Abstract Nanomotors are molecular configurations on the nanoscale which resemble and perform similar tasks as simple Newtonian motors, but are driven and analyzed under principles of quantum mechanics. Nanomotors have many practical applications in novel medical technology and smart materials. One such class of nanomotors, F1F0 ATP Synthase, has particular potential within medical applications. The F0 component of the nanomotor operates on potential differences caused by the proton gradient in between a rotator and stator. Torque-energy efficiency and rotational geometries of such a system may be modeled using Ordinary Differential Equations (ODEs) and the route of protons within the motor can be described by the Langevin Equation of the overdamped diffusion of the rotor. Numerical Analysis computational methods may be used to solve the Langevin equation of proton diffusion within the motor coupled with the Fermi Distributions of the Proton Reservoirs to output the Net Torque of a F0 Motor and delineate the proton gradient over time present within the system. Nine differential equations were directly involved in the solution. Previously modeled nanomotors have been highly torque inefficient and limited to one direction of motion; augmenting stators introduces omnidirectionality and significantly higher torque. Following this, environmental parameters may be altered to ascertain the optimal conditions for the operation of the nanomotors. This is done for the two, three, and four stator cases. The result is a data model which may be used to guide engineering efforts within the field providing definite parameters which to yield distinct torque and direction.

1. A new methodology for a
dramatic increase in efficiency of
nanomotors of varying stators and
system parameters
Michael Shumikhin Bronx High School of Science

2. Abstract
Nanomotors are molecular configurations on the nanoscale which resemble and perform similar
tasks as simple Newtonian motors, but are driven and analyzed under principles of quantum
mechanics. Nanomotors have many practical applications in novel medical technology and smart
materials. One such class of nanomotors, F1F0 ATP Synthase, has particular potential within
medical applications. The F0 component of the nanomotor operates on potential differences
caused by the proton gradient in between a rotator and stator. Torque-energy efficiency and
rotational geometries of such a system may be modeled using Ordinary Differential Equations
(ODEs) and the route of protons within the motor can be described by the Langevin Equation of
the overdamped diffusion of the rotor. Numerical Analysis computational methods may be used to
solve the Langevin equation of proton diffusion within the motor coupled with the Fermi
Distributions of the Proton Reservoirs to output the Net Torque of a F0 Motor and delineate the
proton gradient over time present within the system. Nine differential equations were directly
involved in the solution. Previously modeled nanomotors have been highly torque inefficient and
limited to one direction of motion; augmenting stators introduces omnidirectionality and
significantly higher torque. Following this, environmental parameters may be altered to ascertain
the optimal conditions for the operation of the nanomotors. This is done for the two, three, and
four stator cases. The result is a data model which may be used to guide engineering efforts
within the field providing definite parameters which to yield distinct torque and direction.

3. Introduction Methodology Results Discussion
Conclusion
Background
• Nanomotors are molecular configurations on the nanoscale
which resemble simple Newtonian motors, but are driven and
analyzed under principles of quantum mechanics.
• The F0 component of ATP Synthase is driven by a proton
gradient
• Proton driven nanomotors utilize stators to produce their
rotational motion.
• Stators are pairs of proton sinks and drains which facilitate a
proton gradient to drive rotation. The driving force of the
rotation of the nanomotor rotor is the exchange of protons to a
proton sink previous (clockwise on the rotor), and then a
proton drain (on the stator) exchanges this proton point higher
on the axle receives this proton; such exchanges drive an axel
around.
Introduction Methodology Results Discussion

4. Literature Review
• In 1977, Berg et. Al [7] found that proton exchanges drive bacterial flagellar locomotion.
• Cox et. al [9] observed and modeled a structure of F1F0 ATP synthase.
• Recent publications concerning F1F0 ATP synthase nanomotors ([10], [11], [12], [13]) have failed to
address an explanation for the relatively small torque yield of a single stator nanomotor
• Because of experimental difficulties in creating a lab-isolated nanomotor with let alone one stator,
most torque yields from the quantum process driving the motor have only been calculated
theoretically [14].
• My model is the first to overcome many of the limitations ascribed by previous researchers [19]
Introduction Methodology Results Discussion

5. Research Problem
• The research problem is to build a model which provides for accurate analysis and interpretation
for nanomotors of multiple stators of the F1-F0 ATP synthase nanomotors.
• The model which accounts for various environmental factors in the novel nanomotor system, such
as temperature and source-drain voltage.
Introduction Methodology Results Discussion

6. Significance
• One may develop synthetic flagellar transport mechanisms which may one day be used in
targeted drug delivery systems
• New smart materials can be made to twist, contract, and expand
Hypothesis
• By augmenting stators, the torque produced by the nanomotor will increase correspondingly with
stronger coulomb interactions. This may be modelled across variable environmental parameters.
Introduction Methodology Results Discussion

7. RESEACH HYPOTHESIS
• By augmenting stators, the torque produced by the nanomotor will increase
correspondingly with stronger coulomb interactions. This may be modelled across
variable environmental parameters.
Introduction Methodology Results Discussion

8. • Numerical Analysis computational methods may be used to solve the Langevin
equation of proton diffusion within the motor coupled with the Fermi Distributions of
the Proton Reservoirs to output the Net Torque of a F0 Motor and delineate the
proton gradient over time present within the system.
• My model for the single-stator nanomotor encompasses all of the random fluctuations
and discrepancies that exist within a multi stator system.
• Adding stators requires modifying the core system of equations to include more sites
and in doing so increases the complexity of the problem greatly.
• Many data on dynamic components of the nanomotor may be produced by altering
present environmental factors such as temperature and induced factors such as
source-drain voltage.
• The addition of another stator is represented with the introduction of another proton
source and drain opposite to the original stator
Introduction Methodology Results Discussion

