# Can small-scale magnetic fields be the major cause for the near-surface
effect of the solar p-mode frequencies?

Yan Li  Yunnan Observatories, Chinese Academy of Sciences, Kunming 650216,
China  Key Laboratory for the Structure and Evolution of Celestial Objects,
Chinese Academy of Sciences  Center for Astronomical Mega-Science, Chinese
Academy of Sciences, Beijing 100012, China  University of Chinese Academy of
Sciences, Beijing 100049, China  Qian-sheng Zhang  Yunnan Observatories,
Chinese Academy of Sciences, Kunming 650216, China  Key Laboratory for the
Structure and Evolution of Celestial Objects, Chinese Academy of Sciences
Center for Astronomical Mega-Science, Chinese Academy of Sciences, Beijing
100012, China  Tao Wu  Yunnan Observatories, Chinese Academy of Sciences,
Kunming 650216, China  Key Laboratory for the Structure and Evolution of
Celestial Objects, Chinese Academy of Sciences  Center for Astronomical Mega-
Science, Chinese Academy of Sciences, Beijing 100012, China  Jie Su  Yunnan
Observatories, Chinese Academy of Sciences, Kunming 650216, China  Key
Laboratory for the Structure and Evolution of Celestial Objects, Chinese
Academy of Sciences  Center for Astronomical Mega-Science, Chinese Academy of
Sciences, Beijing 100012, China  Xing-hao Chen  Yunnan Observatories, Chinese
Academy of Sciences, Kunming 650216, China  Key Laboratory for the Structure
and Evolution of Celestial Objects, Chinese Academy of Sciences  Gui-fang Lin
Yunnan Observatories, Chinese Academy of Sciences, Kunming 650216, China  Key
Laboratory for the Structure and Evolution of Celestial Objects, Chinese
Academy of Sciences  Jian-heng Guo  Yunnan Observatories, Chinese Academy of
Sciences, Kunming 650216, China  Key Laboratory for the Structure and
Evolution of Celestial Objects, Chinese Academy of Sciences  Center for
Astronomical Mega-Science, Chinese Academy of Sciences, Beijing 100012, China
University of Chinese Academy of Sciences, Beijing 100049, China  Jie-ying Liu
Yunnan Observatories, Chinese Academy of Sciences, Kunming 650216, China  Key
Laboratory for the Structure and Evolution of Celestial Objects, Chinese
Academy of Sciences

###### Abstract

Small-scale magnetic fields are not only the fundamental element of the solar
magnetism, but also closely related to the structure of the solar atmosphere.
The observations have shown that there is a ubiquitous tangled small-scale
magnetic field with a strength of 60 $\sim$ 130 G in the canopy forming layer
of the quiet solar photosphere. On the other hand, the multi-dimensional MHD
simulations show that the convective overshooting expels the magnetic field to
form the magnetic canopies at a height of about 500 km in the upper
photosphere. However, the distribution of such small-scale “canopies” in the
solar photosphere cannot be rigorously constrained by either observations and
numerical simulations. Based on stellar standard models, we identify that
these magnetic canopies can act as a global magnetic-arch splicing layer, and
find that the reflections of the solar p-mode oscillations at this magnetic-
arch splicing layer results in significant improvement on the discrepancy
between the observed and calculated p-mode frequencies. The location of the
magnetic-arch splicing layer is determined at a height of about 630 km, and
the inferred strength of the magnetic field is about 90 G. These features of
the magnetic-arch splicing layer derived independently in the present study
are quantitatively in agreement with the presence of small-scale magnetic
canopies as those obtained by the observations and 3-D MHD simulations.

Sun: atmosphere — Sun: magnetic fields — Sun: oscillations — Sun: photosphere
— magnetic fields — magnetohydrodynamics

††software: MESA(v11554; Paxton et al. 2011, 2013, 2015, 2018)

##  1 Introduction

In 1962, Leighton and co-workers discovered rich oscillations with periods of
approximately 5 minutes on the surface of the sun (Leighton, Noyes, and Simon
1962). Later in 1975, Deubner (1975) confirmed that these oscillations are
global resonant waves with pressure as the main restoring force, which are now
named as the solar p-mode oscillations. Frequencies of millions of solar
p-mode oscillations have been measured by grand-based networks, for example
GONG (Harvey et al. 1996), and spaced-based telescopes, for example SOHO
(Schou et al. 1997) with extremely high precisions, and have been used to
extract information of the solar interior.

The internal structure of the sun is described by the standard solar models.
They are based on the up-to-date physics, and computed with state-of-the-art
stellar evolution codes, for example the Modules for Experiments in Stellar
Astrophysics (Paxton et al. 2011, 2013, 2015, 2018; referred to hereafter as
MESA). The standard solar models can reproduce, for example, the distribution
of the sound speed in the solar interior that is in agreement with the one
derived from helioseismic inversions with fairly high precision (Christensen-
Dalsgaard et al. 1985). For frequencies of individual solar p-mode
oscillations, however, the standard solar models usually produce theoretical
oscillation frequencies systematically offset from the observed ones. This is
the so-called near-surface effect of solar p-mode oscillations (Christensen-
Dalsgaard, Däppen, and Lebreton 1988; Dziembowski, Paterno, and Ventura 1988),
whose physical nature has not been fully understood up to now.

In order to eliminate the near-surface effect on the applications of the
observed frequencies in explorations of solar and stellar interior, various
fitting formulae have been proposed to correct the regular part of the
frequency offset between observations and model calculations. Kjeldsen,
Bedding, and Christensen-Dalsgaard (2008) proposed a power-law correction,
which takes the form of $\delta\nu=a\nu^{b}$ where $\nu$ is the frequency and
$\delta\nu$ is the frequency correction. According to the solar calibrations,
the parameter $b=4.9$ has been adopted to correct frequencies of pulsating
stars other than the Sun (e.g., ChristensenDalsgaard et al. 2010). Ball and
Gizon (2014) suggested two formulae: one is $\delta\nu=a_{3}\nu^{3}/I$ and the
other is $\delta\nu=(a_{-1}\nu^{-1}+a_{3}\nu^{3})/I$, where $I$ is the
normalized mode inertia. On the other hand, the structure of the sun in the
vicinity of its photosphere has carefully been examined by nonstandard solar
models (Demarque, Guenther, and Kim 1997; Rosenthal et al. 1999) and by
hydrodynamic simulations of 1D, 2D, and 3D configurations (Carlsson and Stein
1994; Leenaarts et al. 2007; Wedemeyer et al. 2004). Recently, modifications
due to interactions between p-mode oscillations and turbulent convection near
the top of the solar convection zone were considered as the possible cause of
the near-surface effect (e.g., Ball et al. 2016; Houdek et al. 2017; Schou &
Birch 2020).

Small-scale magnetic fields of the quiet solar photosphere are an important
component of the solar magnetism. Because of their small sizes on the solar
disk, they cannot be seen on the solar magnetograms and are sometimes referred
to as “hidden magnetic fields”(Stenflo 1987, 2013). Based on 3-D realistic
hydrodynamical simulations of solar surface convection, Trujillio Bueno et al.
(2004) derived an averaged un-signed flux density of at least 60 G with the
aid of the observed Hanle depolarization. With the spatial resolution of
0.3′′, the Solar Optical Telescope onboard the Hinode space observatory allows
for unprecedented polarimetric observations of the magnetic field of the quiet
sun (Kosugi et al. 2007). The observations show vertical field components with
kilo-Gauss strength and concentrated in the intergranular lanes, along with
horizontal field components with ten-Gauss strength and occurring at the edges
of bright granules. Lites et al. (2008) determined for the horizontal field
component a mean apparent field strength of 55 G, while for the vertical field
component a corresponding mean absolute strength of only 11 G. Unlike the
magnetic fields of the sun spots that are isolated on the solar disk and
occupy only a very small fraction of the solar surface, the small-scale
magnetic fields are ubiquitous on the solar surface and thus store up a great
amount of magnetic energy, which can be an important source for the magnetic
coupling to the outer atmosphere and the corona heating (Stenflo 1994;
Schrijver et al. 1998; Schrijver & Title 2003). Furthermore, such “turbulent”
magnetic fields can significantly affect the structure of the solar atmosphere
and the propagations of acoustic waves (Rosenthal et al. 2002; Bogdan et al.
2003; Cally 2007).

