Optimal Control of the F 1-ATPase Molecular Motor Deepak GuptaySteven J. LargeyxShoichi Toyabezand David A. Sivaky

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Optimal Control of the F1-ATPase Molecular

Motor

Deepak Gupta,†,¶Steven J. Large,†,§Shoichi Toyabe,‡and David A. Sivak∗,†

†Department of Physics, Simon Fraser University, Burnaby, British Columbia V5A 1S6,

Canada

‡Department of Applied Physics, Tohoku University, Aoba 6-6-05, Sendai, 980-8579, Japan

¶Institute for Theoretical Physics, Technical University of Berlin, Hardenbergstr. 36,

D-10623 Berlin, Germany

§Current address: Viewpoint Investment Partners, Calgary, Alberta, Canada

E-mail: dsivak@sfu.ca

Abstract

F1-ATPase is a rotary molecular motor that in

vivo is subject to strong nonequilibrium driving

forces. There is great interest in understanding

the operational principles governing its high ef-

ﬁciency of free-energy transduction. Here we

use a near-equilibrium framework to design a

non-trivial control protocol to minimize dissi-

pation in rotating F1to synthesize ATP. We

ﬁnd that the designed protocol requires much

less work than a naive (constant-velocity) pro-

tocol across a wide range of protocol durations.

Our analysis points to a possible mechanism

for energetically eﬃcient driving of F1in vivo

and provides insight into free-energy transduc-

tion for a broader class of biomolecular and syn-

thetic machines.

Graphical TOC Entry

Keywords

molecular motors, stochastic ﬂuctuations,

nonequilibrium thermodynamics, free-energy

transduction, ATP synthase

arXiv:2210.02072v2 [cond-mat.stat-mech] 28 Dec 2022

Nanometer-sized biomolecular machines con-

vert between diﬀerent forms of free energy,

while remaining in contact with a thermal envi-

ronment. Consequently, ﬂuctuations play a cru-

cial role in a machine’s dynamics;1,2 neverthe-

less, a machine achieves (on average) directed

motion that is consistent with the second law

of thermodynamics by transducing free energy,

often stored in nonequilibrium chemical con-

centrations in the surrounding environment.3It

is of paramount interest to unravel the design

principles governing eﬀective free-energy trans-

duction in biomolecular machines.4–6

FoF1-ATP synthase has attracted particular

attention.7–10 This molecular motor is responsi-

ble for ∼95% of the cellular synthesis of adeno-

sine triphosphate (ATP).11,12 ATP production

by the F1subunit is driven by rotation of its

γ-shaft. In vivo, the γ-shaft rotates by utiliz-

ing free energy from proton ﬂux through the

membrane-embedded Fosubunit. The γ-shaft

can rotate as fast as ∼350 revolutions per sec-

ond,9while maintaining high eﬃciency.13 It is

thus of signiﬁcant interest to measure and quan-

titatively understand energy conversion during

rapid mechanical driving of F1to synthesize

ATP.14–16

FoF1can operate in either direction:17 An ex-

cess of ATP drives counter-rotation of the γ-

shaft and transports protons against their con-

centration diﬀerence. This reversibility is also

observed in isolated F1, which under suﬃciently

high torque synthesizes ATP,18,19 but in the ab-

sence of torque hydrolyzes ATP and counter-

rotates the γ-shaft.17,20 Experiments suggest

that this ‘hydrolysis mode’ proceeds via 120◦

rotations of the γ-shaft,21 composed of two sub-

steps: an 80◦step involving ATP binding, and a

40◦step involving ATP hydrolysis (catalysis).22

Overall the γ-shaft rotates 360◦by hydrolyzing

3 ATPs.

Rapidly driving the γ-shaft by applying ex-

ternal torque inevitably produces dissipation,

the diﬀerence between the work performed and

the free energy transduced during synthesis or

hydrolysis. It remains enigmatic how ATP syn-

thase achieves highly eﬃcient energetic conver-

sion despite such rapid operation. In particular,

what manner of rapid forced rotation with the

γ-shaft achieves eﬃcient energy transmission?

Here we theoretically address eﬃcient driving

procedures (protocols) to rapidly force rotation

of the γ-shaft. A near-equilibrium framework23

has proven experimentally useful in designing a

protocol for switching between folded and un-

folded conformations of single DNA hairpins,24

and similarly successful in simulations of bar-

rier crossing,25,26 rotary motors,27 Ising mod-

els,28–30 and other model systems.31–34 In this

Letter we design a protocol that (near equilib-

rium) minimizes dissipation in experimentally

accessible rotation of the γ-shaft driving F1to

synthesize ATP. We ﬁnd that the designed pro-

tocol outperforms the naive (constant-velocity)

protocol for a considerable range of protocol du-

rations, often far from equilibrium. Such pro-

tocols hint at how Fomight rotate the γ-shaft

in an eﬃcient manner.

