King 1
Stability of a Nonequilibrium Biochemical Cycle Revealed
by Single-Molecule Spectroscopy
Saurabh Talele
1,2
and John T. King
1
*
1
Center for Soft and Living Matter, Institute for Basic Science, Ulsan 44919, Republic of Korea
2
Department of Biomedical Engineering, Ulsan National Institute of Science and Technology, Ulsan 44919,
Republic of Korea
Abstract
Biological machinery relies on nonequilibrium dynamics to maintain stable directional fluxes through
complex reaction cycles. In stabilizing the reaction cycle, the role of microscopic irreversibility of
elementary transitions, and the accompanying entropy production, is of central interest. Here, we use
multidimensional single-molecule spectroscopy to demonstrate that the reaction cycle of bacteriorhodopsin
is coupled through both reversible and irreversible transitions, with directionality of trans-membrane H
+
transport being ensured by the entropy production of irreversible transitions. We observe that thermal
destabilization of the process is the result of diminishing thermodynamic driving force for irreversible
transitions, leading to an exponentially increasing variance of flux through the transitions. We show that the
thermal stability of the reaction cycle can be predicted from the Gibbs-Helmholtz relation.
Motor proteins operate through non-thermal motions (nonequilibrium fluctuations) [1-5] induced by the input of
energy by, for example, ATP hydrolysis or photon absorption. Chemically driven motors have been adequately
described using the principle of microscopic reversibility [6], where equilibrium mechanical motions of the protein
are rectified by an asymmetric potential that biases diffusion in one direction [7-9]. Optically driven motors, in contrast,
are thought to operate through a ‘power-stroke’ mechanism, where a series of conformational transitions follow from
a sudden structural change [3, 10, 11]. For this physical mechanism, which is inherently far-from-equilibrium, the
validity of equilibrium notions of time-reversal symmetry and detailed balance, and the stability of the reaction cycle
to external perturbations, are yet to be established.
In this work, we study the reaction cycle of bacteriorhodopsin (bR), an optically-driven H
+
pump found in Archaea
(Fig. 1a,b) [12], at the single-molecule level. Photon absorption induces a trans-cis isomerization of the retinal
chromophore. Steric repulsions between the isomerized chromophore and the protein scaffold initiates a multi-step
reaction cycle involving multiple intermediate species that exist for timescales ranging from μs to ms during the cycle
(Fig. 1c) [13, 14]. Key to our understanding of nonequilibrium reaction cycles are the roles of time-reversal symmetry
and the corresponding entropy production [15]. To characterize these properties on a single-molecule level, we
leverage multidimensional single-molecule fluorescence lifetime correlation spectroscopy (sm-2D-FLCS) [16-19],
which utilizes multiple light-matter interactions separated by a controlled waiting time to monitor structural or
chemical transitions of a molecule. The exchange dynamics are observed through reciprocal time-correlation functions
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King 2
(tcf) for forward and reverse transitions, which appear in opposite quadrants of the 2D spectrum [20] and therefore
allow the forward and reverse rates (k
f
and k
r
) can be measured independently. The violation of time-reversal symmetry
within the reaction cycle is observed from the reciprocal tcf [21], which are equivalent for reversible transitions (obey
time-reversal symmetry) and are not equivalent for irreversible transitions (violate time-reversal symmetry). The
extent to which time-reversal symmetry is violated is a measure of entropy production rate in the cycle [15]. Using
this experimental approach, we are able to experimentally characterize the nonequilibrium thermodynamics and
kinetics of portions of the bR reaction cycle, and quantify the stability of the cycle when subject to kinetic perturbation.
We first characterize the directional flux of the cycle by monitoring the transition kinetics between several
intermediates. The endogenous retinal chromophore serves as the photo-trigger for the reaction cycle as well as a
probe of the intermediates during the cycle. Limitations in the experiment prevent transitions between each
FIG. 1.