More observational studies have been made in recent years to probe the
internetwork magnetic fields. Danilovic et al. (2010) made observations of the
transverse component of the photospheric magnetic fields by use of the
magnetograph SUNRISE/IMAX. Using the infrared spectropolarimeter GRIS equipped
on the GREGOR telescope, Lagg et al. (2016) made highly sensitive full Stokes
measurements in the 1.56$\mu$m Fe I lines and found that about 80% of the
obtained spectra show polarimetric signals. With more detailed analyses,
Martínez González et al. (2016) found that half of the observed region can be
explained by the magnetic fields with spatial scales smaller than the
resolution element. Two magnetic field populations are distinguished: one with
the larger filling factor has weak horizontal fields of about $\sim$150 G, the
other with the smaller filling factor is constrained for the field strengths
above 250 G. Using the high-spatial resolution full-Stokes measurements
obtained by the CRISP equipped on the SST, Morosin et al. (2020) derived the
depth-stratification of the magnetic field in a plage region. The magnetic
fields emanate from the intergranular lanes in the photosphere and spread out
horizontally toward the chromosphere to form a canopy. Based on similar data,
Keys et al. (2020) found evidence for fast amplification of magnetic field
strength in the magnetic bright points during their temporal evolution to
kilogauss field strength. Observations of resonance scattering polarization
with high sensitivity and high spatial resolution are also available to
diagnose small-scale dynamos and magnetoconvection (Zeuner et al. 2018, 2020).

More information about the small-scale magnetic fields is provided by the
multi-dimensional magneto-hydrodynamic (MHD) simulations. In the 2-D MHD
simulations, Grossmann-Doerth et al. (1998) found that the magnetic flux are
quickly swept by the horizontal convective flows into the intergranular
downflow channels, while strong horizontal magnetic fields are pushed upward
by the convective upflows. Horizontal magnetic fields were frequently found in
the 3-D MHD simulations of Schaffenberger et al. (2005, 2006), and often
referred to as “small-scale canopies”. The 3-D MHD simulations of Abbett
(2007) showed horizontal ribbons of the magnetic flux with magnetic voids
located in the below. Schüssler & Vögler (2008) and Rempel (2014) demonstrated
in their radiative MHD simulations that a local dynamo can operate in the
solar photosphere, which maintains a turbulent distribution of small-scale
magnetic fields. In simulations of Steiner et al. (2008), a local maximum is
found for the horizontal field component in the upper photosphere, whose
values can surpass that of the corresponding vertical component by a factor of
2 to 5.6 depending on the different choices of initial states and boundary
conditions. Coincidentally, Rempel (2014) also found that the ratio of the
horizontal to vertical field shows a maximum at similar height as Steiner et
al., with its value from about 3 to 5 depending on the overall field strength
decreasing from 60 G to 20 G.

The coupling between the small-scale magnetic fields and the acoustic waves
mainly has two aspects. The small-scale magnetic fields modify the hydrostatic
structure of the atmosphere, and thus can significantly affect the propagation
of the acoustic waves. On the other hand, the magnetic field can act as an
additional restoring force, and then gradually change the waves from purely
acoustic to magneto-acoustic. Rosenthal et al. (2002) pointed out that due to
rapidly increasing phase speed of the fast magneto-acoustic modes, waves will
be refracted along a magnetic field that is significantly inclined to the
vertical direction, resulting in finally total internal reflection in the
atmosphere. Using the phase lag of the Doppler signal in different layers in
the photosphere and low chromosphere, Finsterle et al. (2004) determined the
3-D topography of the magnetic canopy in and around active regions. Steiner et
al. (2007) found from the direct numerical simulations that a plane-parallel
acoustic wave is partially refracted and reflected by the interaction with the
magnetic field. They pointed out further that the perturbation of the wave
front is not due to the presence of a magnetic field, but rather to the
temperature decreasing associated with the vigorous intergranular downflows.

The accumulation of the horizontal magnetic flux is considered as a direct
result of the convective overshooting beyond the solar convection zone, which
expels the horizontal component of the magnetic field into the upper
photosphere. Because of similar sizes of the individual granules, the resulted
magnetic arches will be formed at a similar height and may be spliced as a
global layer which we refer to hereafter as the magnetic-arch splicing layer.
In this work, we identify the existence of such a magnetic-arch splicing layer
in the solar atmosphere, and find that the inclusion of the reflections of the
solar p-mode oscillations on the magnetic-arch splicing layer significantly
improves the agreement of the theoretical solar p-mode frequencies with the
observed ones. In Section 2, we briefly introduce the input physics and
methods in our calculations of stellar models. We discuss our main results in
Section 3, and summarize our main conclusions in Section 4.

##  2 Input physics and methods of calculations

Frequencies of solar p-mode oscillations are taken from observations of the
Birmingham Solar-Oscillation Network (BiSON, see Broomhall et al. 2009). The
observations ran at 6 stations distributed all around the world and lasted
8640 days from 1985 to 2008. Frequencies of low spherical harmonic degrees
($l$ = 0, 1, 2, and 3) are estimated to unprecedented levels of precision. We
use these p-mode frequencies in the fitting of stellar models to the sun.

The stellar models are constructed by the MESA. The specific version we have
adopted is version 11554, which incorporates the up-to-date input physics in
calculations of stellar structure and evolution. In particular, we adopt the
OPAL equation of state tables (Rogers & Nayfonov 2002). The OPAL opacities
(Iglesias & Rogers 1996) are used in the high temperature region, while the
opacities from Ferguson et al.(2005) are used in the low temperature region to
include contributions from molecules and grains. We use the classical mixing-
length theory (MLT) to treat the convection. In order to satisfy several
observational restrictions (such as the depth of the convection zone, the
surface helium abundance, etc.), we adopt GS98 chemical composition (Grevesse
& Sauval 1998) and include element diffusion from Thoul et al. (1994) in our
solar models. Frequencies of the p-mode oscillations for the considered
stellar models are computed with “adipls” package in MESA, which is developed
and updated by Christensen-Dalsgaard (2008).

In order to approximately include the effects of convection and overshooting
in the the solar atmosphere, we adopt two different $T$-$\tau$ relations in
the atmosphere integration. One is the Krishna Swamy relation (referred to as
the K-S atmosphere, see Krishna Swamy 1966), and the other is the fit given by
Sonoi et al. (2019) to the Vernazza et al. VAL-C model (Vernazza et al. 1981,
referred to as the Hopf-grey atmosphere). As pointed out by Sonoi et al.
(2019), different $T$-$\tau$ relations can significantly affect the entropy
profile in the solar atmosphere, and the VAL-C model actually gives
satisfactory correspondence to the results of the 3-D radiation-coupled
hydrodynamical simulations. The two $T$-$\tau$ relations can be written in a
unified form:

${{T}^{4}}=\frac{3}{4}T_{\text{eff}}^{4}\left[\frac{}{}\tau+q(\tau)\right],$
(1)

where $T$ is temperature, $\tau$ is the optical depth, and $q(\tau)$ is a
Hopf-like function. $T_{\text{eff}}$ is the effective temperature of the star.
It should be noted that we define the base of the photosphere at an optical
depth of $\tau=0.412$ for the solar-Hopf atmosphere (Sonoi et al. 2019) and
$\tau=0.312$ for the Krishna Swamy atmosphere (Krishna Swamy 1966), while in
the 3-D simulations it is defined at about $\tau=1$. These different height
systems in the solar photosphere models may introduce some difference in the
results of MHD simulations and stellar standard models.

We introduce the magnetic field in the solar atmosphere by adding an
additional term in the Hopf-like function:

$q(\tau)={{q}_{\text{ori}}}(\tau)+a\exp\left(-b\sqrt{\tau}\right),$ (2)

where ${{q}_{\text{ori}}}(\tau)$ is the original Hopf-like function that is
given by the two considered $T$-$\tau$ relations, respectively. This
additional term results in an increase of temperature in the upper
photosphere, and the corresponding increase of the radiative pressure takes
the role of the magnetic pressure. $a$ and $b$ are two parameters to be
determined for each specific $T$-$\tau$ relations. In particular, parameter
$a$ mainly determines the strength of the magnetic field, while parameter $b$
specifies the location of the magnetic-arch splicing layer. Table 1 summaries
our main results. In addition to the guess function of square root in Eq. (2),
we have tested a few more guess functions (including linear, square, and
cubic), and the results are similar with only small variations on $\chi^{2}$
(around ten percent), which is defined as the sum of the squared differences
of frequencies between observed and calculated ones weighted by the
observational errors. Therefore we adopt the function of square root that
minimizes the $\chi^{2}$ among the other functions we have tested.