9. Langevin Equation
𝜁𝑟 𝜙 = 𝜉 + 𝒯𝑒𝑥𝑡 −
𝜎
1 − 𝑛 𝜎
𝑑
𝑑𝜙
[𝑈 𝑞 𝜙 + 𝜙 𝜎 + 𝑈𝑐𝑜𝑛 𝜙 + 𝜙 𝜎 ]
Rate equation characterized by the Fermi distribution of the proton sources
𝑛 𝜎 +
𝛼
Γ𝛼𝜎 𝜙 𝑛 𝜎 =
𝛼
Γ𝛼𝜎 𝜙 𝑓𝜎 𝐸 𝜎
Introduction Methodology Results Discussion

10. 1. Derivatives of potential energy over phi for each site were taken (each stator position accounted
for), this was found via solving the composition of the derivative potential energy over site-charge
distances and the derivative of the site-charge distances.
2. The derivatives of the confinement potential (energy penalty included) for each rotator-stator strip
present about the stator were taken. These two derivatives are critical to ascertaining instantaneous
torque.
3. This torque is produced via calculating the summation of the total energy differential between
confinement potential and potential energy at every site within the torque-generating regime (within
the proton gradient potential).
4. Taking the average torque produced within .25 ms at each instance, and the respective
parameters at that instance, a multi-dimensional matrix is constructed, which allows one to visualize
the torque production.
Introduction Methodology Results Discussion

11. UML for one stator
Introduction Methodology Results Discussion

12. Introduction Methodology Results Discussion
At 310 Kelvin (human body temperature) and 200 meV (arbitrary), the nanomotor
produces nearly no effective torque ~1-2 pN nm(minus external torque), but
manages a CW rotation as most of the torque produced is of equal magnitude to
external torque, at higher temperatures, rotation is CCW.

13. Introduction Methodology Results Discussion
The torque produced by the two stator nanomotor is greater than the one
produced by the single stator nanomotor. At 310 Kelvin and 200 meV, effective
torque is ~51 pN nm, a 51x increase.

14. Introduction Methodology Results Discussion
Three stators in a triangular configuration produce significantly more torque than
the two stator case. At 310 Kelvin and 200 meV, effective torque is ~108 pN nm,
a 2.15x increase from two stators.

15. Introduction Methodology Results Discussion
The torque produced by the four stator nanomotor is greater than the one
produced by the three stator nanomotor. At 310 Kelvin and 200 meV, effective
torque is ~146 pN nm, a 1.35x increase from three stators.

16. Introduction Methodology Results Discussion
Results Summary
• The one stator case of the rotator is highly limited, effective torque caps at ~ 1-2
pN nm and doesn’t produce enough torque in operable conditions. The case does
not exhibit controllable omnidirectionality within the torque generating regime.
• The multi-stator cases of the rotator are extraordinary. Within the four stator case
effective torque reaches ~146 pN nm (at operable conditions), this is 146x torque
of the one-stator case.
• Omnidirectionality is inducible within a starting configuration where two pairs of
stators can operate at different times, or with different potentials to operate at
specific torque and rotational direction (a calculable energy and torque penalty
exist).

17. Introduction Methodology Results Discussion
Interpretation of results
• The data indicated a significant increase in produced torque within multi-
stator nanomotors. This is to the extent whereby previously found operable
torque was only around ~1-2 pN nm (Fig B).
• The torque produced by the two stator system was 51 times that (Fig C)
produced by the one stator case.
• The three stator case produced 108 times more torque than the one stator
nanomotor (Fig D)
• The four stator nanomotors produced an impressive 148 times the one stator
nanomotor (Fig E). Augmenting stators the F0 component of the ATP
Synthase nanomotor significantly increases the amount of torque
produced.
• The speed increase pertinent in augmenting stators is very notable, from a
frequency of rotation of 5 kHz within the one stator system, the frequency of
rotation increased to 9.5 kHz within the four stator system, increasing in
between each stator configuration.

18. Introduction Methodology Results Discussion
Discovery
• Augmenting stators within the F0 component of the ATP Synthase
nanomotor significantly increases the amount of torque produced.
• Different multi-stator configurations were modeled at variable parameters
Significance
• When compared to the torque produced by recent lab-produced synthetic
nanomotors, like the NEMS device built at the University of Texas at Austin
[22], the torque produced by the four-stator nanomotor is over a hundred
times greater, is omnidirectional, and rotates at a comparable speed to the
NEMS device.
• Distinguishing these motors however is the greater bio acceptability and
cheaper cost of production of the F0 motor of ATP Synthase.
• The multi-stator nanomotor may be potentially produced cheaply within bacteria
by altering genes to augment another stator. The latter essentially automates
the process of nanomotor production and would have a low cost of mass
production.

19. Introduction Methodology Results Discussion
Limitations
• Since proteins denature at high temperatures, the nanomotor has a limited
operational temperature range.
• The nanomotor has a limited chemical supply available to conduct chemotaxis
and produce torque at an increased rate. The limitation is overcome by
providing a multi stator nanomotor within a bacterium with a free supply of
ATP.

20. Introduction Methodology Results Discussion
Future Research
• The next logical step is to model a system of multiple nanomotors (ATP
Synthase) operating cohesively. If you are able to individually control and
understand components of multiple nanomotors, you are able to apply this
knowledge within nano-scale manufacturing processes towards medical
applications and the nanotechnology industry.
• It would be thus possible to engineer smart materials which may be able to
bend and contract by altering the torque at specific nanomotors within the
material.
• Modelling of multiple bio-mimicking nanomotors at once will be a
milestone in nanotechnology research

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