directly behind the photodiode inside
an electromagnetic protective casing, so that noise picked up by the cabling
to the main board is not amplified and interference is minimized.

###  III.2 Measurement board

Placing the photodiode amplifier directly behind the photodiode is an
effective way to suppress interference. We have the option to transport the
amplified photodiode signal directly to the measurement board or to include
the analog-to-digital (AD) converter next to the amplifier in the measurement
head and convert the analog signal into a digital one. The analog option has
the advantage of a small form factor for the circuit board and better access
for testing. The digital option is more robust against interference and allows
simpler cabling. As the current implementation of the MagOD system is very
much a development instrument, we chose to move the AD converters to a
separate measurement board, together with the microprocessor and other
peripherals.

####  III.2.1 AD converter

The measurement board has two AD converters to read out the various analog
signals on the system. As the measurements are normally performed on a longer
time scale, we chose converters that are able to perform measurements with a
sampling rate of up to $860\text{\,}\mathrm{Hz}$ and have integrated anti-
aliasing filters. A 16-bit resolution provides an upper limit to the
absorbance of 4.8, which is more than sufficient. In practice, the absorbance
range is limited by stray light scattering around the sample.

The AD converters have a free-running mode, which performs measurements at an
internally defined clock rate. A data-ready pin functions as an external
interrupt such that the microcrontroller can be freed for other tasks while
waiting for the AD converter to finalize its acquisition step.

####  III.2.2 Microcontroller

As data acquisition rates are low, the MagOD system can be easily controlled
by a microcontroller ($\mu$C). This allows us to benefit from recent
developments in low-cost, versatile $\mu$C development platforms. Rather than
embedding the $\mu$C directly on the the electronic board, we chose to include
it as a development board. This way, the system can be easily assembled,
debugged and repaired.

The current implementation of the MagOD instrument is built around an ESP32
development board (Espressif Systems, Shanghai, China). The ESP32 $\mu$C has
several characteristics that make it very suitable for this application: It
has a small form factor, a fast 32-bit dual-core processor operating at
$240\text{\,}\mathrm{MHz}$, WiFi and Bluetooth as well as several peripheral
interfaces such as SPI and I2C. This $\mu$C is very popular, and numerous
dedicated libraries, examples and discussions are available on Internet fora.
Additionally, there is a plugin for the Arduino IDE, and many libraries are
natively compatible, so even inexperienced developers can start with little
effort.

####  III.2.3 Display

A resistive touchscreen enables the user to control the system with or without
protective gloves. Additionally, the screen provides the user with information
on the current and past states of the measurement and levels of the signals.
Line drivers on the main board ensure that communication is reliable.

####  III.2.4 Storage

The acquired data and recipes are stored on a secure digital (SD) card. These
cards are readily available in a variety of capacities, are widely applied in
DIY projects, and are replaceable in case of a damaged card. The SD card can
be interfaced to the $\mu$C in the SPI, the 1-bit SD, and the 4-bit SD modes.
Although data transfer is faster in the 4-bit SD mode, we chose the SPI mode
because it is well supported and the write speed is sufficient for our
purpose. However, the write time to an SD card over an SPI interface using the
ESP32 $\mu$C is unpredictable, with SD card-induced peaks in write time of at
least $50\text{\,}\mathrm{m}\mathrm{s}$. Fortunately, the ESP32 has two cores,
so unpredictable processes such as access to the SD card, reaction to
touchscreen input and WiFi file transfers can be moved to a separate core.

####  III.2.5 Current drivers

The current through the coils must be controlled to obtain a specific
magnitude of the magnetic field. We use PWM and benefit from the fact that the
high inductance of the coil provides a low-frequency, low-pass filter for
free. The use of PWM minimizes power dissipation in the supply, but results in
a current ripple and consequently a ripple in the magnetic field. This ripple
can be suppressed by choosing a sufficiently high PMW frequency. We use
commercial motor drivers (Cytron MD13S) because they are specialized for
driving high currents through a coil in two directions based on a simple two-
wire control. The currently employed drivers work with frequencies up to
$20\text{\,}\mathrm{kHz}$, suppressing the ripple by a factor of at least
100\. The drivers can be interchanged by alternative motor drivers with
similar capabilities.

The magnetic field is linearly dependent on the current. However, the current
is not linearly dependent on the PWM duty cycle, as the internal resistance of
the coil will vary due to temperature changes. A precise measurement of the
current is necessary to close the loop and to assess the applied magnetic
field. Therefore, a shunt resistor is placed in series with each coil. The
voltage drop over this resistor is amplified using a current sensing amplifier
and digitized with the AD converter. The measured signal can serve either to
determine the true current or it can be applied in a feedback loop to
compensate for coil heating.

####  III.2.6 Power supply

The measurement board electronics operate at low voltages ($3\text{\,}$ or
$5\text{\,}\mathrm{V}$). However, the magnetic coil system is preferably
operated at higher voltages to limit the currents and subsequent requirements
for cabling and connectors. For reasonable winding wire diameters, the
currents are in the range of a few ampere and the resistance of the coils is
on the order of a few ohm. Therefore, we selected $12\text{\,}\mathrm{V}$ for
the main on-board supply, for which a wide range of external power supplies is
available and that even allows for operation from a car battery while in the
field.

In our MagOD implementation, the three coil sets have a combined resistance of
$4.2\text{\,}\mathrm{\SIUnitSymbolOhm}$ at room temperature. The maximum
current is close to $3\text{\,}\mathrm{A}$ with $12\text{\,}\mathrm{V}$. This
maximum current simultaneously through each coil set would require a maximum
power supply of $120\text{\,}\mathrm{W}$.

The analog and digital circuits have a separate $5\text{\,}\mathrm{V}$ supply
line to prevent noise originating from the switching nature of the digital
circuitry from interfering with the measurement. The analog
$5\text{\,}\mathrm{V}$ supply is built using an ultra-low-noise linear
regulator, whereas the digital $5\text{\,}\mathrm{V}$ is built with a
switching regulator. The latter is more efficient, but produces inherently
more electronic noise. The $3\text{\,}\mathrm{V}$ needed for the $\mu$C
originates from a linear regulator integrated on the development board.

####  III.2.7 Enclosure

The device is enclosed in a laser-cut plastic housing. Plastic was chosen
because it does not block the WiFi signal. There is no need to deal with
interference signals because the measurement signal is amplified in the
measurement head, and the unshielded sections of the leads to the AD converter
are kept very short.

The design is optimized such that no extra materials are needed for assembly.
Additionally, the parts can be manufactured with a 3D printer. The source code
for the enclosure design is available on github.

###  III.3 Cabling

While designing the MagOD system, we envisioned that measurements could take
place inside controlled environments, such as incubators and refrigerators.
Therefore the system was separated into two parts, connected by cabling.
Components that did not have to be on the measurement head were moved to a
separate module. This approach has the additional complication of requiring
cabling and connectors. To mitigate this drawback, we chose commercially
available cabling wherever possible.

For communication with the amplifier boards in the measurement head, we chose
an HDMI cable, which features shielded twisted-pair wires with a separate non-
isolated ground. HDMI cables are ideal for transmitting analog signals with
low interference (5 V, signal, ground). The HDMI interface has evolved through
several standards. The HDMI2.1 + Internet standard has five shielded twisted
pairs that can be used for measurement signals (for instance three
photodiodes, NTC and Hall sensor) and four separate wires that can be used for
control signals (three LEDs). The connectors on the main board, amplifier
boards and motor drives are standard Molex connectors. The coils are connected
to standard measurement leads with banana connectors. The connection from the
banana plugs to the measurement head is based on a Hirose RP 6-pole connector,
which is the only cable that cannot be purchased in assembled form.

