# Using Experimentally Calibrated Regularized Stokeslets to Assess Bacterial
Flagellar Motility Near a Surface

Orrin Shindell  Trinity University, San Antonio, TX, USA<EMAIL_ADDRESS>Hoa Nguyen  Trinity University, San Antonio, TX, USA<EMAIL_ADDRESS>Nicholas Coltharp  Trinity University, San Antonio, TX, USA
<EMAIL_ADDRESS>Frank Healy  Trinity University, San Antonio, TX, USA
<EMAIL_ADDRESS>Bruce Rodenborn  Centre College, Danville, KY, USA
<EMAIL_ADDRESS>
###### Abstract

The presence of a nearby boundary is likely to be important in the life cycle
and evolution of motile flagellate bacteria. This has led many authors to
employ numerical simulations to model near-surface bacterial motion and
compute hydrodynamic boundary effects. A common choice has been the method of
images for regularized Stokeslets (MIRS); however, the method requires
discretization sizes and regularization parameters that are not specified by
any theory. To determine appropriate regularization parameters for given
discretization choices in MIRS, we conducted dynamically similar macroscopic
experiments and fit the simulations to the data. In the experiments, we
measured the torque on cylinders and helices of different wavelengths as they
rotated in a viscous fluid at various distances to a boundary. We found that
differences between experiments and optimized simulations were less than 5%
when using surface discretizations for cylinders and centerline
discretizations for helices. Having determined optimal regularization
parameters, we used MIRS to simulate an idealized free-swimming bacterium
constructed of a cylindrical cell body and a helical flagellum moving near a
boundary. We assessed the swimming performance of many bacterial morphologies
by computing swimming speed, motor rotation rate, Purcell’s propulsive
efficiency, energy cost per distance, and a new metabolic energy cost defined
to be the energy cost per body mass per distance. All five measures predicted
the same optimal flagellar wavelength independently of body size and surface
proximity. Although the measures disagreed on the optimal body size, they all
predicted that body size is an important factor in the energy cost of
bacterial motility near and far from a surface.

##  I Introduction

Living organisms emerge, evolve, and reside within habitats, and the physical
interactions among organisms and their environments impose selective forces on
their evolution. In their low Reynolds number surroundings, bacteria such as
Escherichia coli and Pseudomonas aeruginosa have evolved a mechanical motility
system to propel themselves through fluids. This system consists of one or
more helical flagella, and these flagellar organelles are attached to the cell
body by rotary nanomotors. Flagellar motor rotation is driven by an ion flow
through the motor, causing the flagellum and the bacterial cell body to rotate
in opposite directions [1]. A bacterium swimming through a fluid can be
described as a non-inertial system in which the mechanical power output by the
motor is instantaneously dissipated by fluid drag on the body and flagellar
filaments. The interaction between the bacterium and the fluid generates a
flow that results in the net motion of the bacterium. Different flows can be
more or less favorable to the survival of an organism [2]; and the presence of
a surface introduces boundary effects that modify how a swimming cell
interacts with the fluid. We consider here the example of a unicellular motile
flagellate bacterium swimming through a fluid near to a surface and how the
conformation of the bacterial cell body and the flagellar organelle may be
optimized for such an environment. The efficiency of the bacterial motility
system has been the focus of numerous theoretical [3, 4, 5], computational [6,
7, 8, 9, 10, 11], and experimental works [12, 13, 14]. In an early paper on
swimming efficiency, E. Purcell discussed two measures: the propulsive
efficiency (Purcell efficiency) and the energy consumed during bacterial
motion per body mass [3]. The Purcell efficiency–a specialized form of the
Lighthill efficiency [15] for rotary motor-driven bacterial propulsion–is
defined as the ratio of the least power needed to translate a bacterial body
against fluid drag to the total power output by the motor during motion of the
bacterium. Most work has focused on the Purcell efficiency because it is a
scale-independent function of the geometries of the cell body and flagellum.
One shortcoming of this measure, however, is that it is independent of the
motor’s response to an external load imposed by the environment and therefore
cannot assess the biological fitness of the bacterial motor. Another measure
of bacterial performance used by a few authors is the distance traveled by a
bacterium per energy input by the motor [13, 16], which provides a different
means of evaluating fitness, as explained below. In this work, we investigate
and compare predictions of the optimal bacterial motility system made by five
measures. The first two measures are related directly to the motion of a
bacterium: the swimming speed and the motor rotation frequency. Bacteria live
in an environment where nutrients diffuse on time and length scales comparable
to bacterial motion. To effectively achieve chemotaxis, bacteria must move
quickly enough to sample their chemical environment before it is randomized by
diffusion [3, 11]. The bacterial motor has a characteristic frequency response
that depends on the external torque load [17, 18, 19, 20]. At low frequencies,
small changes in applied load correspond to large changes in operating
frequency, whereas at high frequencies, small changes in load correspond
smaller changes in frequency. In the low speed regime, the motion may be
unreliable because small changes in applied load that occur, for example, by
approaching a boundary could lead to the motor stalling. However, the low
speed regime is more thermodynamically efficient than the high speed regime.
These two competing effects must be balanced to achieve a strong swimming
performance. The other three performance measures we studied are based on the
mechanical energy cost to achieve motility: the Purcell inefficiency (or the
inverse of the Purcell efficiency), the inverse of distance traveled per
energy input, and the metabolic energy cost, which we define to be the energy
output by the motor per body mass per distance traveled. Each of these
measures compares the ratio of the power output of the bacterial motor to the
performance of a particular task. The rationale for introducing the metabolic
cost function is that it measures the actual energetic cost to the organism to
perform a specific biologically relevant task, i.e., translation through the
fluid. Moreover, the metabolic energy cost depends upon the rotation speed of
the motor and, because the bacterial motor has a different responses to
different external conditions, predicts different optimal morphologies based
on the environment than the other measures. To determine the values of
performance measures attained by different bacterial geometries, we employed
the method of regularized Stokeslets [21] and the method of images for
regularized Stokeslets (MIRS) [22], which includes the effect of a solid
boundary. Employing MRS and MIRS requires determining values for two kinds of
free parameters: those associated with computation and those associated with
the biological system. As with any computational method, the bacterial
structure in the simulation is represented as a set of discrete points. The
body forces acting at those points are expressed as a vector force multiplied
by a regularized distribution function, whose width is specified by a
regularization, or “blob” parameter. Though other simulations have produced
numerical values for dynamical quantities like torque [23] that are within a
reasonable range for bacteria, precise numbers are not possible without an
accurately calibrated method. There is no known theory that predicts the
relationship between the discretization and regularization parameters, though
one benchmarking study showed that MRS simulations could be made to match the
results of other numerical methods [24]. To determine the optimal
regularization parameter for chosen discretization sizes, we performed
dynamically similar macroscopic experiments using the two objects from our
model bacterium: a cylinder and a helix, see Fig. 1. Such an approach was
previously used to evaluate the accuracy of various computational and
theoretical methods for a helix [25]. By measuring values of the fluid torque
acting on rotating cylinders near a boundary, we verified the theory of
Jeffery and Onishi [26], which in turn we used to calibrate the ratio of
discretization to regularization size in MRS and MIRS simulations of rotating
cylindrical cell bodies. For helices there are no exact analytical results. To
determine regularization parameters for helices we descretized them along
their centerlines and fit simulation results directly to experimental
measurements. Calibrating our simulations of rotating cylinders and helices
with the experiments allowed us to build a bacterial model with a cylindrical
cell body and a helical flagellum whose discretization and regularization
parameter are optimized for each part. To impose motion on the bacterial
model, we needed only to specify the motor rotation – a consequence of there
being no body forces acting on the bacterium [23]. The motor rotation rate,
however, depends upon the external load [17, 18, 19, 13]. In our simulations,
we ensured the motor rotation rate and the total torque acting on the motor
match a point on the experimentally determined torque-speed response curve
reported in the literature [17, 20]. The dynamical quantities output from the
simulations were then used to compute performance measures for different
bacterial geometries at various distances from the boundary. Our paper is
organized as follows: Sec. II discusses our implementation of the MRS and the
MIRS, our use of dynamically similar experiments to calibrate the simulations,
and our determination of the torque-speed response curve for the motor; Sec.
III compares our five fitness measures: free swimming speed, motor frequency,
inverse Purcell efficiency, energy per distance and metabolic cost per
distance; and Sec. IV discusses the predictions made by each fitness measure
and comments on future directions of our work.

