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arXiv:1508.03219v1 [q-bio.BM] 13 Aug 2015
Enhanced Diffusion of Enzymes that Catalyze Exothermic Reactions
Ramin Golestanian∗
Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3NP, UK
(Dated: August 14, 2015)
Enzymes have been recently found to exhibit enhanced diffusion due to their catalytic activities.
A recent experiment [C. Riedel et al., Nature 517, 227 (2015)] has found evidence that suggests this
phenomenon might be controlled by the degree of exothermicity of the catalytic reaction involved.
Four mechanisms that can lead to this effect, namely, self-thermophoresis, boost in kinetic energy,
stochastic swimming, and collective heating, are critically discussed, and it is shown that only
the last two could be strong enough to account for the observations. The resulting quantitative
description is used to examine the biological significance of the effect.
PACS numbers: 87.14.ej,87.10.Ca,65.80.-g,87.16.Uv
Introduction.—A most fascinating aspect of the
nonequilibrium processes in living cells is active transport
[1]. The basic units of these processes, which could be in
the form of carrying cargo or sliding actin fibres against
one another, are motor proteins that convert chemical
energy directly into useful mechanical work, amidst dom-
inant thermal fluctuations at the nano-scale [2]. Recent
in vitro studies of mixtures of motors and filaments have
revealed their remarkable ability to self-organize into dy-
namic meso-scale structures that resemble those observed
in living cells [3–5]. Much less is known about the na-
ture of the nonequilibrium activity of non-cytoskeletal
elements and how they self-organize in living cells.
It has been recently reported that enzymes undergo
enhanced diffusion, i.e. diffusive motion with an effec-
tive diffusion coefficient Deff that is larger than its equi-
librium value D0, as a result of their catalytic activity
[6, 7]. Considering the enzyme as a sphere of radius R
in a medium with viscosity ηand temperature T, the
Stokes-Einstein relation D0=kBT /ζ gives us the equilib-
rium diffusion coefficient of the enzyme, where ζ= 6πηR
is its friction coefficient. The additional, nonequilibrium,
contribution to the diffusion coefficient, ∆D=Deff −D0,
is found to be proportional to the net rate (or speed) of
the catalytic reaction. The rate has the characteristic
Michaelis-Menten form k=keS/(KM+S), where Sis
the substrate (i.e. reactant) concentration, KMis the
Michaelis constant, and keis the enzyme reaction rate.
Remarkably, ∆Dhas the same order of magnitude as the
equilibrium diffusion coefficient; typically a fraction of it.
It has also been observed that there is a strong correla-
tion between the degree of exothermicity of the catalytic
reaction and the enhancement in the effective diffusion
coefficient of enzymes [8]. In this Letter, I discuss and
critically examine various mechanisms that can lead to
enhanced diffusion for catalytically active enzymes.
Self-phoresis.—Any colloidal particle that actively gen-
erates nonequilibrium phoretic flow in its vicinity exhibits
enhanced effective diffusion at time scales longer than its
orientational persistence time 1/Dr, where Dris the ro-
tational diffusion coefficient [9]. Enzyme activity has the
right ingredients to lead to enhanced diffusion via self-
diffusiophoresis, which takes advantage of gradients in
the concentrations of the chemicals involved in the reac-
tion [10]. However, this contribution is not sensitive to
the degree of exothermicity of the reaction. The appro-
priate mechanism that could account for such an effect is
self-thermophoresis [11, 12]. The heat released from the
chemical reaction during each catalytic cycle, Q, leads to
a temperature difference ∆T≃kQ/(κR) across the en-
zyme, where κis the thermal conductivity of the medium.
For catalase k= 5 ×104s−1,Q= 40 kBT, and R= 4 nm,
which gives D0= 55 µm2s−1and Dr= 2.6×106s−1.
Using these values and κ∼0.6 W/(m ·K) for water, we
obtain ∆T∼10−6K. The self-propulsion velocity is es-
timated as Vst ∼D0ST∆T/R ∼D0STkQ/(κR2) [11],
where STis the Soret coefficient of the enzyme. Using
ST≃0.02 K−1[13], we find Vst ∼10−4µm s−1. The
correction to effective diffusion coefficient due to self-
thermophoresis is ∆D≃V2
st/Dr, for which we find ∆D≃
10−14 µm2s−1, and consequently ∆D/D0≃10−16. This
is fifteen orders of magnitude too small to account for the
observations.
