Can active hydrodynamic fluctuations affect barrier crossing during enzymatic catalysis?
←
→
Page content transcription
If your browser does not render page correctly, please read the page content below
Can active hydrodynamic fluctuations affect barrier crossing during enzymatic catalysis?
Ashwani Kr. Tripathi,1 Tamoghna Das,1 Govind Paneru,1 Hyuk Kyu Pak,1, 2, ∗ and Tsvi Tlusty1, 2, 3, †
1
Center for Soft and Living Matter, Institute for Basic Science (IBS), Ulsan, 44919, Republic of Korea
2
Department of Physics, Ulsan National Institute of Science and Technology, Ulsan, 44919, Republic of Korea
3
Department of Chemistry, Ulsan National Institute of Science and Technology, Ulsan, 44919, Republic of Korea
(Dated: April 30, 2021)
The cellular milieu is teeming with biochemical nano-machines whose activity is a strong source of correlated
non-thermal fluctuations termed “active noise”. Essential elements of this circuitry are enzymes, catalysts that
speed up the rate of metabolic reactions by orders of magnitude, thereby making life possible. Here, we examine
the possibility that active noise in the cell, or in vitro, affects enzymatic catalytic rate by accelerating or decel-
erating the crossing of energy barriers during reaction. Considering hydrodynamic perturbations induced by
arXiv:2104.14064v1 [physics.bio-ph] 29 Apr 2021
biochemical activity as a source of active noise, we attempt to evaluate their plausible impact on the enzymatic
cycle using a combination of analytic and numerical methods. Our estimate shows that the fast component of
the active noise spectrum may enhance the rate of enzymes, by up to 50 %, while reactions remain practically
unaffected by the slow noise spectrum and are mostly governed by thermal fluctuations. Revisiting the physics
of barrier crossing under the influence of active hydrodynamic fluctuations suggests that the biochemical activ-
ity of macromolecules such as enzymes is coupled to active noise, with potential impact on metabolic networks
in living and artificial systems alike.
INTRODUCTION extent and physical nature of this linkage remain open ques-
tions [20, 21, 56]. All this invokes a notion of enzymes as
The idea that enzymes achieve their phenomenal catalytic stochastic molecular machines whose chemical performance
capacity by stabilizing an activated transition state was intro- and evolution are linked to their internal mechanics [50, 57–
duced by Haldane [1] and developed by Pauling [2] who lu- 63].
cidly stated this postulate [3]: “. . . that the enzyme has config- For their nanometric size, these machines are subject to vi-
uration complementary to the activated complex, and accord- olent, thermal and athermal, agitation by the fluctuating en-
ingly has the strongest power of attraction for the activated vironment: Thermal white noise originates from memoryless
complex, means that the activation energy for the reaction is equilibrium fluctuations. Athermal colored noise is generated
less in the presence of the enzyme than in its absence, and by a variety of temporally-correlated active sources, such as
accordingly that the reaction would be speeded up by the en- molecular motors and cytoskeleton rearrangement [9, 11, 13,
zyme.” Electrostatic effects, chiefly the formation of a preor- 64–66], and the dynamics of other cellular machinery, includ-
ganized polar network, were recognized as pivotal in stabiliz- ing enzymes [39–42]. This work lays out a simple model in
ing the transition state [4, 5]. In this extremely fruitful view order to investigate how these thermal and athermal fluctua-
of enzymatic catalysis, the activated complex is jolted past tions, in vivo or in vitro, might affect the catalytic reaction
the transition state’s energy barrier by thermal agitation [6–8]. rate. From a coarse-grained perspective, we treat enzymes as
The cell, however, is bustling with activity that generates sig- stochastic force dipoles [63, 67–70], each made of a pair of
nificant athermal agitation [9–13], provoking the main ques- masses joined by a spring. The relative motion of the masses
tion asked in this paper: how may athermal active noise affect represent conformational changes of the enzyme during the
enzymatic catalysis? catalytic cycle.
During their catalytic cycle, many enzymes undergo con- In essence, the framework developed here is a humble
formational changes, for example to enable substrate binding extension of the classical, thermally-activated transition-rate
and product release [14–25]. Such internal motions and re- theory [71–75], to account for the impact of correlated noise
arrangements are part of essential mechanisms, particularly induced by hydrodynamic fluctuations. In principle, this
induced fit [26], conformational selection [27, 28] and al- method can be applied to other biological processes, such as
lostery [29–33]. The coexistence of multiple conformational unzipping of DNA and RNA hairpins [76–80] or generally,
states [34] may assist evolution to explore new functions [35]. to any other physical process that can be cast as a two-state
Motor proteins operate by converting chemical energy into system with noisy memory. Using this model, we computed
conformational changes and motion [36–38], and recent stud- the reaction rate, relative to a purely thermally-fluctuating en-
ies suggest that similar coupling underlies the boosted diffu- zyme, as a function of the active noise strength and its correla-
sion observed in active enzymes [39–42]. Linkage between tion timescale. Within a biologically relevant parameter range
intrinsic motion and catalysis was reported in adenylate ki- typical to enzymes, we find two potential effects of active
nase (ADK) [43–46], dihydrofolate reductase (DHFR) [47– noise: Strong active noise with long correlation time (relative
53], and other enzymes [50, 54, 55]—though the existence, to the turnover rate) seems to hinder enzymatic activity, and
the reaction slows down compared to a thermally-activated en-
zyme. However, active fluctuations of any strength with short
∗ hyuk.k.pak@gmail.com to intermediate correlation times enhance the catalytic rate.
† tsvitlusty@gmail.com Under the combined influence of thermal and active noise, in2
a biologically relevant regime, we find a potential increase of the force fluctuations scale as
10 – 50 % in the turnover rate for enzymes. Z ∞
2 2
We start by deriving the active noise induced by hydrody- FH ∼ c0 m `40 r−6 d3 r
namic fluctuations, and then model the impact of the noise on `0
the catalytic cycle. Next, we numerically estimate the poten- ∼ c0 `0 m2 ∼ R−3 `0 m2 , (1)
tial effect of the noise strength and correlation time on a re-
action in general. Finally, we present our main results on the where R ≡ c0
−1/3
is the average dipole-dipole separation.
variation of reaction rate in the presence of active noise and The exact expression, derived in Methods, includes a ge-
illustrate it with specific examples of enzymes. We discuss ometric factor of order unity. Eq. (1] preserves the long-
the implications of these findings and conclude with remarks range nature of hydrodynamic fluctuations which decay as
about the general applicability and possible improvements of c0 ∼ R−3 . For typical concentrations of active sources, such
our study. as enzymes or motors, ranging between c0 ∼ 1 µM − 1 mM,
R ∼ 10−100 nm. The dipole moment fluctuation can be ap-
proximated as m2 ∼ (F`)2 , where ` the size of the ac-
MODEL tive elements and F is the net force they generate during their
turnover cycle. For values typical to motor proteins, F ∼ 1 pN
and ` ∼ 5 nm [13, 37, 83], the dipole fluctuations would be
Active noise realisation of hydrodynamic fluctuations. An m2 ∼ 1 (kB T)2 . We shall use the value of hydrodynamic
enzyme in a cellular environment continually experiences cor- fluctuation FH2 as the strength of the active noise.