##  II Materials and Methods

###  II.1 Numerical Methods

Bacterial motility using a helical flagellum often involves multiple flagella,
and bodies may be spherical, cylindrical, or helical [27]. We reduced the
complexity by considering a simpler biomechanical system of a regular
cylindrical body to which a single, uniform flagellum is attached, as shown in
Fig. 1. This simple system, however, contains the same essential geometric
factors as some real bacteria such as E. coli, which have a long rod-shaped
body and helical flagella that bundle together, forming a single helix. Our
goal was to assess how the performance of our model organism changes when its
geometrical parameters and distance to an infinite plane wall are varied in
numerical simulations. We quantified the performance of different models by
computing speed, motor rotation rate, and the three energy cost measures.

Figure 1: Schematic of our model bacterium with flagellar radius $R$,
wavelength $\lambda$, axial length $L$, and filament radius $a$. The body of
the bacterium was modeled as a cylinder with radius $r$ and length $\ell$.
Each flagellum was modeled as a regular helix that tapers to zero radius at
the point it attaches to the body. Our simulations used a surface
discretization of regularized Stokeslets to represent the cylinder and a
string of regularized Stokeslets along the centerline of the flagellum. The
inset represents a radially symmetric blob function described in Sec. II.1.1
that is used to spread the force at a given point on the flagellar centerline.
For the purpose of illustration, we show the blob function of two variables
whose width is controlled by the regularization parameter $\epsilon_{f}$.
Figure 2: Our model bacterium had a cylindrical cell body and a helical
flagellum, and 25 different cell body sizes and eighteen different flagellar
wavelengths were used, as described in Table 1. Three cell bodies with the
smallest, average, and largest volumes, respectively are shown on the right
whereas the three flagella with the shortest, average, and longest wavelengths
are presented on the left. The middle shows an example of one such model,
which has the smallest body and the longest wavelength flagellum.

We composed our model of a bacterium with a cylindrical cell body and a
tapered left-handed helical flagellum as shown in Fig. 1 and Fig. 2. The
flagellar centerline is described by

$\left\\{\begin{aligned} x(s)&=(1-e^{-k^{2}s^{2}})R\sin(ks+\theta)\\\
y(s)&=(1-e^{-k^{2}s^{2}})R\cos(ks+\theta)\\\ z(s)&=s\end{aligned}\right.$ (1)

where $0\leq s\leq L$ and $L$ is the axial length in the $z$-direction. $k$ is
the wavenumber $2\pi/\lambda$ where $\lambda$ is the wavelength. $\theta$ is
the phase angle of the helical flagellum at 16 evenly spaced phases. The
parameter values used for the bacterium models shown in Fig. 2 are given in
Table 1.

Table 1: Parameters used in numerical simulations.

Parameter | Symbol | Value | Unit | Reference
---|---|---|---|---
Dynamic viscosity of the fluid | $\mu$ | $0.93$ | $cP$ |
Cell body (Cylinder) |  |  |  |
length | $\ell$ | (a) | $\mu m$ | [20]
radius | $r$ | (b) | $\mu m$ | [20]
optimal discretization factor | $\gamma_{c}$ | $6.4$ |  |
discretization size | $ds_{c}$ | $0.096$ | $\mu m$ |
regularization parameter | $\epsilon_{c}=ds_{c}/\gamma_{c}$ | $0.015$ | $\mu m$ |
Flagellum (Helix) |  |  |  |
axial length | $L$ | $8.3$ | $\mu m$ | [20]
wavelength | $\lambda$ | (c) | $\mu m$ |
helix radius | $R$ | $0.2$ | $\mu m$ | [20]
filament radius | $a$ | $0.012$ | $\mu m$ | [20]
initial motor frequency | $\Omega_{m}/(2\pi)$ | $154$ | $Hz$ | [20]
optimal filament factor | $\gamma_{f}$ | $2.139$ |  |
regularization parameter | $\epsilon_{f}=\gamma_{f}a$ | $0.026$ | $\mu m$ |
discretization size | $ds_{f}=\epsilon_{f}$ | $0.026$ | $\mu m$ |
distance from flagellar axis to wall | $d$ | (d) | $\mu m$ |

(a) $\ell\in$ {1.9, 2.2, 2.5, 2.8, 3.1} $(\mu m)$
(b) $r\in$ {0.395, 0.4175, 0.44, 0.4625, 0.485} $(\mu m)$
(c) $\lambda\in$ {0.2, 0.5, 0.8, 1.1, 1.4, 1.7, 2.02, 2.22, 2.3, 2.42, 2.6,
2.9, 3.2, 3.6, 4.0, 5.0, 7.0, 9.0 } $(\mu m)$
(d) $d\in$ {0.55, 0.62, 0.71, 0.82, 0.96, 1.12, 1.32, 1.56, 1.85, 2.20, 2.26,
2.52, 2.81, 3.14, 3.5, 3.93, 4.4, 4.93, 5.53, 6.2, 8.2, 10.2} $(\mu m)$

####  II.1.1 Method of regularized Stokeslets

The microscopic length and velocity scales of bacteria ensure that fluid
motion at that scale can be described using the incompressible Stokes
equations. We used the MRS in three dimensions [21] to compute the fluid-
bacterium interactions due to the rotating flagellum in free space at steady
state:

$\begin{split}\mu\triangle\mathbf{u(\mathbf{x})}-\nabla
p(\mathbf{x})&=-\mathbf{F(\mathbf{x})}\\\
\nabla\cdot\mathbf{u(\mathbf{x})}&=0\end{split}$ (2)

$\mathbf{u}$ is the fluid velocity, $p$ is the fluid pressure, and $\mu$ is
the dynamic viscosity. $\mathbf{F}$ is the body force represented as
$\mathbf{f}_{k}\phi_{\epsilon}(\mathbf{x}-\mathbf{x}_{k})$ where
$\mathbf{f}_{k}$ is a point force at a discretized point $\mathbf{x}_{k}$ of
the bacterium model. In our simulations, we used the blob function
$\phi_{\epsilon}(\mathbf{x}-\mathbf{x}_{k})=\frac{15\epsilon^{4}}{8\pi(r_{k}^{2}+\epsilon^{2})^{\frac{7}{2}}}$
where $r_{k}=\left\lVert\mathbf{x}-\mathbf{x}_{k}\right\rVert$. This radially
symmetric smooth function depends on a regularization parameter $\epsilon$
which controls the spread of the point force $\mathbf{f}_{k}$. Given $N$ such
forces, the resulting velocity at any point $\mathbf{x}$ in the fluid can be
computed as

$\mathbf{u}(\mathbf{x})=\frac{1}{8\pi\mu}\sum_{k=1}^{N}\frac{\mathbf{f}_{k}(r_{k}^{2}+2\epsilon^{2})}{(r_{k}^{2}+\epsilon^{2})^{\frac{3}{2}}}+\frac{(\mathbf{f}_{k}\cdot(\mathbf{x}-\mathbf{x}_{k}))(\mathbf{x}-\mathbf{x}_{k})}{(r_{k}^{2}+\epsilon^{2})^{\frac{3}{2}}}=\frac{1}{8\pi\mu}\sum_{k=1}^{N}S_{\epsilon}(\mathbf{x},\mathbf{x}_{k})\mathbf{f}_{k}$
(3)

Evaluating Eq. 3 $N$ times, once for each $\mathbf{x}_{k}$, yields a $3N\times
3N$ linear system of equations for the velocities of the model points. In the
limit as $\epsilon$ approaches 0, the resulting velocity $\mathbf{u}$
approaches the classical singular Stokeslet solution. In practice, the
specific choice of $\epsilon$ may depend on the discretization or the physical
thickness of the structure. In our bacterium model, we discretized the cell
body as $N_{c}$ points on the surface of a cylinder, and we modeled the
flagellum as $N_{f}$ points distributed uniformly along the arclength of the
centerline. In Sec. III, we present the optimal regularization parameter for
the cylindrical cell we obtained by calibrating the simulations based on the
experiments and theory. The regularization parameter for the helical flagellum
was found by calibrating simulations with experiments, since there is no exact
theory for rotating helices, as presented in Sec. II.3.

