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PHYSICAL REVIEW RESEARCH 5, L032018 (2023)
Letter
Diffusion enhancement in bacterial cytoplasm through an active random force
Lingyu Meng ,1Yiteng Jin ,2Yichao Guan ,3Jiayi Xu,4and Jie Lin 1,2,*
1Peking-Tsinghua Center for Life Sciences, Academy for Advanced Interdisciplinary Studies, Peking University, Beijing 100871, China
2Center for Quantitative Biology, Academy for Advanced Interdisciplinary Studies, Peking University, Beijing 100871, China
3The Graduate Program in Biophysical Sciences, University of Chicago, Chicago, Illinois 60637, USA
4Yuanpei College, Peking University, Beijing 100871, China
(Received 5 September 2022; accepted 10 July 2023; published 7 August 2023)
Experiments have found that diffusion in metabolically active cells is much faster than in dormant cells,
especially for large particles. However, the mechanism of this size-dependent diffusion enhancement in living
cells is still unclear. In this Letter, we approximate the net effect of metabolic processes as a white-noise active
force and simulate a model system of bacterial cytoplasm with a highly polydisperse particle size distribution.
We find that diffusion enhancement in active cells relative to dormant cells can be more substantial for large
particles. Our simulations agree quantitatively with the experimental data of Escherichia coli, suggesting an
autocorrelation function of the active force proportional to the cube of the particle radius. We demonstrate that
such a white-noise active force is equivalent to an active force of about 0.57 pN with random orientation. Our
work unveils an emergent simplicity of random processes inside living cells.
DOI: 10.1103/PhysRevResearch.5.L032018
The efficient diffusion of cellular components is crucial
for various biological processes in bacteria since they do not
have active transport systems involving protein motors and
cytoskeletal filaments [1–4]. Meanwhile, bacterial cytoplasm
is highly crowded [5–8]. Particle diffusion inside the bacterial
cytoplasm is significantly suppressed compared with a dilute
solution [9–14]. Interestingly, the diffusion of large cellular
components, such as plasmids, protein filaments, and storage
granules, turns out to be much faster in metabolically active
cells than in dormant cells, i.e., cells depleted of ATP [15–19].
Cellular metabolic activities appear to fluidize the cytoplasm
and allow large components to sample cytoplasmic space.
Another important feature of bacterial cytoplasm is its poly-
dispersity with constituent sizes spanning from subnanometer
to micrometers [20–23]. Intriguingly, diffusion enhancement
of a metabolically active cell relative to a dormant cell is
size dependent as large components’ diffusion constants are
much more significantly increased while small molecules dif-
fuse with virtually the same diffusion constants in active and
dormant cells [16].
The physical mechanism underlying the size-dependent
diffusion behaviors in active cells is far from clear. Because
of the numerous ATP-consuming processes in vivo, finding
the dominant biological processes that speed up diffusion
may be difficult or even impossible. In a passive solution,
particles receive random kicks from neighboring molecules
*Corresponding author: linjie@pku.edu.cn
Published by the American Physical Society under the terms of the
Creative Commons Attribution 4.0 International license. Further
distribution of this work must maintain attribution to the author(s)
and the published article’s title, journal citation, and DOI.
due to thermal fluctuation. Therefore, the thermal noise’s am-
plitude is proportional to the temperature according to the
fluctuation-dissipation (FD) theorem. In contrast, in the cy-
toplasm of a metabolically active cell, particles also receive
random kicks from biomolecules such as ATPs, amino acids,
and other metabolites that do not follow a detailed balance
and are therefore out of equilibrium [24–27]. As a result, the
net effect of their random collision with a particle is a random
force not constrained by the FD theorem.