Table 1: Properties of the best-fit solar models $\text{[Fe/H]}_{\text{ini}}$ | $Z_{\text{ini}}$ | $Y_{\text{ini}}$ | $\alpha$ | $R/R_{\odot}$ | $\bigtriangleup L/L_{\odot}$ | $Y_{\text{s}}$ | $\left(Z/X\right)_{\text{s}}$ | $R_{\text{CZ}}/R_{\odot}$ | $\chi^{2}$
---|---|---|---|---|---|---|---|---|---
| K-S atmosphere | $300\exp\left(-98\sqrt{\tau}\right)$ |
0.08 | 0.01941 | 0.27634 | 2.11610 | 0.999527 | 0.000024 | 0.24899 | 0.02429 | 0.71819 | 240.0
0.09 | 0.01979 | 0.27849 | 2.12535 | 0.999614 | 0.000085 | 0.25116 | 0.02488 | 0.71617 | 219.5
0.10 | 0.02019 | 0.28046 | 2.13595 | 0.999723 | 0.000047 | 0.25318 | 0.02548 | 0.71523 | 212.0
0.11 | 0.02060 | 0.28206 | 2.14650 | 0.999800 | 0.000063 | 0.25481 | 0.02608 | 0.71364 | 220.2
0.12 | 0.02102 | 0.28360 | 2.15490 | 0.999841 | 0.000020 | 0.25636 | 0.02670 | 0.71282 | 243.5
| Hopf-grey atmosphere | $310\exp\left(-86\sqrt{\tau}\right)$ |
0.08 | 0.01941 | 0.27634 | 2.02345 | 0.999489 | 0.000024 | 0.24899 | 0.02429 | 0.71736 | 231.0
0.09 | 0.01979 | 0.27849 | 2.03270 | 0.999579 | 0.000086 | 0.25116 | 0.02488 | 0.71694 | 215.2
0.10 | 0.02019 | 0.28046 | 2.04327 | 0.999677 | 0.000032 | 0.25318 | 0.02548 | 0.71485 | 213.4
0.11 | 0.02060 | 0.28206 | 2.05387 | 0.999763 | 0.000063 | 0.25481 | 0.02608 | 0.71417 | 226.8
0.12 | 0.02102 | 0.28360 | 2.06238 | 0.999801 | 0.000010 | 0.25636 | 0.02670 | 0.71281 | 252.0

An important issue that is related to the reflection and the transmission of
the p-mode oscillations on the magnetic-arch splicing layer is the suitable
form for one of the surface boundary conditions in the calculations of the
linear adiabatic oscillations, i.e., the mechanical surface boundary
condition. In an isothermal atmosphere without the magnetic field, for example
the Eddington gray atmosphere, the mechanical surface boundary condition of
the linear adiabatic oscillations is given by Unno et al. (1989)

$\left(1+\frac{{{H}_{P}}}{r}\left[\frac{l\left(l+1\right)g}{{{\omega}^{2}}r}-4-{{\omega}^{2}}\frac{r}{g}\right]\right){{\xi}_{r}}-\frac{1}{\rho
g}P^{\prime}=0,$ (3)

where ${\xi}_{r}$ is the radial displacement, $P^{\prime}$ is the Eulerian
perturbation of pressure, $\rho$ is density, $g$ is the gravitational
acceleration, $\omega$ is the circular frequency of the pulsation mode, and
$H_{P}$ is the local pressure scale height at radius $r$. For those p-modes
with frequencies that are higher than the acoustic cut-off frequency, the
reflection condition on a free boundary is usually adopted:

$\delta P=P^{\prime}-\rho g{{\xi}_{r}}=0,$ (4)

where $\delta P$ is the Lagrangian perturbation of pressure (for more
discussions, see Unno et al. 1989; Christensen-Dalsgaard 2008). With the
inclusion of the magnetic field in the atmosphere, the magnetic pressure
increases rapidly in the upper photosphere, resulting in a rapid decrease of
density as well as a rapid increase of the sound speed, and thus Eqs. (3) and
(4) no longer hold. Therefore, we need to use a new mechanical surface
boundary condition. For this aim, it is instructive to consider the
propagations of the p-mode oscillations in the upper photosphere. When the
acoustic waves propagate outwardly and finally meet the magnetic-arch splicing
layer, they will be reflected and transmitted. For those waves that are
totally reflected at the magnetic-arch splicing layer, standing waves will be
formed for the specific eigen-frequencies. Those acoustic waves are exactly
what we refer to as the solar p-mode oscillations. Other waves that can
transmit into the solar chromosphere cannot form standing waves, and will not
be observed as global oscillation modes. As the solar p-mode oscillations
should be totally reflected at the magnetic-arch splicing layer, there is no
wave beyond the magnetic-arch splicing layer, and a suitable surface
mechanical boundary condition is

$P^{\prime}=0.$ (5)

The two surface boundary conditions, i.e. Eqs. (4) and (5), are all applied in
our calculations of the solar p-mode oscillations, and comparisons and
discussions are made to illustrate their effects on the frequencies of the
p-mode oscillations.

##  3  Results and discussions

The observed solar p-mode oscillations are eigen-oscillation modes of the sun
with the pressure as the main restoring force. They propagate in the solar
interior and atmosphere, taking the whole sun as a huge resonant cavity. In
order to reproduce the observed frequencies, we need to know the internal
structure of the sun. The so-called standard solar models are based on the up-
to-date physics. By adjusting usually the initial helium abundance and the
mixing-length parameter, a stellar model with exactly one solar mass
($M_{\odot}=1.9892\times 10^{33}$ g) is calibrated to have the radius and the
luminosity of the present sun ($R_{\odot}=6.9598\times 10^{10}$ cm and
$L_{\odot}=3.8418\times 10^{33}$ erg s-1) after evolving from the zero-age
main sequence to the present solar age (4.57Gyr). Based on the standard solar
models, the frequencies of the eigen-oscillation modes are calculated with the
theory of the linear adiabatic oscillations. Compared with the observed
frequencies, we have a good chance to check the theory of stellar structure
and evolution (Bahcall et al. 2005; Basu 2016). However, the standard solar
models always result in the theoretical oscillation frequencies that are
systematically offset from the observed ones. This is the well-known “surface
term” problem (see, e.g., Basu 2016), whose physical nature has not yet been
fully understood up to know.

In order to study reflection and transmission of the solar p-mode oscillations
at the magnetic-arch splicing layer in the upper photosphere, we approximate
the effect of the magnetic field by adding in the solar atmosphere model a
magnetic pressure with two parameters, one determining the location of the
magnetic-arch splicing layer and the other specifying the strength of the
magnetic field. The total pressure is thus composed of the gas pressure and
the magnetic pressure. The increase of the magnetic pressure will result in
the decrease of the gas pressure due to the hydrostatic equilibrium of the
atmosphere, and further the decrease of the density which leads to the
reflections of the solar p-mode oscillations due to the rapid increase of the
sound speed. By adjusting these two parameters of the magnetic field we find
out the best-fit models for the observed solar p-mode oscillations.

Figure 1: The frequency differences between observations and model
calculations of the best-fit solar models with different solar atmosphere
models. The frequencies of the solar p-mode oscillations are taken from BiSON
observations (Broomhall et al. 2009). Results of some recent investigations
(Ball et al. 2016; Houdek et al. 2017; Schou & Birch 2020) are also shown for
comparison.

In the present study, we do not follow the usual procedure to do the
calibration of the standard solar models. But instead, we only match the
stellar models to the present solar luminosity. For the stellar models near
the solar age in each evolutionary sequence, we compare the frequencies of
their p-mode oscillations with the observed values of the corresponding modes
of the sun, and choose the best-fit model by the least of $\chi^{2}$ that is
defined as the sum of the squared differences of frequencies between observed
and calculated ones weighted by the observational errors. We then adjust
values of the mixing-length parameter and the initial helium abundance
iteratively to make the best-fit model being exactly of the solar age. The
differences between the observed frequencies and the corresponding model
calculated frequencies are shown for the best-fit solar models with two
different atmosphere models in Figure 1, i.e., plus sign for the results with
the Hopf-grey atmosphere and the dotted-square sign for the results with the
K-S atmosphere. For comparison, the similar results of some other recent
investigations (e.g., Ball et al. 2016; Houdek et al. 2017; Schou & Birch
2020) are also shown in the same figure, with different approaches to treat
the interactions between p-mode oscillations and turbulent convection. The
turbulent convection was simulated by a 3-D radiation hydrodynamics code MURaM
in Ball et al. (2016) and Schou & Birch (2020). In particular, Ball et al.
(2016) used MESA to calculate the standard solar model and then replaced its
near-surface equilibrium structure with the results of MURaM. The p-mode
frequencies were calculated and compared for the stellar models with and
without the improved near-surface equilibrium structure. In order to evaluate
the frequency perturbations resulted from the dynamical part of the
interaction of the oscillations with near-surface convection, Schou & Birch
(2020) numerically estimated eigenfunctions in the 3-D hydrodynamic
simulations and matched them to the eigenfunctions calculated from the classic
equations applied to the horizontally averaged variables of the stellar
structure. Houdek et al. (2017) also adopted the approach of replacing the
outer layers of a 1-D solar envelope model by the structure of an averaged 3-D
simulation, and moreover they include the effects of non-adiabaticity and
convection dynamics by using a 1-D non-local and time-dependent convection
model. Compared our results with those of other investigations, significant
improvement is achieved on the discrepancy between calculated and observed
solar p-mode frequencies. It should be noticed that the original systematic
offset (a power-law type, e.g. Kjeldsen et al. 2008) between the observed and
the calculated p-mode frequencies is completely eliminated. The residuals now
left behind in our results are still larger than the observational errors. It
can be seen that the residuals show an approximate sinusoidal-type
distribution when the frequency is smaller than about 3700$\mu$Hz, and then
rapidly increase. As pointed out by Gough (1990), a localized feature in the
sound speed inside a star introduces an oscillatory term in the frequencies of
the p-modes as a function of frequency. Based on this conclusion, we can infer
for the sun that the localized feature indicated by the present sinusoidal-
type residuals is in the deep solar convection zone below the solar
photosphere, where the turbulent pressure may contribute a considerable
fraction to the local hydrostatic equilibrium. Finally, it can be noticed that
only a few p-modes with the highest frequencies for each spherical harmonic
degree ($l=0$, 1, 2, and 3) have considerably higher frequencies than the
corresponding calculated ones, which is seen as the steep increase for the
frequency above 3700 $\mu$Hz in Figure 1. Without the magnetic-arch splicing
layer, however, the observed frequencies of those p-modes are significantly
lower than the calculated ones as seen in Figure 4 of Basu (2016). These facts
indicate that those p-modes do not totally reflected at the magnetic-arch
splicing layer and a small fraction of the acoustic waves transmits into the
upper atmosphere. For other p-mode oscillations with even higher frequencies,
most of the acoustic waves will transmit into the upper atmosphere and they
cannot be observed as the eigen-modes of the sun. In particular, we estimate
the acoustic cut-off frequency in the photosphere by

