Physica D 168–169 (2002) 235–243
Diffusion in a model for active Brownian motion
C.A. Condat a,b,∗ , G.J. Sibona c
b
a FaMAF, Universidad Nacional de Córdoba, 5000 Córdoba, Argentina
Department of Physics, University of Puerto Rico, Mayaguez, PR 00981, USA
c Universidad Tecnológica Nacional, 1179 Buenos Aires, Argentina
Abstract
Due to their small mass, microorganisms whose sizes are of the order of the microns, such as marine bacteria, are continuously buffeted by Brownian forces, which can substantially affect their motion. Here we extend the analysis of a model
recently introduced to describe the dynamics of a microorganism moving under the combined influences of its own propulsion
system and of Brownian forces. In the high dissipation regime, the nature of the motion is determined by a synergy between
these two contributions. Although the motion can be extremely directional over short time scales, it is always diffusive when
observed over long times.
© 2002 Elsevier Science B.V. All rights reserved.
PACS: 05.40.Jc; 92.20.Pz
Keywords: Diffusion; Brownian motion; Microorganism
1. Introduction
Many organisms devote a substantial portion of
their energetic resources to motion, often exploring
space in search of the nutrients needed to replenish
their energy stores. The physical properties of animal motion at macroscopic and microscopic scales
have been carefully studied, a particularly nice recent
example being the investigation of the diffusion of
beads due to the motion of bacteria suspended in a
two-dimensional fluid film [1,2]. However, there does
not seem to have been an effort to understand the
relation between the properties of the organism’s motion and its energy budget. In the microscopic world,
it is known that marine bacteria devote a substantial
part of their energetic resources to motion, partly to
∗
Corresponding author. Present address: Department of Physics,
University of Puerto Rico, Mayaguez, PR 00981, USA.
defeat the effect of fluctuational forces, which make
directional motion difficult [3,4]. In the macroscopic
world, it is unlikely that the properties of the wandering albatross’ Lévy flight [5] are unrelated from its
motional energy expenditure. Nutrient availability and
consumption rates determine the energetic balance of
an organism and, therefore, they may determine the
future of a species in a given environment, the growth
of a bacterial colony [6,7] or the growth of a tumor,
considered as the cooperative result of the evolution
of a cancer cell colony [8]. It would be therefore of
interest to have a theoretical framework to study how
organisms may administer their energetic resources in
order to optimize their reproductive efficiency, their
life-span, and their exploration of space.
Three years ago, Schweitzer, Ebeling, and Tilch
(SET) introduced the concept of Brownian motion
with an energy depot to describe the motion of a microorganism that moves under the combined action
0167-2789/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 7 - 2 7 8 9 ( 0 2 ) 0 0 5 1 2 - 2
236
C.A. Condat, G.J. Sibona / Physica D 168–169 (2002) 235–243
of Brownian forces and of its own propulsion system
[9]. Of course, the term “depot” is not meant to convey the existence of a specific storage location inside
the organism, but it is an idealization introduced to
represent the totality of its chemical energy stores.
This model was further explored by Schweitzer and
coworkers [10] in several publications. Recently, we
have pointed out that, under certain circumstances,
Brownian forces can increase the mechanical efficiency of a microorganism’s propulsion system [11].
In this paper, we extend the analysis of the model,
showing that the motion of a microorganism subject
to noise becomes diffusive when observed over long
times, but may be essentially directional over shorter
time scales. We also discuss the differences between
one-dimensional and multi-dimensional motion.
In the next section, we describe the model and its
approximations. The results of the simulations are presented in Section 3, and we summarize our conclusions
and possible future avenues of research in Section 4.