related stochastic forces, as a collective effect of diverse flow- The sources of active noise in the cell have widely varied
generating mechanisms, which we model as sources of ather- correlation timescales, and are interdependent components of
mal active noise. To estimate these stochastic forces, we ap- an intertwined biochemical circuitry. However, the timescales
proximate the active noise sources as an ensemble of force of the network’s collective dynamics are much longer than the
dipoles. This is a valid long-range approximation as force correlation time of a a single source. As suggested by recent
dipoles induce the leading term in the far-field expansion of experimental measurements [9–11, 13, 84], the sources might
momentum-conserving hydrodynamic perturbations [81, 82]. be assumed as independent stochastic processes with inter-
Each force dipole is represented as two equal masses con- mittent bursts of activity, each with its own auto-correlation
nected by a spring of rest length `. The cellular background is statistics. Thus, we consider the background flow as an active
treated as a random ensemble of average concentration c0 of noise ζA (t) with a characteristic correlation time τA realised
such force dipoles whose moments {mi } are randomly dis- as:
tributed at positions {Ri } with randomly isotropic orienta-
tions {ei }. hζA (t)ζA (t0 )i = FH2 exp (−|t − t0 |/τA ) . (2)
As the typical inertial timescale (∼1 ps) is much shorter
than the characteristic timescale of an enzyme, (>1 µs), the Such activity maintains a certain type of fluctuation-
background flow is overdamped. It is therefore convenient dissipation relation, as observed in cells, [11, 65, 84] where
to treat this linear Stokesian flow in terms of its Green func- injection (extraction) of an energy A into (from) the system
tion, the mobility tensor G. A dipole m made of a pair of is compensated by the correlation time such that the noise
opposing point forces will therefore generate a flow field v(r) strength, FH2 = A/τA remains constant. With these
proportional to the gradient of the Green function, v(r) = considerations, we now proceed to derive the reaction rate
∇G(r, r0 ) m(r0 ), where r0 is the position of the source dipole. theory in the presence of such active noise.
A target dipole (i.e., an enzyme) of length `0 subjected to
this flow will experience an internal stress (tension or com- Reaction with hydrodynamic fluctuations. We start by
pression) proportional to the velocity gradient along its axis. writing down the dynamical equation for a reaction occurring
This dipole-dipole force FH will therefore be proportional to in an energy landscape under the influence of a noise ζT (t)
the second derivatives of the mobility, FH ∼ ηw`0 ∇v ∼ of thermal origin, and an active noise ζA (t) resulting from the
ηw`0 m∇∇G, where w is the hydrodynamic diameter of the long-range correlated hydrodynamic fluctuations,
dipole’s beads and η the viscosity (see Methods for a detailed
derivation). As biological flows are typically of low Reynolds γ q̇ = F (q) + ζT (t) + ζA (t) . (3)
number, G can be approximated as the Oseen’s tensor which
scales G ∼ 1/(ηr), where r is dipole-dipole separation. Then, The first term on the right hand side of Eq. (3] is a con-
FH ∼ η`20 m∇∇G ∼ `20 mr−3 (taking w ∼ `0 ). servative force, F (q) = −∂q U (q), exerted by the poten-
Since the force dipoles are randomly positioned and ori- tial U (q) = −(a/2)q 2 + (b/4)q 4 . The reaction potential
ented, ensemble or time averaging forbids the accumula- U (q) ispmade of two wells are positioned symmetrically at
tion of net mean dipole moment, hmi (t)i = 0. Thus, the q0 = ± a/b , separated by an energy barrier EB = a2 /(4b)
average net flow and induced internal forces also vanish, at q = 0. a, b > 0 are phenomenological constants. Phys-
h∇vi ∼ hFH i = 0. What survives averaging are of course ically, a would represent the stiffness of a protein, roughly
the fluctuations experienced by the target dipole (the enzyme), the “spring constant” of the force dipole and b stands for the
(∇v)2 ∼ hFH2 i = 6 0. Summed over the random ensemble, strength of the simplest possible anharmonicity that yields an3
√
activation barrier EB . The internal friction of the landscape γ for τA τ0 . The maximal force, Fmax = (2/ 3 )3 (EB /q0 ),
sets the intrinsic timescale τ0 = γ/a 1 . would be experienced at these inflection points. In the long-
The thermal force ζT (t) is drawn randomly from temporally memory regime, τA τ0 , any force F ≥ Fmax , is likely to
uncorrelated white noise, hζT (t)ζT (t0 )i = 2γkB T δ(t − t0 ) push the reaction to the other potential well by crossing into
with the noise strength fixed by the temperature T , where kB the negative friction region. Thus, Fmax sets an effective force
is the Boltzmann constant. Unlike the thermal noise, the ac- barrier, similar to the energy barrier EB in the short-memory
tive noise ζA (t) is temporally correlated (Eq. (2]), which we regime. As this phenomenology evidently affects the reaction
ensure by modelling it with an Ornstein-Uhlenbeck type evo- rate, we first investigate its asymptotic limits.
lution dynamics [85]: To this end, we turn the Langevin equation (Eq. 4) into the
√ corresponding Fokker-Plank equation for the probability dis-
τA ζ˙A = −ζA + 2A ξW (t) , (4) tribution P (q, v, t), where v = q̇ is the velocity:
where ξW (t) is a standard white noise process. Our main ∂P ∂P F (q) ∂P A ∂2P Γ(q) ∂
objective is to study the effect of active noise ζA (with ther- +v + = 2 + (vP ) .
∂t ∂q γτA ∂v τA ∂v 2 γτA ∂v
mal noise ζT in the background) on the reaction rate, κ =
(6)
1/(2τMFP ), where τMFP is the mean first passage time needed
to cross the energy barrier. For an active noise with long correlation time, τA τ0 and
To gain some intuition, we examine the asymptotic case of strength FH2 Fmax 2
,
negligible thermal noise. Then, Eqs. (3,4), can be recast as an
−1 2
underdamped Langevin equation, κslow ' (2τA ) exp − 12 Fmax / FH2 .
(7)
√
(γτA ) q̈ = −Γ(q) q̇ + F (q) + 2A ξW (t) , (5) This behaviour is specific to the active noise realisation which
relies on Fmax and is markedly different from purely ther-
with the effective friction coefficient Γ(q) ≡ γ − τA ∂q F (q).
mal reaction rate which depends on EB . The importance of
A remarkable feature of this nonlinear dynamical equation is
Fmax in the case of “slow” background has been noted in a
that the evolution of reaction depends, besides on the force
few other recent investigations [90–92]. For the case of “fast”
F (q) itself, also on its gradient, that is the curvature of the
background, τA τ0 and A γEB , the active noise merely
potential, ∂q F = −∂q2 U . Thus, when τA > τ0 , the effective
scales the energy barrier and the reaction rate follows the well-
friction Γ(q) turns negative close to the energy barrier [86–
known thermal behaviour,
89]. The negative friction region—where the motion is accel-
erated past the barrier—grows with τA , until it stretches√be- √ −1
tween the two inflection points of the potential, |q| ≤ q0 / 3 , κfast ' 2 πτ0 exp [−γEB /A] . (8)
We note that while κfast grows with A, i.e., with τA for a fixed
noise strength, κslow decays monotonically with τA , suggest-
ing an intermediate time scale where κ is optimal. Numeri-
10−1 cal simulations of the reaction dynamics for pure active noise
confirms this behavior (Fig 1).