The totally asymmetric allosteric model

(TASAM)35 of F1describes the evolution of a

rotational degree of freedom θ∈[0,2π] obeying

periodic boundary conditions, corresponding to

a bead attached to the γ-shaft (Fig. 1a provides

a schematic of the modeled experiment). The

TASAM is constructed to recover the steady-

state and kinetic behavior of F1hydrolyzing

ATP during single-molecule experiments.36,37

The `= 40◦step is modeled by the system

switching between two harmonic potentials of

the same spring constant k≈20 kBT/rad2, and

with minima oﬀset by the free-energy diﬀerence

∆µ= 5.2kBTbetween the catalytic-dwell and

binding-dwell states. Coarse-graining over the

fast 40◦step gives the eﬀective potential

βUn(θ)≡ −ln e−1

2βk(θ+`−nξ)2−β

∆µ+e−1

2βk(θ−nξ)2,

(1)

for n= 0,±1,±2, . . . .β≡1/(kBT) is the in-

verse temperature. The ﬁrst and second terms

in square brackets respectively indicate the

catalytic-dwell and binding-dwell states. The

80◦step is modeled by a hop between adja-

cent eﬀective potentials Un(θ) and Un±1(θ) an-

gularly separated by ξ= 2π/3 rad [see U1,2(θ)

in Fig. 1b].

Thus, the overall hydrolysis/synthesis of one

ATP molecule is modeled by switching the ef-

Figure 1: a) Schematic of modeled experiment.

b) Eﬀective potential U1,2(θ) (dotted), trap po-

tential Utrap(θ|λ) (black dashed), and total po-

tentials Utot

1,2(θ|λ)≡U1,2(θ) + Utrap(θ|λ) (solid).

c) Potential of mean force UPMF(θ) (dotted),

trap potential Utrap(θ|λ) (black dashed), and to-

tal potential Utot

PMF(θ|λ)≡UPMF(θ) + Utrap(θ|λ)

(solid). Each potential is plotted as a func-

tion of γ-shaft angle θ, for ﬁxed trap mini-

mum λ= 0.41 (vertical line), chemical drive

∆µATP = 18 kBT, and trap strength E‡=

30 kBT/rad2.

fective potential, Un(θ)→Un±1(θ), where the

‘+’ (‘-’) sign is for hydrolysis (synthesis). The

corresponding transition rates obey the local

detailed-balance condition35

R+

n(θ)

R−

n+1(θ)=eβ[Un(θ)−Un+1(θ)+∆µATP],(2)

for chemical-potential diﬀerence (hereafter

chemical drive) ∆µATP ≥0 due to synthesis

of one ATP, favoring ATP hydrolysis and γ-

shaft counter-rotation. The potential switches

with respective forward and backward transi-

tion rates

R+

n(θ)=Γ,(3a)

R−

n+1(θ)=Γe−β[Un(θ)−Un+1(θ)+∆µATP].(3b)

Γ is a rate constant characterizing the chem-

ical reaction (Supporting Information38 and

Fig. S1 relate Γ and [ATP] at ﬁxed ∆µATP).

Among the range of possible splittings of an-

gular dependence between the forward and

backward transition rates that satisfy local

detailed-balance (2), the experimental kinetics

in Refs.36,37 are best ﬁt by this splitting.35

Experimentally, F1is driven by conﬁning the

magnetic bead attached to the γ-shaft in a mag-

netic trap,18,19,39,40 whose minimum is dynami-

cally rotated. Modeling such a magnetic trap,

here we consider a sinusoidal trap potential (see

Fig. 1),

Utrap(θ|λ)≡ −1

2E‡cos 2(θ−2πλ),(4)

with time-dependent control parameter λ∈

[0,1] determining the angle of the two minima

diﬀering by 180◦and separated by barriers of

height E‡(parametrizing trap strength).

Thus, subject to this external trap, the γ-

shaft angle θdynamically evolves according to

θ=−βD ∂θUtot

n(θ|λ) + √2D η(t),(5)

where the dot indicates a time derivative,

Utot

n(θ|λ)≡Un(θ) + Utrap(θ|λ) is the total po-

tential, D= 13.7 rad2/s is the rotational dif-

fusion constant,36 and η(t) is Gaussian white

noise with zero mean and unit variance. For

∆µATP 6= 0, the system eventually reaches a

nonequilibrium steady state.41 Chemical tran-

sition rates (3) and mechanical dynamics (5),

subject to potentials (1) and (4), constitute the

TASAM.