####  II.1.2 Method of images for regularized Stokeslets

We used the method of images for regularized Stokeslets (MIRS)[22] to solve
the incompressible Stokes equations (Eq. 2) and simulate bacterial motility
near a surface. In the method, the no-slip boundary condition on an infinite
plane wall is satisfied by imposing a combination of a Stokeslet, a Stokeslet
doublet, a potential dipole, and rotlets at the image point
$\mathbf{x}^{*}_{k}$ of each discretized point $\mathbf{x}_{k}$. The image
point $\mathbf{x}^{*}_{k}$ is the point obtained by reflecting
$\mathbf{x}_{k}$ across the planar surface. The resulting velocity at any
point $\mathbf{x}$ in the fluid bounded by a plane can be found in Ref.[22]
and written in the compact form similar to Eq. 3:

$\mathbf{u}(\mathbf{x})=\frac{1}{8\pi\mu}\sum_{k=1}^{N}S^{*}_{\epsilon}(\mathbf{x},\mathbf{x}_{k})\mathbf{f}_{k}$
(4)

####  II.1.3 Force-free and torque-free models

On a free-swimming bacterium, the only external forces acting are due to the
fluid-structure interaction. A bacterium is a non-inertial system so the net
external force and net external torque acting on it must vanish. This means
that $\mathbf{F}_{c}+\mathbf{F}_{f}=\mathbf{0}$ and
$\bm{\tau}_{c}+\bm{\tau}_{f}=\mathbf{0}$, where $\mathbf{F}_{c}$ /
$\bm{\tau}_{c}$ and $\mathbf{F}_{f}$ / $\bm{\tau}_{f}$ represent,
respectively, the net fluid forces and torques acting on the cell body and
flagellum. These force-free and torque-free constraints require the cell body
and flagellum to counterrotate relative to each other. In our simulations, the
point connecting the cell body and the flagellum $\mathbf{x}_{r}$ represented
the motor location, and was used as the reference point for computing torque
and angular velocity. Given an angular velocity $\mathbf{\Omega}_{m}$ of the
motor, the relationship between the lab frame angular velocities of the
flagellum and the cell body is
$\mathbf{\Omega}_{f}=\mathbf{\Omega}_{c}+\mathbf{\Omega}_{m}$ [23]. Since
$\mathbf{\Omega}_{m}$ is the relative rotational velocity of the flagellum
with respect to the cell body, the resulting velocity
$\mathbf{\tilde{u}}(\mathbf{x}_{k})$ at a discretized point $\mathbf{x}_{k}$
on the flagellum ($k=1,...,N_{f}$) can be computed as
$\mathbf{\Omega}_{m}\times\mathbf{x}_{k}$ (this velocity is set to zero at a
discretized point on the cell body). Using the MRS (or MIRS) and the six added
constraints from the force-free and torque-free conditions, we formed a
$(3N+6)\times(3N+6)$ linear system of equations to solve for the translational
velocity $\mathbf{U}$ and angular velocity $\mathbf{\Omega}_{c}$ of the cell
body and the internal force $\mathbf{f}_{k}$ acting at the discretized point
$\mathbf{x}_{k}$ of the model:

$\begin{split}\mathbf{\tilde{u}}(\mathbf{x}_{j})&=\frac{1}{8\pi\mu}\sum_{k=1}^{N}G_{\epsilon}(\mathbf{x}_{j},\mathbf{x}_{k})\mathbf{f}_{k}-\mathbf{U}-\mathbf{\Omega}_{c}\times(\mathbf{x}_{j}-\mathbf{x}_{r}),\quad
j=1,...,N\\\
\sum_{k=1}^{N}\mathbf{f}_{k}&=\mathbf{0},\quad\quad\sum_{k=1}^{N}(\mathbf{x}_{k}-\mathbf{x}_{r})\times\mathbf{f}_{k}=\mathbf{0}\end{split}$
(5)

where $G_{\epsilon}$ is $S_{\epsilon}$ from Eq. 3 for swimming in a free space
or $S^{*}_{\epsilon}$ from Eq. 4 for swimming near a plane wall. Each
$\mathbf{f}_{k}$ represents a point force acting at point $\mathbf{x}_{k}$,
which is in principle an internal contact force due to interactions with the
points on the bacterium that neighbor $\mathbf{x}_{k}$. Each $\mathbf{f}_{k}$
is balanced by the hydrodynamic drag that arises from a combination of viscous
forces and pressure forces exerted on the point $\mathbf{x}_{k}$ by the fluid
(Eq. 2). By computing each $\mathbf{f}_{k}$, we were able to deduce the fluid
interaction with each point of the bacterial model. Eq. 5 shows that the
calculated quantities $\mathbf{U}$, $\mathbf{\Omega}_{c}$, $\mathbf{F}_{c}$,
and $\bm{\tau}_{c}$ depend linearly on the angular velocity
$\mathbf{\Omega}_{m}$ since
$\mathbf{\tilde{u}}(\mathbf{x}_{j})=\mathbf{\Omega}_{m}\times\mathbf{x}_{j}$.

###  II.2 Torque-speed motor response curve

The singly-flagellated bacteria we simulated move through their environment by
rotating their motor, which causes their body and flagellum to counter-rotate
accordingly. Drag force from the fluid exerts equal magnitude torques on the
body and the flagellum, and the value of the torque equals the torque load
applied to the motor. The relationship between the motor rotation rate and the
torque load is characterized by a torque-speed curve, which has been measured
experimentally in several organisms [17, 18, 19, 13, 20]. In the context of
motor response characteristics, speed refers to frequency of rotation. We
estimated the torque-speed curve for E. coli with typical values taken from
the literature [17, 20] to match the body and flagellum parameters also taken
from measurements on E. coli [20]. The fluid torque exerted on a rotating
object is proportional to its rotation rate under constant environmental
conditions in Stokes flow, and thus plotting the fluid torque versus rotation
rate in fixed conditions yields a straight line. Fig. 3 shows examples of
these ‘load lines’ computed for our bacterial model at different distances
from the boundary; the shallower blue line is calculated for a bacterium far
from the boundary, and the steeper red line is calculated near the boundary.
The load lines shown in Fig. 3 were computed with typical body and flagellum
parameters for E. coli [20].

Figure 3: Illustration of the estimated torque-speed curve for E. coli [17,
20]. There are two operating regimes: a relatively flat low speed regime
$0\leq\Omega_{m}/2\pi\leq 175$ Hz where the torque drops from its maximum
value of 1300 pN$\cdot$nm at 0 Hz to 1196 pN$\cdot$nm at 175 Hz and a
relatively steep high speed regime $175\leq\Omega_{m}/2\pi\leq 350$ Hz where
the torque drops from 1196 pN$\cdot$nm at 175 Hz to 0 pN$\cdot$nm at 350 Hz.
The insets depict a bacterium model with the average body length $\ell=2.5$
$\mu$m, the smallest body radius $r=0.395$ $\mu$m, and the average flagellar
wavelength $\lambda=2.22$ $\mu$m at different distances from the boundary:
$d=8.2$ $\mu$m (blue), $d=0.71$ $\mu$m (green), $d=0.54$ $\mu$m (red). At
closer distances the torque versus rotation rate load lines are steeper so
that they intersect the torque-speed curve at a slower rotation speed.

The torque-speed curve of the E. coli motor has been determined experimentally
by measuring the rotation rate of a bead attached to a flagellar stub and then
computing the torque on the bead due to fluid drag. By performing the
measurement in fluids of different viscosities, many points on the torque-
speed curve were assembled. It was found that the torque-speed curve of the E.
coli bacterial motor decreases monotonically from a maximum stall torque (i.e.
the zero-speed torque) of about 1300 pN$\cdot$nm to zero torque, which occurs
at a maximum speed of 350 Hz [17, 19, 20]. There are two linear operating
regimes, a low speed regime from 0-175 Hz and a high speed regime 175-350 Hz.
In the low speed regime below 175 Hz, the torque is a relatively flat function
of the motor rotation rate, falling to 0.92 of the stall torque at 175 Hz. In
the high speed regime above 175 Hz, the torque falls steeply to zero at 350
Hz. The torque-speed curve is thus expressed as a piecewise linear function of
the motor rotation rate, $\Omega_{m}$:

$\tau=\begin{cases}\text{$\displaystyle\left(-0.59\left(\frac{\Omega_{m}}{2\pi}\right)+1300\right)$
pN$\cdot$nm}&\text{for $\displaystyle 0\leq\frac{\Omega_{m}}{2\pi}\leq 175$
Hz}\\\
\text{$\displaystyle\left(-6.83\left(\frac{\Omega_{m}}{2\pi}\right)+2392\right)$
pN$\cdot$nm}&\text{for $\displaystyle 175\leq\frac{\Omega_{m}}{2\pi}\leq 300$
Hz}\end{cases}$ (6)

Fig. 3 shows the torque-speed curve as a solid black line. In each of our
simulations, we ensured that the prescribed motor speed and the computed
torque load formed a pair that corresponded to a point on that line.