Reaction Rate, κ
As the semi-analytic Eqs. (7,8) are only valid at the asymp-
10−2 totic limits which are impractical to realize for enzymes, we
now move on to solve Eqs. (3,4) numerically to investigate
the effect of both active and thermal noise on the reaction
10−3 rate. We measure the simulation time and length in units of τ0
and q0 , respectively. Thermal and active fluctuations are also
σA = 0.5 scaled by the relevant force scale as: σT2 = kB T /(2EB ), σA2 =
κfast FH2 /(4aEB ). In the next section, we present behaviour of
10−4 κslow κ in the {σA , τA } plane spanning over orders of magnitude.
Specifically, for each point in the {σA , τA } plane, we generate
10−1 100 101 102 105 independent reaction trajectories starting from an initial
τA position chosen randomly around q0 and evolve the trajecto-
ries till it reaches q = 0 where the energy barrier is maximum.
The reaction rate κ is then computed as inverse of the mean
FIG. 1. Variation of reaction rate κ, for pure active noise of strength
time taken by trajectories to cross the barrier.
σA = 0.5 . κ follows the asymptotic limits κslow (Eq. (7], red curve)
and κfast , (Eq. (8], black curve).
RESULTS
1 Note that Eq. (3] merely assumes that the reaction dynamics is amenable Enhancement of reaction rate under active noise. Active
to stochastic active noise, as it is to thermal noise, but requires no coupling noise changes the magnitude of the overall force that a reac-
of reaction and conformational coordinates. tion experiences in a given reaction energy landscape. It also4
1.2
0.7 A Thermal B
qmax
τA = 0.1 1.1
0.6
τA = 1.0
τA = 10 1.0
0.5
0.8 C
0.4
P (q)
Pmax
0.7
0.3
0.6
0.2 3 D
EBeff
0.1
2
EB = 2kB T
0.0
−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 10−2 10−1 100 101 102
q τA
FIG. 2. Statistical features of the reaction dynamics are presented for an energy barrier EB = 2kB T (i.e., σT = 0.5), and active noise strength
σA = 0.5. (A) The probability density P (q) in the reaction energy landscape for τA = 0.1,1.0 and 10.0. A purely thermal case is shown
for reference. (B) The most probable position in the reaction landscape qmax is monotonically pushed away from the barrier, as compared to
a purely thermal system whose maximum is at q = q0 (dashed line). (C) However, the maximal probability Pmax exhibits a non-monotonic
dependence on τA . For very small τA , Pmax is lower than its expected thermal value (dashed line), and continues to decrease till τA ∼ 1,
and then turns to increase and eventually becomes larger than the thermal value. (D) Similar non-monotonic behavior of the effective reaction
energy barrier EBeff , computed from the distribution (see text). Note that, for τA > τ c , EBeff is larger than the purely thermal case.
changes the persistence of the force direction by introducing observed for small and intermediate τA (Fig. 2D), where en-
a correlation timescale that is absent in purely thermal agita- hancement of κ is naturally expected. While EBeff traces the
tion. As a result of this combined effect, the reaction rate κ is same non-monotonic behaviour of Pmax , it helps us to iden-
expected to change. To investigate, we consider a case where tify a crossover timescale τ c above which EBeff > EB , and
thermal and active noise have equal strength: σA = σT = 0.5, the reaction becomes even slower than a pure thermal case.
and plot P (q), the probability distribution of the reaction tra- Thus, τ c provides us a natural threshold to discern between
jectories in the reaction landscape, for different values of τA “fast” (τA < τ c ) and “slow” (τA > τ c ) active noise, i.e., the
along with the purely thermal case. (Fig. 2A) Note that the background hydrodynamic fluctuations.
probability of forming reactant-substrate activated complex, Next, we examine the effect of the active noise strength σA
P (q = 0), is larger than the thermal case for τA ∼ τ0 but on the reaction rate κ. We find that σA enhances the effect of
becomes smaller for τA = 10τ0 . Following the position of τA on κ as we plot it relative to the thermal reaction rate κT
the most probable value qmax as a function of increasing τA , as a function of the scaled correlation time τA /τ c (Fig. 3A).
we find that qmax gradually moves outwards from its thermal Evidently, κ becomes faster against the “fast” background and
equilibrium position qmax = q0 (Fig. 2B). The movement is slower against the “slow” background. Note that τ c increases
more rapid over an intermediate range of τA while for small with σA (Fig. 3A Inset), demonstrating that larger σA allows a
τA , qmax mostly stays close to q0 and at large τA , it somewhat longer window of τA for reaction rate enhancement.
settles at a certain value of q. However, the maximum value of
the probability Pmax drops below its thermal value even when Most importantly, it is possible to find an optimal corre-
an active noise with tiny correlation is introduced (Fig. 2C). lation time τA∗ for which the enhancement of reaction rate
Pmax continues to decrease for τA ∼ τ0 . As the correlation is maximum. This maximum reaction rate κ∗ is denoted by
time of the active noise grows longer than the thermal cross- a ? for each σA value in Fig. 3A. Notice that the enhance-
ing time, the memory of active noise start to affect the reaction ment of κ is possible for even a tiny value of σA (Fig. 3C),
adversely. The reaction trajectories now stays away from the but that would require a relatively larger optimal correlation
barrier for longer time causing Pmax to increase. Eventually time τA∗ (Fig. 3D, also see Fig. 5). Still, τA∗ is smaller than
Pmax becomes larger than its pure thermal counterpart for ac- τ c by at least one order of-magnitude, as rate enhancement
tive noise with τA & 10 τ0 . can only occur in the presence of a fast hydrodynamic back-
ground. More enhancement is observed with increasing σA
The variation of Pmax indicates that the active noise af- as κ∗ grows in a scale-free fashion with σA > 1. Corre-
fects the reaction by effectively modifying the energy barrier. spondingly, τA∗ decreases in a similar fashion. As an aside,
To confirm this, we construct an effective potential from the we mention that similar behavior is also expected when the
probability distribution as: V (q) = − ln [P (q)/P (0)], and active noise is much stronger than the thermal one and solely
compare the effective barrier EBeff ≡ V (0) − V (qmax ) with dictates the reaction. In this case, κ∗ would decrease exponen-
the thermal barrier. A decrease in EBeff (< EB ) is clearly tially for small σA < 1, markedly different than the more real-5
100
A 40 C
30 10−1
102
τc
Reaction Rate (κ/κT)
Maximum rate, κ∗
20
10−2
10 −1
* 10 100 101
101 10−3
* σA
Active
10−4
* Active + Thermal
*
100 e−1/σA
10−5 σA0.9
σA = 0.3 σA = 2.0
σA = 0.5 σA = 5.0 Thermal
10−6
10−4 10−3 10−2 10−1 100 101 10−1 100 101
c Active Force, σA
Correlation Time (τA/τ )
102
B D σA−2.27
10−1
σA−0.9
Probability, P (τFP)
Optimal time, τA∗
101
10−2
10−3 100
τA = 0.1
τA = 1.0
τA = 10
10−4
τA = 100 10−1
100 101 102 10−1 100 101
τFP Active Force, σA
FIG. 3. (A) The effect of active noise strength σA on reaction rate κ for a fixed barrier EB = 2kB T . The variation of κ relative to thermal
reaction rate κT for different values of σA is plotted as a function of τA scaled by τ c (defined in the text). Dependence of τ c on σA is shown in
the inset. For each value of σA , the maximum reaction rate κ∗ is pointed by a ?. (B) Distribution of first passage time P (τFP ), exponential for
small τA , starts to grow a prominent power-law tail with increasing τA indicating departure from equilibrium. (C) The maximal rate κ∗ (stars)
is never less than κT (dashed line) and increases monotonically with the active noise σA . Also shown are the rates in a purely active system
(crosses). (D) The optimal correlation time τA∗ associated with the maximal rate κ∗ (stars) shows a monotonic decrease with noise strength
σA . Also shown is the purely active case (crosses). The numerical fits in (C) and (D) (black and red lines) are empirical functions described in
the legends.