In this Letter, we simulate a model system of bacterial cy-
toplasm and adopt a coarse-grained approach by introducing
an active random force as the net effect of multiple active
processes in cells. We model this active random force as white
noise with an amplitude independent of temperature. Our
system is highly polydisperse, and according to the Stokes-
Einstein relation, the autocorrelation function of the thermal
random force is proportional to the particle radius. Inspired by
that, we set the autocorrelation function of the active random
force proportional to a power-law function of the particle
radius. We find that in both passive systems without an active
force (corresponding to dormant cells) and active systems
(corresponding to metabolically active cells), the diffusion
constants are reduced under high density compared with the
dilute limit. This diffusion reduction is stronger for larger
particles in both passive and active systems.
Nevertheless, the enhancements of diffusion constants in
active cells relative to dormant cells can be more substantial
for larger particles given an appropriate size-dependent active
random force. Most importantly, the experimentally measured
ratios of diffusion constants between active and dormant E.
coli cells agree quantitatively with our simulations and sug-
gest an autocorrelation function of the active random force
proportional to the cube of the particle radius. We further
demonstrate that such a white-noise active force is equiva-
2643-1564/2023/5(3)/L032018(6) L032018-1 Published by the American Physical Society
MENG, JIN, GUAN, XU, AND LIN PHYSICAL REVIEW RESEARCH 5, L032018 (2023)
FIG. 1. A snapshot of three-dimensional simulations using poly-
disperse spheres as a model system of the bacterial cytoplasm. In
this example, the volume fraction φ=0.58. Particles are colored
according to their sizes.
lent to an active force with a constant magnitude undergoing
rotational diffusion. From the data of E. coli, we infer the
magnitude of this active force as 0.57 pN, consistent with
typical force magnitudes in the cytoplasm [28]. Our results
shed light on the mechanical nature of out-of-equilibrium
processes in the bacterial cytoplasm and unveil an emergent
simplicity in complex living systems.
Size-dependent active random force. We model the various
cellular components by spherical particles with heterogeneous
radii to mimic the polydisperse cytoplasm (Fig. 1) and their
equations of motion using the Langevin dynamics,
ηi
dri,α
dt =−∂U
∂ri,α
+ξi,α +κi,α .(1)
Here, i=1,2,...,Nwhere Nis the number of particles
and α=x,y,zthe directions in the Cartesian coordinate. ηi
is the friction coefficient of the ith particle and obeys the
Stokes’ law ηi=6πνai, where aiis the radius, and νis the
viscosity of the background solvent. Uis the pairwise interac-
tion between particles which we model as U=1
2i= jk(ai+
aj−rij)2(ai+aj−rij) where rij =|ri−rj|and is the
Heaviside step function: Particles repel each other only when
they overlap. ξi,α is the thermal noise, and its autocorrelation
function obeys the FD theorem [29]
ξi,α (t)ξj,β (t)=12πνaikBTδijδαβδ(t−t).(2)
Here, kBis the Boltzmann constant, and Tis the temperature.
We introduce an active random force κi,α as the coarse-grained
outcome of active processes in a metabolically active cell, and
its autocorrelation function is independent of temperature,
κi,α (t)κj,β (t)=2Aaγ
iδijδαβδ(t−t).(3)
Here, we assume that the noise amplitude is a power-law
function of the particle radius where Aand γare constants.
Later, we will show that this size dependence of active random
force is consistent with experimental measurements.
To nondimensionalize the model, we choose the aver-
age particle radius a0, the thermal energy kBT, and t0=
6πνa3
0/kBTas the length, energy, and time unit, respectively.
The dimensionless equation of motion becomes
d˜ri,α
d˜
t=−1
˜ai
∂˜
U
∂˜ri,α
+˜
ξi,α +˜κi,α,(4)
where ˜
ξi,α is the dimensionless thermal noise with its autocor-
relation function ˜
ξi,α (˜
t),˜
ξj,β (˜
t)=2˜a−1
iδijδαβδ(˜
t−˜
t), and
˜κi,α is the dimensionless active noise with its autocorrelation
function ˜κi,α (˜
t),˜κj,β (˜
t)=2˜
A˜aγ−2
iδijδαβδ(˜
t−˜
t). Here, the
dimensionless active noise amplitude ˜
A=Aaγ−1
0/(6πνkBT).