###  III.4 Software

Most modifications to the MagOD system will be made at the software level,
which will be performed primarily by students. Generally speaking,
(electrical) engineering students and many hobbyists are skilled in
programming Arduino development boards. Therefore, the $\mu$C (ESP32) was
programmed in the same way as an Arduino project, using C++ and the native
Arduino IDE both as compiler and uploader. This has the major advantage of
posing a negligible entry barrier for inexperienced $\mu$C programmers.

The disadvantage of the Arduino IDE is that it is not ideal for larger
projects. The current implementation already exceeds 5000 lines of code. To
partially relieve this issue, the code was set up in a highly modular way to
assist new programmers in navigation, using only one main source file (.ino,
.h) of 1000 lines, and several local library source files (src/*.cpp), e.g.
for screen access, readout of the AD converter, writing to Flash memory, and
WiFi access. The source code can be found on github.

The data is collected on the SD card and transferred over WiFi in a format
that can be easily imported and displayed in a spreadsheet program. For more
advanced analysis, Python scripts are available on github.

##  IV Results

We have analyzed the performance of our MagOD system, and shall compare it to
a commercial spectrophotometer in the first part of this results section. To
illustrate the possibilities of our novel instrument, we provide three
examples in Section IV.2.

###  IV.1 Performance

Several iterations of MagOD systems have been realized based on the design
considerations described in Section III. We expect that more iterations will
follow, not only by our team but also by others in the field of magnetotactic
bacteria. The most recent implementation can be found on github. We measured
the performance of our MagOD meter (version 2) with respect to its optical and
magnetic components to provide a baseline for future improvement.

####  IV.1.1 LED and photodetector

##### Photodetector sensitivity.

Our MagOD system is equipped with a three-color LED, which allows selection of
three wavelengths (peak intensities at $645\text{\,}$, $520\text{\,}$ and
$460\text{\,}\mathrm{n}\mathrm{m}$), either individually or in combination.
The LEDs are individually driven by a PWM voltage to adjust their intensity,
for instance to match the transmission of light through the liquid in the
cuvette. A reference photodiode is mounted adjacent to the LEDs, which
captures a small fraction of the LEDs’ light to monitor variations in the
emitted light intensity. Figure 5 shows the signal of the detector and
reference photodiodes as a function of the average LED power for the three
wavelengths. The light pattern is shown in Figure 18 in the appendix, and a
video is provided in the Supplementary Material (MagODLEDProjection.mov).

Space restrictions compelled us to design the two-stage amplifier such that
the output decreases with increasing LED power. The reference photodiode,
which has only one amplifier stage, has an increasing output with increasing
intensity.

The relation between output voltage and intensity is linear for the red and
green LEDs, but not for the blue LED at higher intensities. Measurements with
liquids of different absorbance confirm that the sensitivity to blue light
drops at high intensities of incident light, see Figure 19 in the appendix.
Therefore, the blue LED should be used only for accurate absorbance at low
incident power, i.e. for signals above $2\text{\,}\mathrm{V}$. At low
intensity, the sensitivities of the red and blue channels are approximately
equal, and twice as high as that of the green channel for the chosen
combination of LED and photodetector. However, the sensitivity of the
reference photodiode to red and blue light is clearly different. This again
may be related to the placement of the diodes in the LED housing.

The linear fits to the data are listed in Table 3. The offsets are in
agreement with the manufacturer’s specification of the ADS115 of
$4.096\text{\,}\mathrm{V}$.

Figure 5: Photodiode and reference diode signal versus LED duty cycle (proportional to effective power). Note that the signal on the photodiode decreases with increasing light intensity due to the particular two-stage design of the amplifier. The reference diode amplifier is based on a conventional one-stage design. The photodiode signal is linear with the duty cycle for the red ($645\text{\,}\mathrm{n}\mathrm{m}$) and green ($520\text{\,}\mathrm{n}\mathrm{m}$) LEDs, but not for the blue ($460\text{\,}\mathrm{n}\mathrm{m}$) LED. Similarly, the reference diode signal is linear with the duty cycle of the red and green LEDs, but not for the blue. The parameters for the linear fits are listed in Table 3. Table 3: Linear fits to measurements shown in Figure 5. The blue LED has a nonlinear response and is not tabulated. |  | photodiode | reference diode
---|---|---|---
LED | $I_{\text{max}}$ | offset | slope | offset | slope
| mA | V | V/$I_{\text{max}}$ | V | V/$I_{\text{max}}$
Red | $20(2)\text{\,}$ | $3.159(5)\text{\,}$ | $-15.28(4)\text{\,}$ | $0.545(3)\text{\,}$ | $3.305(5)\text{\,}$
Green | $20(2)\text{\,}$ | $3.168(2)\text{\,}$ | $-6.696(9)\text{\,}$ | $0.516(1)\text{\,}$ | $1.902(2)\text{\,}$
Blue | $20(2)\text{\,}$ |  |  | $0.532(1)\text{\,}$ | $6.152(5)\text{\,}$

##### Absorbance validation.

To validate performance with respect to standard photospectroscopy
measurements, we compared our MagOD system with a commercial optical density
meter (Eppendorf BioPhotoMeter Plus). Figure 6 shows the absorbance
($O_{\text{D}}$) relative to water as a function of the wavelength of the
light for a range of dilutions of a suspension of magnetic nanoparticles
(FerroTec EMG 304). The transmission of light measured by our MagOD meter was
averaged for a range of photodiode intensities ranging from zero to
saturation. For the blue LED, however, care was taken to measure only at low
intensities, where the response is linear, see Figure 5.

As expected, the absorbance increases with increasing nanoparticle
concentration as indicated on the right-hand side of the graphs. The
absorbance increases with decreasing wavelength, which is in agreement with
the observation that the solution has a brown appearance. Care was taken to
determine the accuracy of the measurement as accurately as possible. At this
precision level, it is clear that our novel MagOD meter and the commercial
instrument deviate, albeit never more than $0.2$ for absorbances below $2$.
Above this value, the deviation becomes considerable, see data points inside
dotted loop in Figure 6, probably due to light scattering onto the
photodetector through other paths.

The blue LED seems to systematically underestimate the absorbance, which may
be related to the fact that the response of the detector is ill-defined. The
maximum absorbance is comparable to that of the commercial instrument. We
therefore conclude that our MagOD instrument is satisfactory as a conventional
absorbance meter, especially its red and green channels.

Figure 6: Absorbance relative to water measured with our MagOD meter (solid
circles in the colored bands) for the three LEDs compared to the optical
density measured by an Eppendorf BioPhotoMeter (open squares) measured as a
function of wavelength. We used a range of dilutions of a water-based
ferrofluid (FerroTec EMG 304), the dilution factor of which is indicated on
the right. The data points for every dilution are indicated by a line to guide
the eye. The difference between our MagOD meter and the commercial instrument
is greater than the estimated measurement error, but less than $0.2$ for
absorbances below $2$. Above this value, the estimate is unreliable (data
points inside dotted loop). The absorbance of the blue LED is systematically
lower than that of the commercial instrument. The maximum absorbance measured
was $1.82$, which is slightly lower than for the commercial instrument
($2.14$). The measurement uncertainty is less than the symbol size and is
therefore omitted for clarity.

##### Time response and noise level.

The ADS1115 AD converter has a maximum sampling rate of
$860\text{\,}\mathrm{s}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{/}\mathrm{s}$,
which means a sampling time of $1.2\text{\,}\mathrm{m}\mathrm{s}$. Figure 7
shows a time sequence of the sampled photodiode signal at that rate. The red
LED was switched on and modulated from $46$ to $47$ bits on a full range of
$255$ (relative intensity approximately $0.18$) every $250$ samples. The total
acquisition of $1300$ samples took $4023\text{\,}\mathrm{m}\mathrm{s}$, so the
effective sample rate was only
$323\text{\,}\mathrm{s}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{/}\mathrm{s}$.
The reduction in data rate is due to communication overhead with the AD
converter, and could be optimized.