istic scenario of enzymatic catalysis governed by both thermal i.e., close to the thermal rate. But when the fluctuations be-
and active noise. come more persistent, they approach an adiabatic regime,
τ0 < τA < 1/κT , where each crossing event occurs in a prac-
The non-monotonic behavior of κ (Fig. 3A) is the outcome tically static potential and effective barrier. In this regime,
of the interplay of two competing effects. First, the reaction the average rate will
dynamics changes as fluctuations cross over from a fast to a be the average over the static potentials,
κ ∼ hexp −EBeff i ≥ exp −hEBeff i , which is always larger
adiabatic regime. To understand this effect, note that reac-
than the rate in the fast regime2 , thus explaining the increas-
tion dynamics in the presence of active noise can be consid-
ing part of the curve. The second effect occurs in the large
ered as motion within a fluctuating reaction energy landscape,
correlation limit, when κ is controlled by the maximum force,
U eff (q, t) = U (q) − qζA (t), with a fluctuating effective en-
Fmax rather than the activation barrier. Then, the reaction rate
ergy barrier EBeff (t) ' EB + q0 ζA (t) (akin to Bell’s law [93]).
κ exhibits an inverse dependence on τA (Eq. (7)), due to slow-
The persistence of the fluctuations is controlled by correla-
tion time τA . When the fluctuations of the landscape are much
faster than the enzymatic timescale, τA τ0 , the enzyme
eff
hEB i = EB , and
2
experiences an average effective barrier, This follows fromthe convexityof the logarithm function,
eff = log exp −E eff eff .
eff −EB ≤ log exp −EB
the resulting rate is κ ∼ exp −hEB i = exp(−EB ) ∼ κT , B6
ing down by the increasing effective friction, Γ(q) ∼ τA in Taking a rest length `0 ' 5 nm and internal viscosity of
Eq. (5). η ' 10−2 Pa s [99, 100], we obtain the friction coefficient,
Interpolating these two limits, one expects an optimal γ ' 3πη`0 ' 10−10 kg/s, which sets the thermal timescale
correlation time, where the reaction rate attains a maximum τ0 = γ/a ' 1 µs with a = 0.1 pN/nm. Now we compute
as indeed shown in the simulations. These observations agree the reaction rate for our test enzyme over a range of σA and τA
with the computed distribution of first passage time, P (τFP ), for two different energy barriers, EB = 6 kB T (Fig. 5A) and
the time taken to cross reaction barrier (Fig. 3(B)). The EB = 10 kB T (Fig. 5B). For both cases, a red dashed line is
P (τFP ) distribution follows a non-monotonic dependence drawn to mark the boundary between the “fast” background
similar to that of τMFP . For τA ∼ τ0 , the P (τFP ) shifts to (on the left) and “slow” background (on the right). Within the
shorter τFP values compared to the thermal regime (τA τ0 ), “fast” regime, we always find an enhancement over the ther-
resulting in increasing reaction rate. On the other hand, mal reaction rate, κ/κT > 1. As a crowded solution of dipoles
for τA τ0 , the distribution shifts toward the longer first would correspond to a concentration, (1/`30 ) ' 10 mM, we
passage times, indicating slowing down compared to the consider a moderate regime of 30 – 300 µM. Using the active
thermal rate κT . Interestingly, P (τFP ) crosses over from force map in Fig. 4, we find that an enzyme is expected to
exponential scaling in the fast regime to power law behaviour experience a maximal active force, σA ' 0.003−0.03 (shown
in the slow regime, signaling a transition for equilibrium to as a horizontal dashed line in Fig. 5) over the relevant energy
nonequilibrium behavior. scale, EB = 6−10 kB T. At this limit, we see a maximum en-
hancement of 15 % for EB = 6 kB T (κ/κT ' 1.15) and of
50 % for EB = 10 kB T (κ/κT ' 1.5). Thus, for these typical
parameters, we expect a maximum of 50 % enhancement in
A, σA for a = 0.1 pN nm−1 B, σA for a = 1.0 pN nm−1 the reaction rate.
103
0.1
0.03
0.06
0.0 .014 2
0.0
0.0 0.04
Source Conc. (µM)
CONCLUSION
0
1
3
2
0.0 In summary, we have shown how a fluctuating hydrody-
102
06
0.014 namic background might affect an enzymatic catalysis. Hy-
0.0
04
0.01 0.0 3 drodynamic fluctuations of various origins are considered as
0
0.008 0.0
an outcome of the stochastic oscillations of a random distri-
0.006 2
0.00 bution of force dipoles. Coupled through the flow they gen-
1
0.004 0.00 erate, these force dipoles can be collectively realised as a
101 temporally-correlated athermal noise representing the back-
5 10 15 20 5 10 15 20
Energy Barrier, EB/kBT ground activity. Modelling active noise as an Ornstein-
Uhlenbeck process and numerically solving reaction rate the-
FIG. 4. Variation of scaled active force strength σA as a func- ory, now in presence of both thermal and active noise, reveals
tion of source dipole concentration c0 and barrier height EB /kB T . a special correlation time τ c , above which reaction rate start
(A) and (B) corresponds to the protein stiffness values a = 0.1 and to slow down compared to the bare thermal rate. τ c is of the
1.0 pN/nm, respectively. Solid curves are constant force contours, same order as inverse of thermal reaction rate κT (red dashed
and the dashed lines denote the concentration range, 30 – 300 µM, line in Fig. 5). Further, we find that while a slow background,
which is reasonably accessible in experiments.
τ c < τA , somewhat slows down the catalytic activity, a faster
background, τA < τ c , always enhances the catalytic reaction
The case of enzymes. Finally, we turn to the enzymatic rate relative to the purely thermal case. For example, a suit-
catalysis in the presence of a fluctuating hydrodynamic back- able choice of active noise may result in up to 50 % enhance-
ground realised as a source of active noise. We choose a ment for the typical example of ADK. We note that the present
parameter space, {σA , τA }, to match the experimentally re- model assumes Oseen’s far-field approximation for the mobil-
alisable conditions. The scaled active noise strength, σA =
p ity tensor, and should be modified for densely packed sources.
hFH2 i/(4aEB ) , depends on the reaction energy barrier Once the hydrodynamic interaction is corrected to account for
EB , enzyme stiffness a, and the density of the background near-field effects, our dynamical equations can be solved for
through hFH2 i as in Eq. (1). Catalytic reaction energy bar- in this limit of intense active force.