Boost in Kinetic Energy.—The authors of Ref. [8]
propose a scenario in which the heat released from the
chemical reaction during each catalytic cycle is channeled
into a boost in the translational velocity of the enzyme.
This is not envisaged to be mediated through an effec-
tive temperature increase following the release of heat.
Here I reproduce their analysis using a slightly differ-
ent derivation, to highlight the essence of the proposed
mechanism. Consider the enzyme to be a particle of
mass mwhose stochastic motion satisfies Newton’s equa-
tion md2r
dt +ζdr
dt =f(t), where f(t) is a random force.
The relative significance of the inertial and the dissipa-
tive terms in the equation of motion is characterized by
the time scale τ=m/ζ . Using m=4π
3R3ρpwith a
typical protein mass density ρp= 1.4×103kg/m3and
η= 1 ×10−3Pa·s for viscosity of water at room tempera-
ture, we find τ= 5 ps. Invoking an elegant mathematical
trick that Langevin used in his original 1908 paper [14],
2
we can write the equation of motion as
τd
dt + 1Deff (t) = 2
3ζE(t),(1)
where Deff (t) = 1
6
d
dt hr(t)2iis by definition the effective
diffusion coefficient and E(t) = hm
2dr
dt 2iis the average
kinetic energy of the enzyme. Here we have assumed a
separation of time scales between the random thermal
kicks that the enzyme receives from the medium and the
catalytic cycle, and the averaging is performed over the
thermal kicks. We can write E(t) = 3
2kBT+γQh(t) where
γrepresents the fraction of the released thermal energy
that is converted into the translational boost and h(t)
is a series of spikes of width τbthat appear stochasti-
cally at a rate k, through a Poisson process. Here, τbis
the relaxation time of the boost, which depends on the
specific process that generates it. It is reasonable to as-
sume that τb≈τ. The boost mechanism proposed in
Ref. [8] involves asymmetric excitation of compressional
waves along the enzyme that propagate to the interface
with water and trigger a pressure wave that leads to a
back-reaction on the enzyme itself, giving it a mechani-
cal boost. No evidence is provided in Ref. [8] as to why
the energy is not randomly partitioned between a large
number of possible channels (owing to the large number
of degrees of freedom or normal modes), which would re-
sult in γ≪1, and subsequently dissipated, as opposed to
being channeled to a small number of modes (correspond-
ing to γ∼1). Time averaging gives E=3
2kBT+γQkτb,
and consequently
∆D
D0
Ref.[8]
=2
3
γQ
kBTkτb,(2)
through Eq. (1). Using the above estimate for τband the
values for kand Qcorresponding to catalase, we obtain
kτb= 2.5×10−7and ∆D/D0=γ×10−5from Eq. (2).
Even with the (unrealistic) maximum value of γ= 1, our
estimate from Eq. (2) is four orders of magnitude too
small to account for the observations.
Stochastic Swimming.—We could examine various hy-
drodynamic effects that might contribute towards such
a behaviour. Substrate binding could change the shape
of the enzyme, and consequently its friction coefficient.
Since this process does not require energy input, however,
it cannot be the cause of a nonequilibrium phenomenon.
Such conformational changes are often relatively small,
and will more likely lead to an increase in size that would
result in a decrease in diffusion coefficient, rather than
the other way around. Moreover, it is not clear why such
an effect could lead to universal trends—as it will depend
on specific cases—and how it can correlate with exother-
micity.
As an alternative scenario, it is possible that the cat-
alytic cycle induces conformational changes in the en-
zyme that lead to stochastic swimming [15]. The am-
plitude of these deformations is typically much smaller
than the size of the enzyme, e.g. when they arise from
mechanochemical coupling of electrostatic nature [16]
(analogous to phosphorylation) or structural changes due
to ligand binding [17]. However, local heat release could
have the possibility to transiently disturb the relatively
more fragile tertiary structure of the folded protein [18]
or the state of oligomerization of the enzyme [19], and
produce an amplitude bthat is a fraction of the size R.