The data in Figure 7 shows two clear levels, with no measurement points in
transition from one to the other. Therefore, we can safely conclude that the
response of our MagOD meter at the highest sample rate is better than
$3.1\text{\,}\mathrm{m}\mathrm{s}$. This is in agreement with the filter
applied in the feedback loop of the amplifier, which has a
$-3\text{\,}\mathrm{d}\mathrm{B}$ point at $800\text{\,}\mathrm{H}\mathrm{z}$
($1.25\text{\,}\mathrm{m}\mathrm{s}$).

The ADS1115 has an internal filter that matches the bandwidth, which can be
selected from discrete values of $8$, $32$, $64$, $128$, $250$, $475$ and
$860\text{\,}\mathrm{s}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{/}\mathrm{s}$.
Therefore, the noise should decrease at lower sample rates. Figure 8 shows the
standard deviation of $1000$ samples, which is equal to the root-mean-square
(RMS) noise as a function of sample rate. As expected, noise increases with
increasing sample rate, but much more steeply than can be expected from a
white noise spectrum, i.e. noise proportional to the square root of the
bandwidth. There is a strong jump in noise above
$64\text{\,}\mathrm{s}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{/}\mathrm{s}$,
most likely caused by the presence of a $50\text{\,}\mathrm{H}\mathrm{z}$
cross-talk signal. At
$64\text{\,}\mathrm{s}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{/}\mathrm{s}$
and below, the noise is on the order of
$1\text{\,}\mathrm{b}\mathrm{i}\mathrm{t}$ or
$125\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{V}$. The full range of the
detector circuit is $3.1\text{\,}\mathrm{V}$, which corresponds to a dynamic
range of $88\text{\,}\mathrm{d}\mathrm{B}$ or a theoretical upper limit to the
detectable absorbance of $4.4\text{\,}$. This compares very favorably to the
commercial Eppendorf system, which has an optical-density resolution of
$1\text{\times}{10}^{-3}\text{\,}$ on full range of approximately
$2\text{\,}$. Assuming that the noise level of the Eppendorf system is
comparable to the resolution, this would correspond to a dynamic range of only
$53\text{\,}\mathrm{d}\mathrm{B}$.

At
$64\text{\,}\mathrm{s}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{/}\mathrm{s}$,
the noise level is
$16\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{V}\mathrm{/}\sqrt{Hz}$.
SPICE555Simulation program with integrated circuit emphasis simulations
indicate that the theoretical noise level of the amplifier is on the order of
$0.5\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{V}\mathrm{/}\sqrt{Hz}$,
indicating that we have not yet reached the full potential of the electronics.

Figure 7: Detector photodiode voltage sampled by the AD converter at a rate of
$860\text{\,}\mathrm{s}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{/}\mathrm{s}$
while the red LED power is modulated by $0.4\text{\,}\mathrm{\char 37\relax}$.
The effective sample rate was
$323\text{\,}\mathrm{s}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{/}\mathrm{s}$.
No transitions between the levels can be observed, so the time response of the
detector photodiode is better than $3.1\text{\,}\mathrm{m}\mathrm{s}$. Figure
8: Standard deviation (RMS noise) over $1000$ samples taken by the AD
converter of the detector photodiode signal as a function of sample rate.
Noise increases with increasing sample rate, but not proportional to the
square root of the bandwidth (solid line). There is a strong increase in noise
above $64$ samples/s.

####  IV.1.2 Magnetic field system

Figure 9 shows the magnetic field in the center of the system as a function of
the current through each of the three coil sets. The coils generate
approximately $2\text{\,}\mathrm{m}\mathrm{T}\mathrm{/}\mathrm{A}$, with
around $5\text{\,}\mathrm{\char 37\relax}$ variation between the coils. The
maximum field that can be generated is slightly higher than
$5\text{\,}\mathrm{m}\mathrm{T}$ at full current of approximately
$2.5\text{\,}\mathrm{A}$. The pulse width of the modulation of the driver
circuits can be set with a maximum resolution of
$16\text{\,}\mathrm{b}\mathrm{i}\mathrm{t}$, corresponding to a theoretical
field resolution of about $70\text{\,}\mathrm{n}\mathrm{T}$. In practice, we
operate the PWM at $8\text{\,}\mathrm{b}\mathrm{i}\mathrm{t}$ resolution,
which yields a set-point resolution of about
$20\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{T}$.

As we drive the coils with a PWM signal, the current through the coils is not
constant but follows the modulation frequency. At zero and maximum current,
the ripple is absent. The ripple has a maximum at $50\text{\,}\mathrm{\char
37\relax}$ duty cycle. The filtering action of the coil system dampens the
modulation. At a PWM drive frequency of
$20\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ and $50\text{\,}\mathrm{\char
37\relax}$ modulation, we measured a triangular current signal with a peak-to-
peak amplitude of $24(2)\text{\,}\mathrm{m}\mathrm{A}$ on a mean current of
$1.2\text{\,}\mathrm{A}$. Simulations considering only the low-resolution
nature of the coils, with a corner frequency of
$115\text{\,}\mathrm{H}\mathrm{z}$, yield a theoretical amplitude of
$18\text{\,}\mathrm{m}\mathrm{A}$, so there is probably some additional
capacitive coupling. The current variation corresponds to a maximum field
variation in the field of approximately
$50\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{T}$ or
$1.2\text{\,}\mathrm{\char 37\relax}$.

At the maximum current of $2.5\text{\,}\mathrm{A}$, the coils dissipate about
$13.1\text{\,}\mathrm{W}$ each. As the coil system has no active cooling, the
heating of the sample area can be considerable for prolonged measurement
times. An NTC temperature sensor is mounted on the body of the measurement
chamber to monitor the rise in temperature, see Figure 10. We also measured
the temperature in the chamber with a simple alcohol thermometer for
comparison. The temperature of the coils can be estimated from the increase in
coil resistance, assuming the temperature coefficient of copper
($0.393\text{\,}\mathrm{\char 37\relax}\mathrm{/}\mathrm{K}$).

At a drive current of $0.5\text{\,}\mathrm{A}$ (field strength of
$1\text{\,}\mathrm{m}\mathrm{T}$), the heating of the chamber is barely
noticeable (about $1\text{\,}\mathrm{K}\mathrm{/}\mathrm{h}$). The average
temperature of the coils increases at approximately
$8\text{\,}\mathrm{K}\mathrm{/}\mathrm{h}$. With a drive current of
$1.2\text{\,}\mathrm{A}$, the temperature of the coils increases by
$21\text{\,}\mathrm{K}$. The temperature increase of the chamber is
substantial, with an initial increase of approximately
$0.25\text{\,}\mathrm{K}\mathrm{/}\mathrm{m}\mathrm{i}\mathrm{n}$ and a
subsequent flattening at $7\text{\,}$ to $8\text{\,}\mathrm{K}$ after
$40\text{\,}\mathrm{m}\mathrm{i}\mathrm{n}$.

Figure 9: Magnetic field in the center of the cuvette holder as a function of
average coil current. Fields in $x$, $y$ and $z$ direction with a field-to-
current ratio of $2.175(9)\text{\,}$, $2.113(3)\text{\,}$ and
$2.007(3)\text{\,}\mathrm{m}\mathrm{T}\mathrm{/}\mathrm{A}$, respectively.

Figure 10: Top: Temperature of the coil system as a function of time for
different drive currents. At the maximum drive current of
$2.5\text{\,}\mathrm{A}$, corresponding to $5\text{\,}\mathrm{m}\mathrm{T}$,
the coils heat rapidly, and operation should be limited to a few minutes.
Bottom: MagOD frame temperature (solid symbols) and air temperature in the
chamber (open symbols) as a function of time. At a drive current below
$1.2\text{\,}\mathrm{A}$ ($2.4\text{\,}\mathrm{m}\mathrm{T}$), the temperature
increase of the frame is limited to about $6\text{\,}\mathrm{K}$.