In the following, variables with a tilde above are dimension-
less. We simulate Nparticles in a three-dimensional cubic box
with a dimensionless side length ˜
Lunder the periodic bound-
ary condition. The Nparticles’ radii obey a shifted lognormal
distribution ˜a=˜amin +exp[μ+σN(0,1)], where N(0,1) is
a standard normal random number, and μand σare constants.
The minimum radius ˜amin is 0.1. We set σ=0.85 to mimic
the polydisperse environment and choose μby setting the di-
mensionless average radius ˜amean =˜amin +exp(μ+σ2/2) as
1. We set the physical value of the average radius as a0=10
nm, so the particles’ radii cover two orders of magnitude, from
1 to about 100 nm, consistent with real bacterial cytoplasm
[9,16]. We fix the volume fraction φin each simulation. We
also set T=300 K and choose a large dimensionless spring
constant ˜
k=1000 to mimic a hard-sphere system. The total
simulation time ˜
ttot is 10 000 with a time step d˜
t=2×10−4.
Active noise facilitates diffusion of large particles. We first
investigate the effects of volume fractions on the diffusion
constants relative to the dilute limit for both passive and active
systems. We compute the diffusion constant for each particle
from the time-averaged mean square displacement (MSD)
as D=˜
r2(˜
t)/6˜
twith ˜
t=1 where ˜
r(˜
t)isthe
displacement vector during a time interval ˜
t. We choose
multiple values of φ, including 0.58 and 0.64, the critical
volume fractions of the glass transition, and random close
packing of a monodisperse system in three dimensions [30].
In the dilute limit, collisions between particles are negli-
gible, and the diffusion constant of an active particle with a
dimensionless radius ˜abecomes Ddilute =1/˜a+˜
A˜aγ−2.For
a passive particle with ˜
A=0, D0,dilute =1/˜a. In both pas-
sive and active systems, the reduction of diffusion constants
in high-volume fractions relative to the dilute limit is more
significant for larger particles [Figs. 2(a) and 2(b)]. To demon-
strate the effects of active random force, we compare the
diffusion constants of active and passive systems in the same
volume fraction, which is more biologically relevant. In the
dilute limit, the ratio of diffusion constants between active and
dormant cells is
Ddilute
D0,dilute
=1+˜
A˜aγ−1.(5)
Our simulation results for φ=0.01 confirm Eq. (5)
[Fig. 2(c)].
For systems with high volume fractions, neither Dwith
˜
A>0 nor D0with ˜
A=0 is equal to the dilute limit prediction
[Figs. 2(a) and 2(b)]. Intriguingly, we find that the simulation
results of D/D0−1 with high volume fractions still agree
reasonably well with the theoretical prediction from the dilute
limit, particularly for small particles with ˜a<1 [Fig. 2(d)].
L032018-2
DIFFUSION ENHANCEMENT IN BACTERIAL CYTOPLASM … PHYSICAL REVIEW RESEARCH 5, L032018 (2023)
(a) (b)
(c) (d)
FIG. 2. Diffusion constants of the simulated polydisperse sys-
tems. (a) The diffusion constants of passive systems relative to the
dilute limit. ˜ais the dimensionless radius. (b) The same analysis
as (a) but for an active system. Here, ˜
A=1, γ=3. The diffusion
constants in the dilute limit differ in (a) and (b). (c) D/D0−1for
different γs. Dis the diffusion constant of the active system, and
D0is the diffusion constant of the passive system. Here, φ=0.01,
˜
A=1. The lines with corresponding colors are the predictions in
the dilute limit. (d) The same analysis as (c) but for systems with
φ=0.58. In all panels, N=1000.