riers are measured to typically lie within a range of EB = The proposed physical scenario and the predicted effect of
4−24 kB T [7, 94] and the typical stiffness of enzymes is re- active noise on enzymatic catalysis require cautious exami-
ported to vary within a range a = 0.1−1.0 pN/nm [95– nation. As controlling the background is hard in vivo, we
98]. In Fig. 4, we have charted out the variation of σA over propose a simple in vitro experimental test: Consider a solu-
a wide range of backgrounds with density ranging between tion consisting of two enzymes and their respective substrates
10 – 1,000 µM as a function of EB for two limiting values in an appropriate buffer. Importantly, the two reactions are
of a. As a prototypical example, consider the well-studied chemically orthogonal to avoid any cross-talk. The “source"
enzyme adenylate kinase (ADK) [43, 44]. The energy bar- enzymes generating the active noise are relatively dense to al-
rier EB of ADK at room temperature is about 10 kB T [46]. low strong impact on the “target" enzymes, which are dilute7
A, Relative rate at EB = 6kBT B, Relative rate at EB = 10kBT
10−1
2.50 10−1 24.0
0.60
12.0
1.80
6.0
0.90
Active Force, σA
0.9
1.40
3.0
1.20
1.8
1.0
1.15
1.5
1.10
1.00
0.98
1.06 1.3
1.04 1.2
1.0
1
1.1
10−2 −2 1.02 10−2 −2
10 10−1 100 101 102 103 104 10 10−1 100 101 102 103 104 105
Correlation Time, τA Correlation Time, τA
FIG. 5. The effect of active noise on the catalytic rate of an enzyme. Variation of reaction rate κ, relative to thermal reaction rate κT , as
a function of active noise strength σA and its correlation time τA , for EB = 6 kB T (A) and EB = 10 kB T (B). The red dashed contour
marks the line where κ = κT , even in the presence of active fluctuations. This line separates the slow (right) and the fast (left) background
regimes. Fast background always promotes enhancement of κ, which can be quite substantial depending on the control parameters. In contrast,
slow background somewhat decelerates the reaction but not as spectacularly as in the fast background regime. For enzymatic solutions of
concentration 30 – 300 µM, σA varies between 0.003 – 0.03. Within this range, a maximum enhancement of κ up to 15 % is expected for
EB = 6 kB T (A) and up to 50 % for EB = 10 kB T (B).
to avoid confounding inverse effects. The active noise correla- plan to further extend the present bare-bone model to include
tion time is approximated as the inverse thermal reaction rate the causal dependence of reactions in a network and the
of the source enzyme, τA ' 1/κsourceT . Also, notice that the spatiotemporal heterogeneity of the embedding background.
crossover time τ c remains of the order of the thermal τMFP of We hope the current results would stimulate further study
the target enzyme, τ c ' 1/κtarget
T (red dashed line in Fig. 5). of potential effects of active stochastic environment on
Thus, κsource
T > κtarget
T , implies that the source serves as fast biochemical processes.
background , τA < τ c , and we expect an enhancement in the
reaction rate of target enzyme. Conversely, κsource
T < κtarget
T ,
implies a source that serves as a slow background, τA > τ c ,
and is expected to slow down the rate of target enzyme. In
both cases, there must be a separation of timescales between
the target enzyme (τ c ) and the active noise (τA ) to obtain a
measurable effect on the reaction rate of target enzymes. As
a consequence, a solution of only one enzyme, which serves
as both target and source (τA ' 1/κsource = 1/κtarget ' τ c ), METHODS
T T
is expected to show no self-enhancement or self-reduction of
the rate as concentration varies. The present model should in
principle be applicable to other biologically relevant processes Hydrodynamic forces induced by active processes. We
such as unzipping of DNA hairpins [76–78] for which the re- consider a solution of stochastic force dipoles [63, 69] rep-
ported energy [79] and timescales [80] lie within the range resenting active processes such as enzymatic catalysis and the
explored in the current study. motion of molecular motors. In this coarse-grained view, each
force dipole consists of two beads connected by a spring of
The complex cellular environment is dense in entangled en- equilibrium length `0 . The beads represent the domains of
ergetic processes. A fast-growing bacterium consumes energy the enzyme that move with respect to each other during the
at a power of ∼108 kB T/s, over a volume of ∼1 µm3 [83]. catalysis. Consider a collection of force dipoles {mi } lo-
In the eukaryotic cell, there are high-activity regions and cated at positions {Ri } with random independent orientations
organelles, such as mitochondria and chloroplasts, where the {ei } (3D unit vectors). Our target dipole has its two domains
proposed effects might be significant. One may speculate (i.e., spheres) located at positions R and R0 = R+`e, with an
that molecular motors, whose turnover rate is relatively orientation e and distance ` = |R0 −R|. Following Mikhailov
slow [36, 83], can be accelerated in the presence of high and Kapral [69], we find the velocities Ṙ of the domains—
metabolic activity. To treat such elaborate scenarios, we using the mobility tensor Gαβ —by summing the contributions8
of the velocity fields induced by the surrounding dipoles, Next, we rewrite Eq. (12) in a field-point notation, which
will be convenient for further manipulation,
N
X ∂Gαβ (R − Ri ) Z
∂ 2 Gxβ (r)
Ṙα = eiβ eiµ mi (t) , (9) FH (R, t) = 3πηw`(t) dr
i
∂Riµ ∂rµ ∂rx
N
X
X ∂Gαβ (R0 − Ri ) × eiβ eiµ mi (t) δ (r − Ri + R) .
Ṙ0 α = eiβ eiµ mi (t) , (10) i
i
∂Riµ
The mean force is proportional to the average over the sum
where the Greek indices denote x, y, z components of vectors of dipole moments, which vanishes due to the symmetry
and tensors, and we follow Einstein’s convention of summa- in the homogeneous isotropic solution [69], hFH (R, t)i ∼
tion over repeated indices. The mobility tensor Gαβ is the hmi (t)i = 0. However, the second moment—that is the av-
Green function of the linear Stokes flow, which yields the ve- erage squared force the target dipole experiences due to the
locity field resulting from a localized force [101]. Eq. (10) collective fluctuations of other force dipoles—does not van-
involves spatial derivatives of G, which are the Taylor expan- ish,
sions around each force dipole.
hFH (R, t)FH (R, t0 )i = (3πηw)2 h`(t)`(t0 )i
The time-dependent dipole moment exerted on the target
∂ 2 Gxβ (r) ∂ 2 Gxβ 0 (r)
Z
m(t) is m(t) = `(t)F (t), where `(t) and F (t) are the distance
× dr
and interaction force between the two domains. Thus, for a ∂rµ ∂rx ∂rµ0 ∂rx
target enzyme of length `(t), the relative velocity δv between X
× heiβ eiµ eiβ 0 eiµ0 δ(r − Ri + R)mi (t)mi (t0 )i ,
the two domains is given by
i
N
X Since dipole orientations are uncorrelated with their positions,
δvα = Ṙ0 α − Ṙα = eiβ eiµ mi (t) the last term in above equation can be simplified,
i X
∂Gαβ (R + `(t)e − Ri ) ∂Gαβ (R − Ri )
heiβ eiµ eiβ 0 eiµ0 δ(r − Ri + R)mi (t)mi (t0 )i
− . i
∂Riµ ∂Riµ
= heβ eµ eβ 0 eµ0 i hm(t)m(t0 )i c (R + r) ,
Since the linear extension of the enzyme `(t) is much smaller P
where c(r) = i δ(r − Ri ) is the concentration of force
than the dipole-dipole distances, we can take a far-field ap- dipoles in the solution. Thus, we find that the second moment
proximation by expanding the difference to first order in `, of the force is
2
X ∂ 2 Gαβ (R − Ri ) hFH (R, t)FH (R, t0 )i = (3πηw) h`(t)`(t0 )i
δvα = `(t) eiβ eiµ eν mi (t) .