###  IV.2 Applications

We present four experiments to illustrate the application of our MagOD meter
to analyze magnetotactic bacteria. We measure (1) the scattering of
_Magnetospirillum gryphiswaldense_ (MSR-1) bacteria as a function of their
angle to the incident light, (2) their rotational velocity as a result of a
rotation of the external magnetic field on time scales of seconds and (3) the
development of a culture over a period of several days. The final experiment
measures (4) the velocity distribution of the unipolar _Magnetococcus marinus_
(MC-1) as a function of time.

####  IV.2.1 Transmission as a function of angle (MSR-1)

Using the coil system of our MagOD meter, we can apply a field in any
direction in three-dimensional space. This allows us to study the transmission
of light as a function of the orientation of the bacteria and check the model
presented in Section II.

For this purpose, a cuvette of MSR-1 bacteria, grown according to [18], with
an optical density of approximately $0.1\text{\,}$ was inserted into the MagOD
system. We measured the intensity on the photodetector as a function of the
angle of the magnetic field with steps of approximately
$5\text{\,}\mathrm{\SIUnitSymbolDegree}$. The magnetic field varied with
angle, but was always greater than $1\text{\,}\mathrm{mT}$. As the optical
density of the sample fluctuated continuously due to activity and
sedimentation within the cuvette, we performed the measurement $20$ times. The
resulting curves were normalized to a range of $0$ to $1$ and averaged to
obtain the angle-dependent scattering factor $g(\theta)$ shown in Figure 11.

The simple inverted sine model discussed in Section II fits surprisingly well.
The strongest deviation is around the parallel alignment, which is not
surprising. The MSR-1 are not infinitely thin rods, but spirals. Therefore,
the projected area will be less sensitive to variations in the angle around
the long axis. Additionally, the culture of MSR will have a distribution in
angles (due to Brownian motion and/or flagellar motion), which will round off
the sharp corner at $\theta=0$. The red curve illustrates this effect for
$mB$=$60\text{\,}\mathrm{k}\mathrm{T}$, which still does not fit the
measurement very well. It therefore seems likely that the actual bacteria
shape, and might also be their distribution, should be included in the model.

Figure 11: Scattering of a culture of magnetotactic baceria is dependent on
the orientation of the external magnetic field; it is highest when the field,
and hence the bacteria, are aligned parallel to the light beam and lowest when
the field is aligned orthogonally. By normalizing from $[0,1]$, we obtain the
angle-dependent scattering factor $g(\theta)$, which can be relatively well
approximated by a sine function. The flattening of the curve around
$0\text{\,}\mathrm{\SIUnitSymbolDegree}$ can be partly explained by additional
Brownian motion of the bacteria (red curve).

####  IV.2.2 Response as a function of field strength and time (MSR-1)

In our lab, we most commonly perform measurements in which a sample of MSR-1
is subjected to field switching between parallel and perpendicular alignment
to the light path and varying field strengths. Figure 12 shows the measured
response for a set of field cycles.666In this measurement, the absorbance is
high (transmission of light is low) when the field is aligned along the light
path. This measurement was performed with an older, single-stage photodiode
amplifier, unlike the measurement shown in Figure 5 taken with the new
amplifier, which has an inverted response. At a high field value of
$3\text{\,}\mathrm{m}\mathrm{T}$, the field is switched from parallel to
perpendicular alignment after $10\text{\,}\mathrm{s}$. For the lower field
value of $0.4\text{\,}\mathrm{m}\mathrm{T}$, we can allow longer reversal
times because coil heating is negligible.

From the difference in detector signals, we can calculate $\Delta_{\text{OD}}$
using Eq. (II.1). The signal of the growth medium without bacteria
($I_{\text{ref}}$) was $301(1)\text{\,}\mathrm{m}\mathrm{V}$. We can therefore
calculate $C_{\text{mag}}$ using Eq. (4).

The difference in transmission between in-plane and perpendicular alignment is
higher at $3\text{\,}\mathrm{m}\mathrm{T}$ compared to
$0.4\text{\,}\mathrm{m}\mathrm{T}$. This is in agreement with the predicted
increase of scattering factor with increasing field, see Figure 3. Figure 13
shows the calculated difference in scattering factor as a function of the
magnetic field scaled to $kT/m$. From a previous analysis of MSR-1 [18], we
estimated that the mean magnetic moment $m$ of the magnetosome chain is
$0.25\text{\,}\mathrm{f}\mathrm{A}\mathrm{m}^{2}$, with a
$10-$90\text{\,}\mathrm{\char 37\relax}$$ cutoff of the distribution of
$0.07\text{\,}$ and $0.57\text{\,}\mathrm{f}\mathrm{A}\mathrm{m}^{2}$,
respectively. We can convert these ranges of moments into the energy ratio
$mB/kT$ for the two difference field values. Lines on the top axis of the
graph indicate the ranges, and red circles on the red line indicate the mean
values. The predicted reduction between the average scattering factors
($0.20\text{\,}$) at the two field values is less than that observed in Figure
12 ($0.5\text{\,}$). However, considering that we have a simplified model and
therefore that our previous estimate of the magnetic moment may be different
for the current sample, agreement with the measurement is acceptable. In
addition to a decrease in step height, the time response also decreases with
decreasing field. The time constant is estimated from a fit of Eq. (7) to the
data using the sum of squared errors criterion. The time constant of the
transitions to $3\text{\,}\mathrm{m}\mathrm{T}$ is
$1.7(5)\text{\,}\mathrm{s}$, which is approximately 13 times higher than the
time constant of $5.4(8)\text{\,}\mathrm{s}$ of the transition to
$0.4\text{\,}\mathrm{m}\mathrm{T}$. The ratio is rather high, but still within
measurement error equal to the ratio of the fields, as predicted by the model
discussed in Section II.

Figure 12: Two cycles of a measurement sequence. The magnetic field is
alternatively aligned parallel and perpendicular to the light beam. For high
fields ($3.0\text{\,}\mathrm{m}\mathrm{T}$), we can determine
$\Delta_{\text{OD}}$ from the difference between the averages of the detector
signals (red lines), from which we can calculate $C_{\text{mag}}$. The signal
of the growth medium without bacteria was
$301(1)\text{\,}\mathrm{m}\mathrm{V}$. The difference between the two
directions of the field drops considerably at low field
($0.4\text{\,}\mathrm{m}\mathrm{T}$), whereas the response time increases.
These low fields are suitable for estimating the time constant $\tau$ from a
fit to an exponential (blue line). Using the field magnitude, we can calculate
$\gamma$
($\text{\,}\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{/}\mathrm{m}\mathrm{T}\,\mathrm{s}$).
Figure 13: Calculation of the average scattering factor as a function of the
product of the magnetic moment of the magnetosome chain $m$ and applied
magnetic field $B$ for the magnetic field aligned parallel
($\theta$=$0\text{\,}\mathrm{\SIUnitSymbolDegree}$), and perpendicular
($\theta$=$90\text{\,}\mathrm{\SIUnitSymbolDegree}$) to the light path, see
Figure 3. The difference in the average scattering factor between these
orientations is indicated by a red line ($\Delta g$). From earlier work [18],
we estimated that the mean magnetic moment of the magnetosome chain is
$0.25\text{\,}\mathrm{f}\mathrm{A}\mathrm{m}^{2}$, with a
$10-$90\text{\,}\mathrm{\char 37\relax}$$ cutoff of the distribution of
$0.07\text{\,}$ and $0.57\text{\,}\mathrm{f}\mathrm{A}\mathrm{m}^{2}$,
respectively. The resulting ranges in $mB/kT$ are indicated on the top axis
and by red circles for the mean values of the moments for the fields used in
the experiment illustrated in Figure 12.