2. Formalism
Following SET’s proposal [9], we assume that the
microorganism can take up energy from the environment at a rate q, store it in an internal energy depot
and transform part of it into kinetic energy. The depot
energy e(t) can be also dissipated at a rate G[e(t)],
which represents all the nonmechanical uses of the
available energy. The conversion rate d1 K can depend on the speed v, on the depot energy e, and on
the metabolic state µ, e.g., the stage of the reproductive cycle, or, in the case of a bacterium, the creation
of flagella. Nutrient uptake may depend on position
(through an inhomogeneous nutrient distribution),
speed, and on a manifold of internal signals, which
are likely to be dependent on e(t). The amount of
stored energy is therefore described by the equation
de
= q[r , v, µ; e(t)] − G[e(t)] − d1 (e, µ)K(v). (1)
dt
The total energy of an active particle of mass m
is E(t) = (m/2)v 2 + e(t). To account for the energy reconversion contribution to the microorganism
motion, we follow Ref. [9] and postulate the modified
Langevin equation [11]
m
K(v)
d
v
= −γ v + d1 (e, µ) 2 v + F (t),
dt
v
(2)
where γ is the friction coefficient and F is a stochastic
force satisfying
F (t) = 0,
Fi (t)Fj (t ) = εδij δ(t − t ).
(3)
It is assumed that, on the average, the energy loss
due to friction compensates the energy gain due to
the stochastic force and thus ε = 2γ kB T . Here kB is
the Boltzmann constant and T the temperature (more
general forms for the friction term can also be chosen,
see [10]).
The assumption is now made that the depot energy
reaches its quasi-static equilibrium es fast in comparison to the time scale of the motion. Therefore,
q(es ) − G(es ) − d1 (es , µ)K(v) = 0.
(4)
The validity of this assumption has been verified by
comparing the resulting predictions to the outcome of
simulations of Eqs. (1)–(3). For simplicity, we will further assume that the reconversion and dissipation rates
are both proportional to the depot energy, d1 = d2 e
and G = ce, and that nutrient uptake q is a constant.
Under these conditions, Eq. (2) reads
m
d
v
qd2 K(v)
v
+ γ v −
= F (t).
dt
[c + d2 K(v)]v 2
(5)
Analytical solutions for this equation can be obtained
in the cases of no internal dissipation and of high internal dissipation. If there is no internal dissipation
(c = 0), all the depot energy is transformed into kinetic energy, whatever the form of K(v). Under these
conditions, the F = 0 form of Eq. (5) can be solved
exactly. If v0 is the velocity at t = 0, we obtain for
the one-dimensional deterministic speed
1/2
1/2 γ v02
q
−2γ t/m
vD (t) =
1 + −1 +
.
e
γ
q
(6)
The full (c = 0, F ≡ 0) equation can be solved approximately by replacing the speed in the denominator
C.A. Condat, G.J. Sibona / Physica D 168–169 (2002) 235–243
of its third term by vD (t). We can then calculate the
velocity–velocity correlation function, which yields,
asymptotically
v 2 (t → ∞) =
kB T
q
+
.
γ
m
(7)
For a system of dimension d, we must replace the last
term by dkB T /m. The contributions of the deterministic propulsion mechanism (q/γ ) and of the noise are
additive. The approach to this asymptotic value is exponentially fast, with a time constant τ0 = m/2γ .
If we assume that most of the depot energy is consumed internally without being transformed into mechanical energy [c d2 K(v)], some analytical results can be found. Choosing K(v) = v 2 , Eq. (5) can
be written as
d
v
d2 v 2
m
+ γ v − γ Q 1 −
v = F (t)
(8)
dt
c
with Q ≡ qd2 /γ c being a parameter that is useful to
characterize the dynamics. By neglecting the nonlinearity, this equation can be easily solved. If the solution is then used to calculate the mean square speed,
we see that, at long times it converges asymptotically
to
v 2 (∞) =
dkB T
.
m(1 − Q)
(9)
The mean square displacement can also be calculated.
At long times the motion becomes diffusive, with a
diffusion coefficient given by
D=
kB T
.
γ (1 − Q)2
(10)
The last two equations are valid provided that Q 1.
Contrary to what was found for the c = 0 case, the
microorganism dynamics results from a complex interplay between the noise and the propulsion system.
Indeed, in the absence of noise, v 2 (t → ∞) = 0,
which means that in the steady state all the depot energy ends up being used for nonmechanical purposes
and the microorganism remains at rest.
Again following SET [9], we define the mechanical
efficiency σ of the propulsion system as the ratio of
the kinetic energy increase due to the motor to the
237
instantaneous energy uptake q,
d2 K(v)
eK(v)
=
.
σ =
q
c + d2 K(v)
(11)
The main purpose of Ref. [11] was to discuss the properties of this function. There we showed that, if Q 1, the external fluctuations could increase the mechanical efficiency of the organism propulsion system.
