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JPSJ News and Comments 17, 08 (2020)

https://doi.org/10.7566/JPSJNC.17.08

Brownian Motion Confined in a Brownian Surface
Shigeyuki Komura (Department of Chemistry, Tokyo Metropolitan University)
Published July 27, 2020
A researcher has developed a new theory to describe the Brownian motion of a small object that is confined in a fluctuating random surface, such as a biological membrane. The effective diffusion coefficient is shown to provide a new
dynamical contribution.
Owing to the recent advances in high-precision imaging techniques, scientists are able to directly observe theBrownian motion of a protein molecule embedded in a biological membrane. This type of membrane typically consistsof a thin lipid bilayer, and it can be regarded as a two-dimensional (2D) fluid sheet. Another important feature of this typeof lipid membrane is its high flexibility. Thus, the membrane
can exhibit large out-of-plane shape fluctuations that can beexperimentally measured [1].
What is the lateral diffusion behavior of proteins confined insuch a highly fluctuating environment? Recently, the couplingof protein diffusion to the shape of the membrane surface has
become a subject of active study, and various theoretical andexperimental studies have been performed. Interestingly,
previous theories have predicted that the effective diffusion
coefficient of a particle immersed in a fluctuating membrane
is smaller than that in a flat membrane.
On a fluctuating random membrane, a moving object needs
to travel a larger distance between two points in a threedimensional space because the trajectories are constrained ona rough surface rather than on a flat one (see Fig. 1). This iscalled the “geometrical contribution,” which has been well
explained in the previous studies. In an optical microscopeexperiment, one can only capture a 2D projection of the true
particle motion over the membrane surface.
Recently, Ohta revisited the problem of lateral diffusion ona random surface by using a new theoretical approach [2]. A
key technique in his formulation was to employ a Lagrange
multiplier in the Langevin equation for the particle to take
into account the constraint that it always sits on the fluctuating surface. With this mathematical technique, the theoreticalframework of the problem has been established without any
ambiguity.
In this new formulation, Ohta calculated the mean squaredisplacement for the lateral diffusion of a Brownian particle
and obtained the effective diffusion coefficient. First, theabovementioned geometrical contribution, which results
in a reduced diffusion coefficient on a projected plane, wasreproduced. Second, and most importantly, Ohta also founda new “dynamical contribution” that arises from the velocity
correlation of the fluctuating surface.
In the paper, the two-time velocity correlation function of afluctuating membrane surface has been evaluated in both thepresence and absence of hydrodynamic interactions mediatedby the surrounding bulk fluid. Using this result, Ohta thenderived a self-consistent equation to determine the effectivediffusion coefficient. The main conclusion was that the newly
found dynamical contribution increases the effective diffusion
coefficient as opposed to the geometrical contribution [2].

Fig. 1. Image for Brownian motion of a particle confined in a fluctuating random surface.

What is then the overall effect on the lateral diffusion coefficient owing to these two contributions? According to thenumerical estimate of the effective diffusion coefficient, its
reduction compared to that of a flat surface is diminished.
Nevertheless, the net reduction is still 20–30%, whichshould be taken into account for experimental data analysis.
Importantly, the dynamical contribution also depends on the
dynamical properties of the surrounding bulk fluid, such as
its viscosity.
This study on the Brownian motion on a fluctuating surface
by Ohta opens up several new research areas for the future. In
the recent years, many attempts have been made to quantify
the non-equilibrium properties of biological systems by measuring the fluctuations at different levels and scales [1]. The
work by Ohta clearly shows that the Brownian motion of aparticle and the surface out-of-plane fluctuations are coupled
in a non-trivial and an interesting way.
REFERENCES[1] H. Turlier, D. A. Fedosov, B. Audoly, T. Auth, N. S. Gov, C. Sykes, J.-F.
Joanny, G. Gompper, and T. Betz, Nat. Phys. 12, 513 (2016).
[2] T. Ohta, J. Phys. Soc. Jpn. 89, 074001 (2020).
Shigeyuki Komura obtained a Ph.D. in Physics from The University of Tokyo in 1993
and subsequently spent one year at the Tokyo
Institute of Technology, three years at theKyoto University, and five years at the Kyushu
Institute of Technology. Since 2000, he has
been at the Department of Chemistry, Tokyo
Metropolitan University. The main research
topic of his group is non-equilibrium physicalchemistry of biological and soft matter, such as
microrheology, micromachines, and active matter.
News and Comments on “Brownian Motion on a Fluctuating Random
Geometry” [T. Ohta, J. Phys. Soc. Jpn. 89, 074001 (2020)]© 2020 The Physical Society of Japan