####  IV.2.3 Comparison of $C_{\text{mag}}$ for different instruments

A major advantage of our novel MagOD system is that it allows us to
standardize $C_{\text{mag}}$ measurements obtained at different laboratories.
It is not trivial to calibrate instruments by sending around bacteria cultures
because such cultures develop with time and are not stable. Therefore, a
standard instrument is the better solution. Table 4 compares $C_{\text{mag}}$
with existing instruments in three different laboratories. Researchers at CNRS
used a modified Cary UV at $600\text{\,}\mathrm{n}\mathrm{m}$ wavelength,
those at Aston University used a modified Thermo Evolution at
$565\text{\,}\mathrm{n}\mathrm{m}$ wavelength, and those at Bayreuth
University used a Ultrospec 2100 Pro at $565\text{\,}\mathrm{n}\mathrm{m}$
wavelength as well. The cultures were not the same, so the results obtained at
the above laboratories cannot be compared. Nevertheless, it is very clear that
the difference between the results obtained from commercial instruments and
those obtained from our standardized MagOD vary significantly from laboratory
to laboratory.

Table 4: $C_{\text{mag}}$ of MSR-1. Comparison of MagOD with commercial instruments used at different laboratories (Cary UV: CNRS, Thermo Evolution: Aston, Ultrospec 2100: Bayreuth). The difference between commercial instruments and our MagOD system varies from laboratory to laboratory, which underscores the need for a standardized measurement instrument.  | Cary | Thermo | Ultrospec 2100
---|---|---|---
Commercial Inst. | 2.8 | 3.1 | 2.8 | 2.4 | 2.6 | 2.4 | 2.2
MagOD Green | 1.9 | 2.1 | 1.9 | 2.3 | 1.5 | 2.3 | 1.5
MagOD Red |  |  |  | 2.5 | 1.7 | 1.6 | 1.7

####  IV.2.4 Long-term growth monitoring (MSR-1)

When cultivating magnetotactic bacteria such as MSR-1, it is important to
check regularly whether the bacteria remain magnetic. When bacteria are grown
under laboratory conditions, random mutation may lead to a culture of
magnetotactic bacteria that has lost the ability to form magnetosomes [24]. In
our lab, MSR-1 are grown in $2\text{\,}\mathrm{m}\mathrm{L}$ tubes. The tubes
are closed, and a small headspace of air serves to ensure a proper reduction
of oxygen concentration as the culture grows. Even though this method is
simple, its major disadvantage is that we have no information whether the
magnetosome formation occurs as we expect it to. We cannot open the tubes to
take samples, because that would let oxygen in. The better option would be to
grow the bacteria in bioreactors that allow sampling without disturbing the
oxygen concentration. Unfortunately, bioreactors are complex, costly, and
provide quantities that far exceed what is needed for lab-on-chip experiments.

Our MagOD system offers a solution for monitoring the growth of MSR-1 bacteria
and the magnetosome by keeping cultures in cuvettes inside the MagOD meter for
long periods. Throughout the growth period, we continuously measure the
absorbance during changes of the external magnetic field. In this way, we
obtain information about the total number of bacteria as well as their
magnetic response.

We prepared MSR-1 cultures in the conventional manner [18] but, instead of
tubes, we used quartz cuvettes with a PTFE stopper (Hellma QS 110-10-40) to
avoid oxygen leakage into the cuvette. For the long-term observations shown in
Figure 14, the magnetic field was set to loop through cycles of
$100\text{\,}\mathrm{s}$ consisting of a vertical field of
$1.0\text{\,}\mathrm{mT}$ ($20\text{\,}\mathrm{s}$), a horizontal field of
$2.9\text{\,}\mathrm{mT}$ ($20\text{\,}\mathrm{s}$), and a vertical field of
$0.1\text{\,}\mathrm{mT}$ ($60\text{\,}\mathrm{s}$).

The first transition is at a relatively strong field, thus guaranteeing
reliable estimations of $C_{\mathrm{mag}}$. The second transition guarantees a
relatively high time constant, which is helpful for estimating $\tau$
accurately.

Figure 14 shows the measured parameters of a sample of magnetotactic bacteria
over a period of five days. The optical density, relative ($C_{\text{mag}}$)
and absolute ($\Delta_{\text{OD}}$) magnetic response and relative rotation
velocity ($\gamma$, proportional to the ratio between magnetic moment and
rotational friction coefficient) are plotted from top to bottom. The optical
density is typical for a bacteria growth sequence. After a lag phase L, a
transition into the exponential growth phase E occurs, followed by the
stationary phase S, where the bacteria concentration remains more or less
constant. After three days, however, the density increases unexpectedly as
illustrated in phase X.

During the exponential growth phase, $C_{\text{mag}}$ decreases sharply. As
$\Delta_{\text{OD}}$ remains constant, we assume that the increase in
concentration is entirely due to bacteria without magnetosomes. Only after two
days do we see a gradual increase in magnetic signal M due to an increase in
the proportion of bacteria with magnetosomes. With the increase in magnetic
signal, $\gamma$ also increases, hence the magnetic moment of the magnetosome
increases compared to the average bacteria length. The observation for the
first three days would be consistent with the mechanism that, after seeding
with magnetic bacteria, growth first proceeds by an increase in non-magnetic
bacteria. When that growth stops, the bacteria start to form magnetosomes.
This mechanism contradicts electron microscope observations by Stanisland and
Yang that magnetosome crystals are distributed equally over both parts of the
divided cell [25, 26]. Clarification of this discrepancy will require further
experiments.

A sharp transition in phase X occurs after approximately $3.3$ days. As the
density of the culture increases again, the magnetic response decreases but
$\gamma$ continues to increase. This is a feature we often observe in these
measurements. Additionally, we sometimes observe a cloudiness in the
suspension, which may be caused by aerotaxis or contamination. As we do not
shake the suspensions before measuring like in a standard optical density
meter, these clouds may float in front of the detector and complicate the
analysis. It is possible that, rather than rotating individual bacteria, we
rotate the entire cloud.

Figure 14: Four bacteria suspension parameters measured over a span of five
days: (i) optical density (OD), (ii) $C_{\mathrm{mag}}$, which quantifies the
ratio of magnetic to nonmagnetic bacteria, (iii) $\Delta_{\text{OD}}$, which
quantifies the amount of magnetic bacteria, and (iv) $\gamma$, which
quantifies how strongly the bacteria respond to magnetic fields. The following
phases can be identified: lag L, exponential E, stationary S, magnetic growth
M, and an undefined phase X.

####  IV.2.5 Marathon test: MC-1 velocity measurement

In contrast to MSR-1 bacteria, which reverse frequently, magnetotactic
bacteria of type MC-1 tend to swim for long periods in the same direction [9].
This allows us to collect a large number of bacteria at the bottom of a
cuvette simply by applying a vertical field. After reversing the direction of
the field, all bacteria swim upward in a band-shaped cloud. In the MagOD
system, the passing of this cloud translates to a drop in the photodetector
signal. The time between reversal of the field and the response on the
photodetector is a measure for the velocity of the bacteria. We call this
method the “marathon” test.

To obtain sufficient bacteria for this experiment, we cultivated MC-1 bacteria
in a high-viscosity, agarose-based medium in an oxygen gradient, as described
by Bazylinski [27], but use low-melt agarose instead of bacto agar. The
bacteria form a band in the reaction tube a few millimeters below the surface
of the medium [28]. The easiest way to free the MC-1 from the medium is to
pipette a small amount from the band and insert this into a cuvette filled
with a low-viscosity growth medium from which the agarose has been omitted.
The transfer of some agarose cannot be avoided, especially if large quantities
of bacteria are desired.

Alternatively, one can place a droplet of agarose with bacteria on one side
and a droplet of growth medium with agarose next to it so they merge. One can
then use a magnet to draw the MC-1 out of the agarose and into a clean
droplet. The disadvantage of this method is that only a limited amount of
bacteria can be collected and that it is difficult to avoid admitting oxygen
into the sample.