More information is obtained by solving the
Fokker–Planck equation associated with Eq. (5). If
the conversion factor is a power law, K(v) = v ξ , ξ >
0, a calculation of the steady-state one-dimensional
probability density yields [11]
Ws (v, ξ ) = W0 e
−mv 2 /2kB T
d2
1 + vξ
c
mq/γ ξ kB T
,
(12)
where W0 is a normalization constant. The maxima of
this distribution are located at the values of v that solve
the steady state, deterministic, version of Eq. (5). For
instance, if ξ = 2 the distribution maximum occurs at
v̂ = 0 for Q < 1, but there is a bifurcation at Q = 1,
and Ws has two maxima for Q > 1:
1/2
c
v̂± = ±
(Q − 1)1/2 ,
(13)
d2
which suggest that the presence of the depot has introduced a degree of organization in the motion. To
achieve a complete understanding of the nature of the
motion of individual objects we need computer simulations. These will be discussed in the next section.
Note that v̂ is the deterministic speed obtained by SET
[9]. The odd moments of the distribution (12) are zero,
and it is easy to see that, for a fixed amount of noise,
its even moments will depend on two parameters: c/d2
and q/γ . A simple calculation shows that, to a good
approximation,
v 2 (t → ∞) =
q
dkB T
c
+
−
,
γ
d2
m
(14)
provided that Q 1 and that the noise is not too
intense.
From Eq. (12) we see that the velocity distribution
is very sensitive to the exponent ξ . This sensitivity is
238
C.A. Condat, G.J. Sibona / Physica D 168–169 (2002) 235–243
already evident in the neighborhood of v = 0, where
Ws can be approximated by
m
2
Ws (v, ξ ) ≈ W0 exp −
. (15)
v 2 − Qv ξ
2kB T
ξ
The sign of the exponent indicates that v = 0 is a
minimum if 0 < ξ < 2 and that it is a maximum
if ξ > 2. The crossover occurs at ξ = 2, which has
the behavior described above and has been discussed
before [10].
A detailed examination of Eq. (12) confirms that
two very different pictures emerge, depending on the
value of ξ . Since the effect of the depot is very strong
for small v, the Maxwellian distribution is strongly
deformed in this region, the deformation being naturally milder if the noise is strong or if q is small. The
distribution is therefore always bimodal if 0 < ξ < 2,
although with only a shallow minimum at v = 0 for
small values of q/ε.
Low v modifications are weaker if ξ > 2; in fact,
if Q is small the distribution is unimodal, while for
large Q the strong effect of the depot at high speeds
yields a symmetric trimodal distribution. The active
particle can be moving with a velocity that fluctuates
about one of its two nonzero deterministic values, or
it can be temporarily trapped in a low speed, noise
controlled regime.
3. Simulation results—discussion
The system properties can be analyzed in detail if
we solve Eqs. (1)–(3) by using Monte Carlo simulations. To perform these we used a Heun algorithm in
combination with the Box–Mueller formula [12]. The
time step was decreased until a stable result was obtained. In all simulations we used the special form
K(v) = v 2 .
In Fig. 1 we have plotted the diffusion coefficient as
a function of the parameter Q for a one-dimensional
system and several values of the noise intensity. If Q =
0, the motion is entirely determined by noise, and the
diffusion coefficient has its well-known value. In the
low Q region, the Monte Carlo results agree very well
Fig. 1. One-dimensional motion: diffusion coefficient as a function of Q for several values of the noise intensity. Here we chose q = 1,
γ = 0.1 and c = 0.1. The solid lines correspond to the small Q approximation, Eq. (7).
C.A. Condat, G.J. Sibona / Physica D 168–169 (2002) 235–243
239
Fig. 2. A section of the position versus time simulation output for two different values of the noise: (a) ε = 0.1 and (b) ε = 1. The rest
of the parameters are q = 1, d2 = 0.01, γ = 0.1, and c = 0.01. Between reversals, the speed has, approximately, its deterministic value.