The above observation is nontrivial because all particles’
diffusion constants deviate from the dilute limit regardless of
size [Figs. 2(a) and 2(b)]. Deviations are observed for large
particles, and we define an effective γeff for large particles with
˜a>1.2 such that D/D0−1∼˜aγeff −1(see Supplemental Ma-
terial [31], Fig. S1). Experimentally, the diffusion constants of
small particles are close in active and dormant cells; however,
the diffusions of large particles are much faster in active cells
than in dormant cells [16]. Our simulations in the regime of
γ>1 agree with experiments [Fig. 2(d)].
We also compute the diffusion constant D=
˜
r2(˜
t)/6˜
tusing a longer ˜
twhere ˜
r2(˜
t)= ˜
L2/32
but still short enough to ensure that the finite system size
does not confine the particles’ MSDs, and our conclusions
are equally valid under this definition [Figs. S2(a), S2(b), and
S3]. We also compute the diffusion constants by fitting the
MSDs using ˜
r2(˜
t)=6D˜
tand obtain similar results
[Figs. S2(c) and S2(d)].
We track the trajectories of single particles (Fig. 3). We find
that particles with small radii can equally explore the system
in active and passive systems; however, particles with large
radii can only explore space in active systems and remain
localized in passive systems, consistent with experimental
observations [16].
Comparison with experiments. In Ref. [16], the authors
measured the MSDs of exogenous particles of multiple sizes
in active and dormant cells, including GFP-μNS particles,
which are GFP-labeled self-assembling avian reovirus pro-
teins with changeable sizes, and mini-RK2 plasmid, which is
an engineered low-copy-number plasmid in E. coli. We find
FIG. 3. Single particles’ trajectories projected to two dimen-
sions. (a) The trajectory of a large particle with ˜a=6.75 in a passive
system. The color represents the time elapsed from the beginning of
the trajectory. (b) The trajectory of the same particle in (a) but in an
active system. (c) The trajectory of a small particle with ˜a=2.44 in
a passive system. (d) The trajectory of the same particle in (c) but in
an active system. In (b) and (d), ˜
A=0.42 and γ=3. In all panels,
φ=0.58, N=1000.
that the measured MSDs are subject to large noises, making it
difficult to compute the diffusion constants accurately. To cir-
cumvent this problem, we compute the ratios of MSDs in ac-
tive and dormant cells at every moment and calculate the ratios
of diffusion constants between active and dormant cells D/D0
as the averaged MSD ratios over time. The experimental parti-
cle radius is converted to a dimensionless number using a0=
10 nm.
To compare with experiments, we simulate several dif-
ferent volume fractions since the actual volume fraction of
bacterial cytoplasm is unknown. Intriguingly, the simulated
D/D0−1 with γ=3 nicely matches the experimental data
[Fig. 4(a)], and we will explain the physical mechanism
of γ=3 later. We find that the simulated diffusion en-
hancements D/D0−1 are insensitive to the volume fraction,
although the diffusion constants Dand D0by themselves
change significantly with the volume fraction [Figs. 2(a) and
2(b)]. To confirm the robustness of our results, we also sim-
ulate a narrower distribution of particle radii that is more
Gaussian-like. The agreement between simulations and ex-
periments is equally valid (Fig. S4). Our results are also
independent of the length unit as we choose a different length
unit and obtain the same results (Fig. S5). Using the alter-
native definition of diffusion constants does not affect our
conclusions (Fig. S6).
We note that for a dilute active system with γ=3, the
absolute diffusion constant is a nonmonotonic function of
particle size (Ddilute =1/˜a+˜
A˜a). Nevertheless, we find that
the absolute diffusion constant continuously decreases with
particle size in systems with large volume fractions (Fig. S7),
consistent with the experimental observations [32].
L032018-3
MENG, JIN, GUAN, XU, AND LIN PHYSICAL REVIEW RESEARCH 5, L032018 (2023)
(a) (b)
FIG. 4. Comparison between simulations and experiments. (a) The relative enhancements of diffusion constants (D/D0−1) from the
experimental data match the simulation results of the white-noise active force model. Here, γ=3, ˜
A=0.42. (b) The same analysis as (a) but
for simulations of the self-propelled model with the longer ˜
t. Here, ˜
F=1.38. The results are binned over particles with a bin interval of 0.02
in (a) and of 0.05 in (b) in the log10 scale. In both panels, N=4000.