${{\omega}_{a}}=\frac{{{c}_{s}}}{2{{H}_{p}}}\text{=}\sqrt{{{\Gamma}_{1}}\frac{\mu}{\Re
T}}\frac{g}{2},$ (6)

which is defined in an isothermal atmosphere. In Eq. (6), $c_{s}$ is the sound
speed, $H_{p}$ is the local pressure scale height, $\Gamma_{1}$ is the
adiabatic exponent, $\mu$ is the mean molecular weight, $\Re$ is the gas
constant, and $g$ is the gravity acceleration. The result is about
6400$\mu$Hz, that is in agreement with other results in the literature. This
value is much higher than the frequencies of the considered p-modes, and is
not in agreement with the results in our work (around 3700$\mu$Hz). The reason
most probably comes from the isothermal atmosphere assumption, which is not
valid in our solar models. Due to this reason, it is difficult for us to
theoretically determine the exact cut-off frequency for the considered p-mode
oscillations.

Figure 2: The distributions of the normalized kinetic energy by adopting two
different mechanical surface boundary conditions are shown for a particular
$l=1$ mode corresponding to the observed frequency of 3233.14$\mu$Hz of the
solar model with the inclusion of the magnetic-arch splicing layer. The solid
line represents the one by adopting Eq. (5) (TPP model), while the dotted line
by adopting Eq. (4) (TDP model) at the surface of the atmosphere. The two
lines are matched with each other in the stellar interior, to highlight their
differences in the atmosphere. Figure 3: The distributions of the normalized
radial kinetic energy are shown in the upper panel while the distributions of
the normalized horizontal kinetic energy are shown in the lower panel for the
same oscillation mode as in Figure 2. In the lower panel, numbers on the
ordinate should be multiplied by $10^{-6}$ to obtain the same unit as in the
upper panel.

We illustrate why the reflections of acoustic waves on the magnetic-arch
splicing layer can significantly improve the agreement between observed and
calculated frequencies of the solar p-mode oscillation by examining the
propagations of p-mode oscillations near the magnetic-arch splicing layer. We
consider the distribution of the normalized kinetic energy of an oscillation
mode that is defined by

${{e}_{k}}=\left[\xi_{r}^{2}+l(l+1)\xi_{h}^{2}\right]\rho{{r}^{2}},$ (7)

where $\xi_{h}$ is the horizontal displacement (see Unno et al. 1989; Aerts et
al. 2010 for more details). Figure 2 shows the distributions of the normalized
kinetic energy for a particular $l=1$ mode with the observed frequency of
3233.14$\mu$Hz, computed with different surface mechanical boundary conditions
Eq. (4) (which will be referred to as TDP model) and Eq. (5) (which will be
referred to as TPP model). We obtain the radial and horizontal displacement of
a p-mode oscillation by solving the equations of the linear adiabatic
oscillation (see, e.g., Christensen-Dalsgaard 2008). According to the theory
of linear adiabatic oscillation, the solutions can be multiplied by any
constant, and the usual procedure is to set the normalization condition at the
outer boundary of the considered stellar structure (see Christensen-Dalsgaard
2008 for more details). As a result, different outer boundary conditions will
lead to different values of the radial and horizontal displacement in the
stellar interior. In order to compare the radial and horizontal displacement
under different outer boundary conditions, we match the two distributions of
the normalized kinetic energy in the solar interior. Therefore Figure 2
clearly shows their difference near the magnetic-arch splicing layer. It can
be seen that there is a bump just below the magnetic-arch splicing layer for
the TPP model, which is not seen for the TDP model and for the other cases
that the magnetic-arch splicing layer is not included in the solar models.
Such a bump in the normalized kinetic energy equivalently enlarges the
propagation region of the acoustic waves, which can be recognized easily by
comparing the positions of two corresponding nodes. A larger propagation
region makes the p-mode oscillations to spend a little longer times to travel
across and therefore to have a little lower frequencies.

The outer boundary condition is crucial for the reflection of the p-mode
oscillations at the magnetic-arch splicing layer. We show this by analyzing
the distribution of the pressure perturbation in this near-surface stellar
structure. In Figure 3 we show the distributions of radial kinetic energy,
which is defined by the first term in the right-hand-side of Eq. (7), and of
horizontal kinetic energy which is defined by the second term of Eq. (7) for
the same oscillation mode as in Figure 2. It can be noticed from the lower
panel of Figure 3 that the horizontal kinetic energy of the TPP model is
extremely small around the magnetic-arch splicing layer. As the horizontal
kinetic energy is directly related to the Eulerian pressure perturbation,
there is actually no acoustic wave above the magnetic-arch splicing layer with
the use of the surface boundary condition Eq. (5). As a result, we can see in
the upper panel of Figure 3 that the distribution of the radial kinetic energy
is flat below the magnetic-arch splicing layer, indicating essentially total
reflections of acoustic waves on a node of the Eulerian pressure perturbation.
With the use of the surface boundary condition Eq. (4), however, the Eulerian
pressure perturbation is not zero at the outer boundary of the near-surface
stellar structure due to the normalization condition that demands the radial
displacement to be a finite value at the outer boundary. This is equivalent to
an input of acoustic waves from outside into the stellar interior. As a
result, we can see in the lower panel of Figure 3 that a considerable amount
of the horizontal kinetic energy is present above the magnetic-arch splicing
layer for the TDP model. Consequently, we can see in the upper panel of Figure
3 that the radial kinetic energy decreases rapidly after going outward across
the magnetic-arch splicing layer. This effect considerably reduces the cavity
of the p-mode oscillations, and makes their frequencies higher than the
corresponding ones calculated with the surface boundary condition Eq. (5).

Figure 4: The distributions of the gas pressure and the magnetic pressure, as
well as the strength of the magnetic field for the best-fit model by adopting
the Hopf-grey atmosphere.

The distributions of the gas pressure and the magnetic pressure are shown
respectively for the best-fit solar model in Figure 4. As pointed out by
Rosenthal et al. (2002) and Cally (2007), the reflection and transmission of
the p-mode oscillations occur at the gas pressure roughly equal to the
magnetic pressure ($\beta={{P}_{\text{gas}}}/{{P}_{\text{mag}}}\approx 1$). We
use this condition to infer the strength of the magnetic field that is
necessary for the p-mode oscillations to be reflected in the solar atmosphere.
It can be found in Figure 4 that this condition corresponds to a magnetic
field of about 90 G. Such a magnetic field we have independently derived is in
good agreement with the results of Trujillio Bueno et al. (2004) $60\sim 130$
G, but considerably larger than that of Lites et al. (2008) 55 G. As the
amplitude of the measured polarization signals by the Zeeman effect depends
sensitively on the cancellation of mixed magnetic polarities within the
spatio-temporal resolution of the observations, our result is still consistent
with that of Lites et al. (2008). The origin mechanism of the small-scale
magnetic fields is still debating. A plausible interpretation is that the
tangled magnetic fields are supplied with flux from the decaying flux tube
fields. In particular, convection plays an important role in the expulsion of
the existing magnetic field from the interior of a convective eddy flow like
that of granules. On the other hand, small-scaled dynamos are the key process
to maintain a turbulent magnetic field. However, their distribution in the
photosphere is critical for the reflections of acoustic waves on these
unsigned magnetic fields. Recent studies show that the power spectra of the
internetwork magnetic fields agree well with the results obtained from local
dynamo simulations (see, e.g., Buehler et al. 2013; Hotta et al. 2015;
Danilovic et al. 2016). Our result on the strength of the tangled magnetic
fields (about 90 G) provides a crucial clue on the origin of the small-scale
magnetic fields.