###  II.3 Dynamically similar experiments

Experiments were performed in an 45-liter tank (300 mm $\times$ 500 mm
$\times$ 500 mm high) filled with incompressible silicone oil (Clearco®) with
density 970 kg/m3 and dynamic viscosity $\mu$ = $1.13\times 10^{2}$
kg/(m$\cdot$s) at $22^{\circ}$C, about $10^{5}$ times that of water. The
length and speed scales in the experiment ensured that the incompressible
Stokes equations Eqs. 2 were valid. The viscosity of the oil drifted from the
manufacturer’s stated value ($\mu$ = $1.00\times 10^{2}$ kg/(m$\cdot$s)) very
slowly over a two-year period, so we determined the modified viscosity by
measuring the torque on rotating cylinders at the center of the tank and
recorded data within two months of that measurement. The theoretical value for
torque per unit length on an infinite rotating cylinder in Stokes flow is
$\sigma=4\pi\mu\Omega r^{2}$, where $\mu$ is the dynamic viscosity of the
fluid, $\Omega$ is the angular rotation rate, and $r$ is the cylindrical
radius. We measured the torque $\tau$ on a rotating cylinder with radius
$r=6.35\pm 0.2$ mm and length $\ell=149\pm 1$ mm and, by assuming
$\tau=\ell\sigma$, used the data to solve for the viscosity of the fluid. We
also assumed that the finite size of the tank did not affect the torque value
in the middle, which was more than $20r$ from the nearest boundary. Before
each data collection run, we measured the temperature of the oil with a NIST-
traceable calibrated thermistor (Cole-Parmer Digi-Sense-AO-37804-04 Calibrated
Digital Thermometer) and adjusted the previously determined viscosity using
the manufacturer’s temperature coefficient of viscosity $1.00\times 10^{-6}$
kg/(m$\cdot$s)/∘C. See Sec. II.3.2 for a detailed description of the torque
measurements.

####  II.3.1 Fabricating helices

We fabricated helices of varying wavelengths ($2.26<\lambda/R<11.88$) by
wrapping straight stainless steel welding wire around cylindrical aluminum
mandrels with different helical V-grooves precisely machined using a CNC
lathe. The V-grooves transition to a flat face with a straight groove, to
which the remaining straight section can be clamped; see Fig. 4.

Figure 4: Aluminum rods with helical V-shaped grooves were used to form model
flagella with different wavelengths. After forming, the flagella were annealed
on a precision rod to increase uniformity in the radius ($\Delta R<0.1mm$).
The helical parameters are listed in Table 2.

Mandrels were held on a lathe, and the wire was hand-spun into the V-groove.
The straight sections were secured to the flat faces, which left straight
stems aligned with the axes of the helices to be attached to the motor via a
rigid shaft adapter. Residual tension in the wires caused the wavelengths and
radii to vary after they were removed from the mandrels. The helices were
forced onto a precision stainless steel rod with radius $R=6.350\pm 0.013$ mm
for annealing. The helices on the rod were then placed into a tube furnace
(MTI GSL-1500X) and annealed at 900 degrees Celsius for two hours, which
removed most of the variation in the radii of the helices and fixed the
helical wavelength. The helix parameters used in the experiments are listed
Table 2:

Table 2: Wavelengths and lengths of helices.

$\lambda/R$ | $L/R$
---|---
2.26 $\pm$ 0.13 | 22.3 $\pm$ 0.5
3.88 $\pm$ 0.01 | 24.3 $\pm$ 0.5
5.86 $\pm$ 0.08 | 30.0 $\pm$ 0.5
8.65 $\pm$ 0.01 | 23.3 $\pm$ 0.5
10.91 $\pm$ 0.01 | 24.2 $\pm$ 0.5
11.88 $\pm$ 0.01 | 23.1 $\pm$ 0.5

The helical wavelength $\lambda$ and axial length $L$ are expressed in terms
of the helical radius $R$; $R=6.35\pm 0.10$ mm in all cases. The filament
radius was $a/R=0.111$ for all helices.

####  II.3.2  Axial torque measurements

To measure the dependence of torque on boundary distance, we secured the tank
onto a horizontal stage that allowed for motion in the $x$-direction, as shown
in Fig. 5. The motion of the stage was controlled by a linear guide with a
worm gear screw that advanced the stage 0.3 mm per revolution. The screw was
turned using a computer-controlled NEMA 23 stepper motor with a resolution of
400 steps/rev. This gave better than 100 $\mu$m precision in controlling the
boundary distance, which was necessary: the step size near the boundary was as
small as 0.5 mm. Torque measurements were made for both cylinders and helices
using similar methods. The objects were held in a rigid shaft adapter and then
lowered until centered in the tank using a vertical translation stage built
from 80-20® extruded aluminum. At the beginning of each data set, we first
adjusted the vertical tilt of the object until it was parallel to the
boundary. Next, we manually adjusted the horizontal stage so that the cylinder
or helix touched the front vertical boundary of the tank. We used total
internal reflection to form an image of the object that could be used as a
reference to find where the edge of the object just made contact with the
boundary, which occurs when the image appears to touch the object. The torque
was measured using a FUTEK TFF400, 10 in-oz, Reaction Torque Sensor. The
cylinder and helices were driven by a variable speed DC motor with a magnetic
encoder (Pololu 298:1 Micro Metal Gear Motor with Magnetic Encoder) and housed
inside of a 3D-printed enclosure that included sleeve bearings to minimize
frictional torque. The power and signal wires were fed through a 6.32 mm
opening at the center of the torque sensor. The wires were then fixed to the
outside structure so that they did not create a torque when measurements were
taken. The encoder output was read by the counter input on a National
Instruments USB6211 M series multifunction DAQ. The torque signal was
amplified using an amplifier/driver (Omega DP25B-E-A 1/8 DIN Process Meter and
Controller) and its output fed into the same National Instruments data
acquisition board’s analog to digital input with a resolution of 250 thousand
samples per second, which is much faster than any time scales in the
experiment. Data were taken with the DC motor rotating at varying speeds and
with the objects located at a distance from the boundary set by the horizontal
stage. The torque and motor frequency were simultaneously recorded using
MATLAB to acquire and plot them. We used MATLAB and a motor controller
(ARDUINO MEGA 2560 with an ADAFRUIT Motor Shield v.2) to control the motor.
However, the motor rotation varied depending on the axial load, so we divided
the signal from the torque sensor by the frequency data from the counter input
to get the torque per unit frequency at each boundary distance, see Fig. 6. A
MATLAB data acquisition GUI included the temperature and distance values,
ensuring that the acquisition parameters were stored with the raw data. Data
were taken for approximately 60 rotation periods for both CW and CCW rotation
at each boundary location. The frequency signal occasionally showed large
spikes that affected the average torque-per-frequency value because the torque
signal did not show a corresponding jump. We considered this to be the result
of the encoder miscounting the rotation rate or the counter input in the DAQ
misreading the signal from the encoder. We used MATLAB’s outliers function to
remove such frequency spikes that were more than nine median absolute
deviations from the median calculated in a moving window ten data points wide,
and replaced them with the average of the adjoining data points. The number of
outliers was less than 1% of the data points, so this frequency smoothing
should not have biased the averaging significantly. The difference between
mean CW and CCW rotation values, which should have been the same, was used to
establish the uncertainty in the experimental measurements. An analysis script
read the geometric parameters and data files for a given set of measurements
(cylinder or helix) and plotted the data versus boundary distance. We scaled
the torque using a unit of $[\mu\Omega r^{2}\ell]$ (cylinder) or $[\mu\Omega
R^{2}L]$ (helix), where $\mu$ is the fluid viscosity, $\Omega$ is the angular
speed, $r$ is cylindrical radius, $\ell$ is the cylindrical length, $R$ is the
helical radius, and $L$ is the helical axial length. Plots of the
dimensionless torque for cylinders and helices are shown in Fig. 7 and Fig. 9.
Using these units for the torque allowed for easy comparison between
experiments, numerical simulations, and theory.

Figure 5: Experimental setup showing the tank, translation stage, torque
sensor and a cylinder positioned for measurement. The motor and magnetic
encoder were housed inside a 3D-printed structure that was mounted to the
active side of the torque sensor. Signal wires were run through the center of
the torque sensor for motor control and data acquisition. Figure 6: Example of
data signals from the experimental torque measurements. The frequency and
torque data were read by the DAQ and the torque per unit frequency was
calculated in real time and was smoothed to remove outliers as described in
the text.