∂Riµ ∂Rν ∂ 2 Gxβ (r) ∂ 2 Gxβ 0 (r)
Z
i × dr c(R + r)
∂rµ ∂rx ∂rµ0 ∂rx
Therefore, the relative velocity with which the spring connect- × heβ eµ eβ 0 eµ0 i hm(t)m(t0 )i .
ing the two domains compresses or stretches is the projection
Assuming a uniform concentration, the variance of this force
X ∂ 2 Gαβ (R − Ri ) is
δv · e = `(t) eiβ eiµ eν eα mi (t) .
∂Riµ ∂Rν 2
i FH2 (R, t) = (3πηw) `2 (t) hm2 (t)ic0
∂ 2 Gxβ (r) ∂ 2 Gxβ 0 (r)
Z
Applying Stoke’s law, we find that the deformation forces act- × dr heβ eµ eβ 0 eµ0 i .
ing on the target dipole is ∂rµ ∂rx ∂rµ0 ∂rx
(13)
FH (R, t) = (3πηw) (δv · e) = 3πηw`(t)
Since dipolar orientation is uncorrelated, the 4-point correla-
X ∂ 2 Gαβ (R − Ri ) tion term heβ eµ eβ 0 eµ0 i vanishes unless there are even powers
× eiβ eiµ eν eα mi (t) , (11)
∂Riµ ∂Rν of the components of the orientation vector e. We can there-
i
fore write the 4-point correlation as a sum over products of
where w is the domain size (i.e., its hydrodynamic diameter) δ-functions,
and η the viscosity of the solution. Since enzymes are ran- heβ eµ eβ 0 eµ0 i = Ad [δββ 0 δµµ0 + δβµ δβ 0 µ0 + δβµ0 δβ 0 µ ] ,
domly oriented, then without any loss of generality, we take
the target enzyme oriented along the x-axis, thereby simplify- where Ad = 1/15 for a 3D system.
ing Eq. (11) into To proceed further, we use a far-field approximation for
Gαβ in terms of the Oseen tensor [81, 101], which for a 3D
X ∂ 2 Gxβ (R − Ri ) system is
FH (R, t) = 3πηw`(t) eiβ eiµ mi (t) .
∂Riµ ∂Rx 1 h rα rβ i
i
(12) Gαβ (r) = δαβ + 2 . (14)
8πηr r9
The Oseen approximation is valid as long as the separation where ξW (t) is a white noise source with zero mean and unit
between dipoles is large compared to their size. Substituting variance, A is the energy scale of the active force, and τA cor-
Eq. (14) in Eq. (13) and introducing a scaled coordinate ξ = relation time of the activity. The corresponding active force
r/`0 , we find statistics is given by
0
9 hζA (t)ζA (t0 )i = (A/τA ) e−|t−t |/τA .
FH2 (R, t) = Ad h`2 (t)ihm2 (t)ic0 w2 `−3
0
64
Z ∞
∂ Gxβ (ξ) ∂ Gxβ 0 (ξ)
2 2 For an Ornstein-Uhlenbeck process, the fluctuation-
× dξ dissipation relation implies that A is proportional
1 ∂ξµ ∂ξx ∂ξµ0 ∂ξx to τA . Hence, the variance of the active force,
× (δββ 0 δµµ0 + δβµ δβ 0 µ0 + δβµ0 δβ 0 µ ) , (15) FH2 = hζA2 (t)i = A/τA , remains constant.
To examine the impact of active noise on barrier crossing,
where the scaled Oseen tensor is Gαβ (ξ) = ξ −1 (1+ξα ξβ /ξ 2 ). we numerically solve many realizations of Eq. (17) and an-
Since the mobility tensor diverges as 1/ξ 3 at small distances, alyze the statistics of crossing events. For this purpose, we
we introduce a cut-off in the lower limit of the integration ac- introduce the following scaling
counting for the the finite size of the dipole (i.e., enzyme).
The integral in the Eq. (15) is a dimensionless factor, which t̄ = t/τ0 , q̄ = q/q0 ,
depends on the derivatives of Gαβ (ξ) and the dipole orienta-
tions. A straightforward calculation yields where τ0 = γ/a is the thermal relaxation time of the particle
Z ∞ in the vicinity of the minimum at q0 . Using the above scaling,
∂ 2 Gxβ (ξ) ∂ 2 Gxβ 0 (ξ) we obtain a dimensionless form of the Eqs. (17,18)
dξ
1 ∂ξµ ∂ξx ∂ξµ0 ∂ξx q̄˙ = q̄ − q̄ 3 + ζ¯A (t̄) + ζ¯A (t̄) , (19)
96π
× (δββ 0 δµµ0 + δβµ δβ 0 µ0 + δβµ0 δβ 0 µ ) = . τ̄A ζ̄˙A = − ζ¯A + 2Ā ξW (t̄) ,
p
5 (20)
Finally, substituting the value of integral in Eq. (15), we find with Ā = A/(4γEB ). The corresponding scaled noise statis-
the variance of the hydrodynamic force, tics are
ζ¯T (t̄)ζ¯T (t̄0 ) = σT2 δ (t̄ − t̄0 ) ,
!
9π w2 h`2 i m2
2
c0 `30 . (16)
FH (R, t) = · 2 · 2 2
50 `0 `0 `0 ζ¯A (t̄)ζ¯A (t̄0 ) = σA2 exp (−|t̄ − t̄0 |/τ̄A ) ,
The dependence of the hydrodynamic force on the inter-dipole hξW (t̄)ξW (t̄0 )i = δ (t̄ − t̄0 ) , (21)
distance R arises from the dipole concentration c0 = 1/R3 . where σA2 = FH2 /(4aEB ), and σT2 = kB T /(2EB ) are the
The first three terms on the right-hand side of Eq. (16) are scaled active and thermal noise strength. Eqs. (19,20,21) are
combined into a geometric factor λ = (9π/50)w2 h`2 i/`40 . the central equations in our numerical and analytical study.
This constant is of order λ ' 1/2 since all the three lengths To simplify the notation, we will hereafter omit the overbar
are similar. We have used Eq. (16) to estimate the hydrody- in the scaled variables (so q = q̄ etc.).
namic force generated by an enzymatic solution.
The numerical simulation. We solve Eqs. (19) employing
Barrier crossing under the combined influence of thermal an explicit Euler scheme [102], which yields the following
and active noise. We examine overdamped Langevin dy- iterative dynamics for the reaction coordinate:
namics in a reaction energy landscape U (q) of a symmetric
bistable system, q(t + dt) = q(t)(1 + dt) − q 3 (t) dt + XζT + XζA ,
a b where XζT and XζA are the random processes
U (q) = − q 2 + q 4 .
2 4 Z t+dt
XζT = ζT (u) du , (22)
p
U (q) has two minima at qm = ± a/b , separated by an en-
t
ergy barrier, EB = a2 /(4b). In the overdamped Langevin Z t+dt
framework, the reaction coordinate q evolves according to XζA = ζA (u) du . (23)
t
∂U
γ q̇ = − + ζT (t) + ζA (t) . (17) The Gaussian distribution of the white thermal noise ζT has
∂q 2
zero mean, and a variance √ σT . Therefore, the distribution of
The noise term ζT (t) in Eq.[17] is a standard stochastic ther- XζT is simply XζT = dt σT YT , where YT ∼ N(0, 1) is
mal force with the statistics distributed according to the standard normal distribution with
hζT (t)ζT (t0 )i = 2γkB T δ(t − t0 ) , zero mean and unit variance.