To calculate the contribution of such conformational
changes to effective diffusion coefficient, we use a sim-
ple model in which the conformational change is de-
scribed by one degree of freedom L(t) representing elon-
gation of the structure along an axis defined by a unit
vector ˆ
n(t). To achieve directed swimming, we need
at least two degrees of freedom to incorporate the co-
herence needed for breaking the time-reversal symme-
try at a stochastic level, and we know that realistic con-
formational changes must involve many degrees of free-
dom. The randomization of the orientation, described via
hˆ
n(t)·ˆ
n(t′)i=e−2Dr|t−t′|, will turn the directed motion
into enhanced diffusion over the time scales longer than
1/Dr. Since the same can be achieved through recip-
rocal conformational changes described by one compact
degree of freedom, I will adopt this simpler form. The
stochastic motion of the enzyme can be described by the
Langevin equation v(t)≃αd
dt Lˆ
n(t) + ξ(t) where α
is a numerical pre-factor that depends on the geometry
of the enzyme [20, 21] and ξ(t) is the Gaussian white
noise that will give us the intrinsic translational diffusion
coefficient D0.
We describe the combined mechanochemical cycle us-
ing a two-step process, which takes the enzyme from its
free state to the reaction stage that is followed by the de-
formation with rate k, and a relaxation back to its native
state with rate kr. This is a simplification of a more real-
istic model with three states (free, substrate-bound, and
reacted-deformed) and kis to be understood as the com-
~ l
T
a
Heat Flux
T
real
T
app
FIG. 1. (color online.) Schematic illustration of the essence
of Newton’s law of cooling. The heat generated in the bulk of
a chamber (that is in contact with the environment that has
a fixed ambient temperature Ta), and lost in the form of heat
flux through the boundaries leads to the temperature profile
Treal (solid line), which is approximated by the average value
over the distance ∼ℓ,Tapp (dashed line). This approximation
works best when there is a separation of length scales.
3
0
0.1
0.2
Θ
0.5
1
kkmax
0
0.1
0.2
Θ
0.5
1
Ce
aCe
0.02
0.04
∆
0.02
0.04
Θ
0.02
0.04
∆
0.1
0.2
0.3
DDD0
0.005
0.01
0.015
∆
0.05
0.1
Θ
(a) (b) (c) (d)
FIG. 2. (color online.) (a) The effective rate of catalytic reaction as a function of the reduced temperature θfor θd= 0.1,
g= 50, and ǫ= 7. The inset shows the fraction of catalytically active enzymes as a function of θin each case. (b) The effective
temperature of the catalytically active medium as a function of the coupling strength δ. (c) The corresponding relative increase
in diffusion coefficient. (d) The effective temperature for ǫ= 30, with the dashed region being unstable.
bined catalytic rate that has the Michaelis-Menten form
as defined above. In stationary state, a master equa-
tion formulation can be used to calculate the elongation
speed autocorrelation function as d
dt L(t)·d
dt′L(t′)=
2b2kkr
k+kr[δ(t−t′)−1
2(k+kr)e−(k+kr)|t−t′|]. By com-
bining this with the orientation auto-correlation, we can
calculate the effective diffusion coefficient of the enzyme,
which gives the following correction
∆D=1
3α2b2kkr
k+kr2Dr
2Dr+k+kr
.(3)
Even for the fastest enzymes, we typically have kr≈
Dr≫k. Using an upper bound of b.R, we can ap-
proximate Eq. (3) as ∆D≈kR2. For catalase, we ob-
tain ∆D≈1µm2s−1, which gives an upper bound of
∆D/D0≈10−2. This is one order of magnitude smaller
than the observed values.
Collective Heating.—In Ref. [8], a calculation similar
to what we have above to estimate the relative change
in temperature across the enzyme, ∆T, is used to ar-
gue that heating of the environment by the enzyme is
negligible. This estimate, however, is only correct for an
isolated enzyme. In practice, an experiment is performed
on a solution with a finite concentration of enzyme, Ce.