###  II.4 Summary of algorithms and data analysis

Two separate sets of simulations are presented in this paper. For those with a
helix model or a bacterium model, the results were averaged over 16 evenly
spaced phases as described in Eq. 1 of the flagellar centerline. (i) The goal
of the first set of simulations was to calibrate the MRS and MIRS methods by
finding the optimal factors ($\gamma_{c}$ for a cylindrical cell body and
$\gamma_{f}$ for a helical flagellum) and the optimal regularization
parameters ($\epsilon_{c}$ and $\epsilon_{f}$), as reported in Table 1. Eq. 3
was used to solve for the force $\mathbf{f}_{k}$ at each discretized point
$\mathbf{x}_{k}$ in a free space, whereas Eq. 4 was used for simulations near
a plane wall. The resulting net torque of each rotating structure was then
compared with the results from theory for a cylinder or from experiments for a
helix, as described in Sec. III.1. (ii) The goal of the second set of
simulations was to assess the motility performance of the force-free and
torque-free bacterium models with boundary effects incorporated. Step 1: Eq. 5
was used with $S_{\epsilon}$ (for simulations in a free space) or with
$S^{*}_{\epsilon}$ (for simulations with a plane wall). Different combinations
of the cell body size, flagellar wavelength, and distance to the wall were
simulated. We used five values for the length $\ell$ and five values for the
radius $r$ shown in Table 1. These values are within the range of normal E.
coli [20]. We used $18$ wavelengths $\lambda$ that cover a range of biological
values ($2.22\pm 0.2$ $\mu m$) and values that are shorter and longer than the
biological values (Table 1 and Fig. 2). The set of geometric parameters,
together with $22$ distance values $d$ measured from the flagellar axis of
symmetry to the wall, resulted in $9,900$ simulations. From each simulation,
we obtained the axial component of the translational velocity $U$, the
magnitude of the axial-component of the hydrodynamic drag on the cell body
$F$, and the magnitude of the axial-component of the hydrodynamics torque on
the cell body $\tau$. For each body geometry (450 total), we performed a
simulation in free-space to ensure the convergence of MIRS calculations to MRS
calculations as the distance $d\rightarrow\infty$. Step 2: The torque value
$\tau$ was output from each simulation in Step 1 with the motor frequency set
to $154\,\text{Hz}$. That torque-frequency pair was then used to determine the
load line and its intersection with the torque-speed, as discussed in Sec.
II.2 and shown in Fig. 3. Each motor frequency $\Omega_{m}/2\pi$ on the
torque-speed curve was given as some multiple $q$ of $154\,\text{Hz}$. The
simulation outputs were scaled by $q$, since they were all linear with motor
frequency; i.e., $\left(U,F,\tau\right)\rightarrow q\left(U,F,\tau\right)$.
These scaled quantities were then used calculate the performance measures.
Results are presented in Sec. III.2 and Sec. III.3.

##  III Results

###  III.1 Verifying the numerical model and determining the optimal
regularization parameters

When using MRS or MIRS, the choice of the regularization parameter for a given
discretization (cylinder) or filament radius (helix) of the immersed structure
has generally been made without precise connection to real-world experiments,
because there are large uncertainties in biological and other small-scale
measurements. We therefore used theory, as described below, and dynamically
similar experiments, as described in Sec. II.3, to determine the optimal
regularization parameters for the two geometries used in our bacterial model:
a cylinder and a helix.

####  III.1.1 Finding the optimal regularization parameter for a rotating
cylinder

Jeffrey and Onishi (1981) derived a theory for the torque per length on an
infinite cylinder rotating near an infinite plane wall [28] that was used
previously to calibrate numerical simulations of helical flagella [23]. The
torque per unit length $\sigma$ on an infinite cylinder is given as

$\sigma=4\pi\mu\Omega r^{2}\frac{d}{(d^{2}-r^{2})^{1/2}}$ (7)

where $\mu$ is the dynamic viscosity of the fluid, $\Omega$ is the angular
rotation speed, $r$ is the cylindrical radius, and $d$ is the distance from
the axis of symmetry to the plane wall. We used this theoretical value as a
common reference point between the experiments and simulations to establish
optimal computational parameters, but note that this theory has not been
experimentally tested outside of the present work. We assumed Eq. 7 is valid
for our experiments and simulations, though this assumption as applied to
experiments ignored the finite size of the tank. To control for end effects in
the experiments, we measured the torque with only the first 3 cm inserted into
the fluid and with the full cylinder inserted at the same boundary locations.
We subtracted the torque found for the short section from the torque found for
the full insertion of the cylinder. In simulations, we controlled for finite-
length effects by measuring the torque on a middle subsection of the simulated
cylinder, as discussed below. Our experimental data are shown in Fig. 7, with
the torque made dimensionless using the quantity $\mu\Omega r^{2}\ell$, where
$\mu$ is the fluid viscosity, $\Omega$ is the rotation rate, $r$ is the
cylindrical radius, and $\ell$ is the cylindrical length. The mean squared
error (MSE) between experiments and theory is MSE $\leq 6\%$ when calculated
for the boundary distances where $d/r>1.1$ (i.e. the distance from the
boundary to the edge of the flagellum is $\geq 1$ mm). The theory
asymptotically approaches infinity as the boundary distance approaches
$d/r=1$, which skewed the MSE unrealistically. For the data where $d/r\geq 2$,
the mean squared error is less than 1%. In numerical simulations of the
cylinder, the computed torque value depended on both the discretization and
regularization parameter. Having found good correspondence with the
experiments, we used Eq. (7) to find an optimal regularization parameter for a
given discretization of the cylinder (see Table 1: cylinder part). The
discretization size of the cylindrical model $ds_{c}$ was varied among $0.192$
$\mu$m, $0.144$ $\mu$m, and $0.096$ $\mu$m. For each $ds_{c}$, an optimal
discretization factor $\gamma_{c}$ was found by minimizing the MSE between the
numerical simulations and the theoretical values using the computed torque in
the middle two-thirds of the cylinder to avoid end effects. The optimal factor
was found to be $\gamma_{c}=$ 6.4 for all the discretization sizes. We used
the finest discretization size for our model bacterium as reported in Table 1
since it returned the smallest MSE value of 0.36%.

Figure 7: Dimensionless torque, $\tau/(\mu\Omega r^{2}\ell)$, for a cylinder
versus scaled boundary distance ($d/r$), where $\mu$ is the dynamic viscosity,
and $\ell$ is the length of the cylinder. The boundary distance is scaled by
the cylindrical radius $r$ as measured to the centerline of the cylinder:
theory by Jeffrey and Onishi [28] (solid black line), optimized MIRS
simulations (solid red curve) and dynamically similar experiments (solid blue
circles). The numerical simulations were optimized by adjusting the
discretization factor $\gamma_{c}$ to minimize the MSE between theory and
simulation (the minimum MSE is 0.36%). The MSE between experiments and theory
was large near the boundary because the theory goes to infinity at $d/r=1$.
Outside of the near-boundary region ($d/r\geq 2$) the MSE is less than 1%.

####  III.1.2 Finding the optimal regularization parameter for a rotating
helix far from a boundary

Simulated helical torque values also depend on the discretization and
regularization parameter, but there is no theory for a helix to provide a
reference. Other researchers have determined the regularization parameter
using complementary numerical simulations, but the reference simulations also
have free parameters that may have affected their results [24]. Thus, we used
dynamically similar experiments, as described in Sec. II.3, to determine the
optimal filament factor, $\gamma_{f}=2.139$, for a helix filament radius
$a/R=0.111$. Torque were measured for the six helical wavelengths given in
Table 2 when the helix was far from the boundary. The optimal filament factor
$\gamma_{f}=2.139$ was found by the following steps (i) varying $\epsilon_{f}$
for each helix until the percent difference between the experiment and
simulation was under 5%; and (ii) averaging the $\epsilon_{f}$ values found in
Step (i). In these simulations, the regularization parameter and
discretization size are both equal to $\gamma_{f}a$. The results are shown in
Fig. 8, with the torque values non-dimensionalized by the value $\mu\Omega
R^{2}L,$ where $\mu$ is the fluid viscosity, $\Omega$ is the rotation rate,
$R$ is the helical radius, and $L$ is the axial length. The optimized
simulations returned an average percent difference of $2.4\pm 1.7$% compared
to the experimental values.

Figure 8: Dimensionless torque ($\tau/(\mu\Omega R^{2}L)$ for different
flagellar wavelengths ($\lambda/R$), where $\mu$ is the dynamic viscosity,
$\Omega$ is the angular speed, $R$ is the helical radius and $\lambda$ is the
helical wavelength. Experimental values are solid black circles and solid blue
circles; our MRS simulations with a centerline distribution of regularized
Stokeslets are the solid red and solid green triangles, and MRS simulations
computed with a surface discretization of the helices using the code provided
by Rodenborn et al. (2013) [25] are the blue and black curves.