Integrating Eq. (20), we obtain a formal solution for the
The active force ζT (t) is modeled as an Ornstein-Uhlenbeck active noise,
Process, √ Z t
√ −t/τA 2A
ζA (t) = e ζA (0) + e(u−t)/τA ξW (u) du .
τA ζ̇A = −ζA + 2A ξW (t) , (18) τA 010
Substitution of the latter result into Eq. (23), yields the statis- with the correlations defined in terms of µ = dt/τA as
tics of XζA . To proceed further, we define two Gaussian pro-
τA
Ω20 = 1 − e−2µ ,
cesses [85, 102],
2
3
τ
Ω21 = A 2µ − 3 − e−2µ + 4e−µ ,
2
τ2
hΩ0 Ω1 i = A 1 − 2e−µ + e−2µ ,
2
Z dt
Ω0 = du e(u−dt)/τA ξW (u) , and Y0 ∼ N(0, 1) and Y1 ∼ N(0, 1) are two independent stan-
0 dard Gaussian processes of zero mean and unit variance. With
Z dt Z u the expressions for the stochastic processes, the time update
Ω1 = du dv e(v−u)/τA ξW (v) . algorithm for active noise and reaction coordinate becomes
0 0
√
−µ 2A
ζA (t + dt) = e ζA (t) + Ω0 ,
τA
√
q(t + dt) = q(t)(1 + dt) − q 3 (t) dt + dt σT YT
√
−µ
2A
Solving these equations, we express the Ω0 , Ω1 processes as + τA 1 − e ζA (t) + Ω1 .
τA
To calculate the barrier crossing rate, we consider a parti-
cle, initially positioned at the left minimum q = −1 (i.e., q =
−q0 ). We then monitor the particle trajectory and find the
first passage time—the time when the particle crosses the en-
ergy barrier for the first time. We repeat the process for 105
q
Ω0 = hΩ20 i Y0 ,
s independent noise realizations and average to obtain mean
hΩ0 Ω1 i
2
hΩ0 Ω1 i first passage time τMFP . In a bistable system, the reaction
Ω1 = p 2 Y0 + hΩ21 i − Y1 , rate κ is inversely proportional to the mean first passage time,
hΩ0 i hΩ20 i κ = 12 τMFP
−1
.
[1] J. Haldane, Enzymes (Longmans, Green and Company, 1930). 116, 30008 (2016).
[2] L. C. Pauling, Molecular architecture and biological reactions, [12] C. Battle, C. P. Broedersz, N. Fakhri, V. F. Geyer, J. Howard,
Chem Eng News 24, 1375 (1946). C. F. Schmidt, and F. C. MacKintosh, Broken detailed balance
[3] L. C. Pauling, Chemical achievement and hope for the future, at mesoscopic scales in active biological systems, Science 352,
American scientist 36, 51—58 (1948). 604 (2016).
[4] A. Warshel and M. Levitt, Theoretical studies of enzymic re- [13] W. W. Ahmed, Étienne Fodor, M. Almonacid, M. Bussonnier,
actions: Dielectric, electrostatic and steric stabilization of the M.-H. Verlhac, N. Gov, P. Visco, F. van Wijland, and T. Betz,
carbonium ion in the reaction of lysozyme, J Mol Biol 103, Active mechanics reveal molecular-scale force kinetics in liv-
227 (1976). ing oocytes, Biophys J 114, 1667 (2018).
[5] A. Warshel, Energetics of enzyme catalysis, Proc Natl Acad [14] R. H. Austin, K. W. Beeson, L. Eisenstein, H. Frauenfelder,
Sci 75, 5250 (1978). and I. C. Gunsalus, Dynamics of ligand binding to myoglobin,
[6] J. Kraut, How do enzymes work?, Science 242, 533 (1988). Biochemistry (Mosc ) 14, 5355 (1975).
[7] A. Fersht, Structure and Mechanism in Protein Science (World [15] M. Gerstein, A. M. Lesk, and C. Chothia, Structural mecha-
Scientific, 2017). nisms for domain movements in proteins, Biochemistry (Mosc
[8] A. Kessel and N. Ben-Tal, Introduction to Proteins: Structure, ) 33, 6739 (1994).
Function, and Motion (CRC Press, 2018). [16] G. G. Hammes, Multiple conformational changes in enzyme
[9] M. Guo, A. Ehrlicher, M. Jensen, M. Renz, J. Moore, R. Gold- catalysis, Biochemistry (Mosc ) 41, 8221 (2002).
man, J. Lippincott-Schwartz, F. Mackintosh, and D. Weitz, [17] R. M. Daniel, R. V. Dunn, J. L. Finney, and J. C. Smith,
Probing the stochastic, motor-driven properties of the cyto- The role of dynamics in enzyme activity, Annu Rev Biophys
plasm using force spectrum microscopy, Cell 158, 822 (2014). Biomol Struct 32, 69 (2003).
[10] H. Turlier, D. A. Fedosov, B. Audoly, T. Auth, N. S. Gov, [18] A. Gutteridge and J. Thornton, Conformational changes ob-
C. Sykes, J.-F. Joanny, G. Gompper, and T. Betz, Equilibrium served in enzyme crystal structures upon substrate binding, J
physics breakdown reveals the active nature of red blood cell Mol Biol 346, 21 (2005).
flickering, Nat Phys 12, 513 (2016). [19] D. D. Boehr, H. J. Dyson, and P. E. Wright, An nmr perspec-
[11] É. Fodor, W. W. Ahmed, M. Almonacid, M. Bussonnier, N. S. tive on enzyme dynamics, Chem Rev 106, 3055 (2006).
Gov, M.-H. Verlhac, T. Betz, P. Visco, and F. van Wijland, [20] Z. D. Nagel and J. P. Klinman, A 21st century revisionist’s
Nonequilibrium dissipation in living oocytes, Europhys Lett view at a turning point in enzymology, Nat Chem Biol 5, 54311
(2009). (2018).
[21] D. R. Glowacki, J. N. Harvey, and A. J. Mulholland, Taking [43] M. Wolf-Watz, V. Thai, K. Henzler-Wildman, G. Hadjipavlou,
ockham’s razor to enzyme dynamics and catalysis, Nat Chem E. Z. Eisenmesser, and D. Kern, Linkage between dynamics
4, 169 (2012). and catalysis in a thermophilic-mesophilic enzyme pair, Na-
[22] G. Bhabha, J. T. Biel, and J. S. Fraser, Keep on moving: Dis- ture Structural Molecular Biology 11, 945 (2004).
covering and perturbing the conformational dynamics of en- [44] K. Henzler-Wildman, V. Thai, M. Lei, and et al., Intrinsic mo-
zymes, Acc. Chem. Res. 48, 423 (2015). tions along an enzymatic reaction trajectory, Nature 450, 838
[23] R. Callender and R. B. Dyer, The dynamical nature of enzy- (2007).
matic catalysis, Acc. Chem. Res. 48, 407 (2015). [45] K. A. Henzler-Wildman, M. Lei, V. Thai, S. J. Kerns,
[24] A. G. Palmer, Enzyme dynamics from nmr spectroscopy, Acc. M. Karplus, and D. Kern, A hierarchy of timescales in pro-
Chem. Res. 48, 457 (2015). tein dynamics is linked to enzyme catalysis, Nature 450, 913
[25] M. R. Mitchell, T. Tlusty, and S. Leibler, Strain analysis (2007).
of protein structures and low dimensionality of mechanical [46] U. Olsson and M. Wolf-Watz, Overlap between folding and
allosteric couplings, Proc Natl Acad Sci USA 113, E5847 functional energy landscapes for adenylate kinase conforma-
(2016). tional change, Nat Commun 1, 111 (2010).