For such a sample, the substrate is consumed at the rate
(per unit volume) of keSCa
e/(KM+S), where Ca
eis the
concentration of the catalytically active enzymes. The
exothermic catalytic reaction generates thermal energy
Qper turnover cycle at the location of each active en-
zyme, which then diffuses through the sample container
and escapes via the boundaries. Due to the large number
of heat producing enzymes (of the order of Avogadro’s
number) there will be a significant build-up of thermal
energy in the sample container. To see this, let us define
a length scale ℓthat describes the characteristic distance
heat needs to diffuse until it can exit. ℓis typically set by
the smallest length scale in the geometry of the sample
container. We can estimate a characteristic heat diffu-
sion time τh(ℓ) = ℓ2/χ, using the thermophoretic con-
ductivity χ. For water at room temperature, we have
χ≃105µm2s−1. For ℓ= 10 mm, it takes τh= 1000 s for
the heat released from each enzyme during each catalytic
cycle to leave the container. Given Ce= 1 nM [8], during
this time 1020 units of Qwill have been released into the
chamber (assumed to have a volume ∼ℓ3); this is 10 J of
thermal energy.
The heat diffusion equation is written as
χ−1∂tT− ∇2T=Q
κ·keSC a
e
KM+S−1
ℓ2(T−Ta).(4)
The right hand side of Eq. (4) contains a source term
that couples the catalytic reaction to the production of
heat, and a sink term that approximates the heat loss
through the boundaries by a bulk term in the form of
Newton’s law of cooling [22], where Tais the ambient
temperature. This term is written in terms of the length
scale ℓthat is described above. This approximation has
been widely used in the combustion literature to allow the
temperature-dependent nonlinearities in the source term
to be captured in a manner that does not depend on the
geometric specificities of each experiment [23]. Figure 1
summarizes the essence of this approximation.
Two quantities in the source term of Eq. (4) have sig-
nificant dependence on temperature, which we will rep-
resent using θ= (T−Ta)/Ta. The first quantity is the
turnover rate, which we can assume to have an Arrhenius
form of ke=k∗
0e−Ea/kBT, where Eais the activation en-
ergy. The rate can be rewritten as ke=k0eǫθ/(1+θ),
where ǫ=Ea/kBTaand k0=k∗
0e−ǫ. The second quan-
tity is the concentration of active enzymes. Since en-
zymes are proteins, increase in temperature will eventu-
ally denature them upon approaching the denaturation
temperature, which we denote by θd. We can use a simple
two-state model to account for the denaturation, which
yields Ca
e=Ce/eg(θ−θd)+ 1, where the parameter g
controls the sharpness of the transition. The two effects
enter the heat source term in Eq. (4) via the product
keCa
e, whose temperature dependence is shown in Fig.
4
2(a), for g= 50, θd= 0.1 (that corresponds to Td= 330K
for Ta= 300K), and ǫ= 7, which is a typical value for the
activation energy for enzymes, such as catalase [24]. The
plot shows the two standard regimes of initial increase
in the effective rate due to the Arrhenius temperature
dependence and the sudden decline due to denaturation.
The peak will sharpen with increasing activation energy
ǫ. The inset of Fig. 2(a) shows the fraction of active
enzymes as a function of temperature.
Writing the temperature dependencies in Eq. (4) ex-
plicitly, we find
τh∂tθ−ℓ2∇2θ=δeǫθ/(1+θ)
eg(θ−θd)+ 1 −θ, (5)
where
δ=Qℓ2
κTa
·k0SCe
KM+S,(6)
emerges as a single dimensionless parameter that controls
the strength of collective heating. Assuming a uniform
profile, we can find the stationary state temperature of
the system as a function of δby setting the right hand
side of Eq. (5) to zero. The result is shown in Fig. 2(b).
Note that the temperature dependence of the kinetic rate
provides such a sensitive positive feedback mechanism
that could lead to unrealistically high temperatures, eas-
ily above enzyme denaturation and even above boiling
temperature of water (for a case such as catalase with
the large value of Q= 40 kBT). The presence of de-
naturation provides a negative feedback mechanism that
ensures such dramatic temperature increases are cut off.
The temperature increase can be used to calculate the
relative increase in diffusion coefficient ∆D/D0, by tak-
ing into account the corresponding variations in the fric-
tion coefficient. Denaturation will change the hydrody-
namic radius of the protein from its globular (folded)
form, with size R∼aN 1/3, to its coiled (unfolded) form,
with size Rg∼aN 3/5, where ais the Kuhn length and N
is the polymerization index. This yields Rg∼RN 4/15 .
For catalase, we have a≃1 nm, N≃100, Rg≃14 nm.