We checked whether helices with different filament radii could be accurately
simulated using our optimized $\gamma_{f}$ to scale the regularization
parameter, i.e. ($\epsilon_{f}=\gamma_{f}a$) to account for relative size of
the filament, as is commonly done [29, 30, 31, 32, 33]. We computed torque
values that matched the experimental values given in Rodenborn et al. (2013),
which used a filament radius $a/R=0.063$. The results are also presented in
Fig. 8. The percent difference between our MRS simulations and their data is
2.5$\pm$1.3%. Martindale et al. (2016) [24] used an MRS with a surface
discretization of the flagellum to calibrate their simulation parameters,
whereas our MRS used a string of regularized Stokeslets along the helical
centerline to reduce the computational cost in the MIRS calculations. As a
final test, we used the freely available and calibrated code for the MRS with
surface discretization from Rodenborn et al. to compute torque values for our
$a/R=0.111$ data and for their $a/R=0.063$ data. Fig. 8 shows the torque
comparison of their surface discretized MRS (solid curves), our centerline
distribution MRS (triangles), and the experiments (circles). The percent
difference between their MRS and the experiments is $3.6\pm 3.4$%. The percent
difference between our MRS and theirs is 1.8$\pm$3.7%. Thus, within the range
we tested our MRS with a centerline distribution, the optimal filament factor
$\gamma_{f}$ worked very well for another filament radius and other helical
wavelengths when compared to both the experiments and the surfaced discretized
MRS in Rodenborn et al. (2013) [25] for torques far from the boundary.

####  III.1.3 Torque on rotating helices near a boundary

To determine how boundaries affect bacterial motility, we used our optimized
value for $\gamma_{f}$ in our MIRS simulations to compute the torque as a
function of boundary distance, as shown in Fig. 9. The computed torque values
and measured torque values also show excellent agreement at most boundary
distances, except for the shortest wavelength $\lambda/R=2.26$. We note that
this helix had the largest variation in wavelength, as reported in Table 2.
Furthermore, the torque for short wavelengths is more sensitive to variation
in wavelength as compared to variation at longer wavelengths, which likely
explains the difference between simulation and experiment for this geometry,
whereas for the other wavelengths the simulated values are generally within
the uncertainty in the experiments for all boundary distances.

Figure 9: Dimensionless torque $\tau$ for different helical wavelengths
($\lambda/R$) versus boundary distance ($d/R$) scaled by the helical radius
$R$. The optimized MIRS simulations are the solid curves and the experimental
values are solid circles with vertical error bars. The data also show good
agreement for the far from boundary value at $d/R\approx 20$ (see Fig. 8). The
data show that once the far from boundary distance was properly calibrated,
the MIRS worked very well to represent the effects of the boundary.

###  III.2 Speed Measurements to Assess Performance

The motion of bacteria through their environment enables them to find
nutrients. Indeed, it has been suggested that the purpose of bacterial
motility is primarily to perform chemotaxis [3]. Living in a microscopic
environment where thermal effects are significant, bacteria must be able to
sample chemical concentrations faster than diffusion causes those
concentrations to change [3, 11], so moving faster may confer a survival
advantage. The low speed operating regime of the bacterial motor (below 175
Hz) is thermodynamically more efficient than the high speed regime. A simple
model gives the fraction of energy lost to friction in the motor as
$(\tau_{0}-\tau)/\tau_{0}$, where $\tau_{0}$ is the stall torque and $\tau$ is
the operating torque at a given frequency [13]. In the low speed regime
$\tau\geq 0.92\tau_{0}$, so that the power output of the motor is greater than
92% of the power input. However, the low speed regime may be less
operationally reliable for motility; the flatness of the torque-speed curve
implies that small increases in load correspond to large decreases in motor
rotation rate, so the bacterium risks stalling and may be unable to restart
its motor. Using our simulations, we determined the swimming speed and motor
rotation rate for different bacterial geometries at different distances to a
solid boundary and assessed. the performance of bacterial geometries typically
associated with swimming.

####  III.2.1 Optimal flagellar wavelength

We first consider the effect of different flagellar wavelengths on swimming
speed and motor rotation rate, as shown in Figs. 10a and b. Swimming speed and
motor rotation rate are shown as heat maps for different flagellar wavelengths
at different distances to the boundary. The heat map shows the median values
computed among all 25 bacterial body geometries we investigated (Table 1). The
maximum of all the median swimming speed values is about
$26\,\mu\text{m}s^{-1}$, and it occurs far from the boundary for a wavelength
near $8R$. For long and short flagellar wavelengths, the swimming speed at all
distances is much lower than the maximum. Long wavelengths yield about
$10\,\mu\text{m}s^{-1}$, whereas very short wavelengths give values closer to
$1\,\mu\text{m}s^{-1}$. For the flagellar wavelength of $\lambda/R=11.1$ that
is typical for E. coli, the swimming speed is about $25\,\mu\text{m}s^{-1}$
far from the boundary, whereas it drops to about $20\,\mu\text{m}s^{-1}$ very
near the boundary. Interestingly, the flagellar wavelengths that correspond to
swimming speeds near the maximum in Fig. 10a also correspond to motor rotation
rates in the low end of the high speed regime in the torque-speed curve, so
that the motion is both thermodynamically efficient and operationally
reliable. A wavelength of $\lambda/R=8$ gives 190 Hz and 183 Hz far from and
near to the wall, respectively, which correspond to mechanical energy outputs
of about 84% and 88%. Short and long wavelengths result in a weaker
performance, but for different reasons: short wavelengths operate in the low
speed regime and thus are efficient but unreliable, whereas longer wavelengths
operate farther into the high speed region and thus are reliable but
inefficient.

Figure 10: Swimming speed and motor frequency for different flagellar
wavelengths at different boundary distances. Panels a and b show heat maps of
free swimming speed $U$ and motor frequency $\Omega_{m}/2\pi$ with axes
flagellar wavelength ($\lambda/R$) versus boundary distance ($d/R$), where $R$
is the helical radius. Typical E. coli wavelengths are indicated with the
dashed white lines, which shows this range is near to the peak in swimming
speed. Panels c-e show line plots of speed and motor frequency across
different body sizes far from ($d/R=51.0$) and near ($d/R=2.74$) the boundary.
The solid circles are the simulation data points and the solid curves are
spline fits to the data. The three curves show the maximum (black), the median
(red) and the minimum (blue) among all cell bodies simulated. Panels c and e
show that the peak swimming speed $\lambda/R\approx 8$, which is close to the
range of E. coli wavelengths and the peak has a long “tail” as wavelength
increases. Panels d and f show increasing motor frequency with increasing
wavelength. The trend reflects the plot of the torque-speed curve in Fig. 3.

####  III.2.2 Boundary effects

To illustrate how proximity to the boundary affects swimming speed and motor
rotation rate, we show line plots in Figs. 10c-f of the speed and rotation
rate as functions of the flagellar wavelength both far from and near to the
boundary. The maximum, median, and minimum values among all bacterial body
geometries are shown for each boundary distance. Comparing Figs.10c and e
shows that proximity to the boundary does not appreciably alter the optimal
wavelength: it remains near $8R$ for all body geometries both near and far
from the boundary. However, proximity to the boundary does increase the
difference in the swimming speed among different bodies at a given wavelength.
Far from the boundary, the difference between the maximum and minimum swimming
speeds for the optimal flagellar wavelength is 14% of the maximum value of
$28\,\mu\text{m}s^{-1}$; near the boundary, the difference is 34% of the
maximum value of $26\,\mu\text{m}s^{-1}$. Figs. 10d and f show the motor
rotation rate is less sensitive to the body geometry and proximity to the
surface than the swimming speed. Far from the boundary, the difference between
the maximum and minimum rotation rates for the optimal flagellar wavelength
$8R$ is 6% of the maximum value $198\,\text{Hz}$; near the boundary, the
difference is 6% of the maximum value of $190\,\text{Hz}$. To further probe
the effect of the cell body geometry on swimming speed and motor rotation
rate, we show heat maps of the speed and rotation rate fixed at the typical E.
coli wavelength $\lambda/R=11.1$ as functions of the length and radius of the
cylindrical cell body. Figs. 11a-d show the results. The translational speed
is optimized for short thin cell bodies (lower left-hand corner of Figs. 11a
and c both near and far from the surface. Conversely, the slowest motor
rotation rates (though all higher than 175 Hz), and therefore the most
thermodynamically efficient, occur for long thick cell bodies (upper right-
hand corners of Figs. 11b and d. Taken together, these two results suggest
that balancing the need of a bacterium to move quickly with its need to be
thermodynamically efficient would yield a cell body geometry somewhere between
long, thick cell bodies and short, thin cell bodies. Interestingly, the center
point of the heat maps shown in Fig. 11 corresponds to the mean size of the E.
coli cell body.