Single-molecule spectroscopy of bR. Crystal structure of ground state bR showing (a
) the protein (with the retinal chromophore shown
in green) and (
b) the retinal chromophore and the amino acids involved in the H
+
transport chain of the reaction cycle (PDB: 1KBG). (c
) Photo-
induced reaction cycle of bR, which involves 6 intermediates in addition to the ground state (gs). The subscripts on the labels denote the absorption
maximum of the retinal chromophore for the given conformation. (
d
) Absorption spectrum of retinal embedded in bR shows strong absorption
at λ = 568 nm at pH = 6. (
e
) The fluorescence lifetime histogram measured from a single protein shows multi-component relaxation following
532 nm excitation. A 1D-ILT of the lifetime histogram reveals three relaxation times of τ
1
’ ~ 0.08 ns (K intermediate), τ
2
’ ~ 1.0 ns (L intermediate),
and τ
3
’ ~ 5.0 ns (N intermediate) (
inset). (f
) sm-2D-FLCS spectrum measured at a waiting time of Δt = 10 μs. Forward and reverse transition
cross-peaks are already present between the K intermediate and the L intermediate are already observed at 10 μs, which indicates rapid exchange.
In contrast, no cross-peaks are observed between the N intermediate and either the ground state or the L intermediate.
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intermediate to be directly measured. Instead, portions of the reaction cycle are characterized through select
intermediates to which our experiments are sensitive. The cycle of bR involves six known intermediate structures that
exist for timescales ranging between μs to ms (K, L, M1, M2, N, O, Fig. 1c) [13, 14]. In the ground state structure,
the retinal chromophore has an absorption maximum at λ
max
= 568 nm (Fig. 1d). Fluorescence emission from a single
monomeric bR protein, confirmed by diffraction limited emission spots observed in confocal microscopy and single-
step photo-physics (Fig. S1a-c), shows multi-component relaxation following a 532 nm excitation (Fig. 1e, Fig. S1d).
An inverse Laplace transform (ILT) [22] of the cumulative photon histogram measured from more than 100 single
protein molecules (Fig. 1e) gives a 1D fluorescence lifetime spectrum that contains three distinct relaxation peaks
(Fig. 1e, inset). We do not anticipate signal from the M1, M2, and O intermediates due to low absorption at the 532
nm excitation [13, 14]. Furthermore, fluorescence from the ground state structure is unlikely due to the ultrafast
isomerization reaction that occurs in the excited state [23]. Therefore, the signals arise from the K intermediate, the L
intermediate, and the N intermediate.
FIG. 2. Microscopic irreversibility of bR catalytic cycle. (a) sm-2D-FLCS spectra of bR shown for waiting times ranging from 0.01 to 50 ms.
Exchange kinetics for waiting times Δt = 0.01 – 200 ms between (
b) K intermediate to L intermediate, (c
) L intermediate to N intermediate, and
(d) K intermediate to N intermediate. Forward transitions, represented by solid symbols and solid lines, are measured from the upper quadrant of
the 2D spectra. Reverse transitions, represented by open symbols and dashed lines, are measured from the lower quadrant of the 2D spectra. The
forward (k
f
) and reverse (k
r
) transition rates between the ground state and the L intermediate, measured by fitting the cross-correlation functions,
are comparable, suggesting reversible dynamics and equilibration between these two states. In contrast, k
f
between the L intermediate and the N
intermediate are two orders of magnitude larger than k
r
, indicating a directional transition in the reaction cycle. The asymmetry in transition cross-
correlation functions (
Eq. 1
) indicates microscopic irreversibility of conformational transitions between the L and the N intermediates. This
irreversible transition corresponds to the reprotonation switch step of the cycle.
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The 2D-FLCS spectrum generated from a 2D-ILT of the photon histogram [22] calculated at a waiting time of Δt
= 10 μs shows three distinct diagonal peaks (Fig. 1f). The cross-peaks between the signals at τ’ ~ 0.08 ns and τ’ ~ 1.0
ns indicate rapid exchange between the two states represented by these signals. However, no cross-peaks are observed
between the diagonal signal at τ’ ~ 5.0 ns and the other two states. This is consistent with ensemble experiments which
have measured the formation time of the N intermediate to be on the ms timescale [13, 14]. From these observations
we can assign the diagonal peaks at τ’ ~ 0.08 ns, τ’ ~ 1.0 ns, and τ’ ~ 5.0 ns to the ground state, the L intermediate,
and the N intermediates, respectively (Fig. 1c, 1f).