To calculate the diffusion coefficient within our simple
two-state model, we need to use the ensemble average
of the inverse size, namely, use 1/R with the probabil-
ity p= 1/eg(θ−θd)+ 1and 1/Rgwith the probabil-
ity 1 −p. We can use the following expression for the
temperature dependence of the viscosity of water η(θ) =
η0exp n(B/Ta)
1+θ−(T0/Ta)owhere B= 579K, T0= 138K, and
η0= 2.41 ×10−5Pa·s. The result is plotted in Fig. 2(c),
showing a similar pattern of behaviour as seen in θ. The
values obtained for ∆D/D0are of the same order of mag-
nitude as the experimentally observed values, while the
actual temperature increase in the solution is relatively
modest. Moreover, the trend in Fig. 2(c) resembles the
experimental results reported in Refs. [6–8].
To estimate typical experimental values for δ, we need
a typical length scale from the sample container and the
smallest conductivity involved in the geometry of the sys-
tem. For Fluorescence Correlation Spectroscopy (FCS)
experiments, a droplet of the solution is placed over a
glass slide. Using κ∼0.02 W/(m ·K) for air, ℓ= 5 mm,
and Ce= 1 nM (as used in Ref. [8]), k0= 5 ×104s−1,
and Q= 40 kBT, we find δ≃0.02 at substrate satu-
ration. This estimate appears to provide a rough order
of magnitude agreement with the observations of Ref.
[8]. Note that due to our approximate treatment of the
boundary conditions, order unity differences are to be ex-
pected when comparing to the experimental results. The
experiments in Refs [6, 7] use higher concentrations of
enzymes and similar sample sizes, and thus comfortably
fall in the regime described by the collective heating sce-
nario.
Discussion.—The collective heating model leads to
very specific predictions that could be experimentally
tested. The near linear dependence of the relative en-
hancement in diffusion coefficient on δ[Fig. 2(c)] and
the definition of δ[Eq. (6)] suggest a linear dependence
on the enzyme concentration and a quadratic dependence
on the size of the container. In practice, protein denat-
uration is irreversible, which suggests that recording ex-
perimental data over a long time scale would presumably
lead to a systematic reduction in the magnitude of the en-
hanced diffusion, provided the experiment is done under
the condition that the substrate concentration is main-
tained at a constant level.
The heating mechanism can also lead to interesting
nonlinear phenomena. While at small values of ǫ, we
observe a near-linear dependence of temperature on δ
[Fig. 2(b)], upon increasing ǫthe curve takes an S-shape
that develops an instability at sufficiently large values of
ǫ, as shown in Fig. 2(d). The instability will lead to
the formation of waves, which will dissipate in a sealed
sample container when all the fuel molecules are con-
sumed, following closely the phenomenology of flames in
combustion [25]. Moreover, collective heating could have
synergistic influence on the other mechanisms: while the
increase in temperature could facilitate the emergence of
large conformational changes in the tertiary structure or
the oligomerization state during the enzymatic turnover,
phoretic collective heating can lead to further instabili-
ties [26] that could accentuate the degree of fluctuations
in the system.
Finally, let us examine whether the total heat gen-
erated in a cell could be sufficient to trigger this ef-
fect. Considering a cell of size ℓ= 10 µm that is fully
packed with enzymes similar to catalase (that gives us
Ce∼1 mM), we find δ∼0.01. While this is an upper
limit, it certainly points to a strong possibility that the
enhanced diffusion via collective heating could be a con-
tributing factor to non-directed intracellular transport in
living cells.
In conclusion, enhanced diffusion of enzymes that cat-
alyze exothermic reactions could be explained by a com-
5
bination of global temperature increase in the sample
container, and possibly enhanced conformational changes
that can lead to a hydrodynamic enhancement of effec-
tive diffusion coefficient. Self-thermophoresis and boost
in kinetic energy as suggested by Ref. [8] are too weak to
account for the experimentally measured values of effec-
tive diffusion. Although the primary focus of this work
has been on enhanced diffusion of enzymes, the theoret-
ical description should be relevant to the study of any
class of thermally activated microswimmers.
I have benefitted from discussions with Carlos Busta-
mante, Krishna Kanti Dey, and Ayusman Sen.
∗ramin.golestanian@physics.ox.ac.uk
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