Figure 11: Free-swimming speed $U$ and motor frequency $\Omega_{m}/2\pi$ shown
as heat maps with axes cylindrical radius ($r/R$) versus body length
($\ell/R$). The data are for a fixed flagellum wavelength, $\lambda/R=11.1$,
where $R$ is the helical radius. The top row a and b is far from the boundary
$d/R=51.0$ data where boundary effects are minimal, and the bottom row c and d
are data close to the boundary $d/R=2.74$. The swimming speed data in a and c
show that short thin bodies result in higher swimming speed both near and far
from the boundary, though near the boundary the swimming speed is lower for a
given body geometry. Therefore the swimming speed measure predicts short thin
bodies far from the surface result in a better motility performance. The motor
frequency data in b) and d) show long thick bodies result in a slower motor
frequency near and far from the boundary, though far from the boundary the
motor frequency is higher. Therefore, the motor frequency measure predicts
long thick bodies near the surface result in better motility performance.

###  III.3 Energy Cost Measures to Assess Performance

The energy cost required to move is another way to assess the performance of
the bacterial motility system. Here we present simulation results of three
different energy cost measures. The first measure we consider is what we term
the Purcell inefficiency $\mathcal{E}_{Purcell}^{-1}$ given by,

$\mathcal{E}_{Purcell}^{-1}=\frac{\tau\Omega_{m}}{FU},$ (8)

where $\tau$ is the motor torque (or the torque on the cell body or the
flagellum), $\Omega_{m}$ is the motor rotation rate, $F$ is the drag force on
the cell body (or on the flagellum), and $U$ is the swimming speed of the
bacterium. Thus, the Purcell inefficiency measures the mechanical energy
($T\Omega_{m}$ ) required to swim at speed $U$ relative to the least amount of
energy ($FU$) needed to translate the cell body at speed $U$. The Purcell
inefficiency is useful because, under certain simplifying assumptions [34], it
can be expressed as a function of the geometry of the cell body and the
flagellum alone. The difficulty with this measures is that it does not depend
on the rotation rate of the motor because all four quantities appearing in Eq.
8 scale with the motor frequency (see Eq. 5). Therefore, the Purcell
inefficiency cannot assess how swimming performance depends on the torque-
speed characteristics of the motor, and thus omits an important element of the
bacterial motility system that is subject to selective forces. The second
measure is the energy cost $E$ to travel a unit distance $d$ given by

$\frac{E}{d}=\frac{\tau\Omega_{m}}{U}.$ (9)

Several authors [13, 16] have considered the distance traveled per energy
output by the motor, which is the inverse of the measure we consider here. The
merit of the energy cost per distance measure is that it expresses the amount
of energy used by the bacterium to perform a biologically relevant task;
namely, to swim one unit distance. Another advantage is that it depends on the
motor rotation rate and thus can probe the effect of the torque-speed
characteristics of the motor. However, it does not account for the size of the
bacterium, and thus does not measure the energy cost relative to the overall
metabolic budget of the organism. To account for the metabolic energy cost
required to swim a unit distance, we introduce a third measure,

$\frac{(E/m)}{d}=\frac{\tau\Omega_{m}}{mU}.$ (10)

The mass $m$ associated with each bacterial model is $m=1.1\times
10^{-15}\left(\pi r^{2}l\right)\,\text{kg}$, where $r$ is the body radius and
$\ell$ is the body length, both measured in $\mu\text{m}$. Though this energy
cost measure has not been considered in the literature, it was suggested
earlier by Purcell [3].

####  III.3.1 Optimal wavelength

We first consider the optimal flagellar wavelength predicted by the three
energy cost measures, as shown in Figs. 12. The top row a-c shows heat maps of
the three energy cost measures as functions of flagellar wavelength and
boundary distance, which correspond to the median values computed for all body
geometries listed in Table 1. All three measures give an optimal wavelength
near $\lambda/R=8$ (where each energy cost measure is minimal). However, the
three measures differ in other ways. The Purcell inefficiency predicts that
swimming near the boundary is less inefficient than swimming far from the
boundary, whereas the opposite is true for the energy per distance and
metabolic cost measures. At a wavelength of $8R$, the minimum Purcell
inefficiency value is about 84 (or $1/84=1.2\%$ if calculated as Purcell
efficiency), the minimum energy per distance measure is $5.0\times
10^{-11}\,\text{Jm}^{-1}$, and the minimum metabolic energy cost is $3.1\times
10^{4}\,\text{J}\text{m}^{-1}\text{kg}^{-1}$.

####  III.3.2 Boundary effects

To evaluate how proximity to the surface affects the predictions of the energy
cost measures, we show line plots in Figs. 12 of the measures as functions of
flagellar wavelength far from ($d/R=51$) in d-f and near the boundary
($d/R=2.74$) in g)-i). The maximum, median, and minimum values among all body
geometries are shown for each wavelength. The Purcell inefficiency is the
least sensitive of the measures to changes in the body size. For a wavelength
of $\lambda/R=8$, the difference between the maximum and the minimum is 8% of
the maximum (110 vs 101). Near the boundary, the difference increases to 13%
of the maximum value (94 vs 82). The energy per distance measure is more
sensitive to the body size, and the sensitivity increases near the boundary.
For a wavelength of $\lambda/R=8$, the difference between the maximum and
minimum values is 16% of the maximum value of $5.5\times
10^{-11}\,\text{J}\text{m}^{-1}$ far from the boundary. Near the boundary the
difference increases to 35% of the maximum value of $7.5\times
10^{-11}\,\text{J}\text{m}^{-1}$. The metabolic energy cost is the measure
most sensitive to the body size, though interestingly the sensitivity
decreases with proximity to the boundary. At a wavelength of $\lambda/R=8$,
the difference between the maximum and minimum value far from the boundary is
51% of the maximum value of $4.5\times
10^{4}\,\text{J}\text{m}^{-1}\text{kg}^{-1}$. Near the boundary the difference
decreases to 38% of the maximum value of $5.0\times
10^{4}\,\text{J}\text{m}^{-1}\text{kg}^{-1}$.

Figure 12: Energy cost as a function of wavelength and boundary distance. The
top row shows three energy cost measures as a function of helical wavelength
$\lambda/R$ and boundary distance $d/R$, where $R$ is the helical radius.
Typical E. coli wavelengths are indicated with the dashed white lines whose
range is close to the optimal wavelength predicted by these energy cost
measures. The second and third rows show line plots at distances far from
($d/R=51.0$) and near ($d/R=2.74$) the boundary to assess the wavelength
dependence of each measure at those distances. The solid circles are numerical
simulations and the solid curves are spline fits to the numerical data. The
three curves show the maximum (black), the median (red) and the minimum (blue)
among all cell bodies simulated. All these plots have the optimal flagellar
wavelength $\lambda/R\approx 8$.

Finally, we consider how the energy cost measures depend on body radius and
body length at different distances to the boundary. In Fig. 13 we show heat
maps of the three energy cost measures fixed at the typical E. coli wavelength
$\lambda/R=11.1$, as functions of the radius and length. The Purcell
inefficiency shown in Figs. 13 gives different optimal body geometries near
and far from the boundary: far from the boundary short, thick cylinders (top
left corner of Fig. 13a) are the least inefficient; near the boundary short
thin cylinders (bottom left corner of Fig. 13d) are the least inefficient. The
energy per distance measure gives the same optimal body far from and near to
the boundary: the lowest energy per distance cost measure is given by short,
thin cylinders (bottom left corners of Figs. 13b and e. The metabolic cost
measure gives the same optimal body near and far from the surface, though it
is opposite of the optimal body predicted by the energy per distance measure:
the lowest metabolic cost measure occurs for cylinders that are long and thick
(top right corners of Figs. 13c and f.