To characterize the kinetics of exchange between different intermediates of wt-bR reaction cycle, we measure
kinetic traces for waiting times
t ranging from 0.01 to 200 ms (Figure 2a). In the 2D spectrum, forward and reverse
transitions between states i and j appear in the upper and lower quadrant of the spectrum, respectively [20]. The time-
dependent amplitudes of the spectrum are given by the tcfs [16, 17],
ij i j
C t S t S t t
(1a)
and
ji j i
C t S t S t t
(1b)
where S
i
and S
j
are the probabilities the system is in state i and j, respectively, measured in our experiment through
the unique fluorescence lifetimes τ
i
and τ
j
of each state. The equivalency of Eq. 1a and Eq. 1b under the condition of
microscopic reversibility is given by detailed balance and time-reversal symmetry [21]. Therefore, for reversible
transitions, the 2D spectra are symmetric and the dynamics observed through the cross-peaks are identical. Cross-
peaks between the K intermediate and the L intermediate show rapid equilibration of both the forward (Fig. 2b, solid
symbols) and reverse reaction (Fig. 2b, open symbols). The k
f
and k
r
values for individual transitions are determined
from exponential fits of the cross-peak amplitudes. The symmetry of the transitions in the spectra and the equivalency
of the kinetic traces suggest microscopically reversible dynamics between the K and L intermediate.
Under the condition of microscopic irreversibility, violation of detailed balance and time-reversal symmetry
manifests in asymmetric 2D spectra as the cross-correlation functions in Eq. 1a and Eq. 1b are no longer equivalent
[21]. At 0.20 ms, a cross-peak emerges between the L and N intermediates that is asymmetric (no cross-peak for the
reverse transition) (Fig. 2a), indicating a microscopically irreversible step in the reaction cycle. The forward transition
occurs on a timescale of ~ 0.50 ms (k
f
= 2.0x10
3
s
-1
), while the reverse transition occurs on a timescale over 200 ms
(k
r
= 5.0 s
-1
) (Fig. 2c). This transition represents reprotonation of the Asp96 residue on the cytoplasmic side of the
protein (M intermediate) and formation of a Grotthuss-like H
+
wire between the Asp96 residue and the Schiff base of
retinal [24-26]. Proton uptake and reprotonation of the retinal Schiff base (formation of N intermediate) from the
cytoplasmic side is thought to be the switch step of the photo-cycle [27-29]. At slightly longer times, we observe
asymmetric cross-peaks between the K intermediate and the N intermediate (Fig. 2a, d).
The properties of the measured time-correlation functions demonstrate that the reaction cycle of bR is coupled
through both microscopically reversible and irreversible transitions. The presence of microscopically irreversible
transitions implies that nonequilibrium transitions are inherent to the protein reaction cycle and are required to
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maintain stable fluxes through portions of the reaction cycle, even at steady-state. The results also suggest that the
stability of the reaction cycle is governed by entropy production associated with irreversible transitions.
Next, we analyze the nonequilibrium thermodynamics of the bR reaction cycle as a function of temperature. In
general, the efficiency of a motor protein that relies on microscopic irreversibility should be inversely proportional to
temperature, ultimately failing at the point where thermal energy is comparable or larger than the highest activation
barrier of the cycle, resulting in zero net flux. Therefore, k
f
and k
r
should converge at high temperature for each
transition of the reaction cycle.
The k
f
and k
r
for an elementary transition are given by Arrhenius expressions,
†
B
f
G k T
f
k ae
(2a)
and,
†
†
trans B
f
rB
G G k T
G k T
r
k ae ae
(2b)
FIG. 3. Thermal stability of the bR reaction cycle. Eyring plots for the K-L (a) and L-N transition rates (b). The k
f
and k
r
values are shown in
green and blue, respectively. For the reversible K-L transition, the k
f
and k
r
values are equal and have the same temperature dependence (
a). (
c)
The affinity A of the K-L transition plotted vs T
-1
. As the transition is microscopically reversible, the A value is ~ 0 kJ mol
-1
at all temperatures
measured. For the irreversible L-N transition, the k
f
and k
r
values are unequal, indicating a net flux through the transition, and have different
temperature dependences (
b). (d
) The A of the L-N transition plotted vs T
-1
. The microscopically irreversible transition has an A value of ~ 10 kJ
mol
-1
at 291 K that decreases linearly towards 0 kJ mol
-1
with increasing temperature. The temperature dependencies of A are fit to the Gibbs-
Helmholtz equation (red line,
Eq. 6), highlighting the connection between A and
G
rxn
for nonequilibrium systems.
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was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made
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