[26] D. Koshland, Application of a theory of enzyme specificity to [47] J. R. Schnell, H. J. Dyson, and P. E. Wright, Structure, dynam-
protein synthesis, Proc Natl Acad Sci 44, 98 (1958). ics, and catalytic function of dihydrofolate reductase, Annu
[27] B. Ma and R. Nussinov, Enzyme dynamics point to stepwise Rev Biophys Biomol Struct 33, 119 (2004).
conformational selection in catalysis, Curr Opin Chem Biol [48] R. P. Venkitakrishnan, E. Zaborowski, D. McElheny, S. J.
14, 652 (2010). Benkovic, H. J. Dyson, and P. E. Wright, Conformational
[28] B. G. Vértessy and F. Orosz, From “fluctuation fit” to “confor- changes in the active site loops of dihydrofolate reductase
mational selection”: Evolution, rediscovery, and integration of during the catalytic cycle, Biochemistry (Mosc ) 43, 16046
a concept, Bioessays 33, 30 (2011). (2004).
[29] J. Monod, J. Wyman, and J.-P. Changeux, On the nature of [49] D. D. Boehr, D. McElheny, H. J. Dyson, and P. E. Wright, The
allosteric transitions: A plausible model, J Mol Biol 12, 88 dynamic energy landscape of dihydrofolate reductase cataly-
(1965). sis, Science 313, 1638 (2006).
[30] M. F. Perutz, Stereochemistry of cooperative effects in [50] S. Hammes-Schiffer and S. J. Benkovic, Relating protein mo-
haemoglobin: Haem-haem interaction and the problem of al- tion to catalysis, Annu Rev Biochem 75, 519 (2006).
lostery, Nature 228, 726 (1970). [51] G. Bhabha, J. Lee, D. C. Ekiert, J. Gam, I. A. Wilson, H. J.
[31] N. M. Goodey and S. J. Benkovic, Allosteric regulation and Dyson, S. J. Benkovic, and P. E. Wright, A dynamic knockout
catalysis emerge via a common route, Nat Chem Biol 4, 474 reveals that conformational fluctuations influence the chemical
(2008). step of enzyme catalysis, Science 332, 234 (2011).
[32] H. N. Motlagh, J. O. Wrabl, J. Li, and V. J. Hilser, The ensem- [52] L. Y. P. Luk, J. Javier Ruiz-Pernía, W. M. Dawson, M. Roca,
ble nature of allostery, Nature 508, 331 (2014). E. J. Loveridge, D. R. Glowacki, J. N. Harvey, A. J. Mulhol-
[33] K. H. DuBay, G. R. Bowman, and P. L. Geissler, Fluctuations land, I. Tuñón, V. Moliner, and R. K. Allemann, Unraveling
within folded proteins: Implications for thermodynamic and the role of protein dynamics in dihydrofolate reductase catal-
allosteric regulation, Acc. Chem. Res. 48, 1098 (2015). ysis, Proc Natl Acad Sci USA 110, 16344 (2013).
[34] B. P. English, W. Min, A. M. van Oijen, K. T. Lee, G. Luo, [53] P. Hanoian, C. T. Liu, S. Hammes-Schiffer, and S. Benkovic,
H. Sun, B. J. Cherayil, S. C. Kou, and X. S. Xie, Ever- Perspectives on electrostatics and conformational motions in
fluctuating single enzyme molecules: Michaelis-menten equa- enzyme catalysis, Acc. Chem. Res. 48, 482 (2015).
tion revisited, Nat Chem Biol 2, 87 (2006). [54] E. Z. Eisenmesser, O. Millet, W. Labeikovsky, D. M. Korzh-
[35] E. Campbell, M. Kaltenbach, G. J. Correy, P. D. Carr, B. T. nev, M. Wolf-Watz, D. A. Bosco, J. J. Skalicky, L. E. Kay, and
Porebski, E. K. Livingstone, L. Afriat-Jurnou, A. M. Buckle, D. Kern, Intrinsic dynamics of an enzyme underlies catalysis,
M. Weik, F. Hollfelder, N. Tokuriki, and C. J. Jackson, The Nature 438, 117 (2005).
role of protein dynamics in the evolution of new enzyme func- [55] S. Kale, G. Ulas, J. Song, G. W. Brudvig, W. Furey, and F. Jor-
tion, Nat Chem Biol 12, 944 (2016). dan, Efficient coupling of catalysis and dynamics in the e1
[36] J. Howard, Molecular motors: structural adaptations to cellu- component of escherichia coli pyruvate dehydrogenase mul-
lar functions, Nature 389, 561 (1997). tienzyme complex, Proc Natl Acad Sci USA 105, 1158 (2008).
[37] R. D. Vale, The molecular motor toolbox for intracellular [56] S. C. L. Kamerlin and A. Warshel, At the dawn of the 21st cen-
transport, Cell 112, 467 (2003). tury: Is dynamics the missing link for understanding enzyme
[38] N. Kodera, D. Yamamoto, R. Ishikawa, and T. Ando, Video catalysis?, Proteins 78, 1339 (2010).
imaging of walking myosin v by high-speed atomic force mi- [57] Y. Togashi and A. S. Mikhailov, Nonlinear relaxation dynam-
croscopy, Nature 468, 72 (2010). ics in elastic networks and design principles of molecular ma-
[39] H. S. Muddana, S. Sengupta, T. E. Mallouk, A. Sen, and P. J. chines, Proc Natl Acad Sci 104, 8697 (2007).
Butler, Substrate catalysis enhances single-enzyme diffusion, [58] H. Flechsig and A. S. Mikhailov, Tracing entire operation cy-
J Am Chem Soc 132, 2110 (2010), pMID: 20108965. cles of molecular motor hepatitis c virus helicase in struc-
[40] K. K. Dey, X. Zhao, B. M. Tansi, W. J. Méndez-Ortiz, U. M. turally resolved dynamical simulations, Proc Natl Acad Sci
Córdova-Figueroa, R. Golestanian, and A. Sen, Micromotors 107, 20875 (2010).
powered by enzyme catalysis, Nano Lett 15, 8311 (2015). [59] D. R. Hekstra, K. I. White, M. A. Socolich, R. W. Henning,
[41] A.-Y. Jee, S. Dutta, Y.-K. Cho, T. Tlusty, and S. Granick, En- V. Srajer, and R. Ranganathan, Electric-field-stimulated pro-
zyme leaps fuel antichemotaxis, Proc Natl Acad Sci 115, 14 tein mechanics., Nature 540, 400 (2016).
(2018). [60] X. Ma, A. C. Hortelão, T. Patiño, and S. Sánchez, Enzyme
[42] A.-Y. Jee, Y.-K. Cho, S. Granick, and T. Tlusty, Catalytic catalysis to power micro/nanomachines, ACS Nano 10, 9111
enzymes are active matter, Proc Natl Acad Sci 115, E10812 (2016).You can also read