Figure 13: Comparison of Purcell inefficiency, energy per distance, and
metabolic energy cost with respect to body geometry at the typical wavelength
of E. coli ($\lambda/R=11.1$). The top row shows results far from the boundary
($d/R=51.0$) and the bottom row shows results near the boundary ($d/R=2.74$).
In panels a and d, the Purcell inefficiency shows that short thick bodies are
most efficient (i.e., least inefficient) far from the boundary but short thin
bodies are most efficient near the boundary. In panels b and e, short thin
bodies require the least energy cost per distance both far from and near the
boundary. In panels c and f, long thick bodies require the least metabolic
energy cost per distance traveled both far from and near the boundary

##  IV Discussion

In this work we used the method of images for regularized Stokeslets (MIRS) to
simulate a motile flagellate bacterium moving near a solid boundary. We
determined the regularization parameter in the method by conducting
dynamically similar macroscopic experiments with rotating cylinders and
rotating helices near a solid boundary and comparing the results to equivalent
simulations. By varying the regularization parameters, we were able to find
optimal values that matched the experimental results within 5%. Having
calibrated MIRS, we simulated various bacterial morphologies to assess their
swimming performance. We assessed swimming performance using multiple
measures: swimming speed, motor rotation rate, the Purcell inefficiency,
energy cost per distance, and metabolic energy cost per distance. As a
important and novel addition to our simulations, we incorporated the
experimentally measured torque-speed response curve [17] by ensuring that the
torque and motor rotation rate matched a point on the curve in all our
calculated measures. Using our MIRS calibration method, we found that the
optimal discretization factor for a cylinder is $\gamma_{c}=6.4$ for the
surface discretizations we used, which may be used as a reference value for
other researchers who simulate rotating cylinders using MRS or MIRS. We also
found an optimal filament factor $\gamma_{f}=2.139$ when using MRS and MIRS
with each helix modeled as a string of regularized Stokeslets along the helix
centerline. Selecting an appropriate regularization parameter for a center-
line discretization of helices MIRS has been considered by other researchers.
Martindale et al. (2016) [24] benchmarked their center-line discretization of
a helix with a surface discretization model. They reported that the optimal
filament factor should be in the range $1\leq\gamma_{f}\leq 3$ to keep the
percent difference less than about 10% in their simulations, which is
consistent with our results. In our work, we calibrated simulations that used
a centerline discretization of helices by fitting the regularization parameter
directly with experimentally measured values of torque. These MIRS
computations showed excellent agreement with the experimental torque values at
most boundary distances (Fig. 9). In MRS/MIRS, using a centerline distribution
for a model helix (or flagellum) with a calibrated regularization parameter is
more useful than a surface discretization for several reasons: (i) the
computational cost is significantly reduced because the matrix system for the
centerline distribution is much smaller than for a surface discretization;
(ii) simulations of very short helical wavelengths using a centerline
distribution do not encounter discretization issues such as overlapping cross-
sections; (iii) in a centerline distribution, the point connecting the
cylindrical cell body and the tapered helical flagellum can be considered as
the motor location, whereas the motor location in a surface discretization is
hard to define because of the small gap between the cell body and the
flagellum needed to allow counter-rotation between the cell body and the
flagellum. Interestingly, all five performance measures we computed with our
calibrated model – swimming speed, motor speed, Purcell inefficiency, energy
per distance, and metabolic energy cost – predict an optimal flagellar
wavelength of $\lambda/R\approx 8$, where $R$ is the helical radius of the
flagellum. This result agrees with the work of Zhang et al. (2014) [9] who
studied the Purcell efficiency of a rotating helix, whereas our model includes
a cell body with rotation and translation. Furthermore, this prediction occurs
both near and far from the surface and for all body geometries, which suggests
that the bacterial wavelengths may be selected independently of body shape or
surface proximity. Therefore, none of the five measures can be distinguished
by their predictions of flagellar wavelength, but they are distinguished by
their predictions for the optimal body size. Further analysis showed that the
swimming speed is optimal (i.e. fastest) for bodies that are short and thin,
both near and far from the surface. The structure of the torque-speed curve
imposes two competing conditions that need to be balanced to achieve
optimality: at low speeds the torque-speed curve is flat and therefore
thermodynamically efficient, but in that regime small increases in applied
load result in large decreases in motor rotation rate that could cause the
motor to stall. We therefore suggest that the optimal speed is higher than the
175 Hz knee speed (see Fig. 3) so that the motor operates in the reliable
regime, but not much higher so that it remains thermodynamically efficient.
The lowest motor speed that is still above the knee speed for typical
bacterial wavelengths occurs for long and thick bacterial bodies, both near
and far from the surface. It is tempting to suggest that balancing the short,
thin bodies needed for optimal speed and long, thick bodies needed for optimal
motor operation yields the average body size. However, we do not infer too
much from this result because we do not have a principled way of performing
the balancing needed to draw a definitive conclusion. The three energy cost
measures also make different predictions about body shape. The Purcell
inefficiency is relatively insensitive to differences in body shape,
especially far from the wall. However, based on the small differences
($\approx 8\%$), the optimal body far from the boundary is short and thick,
whereas the optimal body near the wall is short and thin. The Purcell
inefficiency is the only quantity that makes different predictions about the
optimal body near and far from the boundary. The Purcell inefficiency also
predicts that bacterial motility systems become generally more efficient near
the boundary, which would suggest a natural benefit for all bacteria to move
near boundaries that are independent of any other biologically relevant
activities. Unlike the Purcell inefficiency, the energy cost per distance
traveled and the energy cost per body mass per distance traveled (metabolic
energy cost) both predict larger energy costs for moving near a surface.
However, they make opposite predictions about the optimal body size. The
energy cost per distance suggests short and thin cell bodies are most
efficient, and the metabolic energy cost suggests long and thick cell bodies
are most efficient. Though increasing body size results in a greater energy
cost for moving a given distance, the increase in body size results in a
smaller relative energy expenditure. The energy per distance predicts the same
optimal body as predicted by the fastest swimming speed, and metabolic energy
cost predicts the same optimal body as predicted by motor rotation rate. Only
the Purcell efficiency predicts a short, thick body is optimal, and this
occurs only far from the boundary. Although the Purcell efficiency has been a
popular quantity of analysis, we believe it has several important shortcomings
that warrant discussion, at least one of which was anticipated by Purcell.
First, the Purcell efficiency is dependent only on the geometry of the body
and flagellum and not on the motor’s torque-speed response characteristics.
From a physical standpoint, it is interesting to find such an invariant
quantity, but from a biological standpoint, it does not assess the bacterial
motility system’s thermodynamic efficiency because it ignores motor mechanics.
Second, the Purcell efficiency is defined to be the ratio between the minimum
power required to translate the cell body and the power actually dissipated
during the bacterial motion. In our simulations, we find the maximal
efficiency is in the range of 1-2%, similar to what others have found [3, 12,
9]. These two quantities (the minimum power vs the actual power) are clearly
of very different orders, which suggests that least power needed may not be an
appropriate reference quantity.To give a biophysical interpretation to the
least power needed to translate the cell body, some authors have suggested
that it represents the “useful” portion of the power dissipated during motion,
[12, 8] but we believe this is a misconception. The bacterium is non-inertial;
therefore, the force acting on the cell body by the fluid is exactly balanced
by the force acting on the flagellum by the fluid (assuming no net body
forces). Both the bacterial body and the flagellum have the same axial
velocity (in a rigid model); therefore the power dissipated due to the axial
fluid drag on the body is exactly compensated by the power input by the axial
fluid force exerted on the flagellum. Finally, as Purcell noted in 1977, the
efficiency of the bacterial motility system is probably best characterized by
the energy consumption relative to the overall metabolic budget of the
organism [3]. This suggestion led us to consider the metabolic energy cost
introduced in this paper. The actual amount of that metabolic budget used for
motility is a small fraction, which led Purcell [3] to suggest that bacterial
motility is not really subject to strong selective forces toward optimal
efficiency. Our data do not say whether evolutionary processes tend to
minimize the energy cost of bacterial motility, but a plausible
counterargument is that the bacterium needs to consume most of its energy for
other biological functions and has only a small fraction available for
motility. Thus, small absolute changes in energy consumption correspond to
large relative changes in the energy available for motility, resulting in a
significant selective pressure to make the motility system as efficient as
possible. Many research questions about how physical interactions between
bacteria and their environment result in selective pressures in evolutionary
processes remain open, despite significant progress in the field. Modern
computational simulations and methods such as MRS and MIRS will remain
important tools for quantifying microscopic bacterial motion with precision.
In this work, we presented for the first time in the literature a procedure
for calibrating MRS and MIRS using macroscopic dynamically similar
experiments. Calibrating models in this way helps to ensure simulations give
accurate quantitative results. In future work, we will extend the macroscopic
experimental system to consider a wider variety of possible geometries
relevant to bacterial motility and make comparisons with biological
measurements. Funding: This research was partially funded by NSF DMS-1720323
to H.N. and N.C., and NSF MRI-1531594 to H.N. We thank Trinity University for
the Summer Research Grant to O.S. and the provision of computational
resources. We would also like to thank the Faculty Development Fund at Centre
College for research support to B.R.

###### Acknowledgements.

We wish to thank undergraduate students Asha Ari, Alexandra Boardman, Tanner
May and Mackenzie Conkling and Prof. Philip Lockett for their assistance in
collecting experimental data at Centre College. We also acknowledge the
contributions of Mica Jarocki and David Clark at Trinity University to the
initial implementation of the model bacterium. We would also like to thank
Deon Lee for her support by editing the manuscript.

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