It can be found in Figure 4 that the magnetic-arch splicing layer is located
at a height of about 630 km in the photosphere. It is interesting to notice
that a magnetic layer dominated by the horizontal fields has been revealed by
the 3-D radiative MHD simulations. Steiner et al. (2008) found a local maximum
of the horizontal field component at a height of around 500 km , while Rempel
(2014) found that the ratio of the horizontal to vertical field shows a
maximum at a height of about 450 km in the photosphere. The magnetic-arch
splicing layer we have found has a similar location as those demonstrated by
the above 3-D simulations. It should be noticed that the radiative MHD
simulations use frequency-dependent opacity while we just adopt the Rosseland
mean opacity from OPAL (Iglesias & Rogers 1996) for the high temperature
region and Ferguson et al.(2005) for the low temperature region. This
difference may contribute to the uncertainties in the evaluation of the
location of the magnetic-arch splicing layer. As pointed out by Galloway &
Weiss (1981) and Steiner et al. (2008), the magnetic field tends to be
expelled by the convective motion not only in the lateral direction to the
intergranular lanes but also in the vertical direction to the upper level of
the photosphere. Therefore, the location of the magnetic-arch splicing layer
directly corresponds to the accumulation of the horizontal magnetic flux that
is expelled by the convective motions to the upper layer of the photosphere.
In addition, the rapid increase of the magnetic pressure in Figure 4
corresponds closely to a configuration of magnetic arches with magnetic voids
just in the below, which is often seen in the MHD simulations as “small-scale
canopies”. And that is exactly why we refer to this magnetic configuration as
the magnetic-arch splicing layer.

The heavy-element abundance is one of the fundamental ingredients to determine
the structure and evolution of the sun (Basu 2016). The heavy elements
increase the opacity, and thus significantly influence the internal structure
of the sun. Besides, the heavy elements directly participate in the central
hydrogen burning process through the CNO cycle, and therefore contribute to
the solar neutrinos. We examine the matches of our best-fit models with
different heavy-element abundance, and the result is shown in the left panel
of Figure 5. It can be found that the solar model with the heavy-element
abundance $(Z/X)_{\rm surf}$ around 0.0255 can best fit the p-mode frequencies
of the sun, which is in agreement with the standard solar models (Basu 2016).

Figure 5: In the left panel the variations of $\chi^{2}$, which is defined as
the sum of the squared differences of frequencies between observed and
calculated ones weighted by the observational errors, are shown with the
initial metallicity for the best-fit models by adopting different atmosphere
models. In the right panel, the radius differences between the best-fit model
and the sun are shown with the initial metallicity.

As an important quantity that can be observed accurately, the solar radius is
used to probe the static and dynamic structure of the solar interior and
atmosphere (Kiliç & Golbasi 2011). Moreover, the solar diameter is also
adopted in the climate model of the earth to get a realistic energy input
reconstruction (Thuillier 2005). In the present study, we determine the best-
fit solar models by matching the solar luminosity and the frequencies of the
p-mode oscillations, leaving the stellar radius as a resulted parameter. In
the right panel of Figure 5 we show the radius difference of our best-fit
solar models with different heavy-element abundance to the sun. It can be
found that all of our solar models have smaller radii than that of the sun
with a few tenths in a megameter. Haberreiter et al. (2008) pointed out that
this difference can be understood by the difference between the height at
solar disk center where the optical depth $\tau_{\rm Ross}=2/3$ and the
inflection point of the solar intensity profile on the limb, and they
suggested that this difference is about 0.333Mm as shown in the right panel of
Figure 5. However, they adopted two solar atmosphere models that do not
include the magnetic field, and the seismic solar radius they adopted was
inferred from only the observed f-mode frequencies (Schou et al. 1997). We
include the magnetic field in the solar atmosphere and infer the seismic
radius from all observed p-mode frequencies with the spherical harmonic degree
($l=0$, 1, 2, and 3), and our result is in good agreement with theirs.

##  4  Conclusions

Small-scale magnetic fields are an important component in the solar
photosphere. They contribute a significant amount of magnetic energy in the
solar atmosphere, which can not only be responsible for the corona heating,
but also influence the hydrostatic equilibrium in the photosphere itself.
However, the distribution of the small-scale magnetic fields in the solar
photosphere is not clear, i.e., randomly distributed or well-organized? In the
present work, we identify that the small-scale canopies within the individual
granules can act as a global magnetic-arch splicing layer, and investigate the
effect of this magnetic-arch splicing layer on the propagations of the solar
p-mode oscillations. Our main conclusions are summarized as follows.

1\. The solar p-mode oscillations are totally reflected at the magnetic-arch
splicing layer, resulting in tremendous improvement on the discrepancy between
the observed and calculated p-mode frequencies.

2\. The strength of the magnetic fields derived independently in our study is
about 90 G, and the location of the magnetic-arch splicing layer is 630 km
high in the upper photosphere. These results are quantitatively in agreement
with the observations and 3-D MHD simulations.

3\. The radii of the best-fit models are a few tenths in a megameter smaller
than that of the sun, which is in agreement with Haberreiter et al. (2008).
Our best-fit model has a metal abundance of Grevesse and Sauval (1998), which
is in agreement with the standard solar models (Basu 2016).

This work was supported by the National Natural Science Foundation of China
(Nos. 11333006, 11973079, and 11521303). Fruitful discussions with Zhong Liu,
Xiaoli Yan, and Mingde Ding are highly appreciated. We thank two anonymous
reviewers for their productive comments and fruitful suggestions.

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# Load-dependent adaptation near zero load in the bacterial flagellar motor

Jasmine A. Nirody  Clarendon Laboratory, Department of Physics, University of
Oxford, Oxford, UK  Center for Studies in Physics and Biology, The Rockefeller
University, New York, USA  Ashley L. Nord  Centre de Biochimie Structurale,
INSERM, CNRS, Université de Montpellier, Montpellier, France  Richard M. Berry
Clarendon Laboratory, Department of Physics, University of Oxford, Oxford, UK

###### Abstract

The bacterial flagellar motor is an ion-powered transmembrane protein complex
which drives swimming in many bacterial species. The motor consists of a
cytoplasmic ‘rotor’ ring and a number of ‘stator’ units, which are bound to
the cell wall of the bacterium. Recently, it has been shown that the number of
functional torque-generating stator units in the motor depends on the external
load, and suggested that mechanosensing in the flagellar motor is driven via a
‘catch bond’ mechanism in the motor’s stator units. We present a method that
allows us to measure – on a single motor – stator unit dynamics across a large
range of external loads, including near the zero-torque limit. By attaching
superparamagnetic beads to the flagellar hook, we can control the motor’s
speed via a rotating magnetic field. We manipulate the motor to four different
speed levels in two different ion-motive force (IMF) conditions. This
framework allows for a deeper exploration into the mechanism behind load-
dependent remodelling by separating out motor properties, such as rotation
speed and energy availability in the form of IMF, that affect the motor
torque.

Correspondence<EMAIL_ADDRESS>

The bacterial flagellar motor (BFM) is an ion-driven nanomachine that drives
swimming in a variety of bacterial species. The BFM couples the flow of
cations (protons, in Escherichia coli) across the bacterial membrane to induce
rotation in the flagellum, spinning the filament like a propeller to move the
bacterium forward [1, 21]. The flagellar motor generates torque through
interactions between the motor’s stator and rotor; specifically torque is
generated via an interaction between a stator unit (in E. coli, comprising the
proteins MotA and MotB) and FliG protein ‘spokes’ that line the rotor’s
cytoplasmic C-ring [18, 8] (Figure 1a). The BFM’s stator can be composed of
between 1 and at least 11 independent units [6, 26, 14, 24].

The flagellar motor plays a crucial role in bacterial pathogenicity and in
several processes like chemotaxis and biofilm formation. Though understanding
its function has been a long-standing problem in biophysics, its size and
localisation to the membrane have made this challenging. The development of
the tethered cell assay definitively confirmed that the BFM was a rotary motor
[29]. In this assay, cells were tethered to a surface by a single filament,
and the motor rotation was characterised via rotation of the tethered cell.
This assay is ideal for direct observation of rotation by eye, as the load of
the cell body is such that the motor can rotate at only a few Hertz. However,
experiments using tethered cells limited measurements of motor rotation to
high loads. Later, the bead assay, in which beads of varying sizes, and
accordingly varying drags, are attached to flagellar stubs of adhered cell
bodies, allowed motor rotation to be studied across a wider range of loads
[25].

