A new methodology for a

dramatic increase in efficiency of

nanomotors of varying stators and

system parameters

Michael Shumikhin Bronx High School of Science

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.

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

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

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

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

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

• 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

Langevin Equation

𝜁𝑟𝜙 = 𝜉 + 𝒯𝑒𝑥𝑡 −

𝜎

1 − 𝑛𝜎𝑑

𝑑𝜙[𝑈𝑞 𝜙 + 𝜙𝜎 + 𝑈𝑐𝑜𝑛 𝜙 + 𝜙𝜎 ]

Rate equation characterized by the Fermi distribution of the proton sources

𝑛𝜎 +

𝛼

Γ𝛼𝜎 𝜙 𝑛𝜎 =

𝛼

Γ𝛼𝜎 𝜙 𝑓𝜎 𝐸𝜎

Introduction Methodology Results Discussion

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

UML for one stator

Introduction Methodology Results Discussion

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.

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.

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.

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.

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).

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.

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.

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.

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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