240
C.A. Condat, G.J. Sibona / Physica D 168–169 (2002) 235–243
with the approximate values given by Eq. (10). The
rapid increase of the diffusion coefficient is a manifestation of the synergy between noise and propulsion
system that is responsible for the enhanced mechanical efficiency of the system. This synergy weakens
for large values of Q, for which the propulsion system is already very efficient by itself, and D increases
at a lower rate. In this region, the distance spanned
by the object in each diffusional step grows very fast
as a function of Q: the deterministic speed is approximately equal to (q/γ )1/2 and reversals are rare. A
consequence is that the motion has to be recorded over
very long times to observe the diffusive regime.
The nature of one-dimensional motion can be better appreciated in a plot of x(t) for a single run. If the
energy drawn from noise is of the order of or larger
than the energy effectively used for propulsion (say,
kB T /m q/γ ), there are frequent noise-induced
direction reversals and the motion becomes rapidly
diffusive. On the other hand, if the relative contribution of the noise is weak, the motion is mostly
unidirectional, with long quasi-ballistic stretches.
During these stretches the speed fluctuates about its
deterministic value. Fluctuations are seldom strong
enough to reverse the direction of motion. The distribution Ws has two sharp peaks located at v̂± . Fig. 2
shows two instances of this “weak noise” regime: the
position versus time plots indicate that the motion is
essentially deterministic between reversals. Only the
strongest fluctuations can cause direction reversals.
Other fluctuations are rapidly dampened out and can
be observed only on a finer scale. The same behavior
is observed by plotting the simulation output for v(t).
The probability of a direction reversal can be estimated by noting that the motion of a particle moving
with speed v0 can only be reversed if the impulse provided by the external force is opposed to the direction
of motion and of a magnitude at least equal to the
linear momentum of the particle. Since the probability distribution of having a fluctuation of a given size
at any time step is a Gaussian with second moment
ε, the probability that the fluctuation will provide an
impulse larger than u = mv√
0 , i.e., the probability for a
reversal, is P = erfc(mv0 / 2ε), where erfc(x) is the
Fig. 3. Two-dimensional motion: probability density for a directional change of θ (rad) after five time steps. Here q = 1, d2 = 0.01,
γ = 0.1, c = 0.01, and ε = 0.1.
C.A. Condat, G.J. Sibona / Physica D 168–169 (2002) 235–243
241
Fig. 4. Two-dimensional motion: (a) mean square change-of-heading speed as a function of time for the indicated values of γ and (b)
steady-state mean square change-of-heading speed as a function of γ . The rest of the parameters are q = 1, d2 = 0.01, c = 0.01, and
ε = 0.1.
complementary error function. For large q/γ and
small ε, this probability can be approximated by
exp(−m2 q/2εγ ).
In higher dimensions, completely unidirectional
motion is not possible, because even small fluctuations
modify the particle heading, as observed in the case of
marine bacteria [3,4]. The dependence of the diffusion
coefficient with Q is qualitatively similar to that shown
in Fig. 1. Fig. 3 shows the probability density f (θ ) that
the direction of the motion has changed by an angle
of θ (rad) after five time steps when the motion occurs
in a two-dimensional space. Even small fluctuations
242
C.A. Condat, G.J. Sibona / Physica D 168–169 (2002) 235–243
will affect the direction of motion, but for large values of Q a long time must elapse before the particle
completely loses memory of its original heading. The
average mean square change-of-heading is plotted in
Fig. 4a as a function of time for several values of the
friction coefficient γ . The particle is supposed to start
from rest, and therefore it changes its heading very
fast at short times. After the steady-state translational
speed is reached, the mean square change-of-heading
stabilizes to its steady-state value, shown in Fig. 4b.
This steady-state value is a monotonically increasing
function of γ because an increase in γ decreases the
translational speed, allowing for larger directional
changes. If γ is large, however, the average translational speed is already low and the growth in the
mean change-of-heading speed saturates. Note that,
for the parameters of the figure, Q = 1/γ , so that
this saturation is expected to occur at about γ = 1.
The analysis of the time dependence of the mean
square speed reveals that, unless Q is large, the object
departs from rest, with v 2 (t) increasing smoothly
towards its stationary value. Plots of individual runs
for v(t) (not shown here) look qualitatively similar to
those obtained by Berg and Brown in their classical experiment on the motion of Escherichia coli (see Fig. 2
in Ref. [13]), although it must be remarked that the
relatively large size of this bacterium makes Brownian forces less important than for smaller organisms.