Beyond the torque-speed relationship in ‘fully assembled’ motors, the
discovery that motor components are constantly being turned over opened up a
new avenue of scientific questions. In particular, stator units have been
shown to continuously exchange with a large pool of membrane-bound ‘spares’
[14, 30]. Further evidence that motors at high loads are able to maintain a
much higher number of engaged stator units than motors at low load may suggest
that bacterial mechanosensing, utilised in the transition from liquid to
surface living, may arise from the flagellar motor [15, 32] .

This discovery both pointed to an exciting new line of research to be explored
in the future, and brought into question the results of several experiments
from the past. For one, it meant that all previously measured torque-speed
relationships were likely made on motors with varying numbers of stator units.
This has been foundational to the development of theoretical models developed
since [18, 20]. As the motor’s structure and function was shown to vary with
external conditions, tools to manipulate the load on single motors proved
vital to properly understand remodelling in the BFM. This has been achieved
through monitoring the process of cell tethering [15], which dramatically
increases the load on the motor, and electrorotation [27, 34], which uses a
rotating electric field to apply external torque on the motor.

In a recent paper, the molecular mechanism behind stator remodelling was
investigated using an external magnetic field and magnetic microbeads of
different sizes bound to the hook of individual E. coli motors [22]. Load-
dependent remodelling in the BFM can ostensibly manifest in several ways: by
stator units engaging at a higher rate ($k_{\text{on}}$) or disengaging at a
lower rate ($k_{\text{off}}$) as a function of the load, or by some
combination of the two. Nord et al. showed evidence that $k_{\text{off}}$ (and
thus unit lifetime) is load-dependent, suggesting that the dynamics behind
stator assembly are consistent with a catch bond [22]. This somewhat
counterintuitive bond is one that is strengthened, rather than weakened, as
more force is applied to it — stator units stay engaged onto the motor longer
at higher loads.

Nord et al. used permanent magnets to stall motors with magnetic microbeads
attached to their hooks. During the period when the motor is stalled,
additional stator units are recruited to the motor. Upon release, units
disengage and the motor relaxes to its steady state structure, determined by
the load imposed on the motor by the drag of the bead [22]. This method
allowed for exploration of stator dynamics at the full range of loads
accessible by the bead assay, but not those near vanishing load. In the
following, instead of permanent magnets, we use a rotating electromagnetic
field, which allows us to rotate the motor across the entire range of speeds
achievable by the BFM. Our experiments were contemporary with, and are
complementary to, a similar investigation using electrorotation [34]. We
discuss electrorotation as an alternate method to manipulate motor torque on
the flagellar motor in the Discussion.

Motor torque is a complex function of several factors, including motor speed
and ion-motive force (IMF) [10, 17, 32]. Here, we probe the dynamics behind
load-dependent motor remodelling, focusing on the low-load regime, where motor
torque varies most dramatically with speed (Figure 1). We also investigate how
these dynamics depend on ion-motive force, by measuring stator remodelling at
the same speeds as IMF is varied, and provide support that mechanosensing in
the BFM shows behaviour akin to that of a catch bond across a wide range of
motor operating conditions [22].

## Manipulating motor speed using rotating magnetic field

Using a rotating magnetic field, we control the speed of single motors by
exerting torque on a superparamagnetic bead attached to a flagellar hook
(Figure 2a). The torque experienced by the bead from a magnetic field
$\overrightarrow{B}$ is given by

$\overrightarrow{\tau_{B}}=\overrightarrow{m}\times\overrightarrow{B}.$ (1)

The applied torque $\overrightarrow{\tau_{B}}$ on the bead will be nonzero
only if the magnetic dipole moment $\overrightarrow{m}$ is not aligned with
$\overrightarrow{B}$. These beads comprise randomly oriented anisotropic
magnetic nanoparticles (NP), and $\overrightarrow{m}$, the vector sum of the
dipole moments of the individual NPs, is in general not parallel with
$\overrightarrow{B}$ allowing for torque generation [33]. When the magnetic
field $\overrightarrow{B}$ is rotated at speed $\omega_{B}$, the
superparamagnetic bead will follow at the same speed so long as the maximum
applied torque $\overrightarrow{\tau_{B}}$ is sufficient to satisfy the
balance between $\overrightarrow{\tau_{B}}$, the motor torque, and the drag of
the bead (Figure 2b).

To generate the rotating field $\overrightarrow{B}$, a three-pole
electromagnet was positioned over a sample on the stage of an inverted
microscope; this design is a modification of the six-pole magnet system
constructed in [11]. The core of the magnet was constructed of a highly
magnetic-permeable ferrite core. Three removable soft-iron poles were attached
to slots within an iron block, which was attached to the core; this
configuration allowed us to vary the spacing of the gap between the pole
pieces (and thus the magnitude of the magnetic field at the centre of the
magnet) while strictly maintaining a three-fold symmetry, which allows
rotation of the direction of the magnetic field and therefore of the magnetic
particle [11].

An enamelled 20 British Standard Wire Gauge (SWG) copper wire (diameter$\sim$1
mm) was used to wind three separate coils, each composed of 108 turns, around
the electromagnet. A polyimide film (Kapton, DuPont) was used between wire
layers and at the ends of each coil for electrical and thermal insulation.
Each coil was driven by a voltage-to-current amplifier. Voltage-output modules
(N19263, National Instruments) were used as a signal generator to modulate
inputs into the amplifiers; inputs to the three coils were sinusoids of equal
amplitudes and 120∘ out of phase. Further details on magnet construction and
calibration can be found in [11] and [16].

In our assay we use a non-switching strain of E. coli (MTB32 + $\Delta$CheY)
with an endogenously biotinylated hook [7, 22]. Streptavidin-coated
superparamagnetic beads (MyOne DynaBeads T1, diameter 1$\mu$m) were attached
to the biotinylated hooks after cells were immobilised onto tunnel slides
using poly-L-lysine. Bead position was determined at 10k Hz using back-focal
plane interferometry and a quadrant photo diode (QPD; see, e.g., [31, 23]).
Bead $(x,y)$ positions from the QPD were converted to speed vs time traces
using a Fast Fourier Transform (FFT) with a sliding zero-padded window of
length 0.2s. A median filter with window 0.5s was then applied to the speed
traces. All code was written in Python and is available at
http://users.ox.ac.uk/$\sim$phys1213.

Motors were allowed to rotate for at least 60s before current was passed
through the magnets and torque was applied. Stator unit number was determined
from jumps in the first 60s of speed traces as in [22]; the single-unit speed
(3.7 $\pm$ 1.0 Hz in the normal condition and 1.9 $\pm$ 0.9 Hz in the butanol
condition) determined from these discontinuities was used to quantify stator
unit stoichiometry for the remainder of the trace. This assumption, that the
motor speed increases linearly with stator number, and thus that stator number
can be quantified by the speed of a single step, has been generally accepted
at high loads [30, 20, 23], where our stoichiometry estimates are made. An
example trace is shown in Figure 3a.

Motors were then rotated at fixed speed by flowing a current through the
three-pole electromagnet, as shown in Figure 1. Each ‘magnet sequence’
consists of ten 5s intervals of forced rotation by the electromagnet at a
fixed speed, separated by 0.2s intervals with the magnets off, during which
the number of stators was determined (Figure 3a,b). Rotation speeds were
chosen as ($nx+1$) Hz, ${n=1,2,4,6\\}}$, where $x$ is the acquisition rate of
the camera (50 Hz) used to observe the brightfield video. This allowed easy
visual determination of whether the bead was rotating at the same speed as the
magnetic field: such rotation was aliased to 1 Hz by the camera sampling (see
supplementary video).

Currents of amplitude 1.5A were used for all experiments; this strength
allowed us to reach zero-torque speeds at a sufficient yield using
commercially available beads while minimising any force on the motor [11]. To
further assure that the beads’ moments were aligned with the magnetic field,
beads were flowed into the tunnel slide and left to settle in the presence of
a magnetic field of low strength, generated using a pair of Helmholtz coils.
This ensured that free beads were reasonably well-aligned before attaching to
a flagellar hook. All experiments were performed at 25∘C in motility buffer
(10mM potassium phosphate, 0.1mM EDTA, pH 7.0) with or without the addition of
0.5% butanol by volume.