In the limit of fast switching between eﬀective

potentials when chemistry is fast compared to

mechanics (Γ → ∞, e.g., at high [ATP]),35 the

dynamics (5) reduce to

θ=−βD ∂θUtot

PMF(θ|λ) + √2D η(t),(6)

for total potential energy Utot

PMF(θ|λ)≡

UPMF(θ) + Utrap(θ|λ) and potential of mean

force, averaging over all eﬀective potentials

(see Fig. 1c):

βUPMF(θ)≡ −ln

+∞

n=−∞

e−β[Un(θ)−n∆µATP].(7)

We seek a driving protocol that minimizes

dissipation while rotating the γ-shaft (for the-

oretical developments when control is much

ﬁner-grained, see Refs.42,43). For a harmon-

ically conﬁned Brownian particle, minimum-

dissipation protocols have been analytically

solved for arbitrary protocol duration.44,45 For

more complicated scenarios such as this model,

no analytical solution is known; nevertheless,

linear-response theory gives an approximately

dissipation-minimizing protocol.23 Up to the

linear-response approximation, the instanta-

neous excess power (that exceeding the qua-

sistatic power) during dynamic variation of con-

trol parameter λis

Pex(t)≈ζ(λ)dλ

dt2

.(8)

Its time integral over protocol duration tprot

gives the excess work Wex ≡W−∆F=

Rtprot

0dt Pex(t), for protocol work Wand equi-

librium free-energy change ∆Ffrom initial to

ﬁnal control-parameter values. ζ(λ) is a gener-

alized friction coeﬃcient obtained here by inte-

grating the equilibrium torque autocovariance:

ζ(λ)≡βZ∞

dthδτ(0) δτ (t)iλ.(9)

Angle brackets h. . . iλindicate a steady-state

average at ﬁxed λ.δτ (t)≡τ(t)− hτiλis

the deviation of the conjugate torque τ≡

−∂λUtrap(θ|λ) = 2πE‡sin 2(θ−2πλ) from its

equilibrium average. The generalized friction

coeﬃcient can be decomposed as

ζ(λ) = βh(δτ)2iλtrelax(λ),(10)

the product of the torque variance h(δτ)2iλand

the torque relaxation time

trelax(λ)≡Z∞

dthδτ(0) δτ (t)iλ

h(δτ)2iλ

.(11)

In the linear-response regime, the minimum-

dissipation protocol proceeds with velocity in-

versely proportional to the square root of the

generalized friction coeﬃcient:23

dλdes

dt∝[ζ(λ)]−1/2.(12)

Dissipation is reduced by driving slower where

system resistance is greatest (due to large ﬂuc-

tuations and/or slow relaxation) and compen-

sating by driving faster where system resis-

tance is least. Imposing boundary conditions

λ(0) = λiand λ(tprot) = λfﬁxes the propor-

tionality constant. Such a designed protocol

λdes(t) gives (up to linear response) constant

excess power.

Figure 2a shows Utot

PMF(θ|λ) for diﬀerent trap

minima λand fast switching. For some λ’s, the

total potential has two metastable states, with

a small (<0.5kBT) barrier.

The friction coeﬃcient (9) is obtained

through measurement of the equilibrium torque

autocovariance for a stationary trap (i.e., ﬁxed

λ) and hence ﬁxed total potential-energy land-

scape (Fig. 2a). Figure 2b shows this torque

autocovariance function. For those trap min-

ima λgiving two metastable states (Fig. 2a),

the torque autocovariance relaxes particularly

slowly.

Figure 2c shows the torque variance com-

puted from the equilibrium torque ﬂuctuations.

The torque variance is independent of Γ, as

expected since varying Γ modiﬁes system re-

laxation between landscapes but not the land-

scapes themselves. Similarly, Fig. 2d shows

the torque relaxation time (11), trelax(λ). The

product of the torque variance and torque re-

laxation time gives the generalized friction co-

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OptimalControloftheF1-ATPaseMolecularMotorDeepakGupta,y,{StevenJ.Large,y,xShoichiToyabe,zandDavidA.Sivak,yyDepartmentofPhysics,SimonFraserUniversity,Burnaby,BritishColumbiaV5A1S6,CanadazDepartmentofAppliedPhysics,TohokuUniversity,Aoba6-6-05,Sendai,980-8579,Japan{InstituteforTheoreticalPhysics,Techn...

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