The method we prefer is to pass the mixture of bacteria and agarose through a
Pasteur pipette filled with a small plug of cotton. Our assumption is that the
cotton breaks up the agarose matrix and perhaps even captures it. By using
compressed nitrogen to push the growth medium with bacteria through the plug,
exposure to oxygen can be avoided. To reduce oxygen exposure even further, we
performed this procedure in a nitrogen atmosphere. For this purpose, we simply
use a glass beaker with a paraffin cover through which the Pasteur pipette is
inserted into the cuvette.

The MagOD system is equipped with a 3D magnetic coil configuration, which
makes it simple to apply a vertical field along the cuvette. A field of
$1\text{\,}\mathrm{m}\mathrm{T}$ is applied in positive $z$-direction for
$220\text{\,}\mathrm{s}$ to collect south-seeking bacteria at the bottom of
the cuvette. Then the field is reversed so that the collected bacteria swim
upward towards the photodetector. We allow the bacteria to swim upward for
$200\text{\,}\mathrm{s}$, after which the sequence is repeated. The asymmetry
in time ensures that a sufficient amount of bacteria can reassemble at the
bottom of the cuvette. The cloud of bacteria that leaves the bottom of the
cuvette disperses due to a distribution of bacteria velocities. To keep the
peak sharp and intensity variation high, we reduce the distance between the
bottom of the cuvette and the light path to $2.5\text{\,}\mathrm{m}\mathrm{m}$
by using a special insert.

Figure 15 shows the output of the photodetector as a function of time elapsed
after the magnetic field reversal. A series of eight experiments are shown.
For each experiment, the cloud reaches the light path after approximately
$30\text{\,}\mathrm{s}$ with a maximum density at about
$90\text{\,}\mathrm{s}$.777These experiments are performed with our novel
amplifier. Lower intensity results in a higher detector voltage, see Figure 5.
As time progresses, the curves have a similar shape, but with a lower
amplitude. Apparently, less and less bacteria collect at the bottom of the
cuvette. The decrease in amplitude shown in Figure 16 agrees very well with an
exponential decay $\exp(-t/\tau)$ and a time constant of approximately 30 min.
This suggests that a fixed amount of bacteria is lost per iteration. The
reason for the loss is yet unclear; this remains a topic for further
investigation.

The arrival time $t$ (s) of MC-1 at the detector agrees well with a log-normal
distribution (shown as red curves in Figure 15),

$f_{\text{t}}(t)=\frac{1}{t\sigma\sqrt{2\pi}}\exp\left(\frac{-\left(\ln(t)-\mu\right)^{2}}{2\sigma^{2}}\right)\text{
(1/s)}\,,$ (9)

where $\mu$ (with unit $\ln(s)$) and $\sigma$ (unitless) are the mean and
standard deviation of the natural logarithm of $t$, respectively. From the
distance of $2.5\text{\,}\mathrm{m}\mathrm{m}$ from the bottom of the cuvette
to the light beam $a$, we can calculate the distribution of the velocities $v$
(m/s) as follows:

$f_{\text{v}}(v)=\frac{1}{v\sigma\sqrt{2\pi}}\exp\left(\frac{-\left(\ln(\frac{a}{v})-\mu\right)^{2}}{2\sigma^{2}}\right)\text{
(s/m)}\,.$ (10)

The most likely velocity, or mode of this distribution, is888Note that the
most likely arrival time is $\exp(\mu-\sigma^{2})$. Therefore, one cannot
simply divide the distance travelled by the most likely arrival time to obtain
the most likely velocity.

$v_{\text{m}}=a\exp\left(-(\mu+\sigma^{2})\right)\text{ (m/s)}\,.$ (11)

The resulting velocity distributions are shown in the lower graph of Figure
15. The curves are offset vertically for clarity; the top curve is the first
measurement. This figure shows clearly that the velocity distribution of the
bacteria does not change significantly with time. As the experiment duration
was limited to $200\text{\,}\mathrm{s}$, the minimum velocity that can be
determined is
$12.5\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}\mathrm{/}\mathrm{s}$. The
most likely velocity is on the order of
$20\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}\mathrm{/}\mathrm{s}$, and
the fastest bacteria swim a rate of approximately
$80\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}\mathrm{/}\mathrm{s}$.

Figure 15 is a typical example; we have measured both faster and slower
average velocities. The measured velocity is considerably lower than that
observed by Lefèvre and colleagues using high-speed microscopy [3]. We noted
from experiments with microfluidic chips that the velocity distribution is
strongly dependent on the duration the MC-1 have been growing in the semi-
solid medium, temperature (both too high and too low reduced velocity) and
oxygen concentration. Further experiments are required to determine the
relation between the velocity and these environmental conditions.

Figure 15: Photodetector output as a function of time elapsed since reversal
of the magnetic field. After approximate $30\text{\,}\mathrm{s}$, the first
MC-1 pass the light beam and scatter the light. Maximum light extinction is
reached about $90\text{\,}\mathrm{s}$. The experiment is repeated eight times
in intervals of $440\text{\,}\mathrm{s}$ between measurements. The responses
fit relatively well to a log-normal distribution (red lines). These fits can
be inverted to obtain the velocity distribution of the MC-1 (bottom curve).
For clarity, these curves are offset by
$0.005\text{\,}\mathrm{s}\mathrm{/}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ from
top to bottom. The velocity distribution remains more or less constant. Figure
16: Decrease in amplitude of the marathon curves in Figure 15 with increasing
time agrees very well with an exponential decay.

##  V Discussion

Our novel MagOD magnetic absorbance instrument has proved to be a versatile
system that can be successfully applied to research on magnetotactic bacteria.
All designs and source codes have been made available online, so that the
system can be easily replicated, modified and improved. The data presented in
this paper is intended to serve as a benchmark for future systems. We hope our
efforts will inspire colleagues to improve and apply the MagOD system for
their own research. In the following, we address possible improvements and
suggestions for further research.

###  V.1 Possible improvements

There are a few issues regarding the measurement head, measurement board and
software that deserve attention for future iterations of the system.

####  V.1.1 Measurement head

The response of the photodiode to the blue LED is nonlinear, see Figure 5, for
which we do not have a satisfactory explanation. Furthermore, the fact that
the signal is inversely proportional to the light intensity is
counterintuitive. It may be possible to redesign the amplifier while
maintaining the required footprint. The noise floor in the current design is
still a factor of 30 above the theoretical limit, and there are signs that
$50\text{\,}\mathrm{H}\mathrm{z}$ crosstalk deteriorates the signal, see
Figure 8. One might consider moving the AD converter from the measurement
board to the measurement head to better protect the signal from interference.
To compensate for drift, automatic offset correction can be applied by
modulating the LED intensity periodically.

One might consider moving the LED drivers to the measurement head, so that the
high-frequency PWM signal does not have to be transported over the HDMI cable.
This would free up ports on the ESP32. For this purpose, RGB LED drivers that
communicate over I2C are readily available. Care should be taken that the I2C
clock signal does not interfere with the detection electronics.

If suspensions with higher densities are to be observed, one might consider
using solid-state lasers that offer at least $100\text{\,}$ times higher light
intensity.

In contrast to commercial absorbance meters, our MagOD system does not have a
piezoelectric actuator to disperse the suspension prior to measurement. One
might consider integrating such a component. Alternatively, one might make use
of the existing coils and attempt a voice coil actuation principle using a
soft magnetic element, an additional small coil or even a small permanent
magnet mounted such that it does not interfere with the field.

####  V.1.2 Measurement board

In future designs of the measurement board, a number of improvements could be
made as well. Even though the AD converter is capable of data acquisition at
$860\text{\,}\mathrm{s}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{/}\mathrm{s}$,
we achieve only
$323\text{\,}\mathrm{s}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{/}\mathrm{s}$
in practice. We assume this discrepancy is caused by communication overhead
that could be optimized.

The current implementation of the measurement circuits allows only for
positive currents. Modifying the circuits to allow for bi-directional currents
is straightforward, for instance by applying an INA266 integrated current
monitor.