Within each magnet sequence, periods of applied torque are called ‘magnet on’
periods, and intervals in between (during which the motor is allowed to rotate
freely) are called ‘magnet off’ periods. The motor speed was counted as the
average speed in the 0.2s directly after the current to the magnets was shut
off (see Figure 3a). Recordings from ‘magnet off’ intervals were discarded and
treated as missing data if the motor did not maintain a speed within 5 Hz of
the speed of the rotating magnetic field for at least 4 consecutive seconds in
the preceding ‘magnet on’ interval (i.e., segments in which the bead did not
sufficiently follow the magnetic field’s rotation). Motors were allowed to
return to within 5 Hz of baseline speed after each magnet sequence before
beginning subsequent magnet sequences; we stopped recording from motors which
did not resurrect fully (on average, motors were run 3.8 $\pm$ 2.2 times at
each speed before failure).

## Unit unbinding varies with speed near zero torque; binding does not

We use a simple Hill-Langmuir model for stator assembly [22]. We consider the
rotor to be surrounded by $N_{\text{max}}$ independent, noninteracting binding
sites, each of which can house a single stator unit. A freely diffusing unit
binds to an empty binding site with rate $k_{\text{on}}$, and a bound stator
unit unbinds with rate $k_{\text{off}}$. We note quickly here that there is no
convincing evidence that fixed binding sites exist along the rotor’s
circumference; a more generalised model could be considered in place of the
chosen Hill-Langmuir adsorption. In this model, the random sequential
absorption (RSA) model, the periphery of the rotor has no discrete, separated
binding spots. This can lead to suboptimal packings when many stators are
present – think, for instance, of a kerb without designated parking spots. At
high loads (and high stator unit occupancies), a new stator unit can be
blocked from engaging the motor if it is not allotted enough contiguous space
on the ring (e.g., due to suboptimal packing of already docked units). As unit
occupancy near the zero-torque limit is low, the probability of such a
situation is vanishing in this regime; further, previous simulations suggest
that differences in relaxation dynamics are negligible between these two
models [22]. To this end, we focus primarily on the Hill-Langmuir model for
ease of analysis.

Under this model, the stator unit occupancy $N(t)$ follows

$\frac{dN}{dt}=k_{\text{on}}(N_{\text{max}}-N)-k_{\text{off}}N.$ (2)

Motivated by observations that the pool of membrane-bound stator unit ‘spares’
is large [14] ($\sim$200 vs our $N_{\text{max}}$=14 [22]), we consider their
concentration to be unaffected by motor kinetics, and therefore
$k_{\text{on}}$ to be independent of $N$.

At steady state, $\frac{dN}{dt}=0$, and we solve for the steady state stator
unit occupancy $N_{\text{ss}}$ as

$N_{\text{ss}}=\frac{N_{\text{max}}}{1+K_{D}}.$ (3)

Here, $K_{D}=k_{\text{off}}/k_{\text{on}}$ is the disassociation constant.
Given evidence that $N_{\text{ss}}$ decreases with decreasing external viscous
load, $K_{D}$ accordingly increases [22]. A load-dependent $K_{D}$ can arise
from a load-dependent $k_{\text{on}}$, a load-dependent $k_{\text{off}}$, or
both.

Looking to the solution of (2), together with (3), leads to an expression for
the time course of stator occupancy:

$N(t)=N_{\text{ss}}+(N_{0}-N_{\text{ss}})e^{-(k_{\text{on}}+k_{\text{off}})t},$
(4)

where $N_{0}$ is the observed stator unit occupancy before manipulation. In
fitting Equation 4 to our data in Figure 4, we make the assumption that
$k_{\text{off}}$, as well as $k_{\text{on}}$, is not a function of stator unit
number. This is consistent with the assumption that $k_{\text{off}}$ is a
function only of the mechanical strain in each unit, proportional to the
torque generated per unit, which is not expected to change in our experiments
(see Figure 3b). While there have been conflicting results on this in the low
load regime [23, 35], we are encouraged in our selection of a simple model as
our data (apart from at 51 Hz, which does not result in a notable change in
$N$) is well fit by a single exponential (Figure 4).

We note that this assumption holds across the torque-speed curves on the right
of Figure 3b, but fails for the set of curves on the left when the torque per
stator changes drastically at low stator number and high speed — for instance,
from two stator units to one. More detailed exploration in this regime, likely
at finer time resolution than is sampled here, will be needed to concretely
distinguish between these two scenarios.

We define $t_{c}=1/(k_{\text{on}}+k_{\text{off}})$ as the fitted decay time
constant (Figure 4). From here, binding $k_{\text{on}}$ and unbinding
$k_{\text{off}}$ rates are given by:

$k_{\text{on}}=\frac{N_{\text{ss}}}{t_{c}N_{\text{max}}},$ (5)
$k_{\text{off}}=\frac{N_{\text{max}}-N_{\text{ss}}}{t_{c}N_{\text{max}}}.$ (6)

In the ranges of load considered by the authors, Nord et al. found that
$k_{\text{off}}$ increases with decreasing load, while the binding rate
$k_{\text{on}}$ stays largely constant. Stator number did not change
appreciably in the 51 Hz condition (Figure 4), and so this time course could
not be fit well to Equation 4. Therefore, in this condition, we chose
$k_{\text{on}}$ to be the average of the binding rates at higher speeds (101
Hz, 201 Hz, 301 Hz). This was done because our calculated binding rate seemed
to remain constant across our measurements. Then, knowing $N_{\text{ss}}$, we
used Equations 5 and 6 to solve for $k_{\text{off}}$. These calculations are
as performed in [22].

We explore this trend in the low load regime and find, consistent with Nord et
al., that the unbinding rate increases as motor speed increases (and
accordingly, load decreases; see Figure 5c). The binding rate $k_{\text{on}}$
does not change significantly with load (Figure 5b). This is consistent with
that of a catch-bond: as the external load decreases, so decreases the
stability and the lifetime of the bond.

## IMF likely works via motor torque to affect stator dynamics

Though much of the recent work on remodelling in the motor has focused on
external load, early indications of the dynamic nature of the BFM’s stator
came courtesy of investigations into IMF effects [12, 19, 10, 32]. Here, we
revisit the role of IMF (here, PMF, as we are dealing with wild-type H+-driven
E. coli motors) in stator unit kinetics. We look at remodelling kinetics at
the same magnet-driven motor speeds in a low-concentration butanol solution to
lower the PMF as in [13]; in this way, we consider the question of whether IMF
works independently of load to affect motor assembly, or if motor energetics
act indirectly through mechanics in the remodelling process.

To lower the IMF is to take available energy away from the motor, and a
natural effect of de-energisation is a lowering of motor speed and torque
[17]. We observe this directly: the single-stator speed in butanol solution is
significantly lower than that in standard motility buffer (1.9 $\pm$ 0.9 Hz vs
3.7 $\pm$ 1.0 Hz in plain motility buffer). For a given rotational speed
imposed on the motor via the electromagnets, we expect the motor’s torque in a
butanol solution to be lower than that in motility buffer; indeed, we observe
higher unbinding rates $k_{\text{off}}$ in butanol than in plain motility
buffer for all speeds; these differences are more dramatic until the zero-
torque speed is reached (likely between 200 and 300 Hz; Figure 5a,c). Changes
in the binding rate $k_{\text{on}}$ are not significant (Figure 5b).

Our results, showing that motors rotated at the same speed at low IMF exhibit
higher $k_{\text{off}}$ rates (Figure 5c), are in line with the pattern
expected for IMF working via motor torque to affect stator dynamics. Further
characterisation of the effect of butanol on motor torque is required
(including, for instance, full torque-speed curves in butanol) to fully
understand the effect of IMF on stator dynamics; we leave this for future
work.

## Discussion

Developing methods to manipulate the external torque on the flagellar motor is
necessary to explore several aspects of the motor’s functionality fully. The
first enquiries into the dynamics of the BFM were performed using tethered
cell and bead assays, both of which do not allow, on their own, for a wide
range of loads to be studied on a single motor [29, 25]. The need for a method
to manipulate the external load on individual motors became stronger with the
discovery that the stoichiometry of the motor’s stator is load-dependent: at
higher loads, the motor contains a higher number of engaged stator units (up
to at least 11), but can maintain only one or two at low loads [15, 32].

We have presented an experimental framework capable of probing dynamic load-
dependent remodelling in the bacterial flagellar motor across the entire range
of loads the motor can experience, in particular at low load, a region of the
torque-speed curve which has thus far proven difficult to reach [17, 23]. Our
work was performed concurrently with an investigation of this regime using
electrorotation [34]. The results of this work and those of Wadhwa et al. are
complementary: we both observe a load-dependent $k_{\text{off}}$, which is
consistent with the catch bond hypothesis [22]. Quantitatively, their
estimated ‘knee’ and zero-torque speeds are both higher than we observe (e.g.,
we report stator unit drop-off at 101 Hz, suggesting a ‘knee’ somewhere
between 51-101 Hz); the torque-speed relationship is dependent on several
complex factors [9, 1, 23], and these differences can arise from several
origins, including bacterial strain choice. Electrorotation and speed
manipulation using an external magnetic field are complementary methods to
explore the BFM’s dynamics across its full range of operating conditions:
electrorotation works by directly varying the torque on the motor [5], while
magnetic rotation works by controlling the motor speed. In conjunction, these
methods have the ability to allow us to investigate, from both sides, the
BFM’s torque-speed relationship.