We also calculate the radius of gyration, the root-mean-
square distance from the center of the trajectory, for both
passive and active systems [Fig. S8(a)]. The radii of gyration
from simulations decrease linearly with the particle radius in
both passive and active systems. The ratio between the passive
and active systems also has a linear relationship with the par-
ticle radius [Fig. S8(b)]. These results agree with experiments
[16], further supporting the validity of our simulations.
In Ref. [16], the authors observed much stronger glassylike
properties in dormant cells than in living cells. We find similar
hallmarks of a glass transition in our simulations, includ-
ing dynamic heterogeneity and non-Gaussian displacements
[33–35]. The MSDs of particles with similar radii in the same
passive system vary over two orders of magnitude [Fig. 5(a)],
showing significant dynamic heterogeneity. Meanwhile, this
dynamic heterogeneity is much weaker in the corresponding
active system. Furthermore, the displacement distributions
have a non-Gaussian tail in the passive systems while the devi-
ation from Gaussian distribution is much less significant in the
(a) (b)
FIG. 5. Activity fluidizes the glassy polydisperse system under
a high volume fraction. (a) Violin plot of MSDs of particles whose
˜a≈4 for passive and active systems under φ=0.75. The MSDs
within a dimensionless time of 1000 are shown in the log scale
and normalized by the average value. (b) The non-Gaussian pa-
rameter α2of the displacement distribution as a function of the
particle radius for passive and active systems under φ=0.75. α2=
˜
r(˜
t)4/[3˜
r(˜
t)22] and it equals 1 if the displacements obey
the Gaussian distribution. The results are averaged with 50 particles
in each bin. The maximum α2is shown from ˜
t=50 to 2500. For
the active systems, ˜
A=0.42 and γ=3. In both panels, N=4000.
active systems (Fig. S9). Indeed, the non-Gaussian degree of
displacement distributions for large particles is much weaker
in the active system than in the passive system [Fig. 5(b)].
Our results show that activity fluidizes the glassy polydisperse
system, in agreement with the experimental observations [16].
We find that a higher volume fraction is needed to observe
the hallmarks of a glass transition in polydisperse systems
(Fig. S10) than in monodisperse systems (Fig. S11), consis-
tent with observations that polydispersity can smear out a
glass transition [36].
Mechanism of the cubic scaling γ=3.In the follow-
ing, we explain the cubic scaling of the white-noise active
force with the particle radius. We consider a self-propelled
model in which an active force with a constant magnitude
is exerted on each particle [37–40]. The orientation of this
active force is random due to the rotational diffusion of the
particle. The equation of motion for the ith active particle
becomes
ηi
dri,α
dt =−∂U
∂ri,α
+ξi,α +Fn
i,α ,(6)
where Fis the magnitude of the active force and ni,α is the
orientation vector of the active force in the direction α.The
orientation vector niobeys dni/dt =Ti×ni, where Tiis
the thermal random torque. Its autocorrelation function satis-
fies the FD theorem, Ti,α(t),Tj,β (t)=2DR,iδijδαβδ(t−t).
Here, DR,i=kBT/8πνa3
iis the rotational diffusion constant
for a spherical particle with radius ai.
At long times, the additional diffusion constant due to
activity Dactive =(F/6πνa)2×(1/2DR)/3, where F/6πνais
the speed of the active particle and 1/2DRis the time for
the active force to change its orientation in three-dimensional
space. Therefore, the diffusion enhancement of an active par-
ticle relative to a passive particle in the dilute limit becomes
Ddilute
D0,dilute
=1+2˜
F2
9˜a2.(7)
Here, ˜
Fis the dimensionless active force with unit kBT/a0.
Comparing Eqs. (5) and (7), we find that the two models are
equivalent in terms of diffusion enhancement when γ=3.