For large values of Q (weak dissipation), v 2 (t) exhibits some oscillations before reaching its stationary
value.
4. Conclusions
In the low dissipation (or high Q) problem, the
depot runs efficiently (it stores only a small amount
of energy in the steady state) and both energy sources
(depot and noise) contribute independently to the
mean square speed and to the mechanical efficiency.
If we observe the motion over short time spans, motion is directional, appearing to be almost ballistic.
However, observation over very long times reveals
that, in the one-dimensional problem, the occasional
peaks in the noise strength that cause direction
reversals will eventually generate diffusive motion.
Since a progressive change-of-heading is possible,
the transition to the diffusive regime is smoother in
higher-dimensional systems.
Under appropriate conditions (Q 1), there can
be an effective synergy between Brownian forces and
molecular motors. This may reduce the amount of
stored energy required to move at a certain speed
or to explore a certain region of space. We do not
know if nature does indeed make use of noise to
optimize energy utilization in small microorganisms,
but we suggest that this is a subject that deserves
to be explored experimentally. We remark that there
is another problem in which noise can contribute to
organize motion: as first noted by Magnasco [14],
energy can be usefully extracted from a thermal bath
by a stochastically driven ratchet.
The model discussed in this paper could be helpful
in the study of the dependence of the dynamics of a
small microorganism on the distribution of available
energy resources, suggesting the best strategies for the
microorganism well-being and survival. One possibility would be to find the optimum strategy to reach one
or more targets as fast as possible, while keeping nutrient consumption as low as possible. Targets could be
proteins to be activated, other members of the species
to be fecundated, or food deposits. Inhomogeneities in
the diffusive space may be included to represent, for
instance, geometric or anatomic constraints. A simple
but instructive case is that of a “bacterium in a hostile
medium”: a Brownian microorganism is released with
a full depot in a region where q = 0, and we calculate
its mean square displacement before the depot energy
falls below a death threshold ed (if e < ed , the organism cannot maintain its primary metabolic functions
and dies). This procedure allows us to find the parameter values that optimize spatial exploration, yielding
some interesting surprises [15].
Acknowledgements
This research was supported by grants from CONICET and SECYT-UNC, Argentina. CAC is a member
of CONICET.
C.A. Condat, G.J. Sibona / Physica D 168–169 (2002) 235–243
References
[1] X.-L. Wu, A. Libchaber, Phys. Rev. Lett. 84 (2000) 3017.
[2] G. Grégoire, H. Chaté, Y. Tu, Phys. Rev. E 64 (2001)
11902.
[3] J.G. Mitchell, Microb. Ecol. 22 (1991) 227.
[4] J.G. Mitchell, et al., Appl. Environ. Microbiol. 61 (1995) 877.
[5] G.M. Viswanathan, et al., Physica A 295 (2001) 85 and
references therein.
[6] E. Ben-Jacob, et al., Physica A 282 (2000) 247.
[7] T. Vicsek (Ed.), Fluctuations and Scaling in Biology, Oxford
University Press, Oxford, 2001.
[8] P.P. Delsanto, G. Pescarmona, E. Romano, M. Scalerandi,
C.A. Condat, Phys. Rev. E 59 (1999) 2206;
B. Capogrosso Sansone, C.A. Condat, M. Scalerandi, Phys.
Rev. E, in press.
243
[9] F. Schweitzer, W. Ebeling, B. Tilch, Phys. Rev. Lett. 80
(1998) 5044.
[10] B. Tilch, F. Schweitzer, W. Ebeling, Physica A 273 (1999)
294;
W. Ebeling, F. Schweitzer, B. Tilch, BioSystems 49 (1999)
17;
F. Schweitzer, B. Tilch, W. Ebeling, Eur. Phys. J. B 14 (2000)
157.
[11] C.A. Condat, G.J. Sibona, Physica A, submitted for
publication.
[12] M.O. Cáceres, C.E. Budde, G.J. Sibona, J. Phys. A 28 (1995)
3877.
[13] H.C. Berg, D.A. Brown, Nature 239 (1972) 500.
[14] M.O. Magnasco, Phys. Rev. Lett. 71 (1993) 1477;
K. Sumithra, T. Sintes, Physica A 297 (2001) 1.
[15] G.J. Sibona, C.A. Condat, Unpublished.