The development of these methods allow us to dynamically manipulate single
motors, and therefore to remove some of the variability that arises from
making population measurements. Electrorotation has been shown to be quite
harsh on cells, due to heating and large lateral forces on the motor [2, 5].
Though heating likely can be mitigated using a cooling system [34], magnetic
rotation setups are more easily constructed and replicated. The future of such
systems is promising: as superparamagetic microbeads with higher
magnetisability are developed [33], magnetic rotation can be used to push the
motor far past its zero-torque speed, as well as in the reverse CW direction,
while avoiding the pitfalls that have previously been demonstrated with
electrorotation [2, 3, 4].

Previous studies of remodelling have shown that mechanosensitivity in the
flagellar motor is consistent with that of a catch bond, a bond whose lifetime
increases with external load [22]. In the BFM, this manifests as a load-
dependent unbinding and load-independent binding rate. We observe similar
trends in the zero-torque regime (Figure 5); our investigation into the
effects of IMF provide further initial support for a load-dependent catch bond
mechanism. The rates we calculate (both for unit binding and unbinding) are
significantly higher than those calculated in [22]. It is likely that at the
small timescales considered here, disengaged stators have not yet diffused
away from the rotor and may rebind often and transiently [28]. The effects of
this process are likely also more easily observed in the low load regime, when
there is more ‘open space’ on the rotor’s periphery.

A natural extension of our work is to relate flagellar motor speed to
bacterial swimming speed. While studies at higher load lend themselves
naturally to understanding surface sensing [15], understanding motor
adaptation at low loads may give us further insight into the torque regime in
which free swimming bacteria reside.

## Author Contributions

J.A.N., A.L.N., and R.M.B. formulated the hypotheses and designed the
experiments; J.A.N. carried out the experiments, analysed data, and wrote the
manuscript; J.A.N., A.L.N., and R.M.B. edited the manuscript.

## Acknowledgments

We thank C.J. Lo and T.S. Lin for help with initial experiments, S.E. Tusk for
assistance in microscope assembly and technical troubleshooting, and all the
above for useful discussion and comments.

## Data Accessibility

All data and analysis code is available at http://users.ox.ac.uk/ phys1213
and/or on request from the authors.

## Funding

J.A.N. was supported by a Fellowship from All Souls College at the University
of Oxford, a James S. McDonnell Foundation Fellowship for Studying Complex
Systems, and a Fellowship from the Rockefeller University. A.L.N. acknowledges
funding from the European Research Council under the European Union’s Seventh
Framework Programme (FP/2007â€“2013)/ERC Grant Agreement No. 306475. CBS is a
member of the France-BioImaging (FBI) and the French Infrastructure for
Integrated Structural Biology (FRISBI), 2 national infrastructures supported
by the French National Research Agency (ANR-10-INBS-04-01 and ANR-10-INBS-05,
respectively). R.M.B was supported by the Biotechnology and Biological
Sciences Research Council grant BB/N006070/1.

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Figure 1: BFM structure and dynamics. (a) The flagellar motor’s rotor consists
of a series of large co-axial rings that attach to a flagellar filament via a
flexible hook. An active motor can have up to at least 11 torque-generating
stator complexes. Stators interact with proteins (FliG) along the rotor’s edge
to drive motor rotation. (b) Experiments in recent years have established that
the number of torque-generating units varies with external load on the motor
(among other possible factors, including IMF). Points in the high-load regime
correspond to motors near full occupancy and points at low loads to motors
with only one or two. Data shown from [9]. Solid lines are included to guide
the eye. Figure 2: Manipulating motor speed using a rotating magnetic field.
(a) Experimental setup: Bacteria are immobilised on a coverslip, and a
superparamagnetic bead is attached to the motor’s hook; (x,y) positions of the
bead are tracked and used to obtain speed vs time traces. Current flowing
through a three-pole electromagnet generates a rotating magnetic field $B$ to
manipulate the speed of the motor. (b) When the magnet is turned off (left),
motor torque (green) must balance viscous drag from the bead (red), resulting
in a low rotation speed. When the magnetic field is on and rotating (right),
the magnetic torque (blue) takes on most of the drag from the bead, allowing
for faster rotation with minimal contribution from motor torque. View is from
the top of a CCW rotating motor. Figure 3: (a) An example experimental trace.
Motors are allowed to freely rotate for at least 1 minute, during which time a
step detection algorithm is used to calculate single stator speeds and
determine stator unit stoichiometry (red lines). An inlay shows a histogram of
single stator speeds for the plain motility buffer condition. Motors are then
rotated at a fixed speed (201 Hz in this example) for 5 second intervals with
0.2s ‘off’ periods during which the motor speed is recorded. ‘Magnet on’
periods are shown as lines at 201 Hz; red circles depict an example time
course of motor remodelling;each red circle contains a single speed calculated
within one FFT window. The results of the first ‘magnet on’ period are shown
by points 1,2,3,4. At point 1, the motor has 7 stator units. When the magnetic
field rotates the motor at 201 Hz (blue arrow to point 2), the motor loses 2
stator units (green arrows) between points 2 and 3 (open circles, unobserved).
This loss is observed as a decrease in speed during the first ‘magnet off’
period at point 4. (b) Two possible torque-speed curves for a motor with 7
stator units. The left depicts results by [23], which showed that the zero-
torque speed of the BFM increased with the number of engaged stator units; the
right depicts results by [25], which suggested that the limiting speed is
independent of stator number. Solid lines are based on data; dashed lines are
extrapolated as in [23] and [25]. Dashed grey lines are load lines for the
passive beads; solid grey lines show constant speed across stator number
during magnet-assisted rotation. Recent experiments by [27] have supported the
results of [23] (left). Points 1,2,3,4 depict the same cycle of the magnet
from the experimental trace shown in (a). Our assumption that unit binding and
unbinding rates are not a function of stator number holds across the torque-
speed curves on the right. Considering the left set of curves, this assumption
fails when the torque per stator changes drastically at low stator number and
high speed — for instance, from two stator units to one (see solid grey line).
Further exploration in this regime, likely at finer time resolution than is
sampled in this work, will be needed to distinguish between these two
scenarios. Figure 4: Motor remodelling following a sudden change in load in
plain motility buffer (left) and at low PMF (right). Low PMF environment is
attained by mixing 0.5% butanol, shown to act as an ionophore [13], into
motility buffer (MB). Time traces show stator unit stoichiometry after magnets
are turned on at (a) 51 Hz (red); (b) 101 Hz (blue); (c) 201 Hz (orange) and
(d) 301 Hz (green). Circles and error bars show mean $\pm$ standard deviation
for experimental data in each condition; filled circles represent data
collected in plain MB and open circles data collected in low PMF buffer. Lines
(solid for MB condition; dashed for low PMF) and shaded regions depict fits of
Equation 4 to data $\pm$ 3 standard deviations. We were not able to fit the
traces collected at 51 Hz in plain MB (top left, red), which did not lose
stators appreciably during magnetic rotation, to this model. Figure 5:
Comparison of motor remodelling dynamics at different speeds and IMF levels.
(a) A zoomed-in look at the early, most dynamic region from the plots in
Figure 4; lines shown are fits of Equation 4 to the data in that figure. As
previously, dashed lines represent results in the lowered PMF condition and
solid lines in plain MB. The same colour scheme for magnet speed is used
throughout this manuscript. We extrapolate the zero-torque speed in the plain
MB condition to vary between 201 and 301 Hz, and between 101 and 201 Hz in the
butanol condition; dynamics at these speeds are similar, while those at
appreciably higher loads (i.e., 101 Hz in MB, solid blue line) and appreciably
lower loads (i.e., 301 Hz in butanol + MB, dashed green line) diverge. (b)
Binding rates $k_{\text{on}}$ remain relatively constant in both IMF
conditions across all four speed levels while (c) unbinding rates
$k_{\text{off}}$ vary as speed is increased and are on average higher in the
lowered IMF condition as compared to in plain MB. Rates $k_{\text{on}}$ and
$k_{\text{off}}$ were calculated from exponential fits to the data of Figure
4, except for the points outlined in red (see text for details). These
independent effects suggest that the effect of both speed and IMF on stator
dynamics in the BFM quite likely works via the relationship of these factors
to the load on the motor. Our results also show behaviour consistent with that
of a catch bond, as did those of a prior investigation in the higher load
regime [22].