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DIFFUSION ENHANCEMENT IN BACTERIAL CYTOPLASM … PHYSICAL REVIEW RESEARCH 5, L032018 (2023)
This conclusion applies to the dilute limit, and we hypothesize
that the equivalence of the two models is still valid under
high-volume fractions.
To test our hypothesis, we simulate the self-propelled
model with ˜
Fsatisfying ˜
A=2˜
F2/9 so that the two models
lead to the same diffusion enhancement in the dilute limit.
The agreement between simulations and experimental data
also holds for the self-propelled model [Fig. 4(b)]. We use
the longer ˜
tand confirm that the ˜
tcalculated in this
way is longer than the rotational relaxation time. We find
that the magnitude of the active force F=0.57 pN in the
physical unit, consistent with typical force magnitudes in the
cytoplasm [28]. We remark that the constant magnitude of
the active force is crucial to obtain the correct scaling of
diffusion enhancement. In an alternative model with a con-
stant active speed, Ddilute/D0,dilute −1∼˜a4, inconsistent with
experiments. In a more biologically plausible scenario, the
magnitude of the active force can be random among differ-
ent particles [41]. Therefore, we also simulate a modified
model in which the active force obeys a size-independent nor-
mal distribution and obtain similar results (Fig. S12), further
strengthening our proposed mechanism.
Discussion. In this Letter, we introduce a white-noise ac-
tive force to a highly polydisperse system to mimic bacterial
cytoplasm. While prior works have investigated the effect of
activities on particle mobility [42–44], our work simultane-
ously incorporates crowding by different particle sizes and
active forces. Due to its out-of-equilibrium nature, the FD the-
orem does not constrain the active random force. Surprisingly,
a white-noise active force reproduces the experimentally mea-
sured ratios of diffusion constants between living and dormant
bacteria with its autocorrelation function proportional to the
cube of the particle radius. We note that an active random
force generally generates an additional friction coefficient that
is inversely proportional to an active temperature [27,45–47].
We show that the additional friction coefficient does not affect
our conclusions because the active temperature is typically
much higher than the thermal temperature (see Supplemental
Material [31]).
We further demonstrate that the white-noise active force
model with γ=3 is equivalent to the self-propelled model
with a constant-magnitude active force regarding the diffu-
sion enhancement. Our results suggest an emergent simplicity
when many active processes are averaged simultaneously.
Importantly, we identify the magnitude of the active force,
F=0.57 pN. While the origin of this active force is still
unclear [16], we hypothesize that it may come from the colli-
sion of small molecules with proteins, e.g., amino acids and
ions, which are out of equilibrium [45]. We note that the
active force can be estimated as the thermal energy divided
by typical protein sizes, F=kBT/a≈0.4 pN, where ais the
typical size of proteins around 10 nm. The active force applied
to proteins is just enough to overcome thermal fluctuation.
This particular magnitude of active force allows proteins to
move or change their configurations according to some spe-
cific intracellular signaling. On the other hand, it lets proteins
quickly change their dynamics when the signaling changes.
Therefore, the magnitude of the active force around the pN
range may be evolutionarily selected.
We note that the time-averaged MSDs of particles of the
same size differ among independent simulations. Neverthe-
less, the MSD averaged over the time-averaged MSDs of inde-
pendent simulations are close to the ensemble-averaged MSD
(Fig. S13), suggesting a weak nonergodicity effect, presum-
ably because polydispersity smears out the glass transition
(Fig. S11). Finally, we remark that while a hydrodynamic
interaction has been shown to reduce the diffusion coefficient,
its effect may be negligible for particles with a radius above
25 nm that we use to compare with experimental data [22].
Acknowledgments. We thank Yiyang Ye, Hua Tong, Sheng
Mao, and Ming Han for helpful discussions related to this
work. The research was funded by National Key R&D Pro-
gram of China (2021YFF1200500) and supported by grants
from Peking-Tsinghua Center for Life Sciences.
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