Finally, the small form factor of micro-SD cards poses a problem in biolab
environments because they are easily lost. Removal of the SD card can be
avoided if WiFi access is available, but a USB stick may be a better option.

####  V.1.3 Software

We expect that the software of our MagOD system will undergo the most
development. In addition to improvements of the user interface, the main
restriction is currently that measurement recipes are based only on feed-
forward instructions, i.e. iterations of a specified amount of time, field
settings and LED color. There is currently no capability to react to changes
in the detected signal. For instance, it would be very useful if the LED
intensity could be adjusted automatically to the absorbance of the suspension
under investigation. In marathon experiments, it would be convenient if the
field reversal took place at a fixed delay after the occurrence of the peak.
The current recipe language definition is not capable of handling this type of
feedback. We suspect a complete redesign of the software is required, taking
full advantage of the EPS32 capabilities. This would be a very interesting
task for a (software) engineering student.

###  V.2 Possible future applications

The four experiments we presented are merely a selection of the many
possibilities offered by our novel MagOD system. Even without additional
modification, there are numerous possible experiments to inspire future work.

####  V.2.1 Flagellar motion

As the MagOD system has precise field control, it allows a simple study of the
relation between field strength and $C_{\text{mag}}$. It would be interesting
to check whether the swimming activity of the bacteria themselves contributes
to their random motion, which effectively would increase $kT$ and could
explain the observed difference. For instance, it would be sufficient to
measure $C_{\text{mag}}$ as a function of field before and after killing the
MSR-1 (for example by exposure to intense UV light or formaldehyde).

####  V.2.2 Multi-color optical density

So far, we have measured the transmission through MSR-1 cultures only under
green light. However, we noticed that the color of cultures changes with
elapsed time. We speculate that these color changes may be caused by an
increase in bacteria size and/or formation of magnetosomes. For long-term
analysis as illustrated in Figure 14, it may therefore be useful to measure
this at different wavelengths. The MagOD system can easily achieve this by
measuring iteratively with red, green and blue LEDs. Multiple wavelengths may
be combined with the addition of an indicator agent that changes its
absorbance spectrum based on changed conditions.

An example of such an indicator is Resazurin, which reacts to an increase in
oxygen concentration with a shift in the absorbance spectrum toward red. The
ratio between the absorbance in the red and green channels could therefore be
a measure of the oxygen concentration in the culture, using the blue channel
for calibration.

####  V.2.3 Modulated light intensity

The intensity of the LEDs can be varied rapidly, as illustrated in Figure 7.
One can use this modulation for differential measurements to correct for
interference signals due to changes in environmental light or crosstalk on the
analog signal wiring.

Modulation of the light intensity would also provide information about the
photosensitivity of the bacteria. For instance, one could measure
$C_{\text{mag}}$ in the red channel before and after a pulse with intense blue
light.

####  V.2.4 Combined marathon and $C_{\text{mag}}$

We demonstrated $C_{\text{mag}}$ measurements on MSR-1 bacteria and marathon
tests on MC-1 bacteria. It is straightforward to combine the marathon test
with $C_{\text{mag}}$ measurements. The vertical field ($z$-direction) should
then be switched between zero and, for instance, a positive value, whereas the
field along the light path ($x$-direction) should be switched between zero and
alternatively positive and negative values, hence
$(x,z)={(0,1),(1,0),(0,1),(-1,0)}$, etc. Such an experiment might reveal
whether the velocity distribution is related to a distribution of magnetosome
strength as well.

####  V.2.5 Sedimentation

We often observed an initial increase of light transmission after loading a
sample with bacteria. We therefore usually waited until the signal settled.
However, there may be information to be extracted from this behavior. We
suspect the increase in transmission is caused by sedimentation of debris,
such as dead bacteria. If the dead bacteria contain magnetosomes, they will
still rotate in the magnetic field. Therefore a measurement of
$C_{\text{mag}}$ during sedimentation might provide additional information
about the status of the culture.

Moreover, it is very simple to drive only one coil of the vertical coil set.
In this way, one can generate field gradients that would pull magnetic debris
either up or down, thus decelerating or accelerating the sedimentation
process.

####  V.2.6 Suspensions of magnetic nanoparticles

We often use our MagOD system with a suspension of magnetic micro- and
nanoparticles. This works particularly well for magnetic needles [29] and
magnetic discs [30]. In principle, spherical particles should not show a
change in transmission under rotation of an external field. However, magnetic
particles have a tendency to form chains that align with the field, see for
instance the work by Gao [31]. Angle and field-dependent transmission
measurements in the MagOD could therefore provide information about the
dynamic interaction between particles. Use of our MagOD system could perhaps
be extended even beyond the magnetotactic research community.

##  VI Conclusion

We constructed a magnetic spectrophotometer (magnetic optical density meter,
or MagOD) that analyzes the amount of light transmitted through a suspension
of a magnetotactic bacteria in a transparent cuvette under application of a
magnetic field.

Light transmission measurements with our novel MagOD system were compared with
those obtained with a commercial instrument (Eppendorf BioPhotoMeter) using a
dilution series of a magnetic nanoparticle suspension. The deviation between
our MagOD system and the commercial instrument is less than $0.2\text{\,}$ in
terms of relative absorbance for wavelengths ranging from $460\text{\,}$ to
$645\text{\,}\mathrm{n}\mathrm{m}$. However, the blue channel suffers from
nonlinearity and should only be used at low intensities. The dynamic range
(from noise level to maximum signal) of our novel MagOD system is
$88\text{\,}\mathrm{d}\mathrm{B}$ (optical density of $4.4\text{\,}$), whereas
the commercial system reaches $53\text{\,}\mathrm{d}\mathrm{B}$ (optical
density of $2.6\text{\,}$). In addition, our MagOD system is considerably
faster, with a sample rate of
$323\text{\,}\mathrm{s}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{/}\mathrm{s}$,
whereas the commercial instrument has a sampling time in excess of
$1\text{\,}\mathrm{s}$.

The magnetic field can be applied in three directions, with a set-point
resolution of $70\text{\,}\mathrm{n}\mathrm{T}$ and a ripple of less than
$50\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{T}$. The maximum field is
$5.1(1)\text{\,}\mathrm{m}\mathrm{T}$, but is limited in duration due to coil
heating. When a field of $1.0\text{\,}\mathrm{m}\mathrm{T}$ is continuously
applied, the temperature increase of the cuvette is approximately
$1\text{\,}\mathrm{K}\mathrm{/}\mathrm{h}$ and limited to
$2.1(3)\text{\,}\mathrm{K}$.

The MagOD system was used to characterize various aspects of MSR-1 and MC-1
magnetotactic bacteria. By means of the magnetic field, MSR-1 bacteria were
oriented at different angles with respect to the light path. The transmission
rate is high when bacteria are aligned along the light beam and lower when the
bacteria are aligned perpendicular to the light path. The relation between
angle and optical density can be approximated relatively well by a sine
function.

The difference in transmission rates allows us to derive a measure for the
amount of magnetic bacteria. This amount is commonly expressed as a ratio
$C_{\text{mag}}$, which is a parameter that increases with the relative
fraction of magnetic bacteria compared to the total number of bacteria. It can
also be expressed as a difference $\Delta_{\text{OD}}$, which is a measure for
the absolute amount of magnetic bacteria. Both parameters increase with
applied field strength in a way that is in agreement within measurement error
with a simple model based on Brownian motion.

We used our novel MagOD system to continuously monitor the development of a
culture of MSR-1 magnetotactic bacteria for
$5\text{\,}\mathrm{d}\mathrm{a}\mathrm{y}\mathrm{s}$. We recorded the optical
density $O_{\text{D}}$, change in light transmission under rotation of the
magnetic field $C_{\mathrm{mag}}$ and $\Delta_{\text{OD}}$, and the rotation
velocity of the bacteria $\gamma$. We were able to distinguish clearly the
separate growth phases (lag, exponential growth, and stationary). The increase
in magnetic response $C_{\mathrm{mag}}$ and $\Delta_{\text{OD}}$ takes place
during the stationary phase.

Unipolar bacteria such as MC-1 can be collected at the bottom of the cuvette
with a vertical magnetic field. Upon reversal of the field, the entire group
departs from the bottom and will arrive at the light beam, causing a dip in
the transmitted light. This “marathon” test allows us to measure the velocity
distribution.

The arrival times can be accurately described by a log-normal distribution,
with a mode (most frequently occurring velocity) of
$20\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}\mathrm{/}\mathrm{s}$. The
maximum velocity we observed is on the order of
$80\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}\mathrm{/}\mathrm{s}$. The
amount of bacteria participating in the marathon test decreases exponentially
with each test with a time constant of approximately 30 min.

The dedicated magnetic optical density meter presented here is relatively
simple and inexpensive, yet the data that can be extracted from magnetotactic
bacteria cultures is rich in detail. All information for the construction of
the device, including 3D print designs, printed circuit board layouts and code
for the microprocessor, has been made available online. The authors trust that
the magnetotactic bacteria community will benefit from our work, and that the
MagOD instrument will become a valuable tool for research in this field.

## Acknowledgments

The authors thank Christopher Lefèvre for teaching us how to grow MSR-1 and
MC-1 bacteria and for discussions and feedback on the manuscript, Annissa
Dieudonne for testing the first version of the MagOD meter and for valuable
feedback on the manuscript, Andreas Manz for beneficial discussions and
guidance, Prof. Long Fei Wu and Stefan Klumpp for their excellent feedback on
the manuscript, and Sander Smits for 3D prints and laser cutting.

##  Appendix A $C^{*}_{\text{mag}}$ and $C_{\text{mag}}$ approximations

The effect of a magnetic field rotation is usually small. It is therefore
useful to express the variation with respect to the average intensity or
absorbance999$A$=$O_{\text{D}}$

$\displaystyle I_{\text{s}}=\frac{I(0)+I(90)}{2}$ (12) $\displaystyle
A=\log(I_{\text{ref}}/I_{\text{s}})$ (13)

by a small deviation $\alpha$

$\displaystyle A(0)=(1+\alpha)A$ (14) $\displaystyle A(90)=(1-\alpha)A$ (15)

so that

$\displaystyle\Delta_{\text{A}}=2\alpha A$ (16)

and

$\displaystyle C_{\text{mag}}=\frac{(1+\alpha)}{(1-\alpha)}\approx
1+2\alpha=1+\frac{\Delta_{\text{A}}}{A}\,.$ (17)

The approximation is better than $5\text{\,}\mathrm{\char 37\relax}$ in terms
of $C_{\text{mag}}-1$ for $C_{\text{mag}}<1.1$.

Similarly, to estimate $C^{*}_{\text{mag}}$, we can define

$\Delta I=2\beta I_{\text{s}}$ (18)

so that

$C^{*}_{\text{mag}}\approx
1+\frac{I_{\text{s}}}{I_{\text{ref}}-I_{\text{s}}}2\beta=\ 1+\frac{\Delta
I}{I_{\text{ref}}-I_{\text{s}}}\,.$ (19)

Both definitions of $C_{\text{mag}}$ can be related by realizing that

$\displaystyle\frac{I(0)}{I_{\text{ref}}}=\left(\frac{I_{\text{s}}}{I_{\text{ref}}}\right)^{1+\alpha}\approx\
\frac{I_{\text{s}}}{I_{\text{ref}}}\left(1+\alpha\ln(I_{\text{s}}/I_{\text{ref}})\right)$
(20)

and similarly for $I(90)$ with $-\alpha$, so that

$\Delta I=-2\alpha I_{\text{s}}\ln(I_{\text{s}}/I_{\text{ref}})\,.$ (21)

Therefore, in the approximation for $C_{\text{mag}}$ close to unity, the
relation between these two definitions is

$\displaystyle\frac{C_{\text{mag}}-1}{C^{*}_{\text{mag}}-1}\approx\
\frac{\Delta_{\text{A}}}{A}\frac{I_{\text{ref}}-I_{\text{s}}}{\Delta I}=$ (22)
$\displaystyle\frac{I_{\text{ref}}-I_{\text{s}}}{I_{\text{s}}\ln(I_{\text{ref}}/I_{\text{s}})}=\frac{(I_{\text{ref}}-I_{s})\log(e)}{I_{\text{s}}A}.$
(23)

The definitions converge for $I_{\text{s}}\to I_{\text{ref}}$ for samples with
very low optical density.

##  Appendix B Cotangent approximation

For fitting purposes, the rather complicated cotangent expression of Eq. (7)
can be approximated by a much simpler exponential function. The fit was
performed in gnuplot, resulting in a fit parameter of $0.85(1)\text{\,}$. The
error is less than $0.065\text{\,}\mathrm{r}\mathrm{a}\mathrm{d}$ as shown in
Figure 17.

Figure 17: Approximation by an exponential function of the exact solution to
the differential equation for the rotation of the bacterium as a function of
time, see Eq. (7). The optimal fit is for a prefactor $0.85(1)\text{\,}$, in
which case the error is less than
$0.065\text{\,}\mathrm{r}\mathrm{a}\mathrm{d}$.

##  Appendix C Measurements

Figure 18 shows the projection onto a sheet of white paper of the light
exiting from the measurement head (with the photodetector circuit board
removed). The images are snapshots of a video taken with an iPhone camera for
a range in LED duty cycles (measurement of Figure 19). The full video is
available in the Supplementary Material, see MagODLEDprojection.mov. The
opening behind the cuvette is a square hole of
$3\text{\times}3\text{\,}\mathrm{m}\mathrm{m}$, which is clearly visible. The
photodetector itself has an area of
$2.7\text{\times}2.7\text{\,}\mathrm{m}\mathrm{m}$, hence it collects the
inner portion of this pattern. For the green and blue LEDs, echo images can be
observed. The three patterns do not align, which is most likely caused by the
fact that the three LEDs in the WP154 housing are not centered on the axis of
the front lens. From the distance between the projected image and the LED
(approximately $15\text{\,}\mathrm{c}\mathrm{m}$), with a maximum shift of
about $5\text{\,}\mathrm{m}\mathrm{m}$, we estimate that the misalignment is
on the order of $2\text{\,}\mathrm{\SIUnitSymbolDegree}$. As the photodetector
is mounted directly behind the opening behind the cuvette, this misalignment
is of no consequence.

Figure 18: Projection of the light pattern of the three different LEDs. The
$3\text{\times}3\text{\,}\mathrm{m}\mathrm{m}$ opening at the back of the
cuvette can clearly be seen. The green and blue LEDs show some reflection, and
the patterns are not aligned. The estimated misalignment is on the order of
$2\text{\,}\mathrm{\SIUnitSymbolDegree}$. As the photodetector has a sensitive
area of $2.7\text{\times}2.7\text{\,}\mathrm{m}\mathrm{m}$ and is mounted
directly behind the opening in the holder, we expect no adverse effects from
the reflections of misalignment. A full video of the pattern shape as a
function of intensity is available in the Supplementary Material.

Figure 19 shows the intensity on the photodetector as a function of the duty
cycle of each of the three LEDs for a cuvette filled with water and three
different dilutions of a EMG304 magnetic nanoparticle suspension. The
absorbance relative to the water-filled cuvette is determined from the
difference in slopes. This measurement is used for the MagOD data points in
Figure 6. The blue LED suffers from artefacts. The slope is not constant, but
lower at high intensities. Furthermore, there is a small step at an intensity
of about $0.6$. For the estimate of absorbance of the blue LED, only the
linear region at low intensity was used.

Figure 19: Signal as a function of the LED duty cycle for the three different
LED wavelengths as well as for a water reference and three different dilutions
of a EMG304 magnetic nanoparticle suspension.

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