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\begin{document}

\title{~~~ Mechanics of active membranes~~~}



\author[1]{Timo Betz}%
\author[1]{herve.turlier}%
\affil[1]{Affiliation not available}%


\vspace{-1em}



  
  \date{May 04, 2024}


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\section*{Introduction (History):~}

{\label{introduction-history}}

The quantitative understanding of active materials far from the
thermodynamic equilibrium remains one of the most fascinating and moving
topics in modern physics~\cite{Ramaswamy_2010,Menon_2010,Saintillan_2018,Gross_2017,Ramaswamy_2017,Marchetti_2013}. While active 3D materials
have been studied in great detail, the mechanical description of 2D
membranes has only received limited attention~\cite{Ramaswamy_2001,Cornelius_2011,D_bereiner}. This
is in sharp contrast to the outstanding relevance of active membranes,
especially in the description of living and biomimetic systems where
membranes are the key element separating compartments and hence chemical
environments.~ Historically, the already early observed dynamical
movement of membranes has been interpreted as purely passive, Brownian
like fluctuations that are driven by thermal motion, even in the case of
living systems that obviously have the capacity to provide additional
chemically driven forces on the membrane~\cite{Browicz1890}. A beautiful
example illustrating how difficult it can be to make the differences
between random thermal and random forces actin on membranes is provided
by the red blood cell (RBC) membrane flickering. First reported as
Brownian motion like movement in the year 1890~\cite{browicz_notitle_1890}, the
finding that flickering amplitudes depend on ion transport suggested an
active metabolic origin of the flickering more than 60 years
later~\cite{BLOWERS1951}, only to be soon again challenged by the
observation that the flickering is present even in the absence of any
energy source~\cite{PARPART1956}. While all these observations had
mainly qualitative character, the first quantitative approach to measure
RBC membrane flickering suffered from low resolution\cite{Burton1968}.
In 1975 Brochard and Lennon observed in a landmark paper that the
frequency resolved flickering spectrum show a power-law
behavior~\cite{Brochard_1975}. This analysis was the starting point of a
still ongoing detailed biophysical quantification of the red blood cell
membrane flickering which also triggered a full field in biophysics
focusing on fluctuations analysis of artificial and biological membranes
(recently reviewed in~\cite{Monzel2016}). In these early days both, the
static and dynamic analysis of membrane movement was modeled by a purely
passive theory exclusively using the thermal energy~\(k_BT\)to
excite the fluctuations~\cite{Brochard1975,Strey_1995}. It took more than 20 years
for the first experimental signatures of possible active contributions
to the RBC flickering. Using a special kind of dark field microscopy,
Tuvia observed ATP dependent fluctuation amplitudes and suggested that
an an active metabolic contribution to the flickering is the reason for
this~\cite{Tuvia1997}.~ This observation can be seen as the starting
point of a rich scientific debate about the importance of metabolic
contributions, where a series of contradicting results where
reported~\cite{Evans2008,Evans2017,Park2010,Betz2009,Szekely_2009,Puckeridge_2014}. Recently, a direct violation of
equilibrium thermodynamics has been observed, which shows that active
processes are required to explain the observed
fluctuations\cite{Turlier2016}. Due to its importance we will discuss
the issue of RBC flickering below in an extra chapter of this review.~

\par\null

But active membranes have not only been an interest in understanding the
mechanics of the red blood cells. In a seminal paper Prost and
Bruinsma~\cite{Prost_1996}~introduced the first theoretical
descriptions of membranes driven by an active component, just before the
first experimental signs were published. It is remarkable that already
in~ this initial paper, a separation of scales between active and
passive processes was predicted, which turns out to be a common
characteristics of actively driven membranes\cite{Faris2009,Turlier2016}.~ This
first model focused on activity driven by membranes by embedded ion
channels, and was soon further developed to describe more complex
scenarios, such as fluctuations close to walls~\cite{Prost_1998},
rotational activity~\cite{Lenz2003}, diffusion of active
channels~\cite{Granek_1999}, active forces by membrane bound
pumps~\cite{Gov_2004,Lacoste_2005,Lin_2006}and single forces acting on the membrane by a
supporting structure such as a cytoskeleton\cite{Gov_2005,Auth_2007,Gov_2003}.~
Experimentally, the first direct proof of an active membrane was given
by changes in membrane area observed in micropipette aspiration essays
of vesicles containing the light driven proton pump
bacterio-rhodopsin\cite{Manneville1999}.~ In this article we will quickly
review the mechanics of passive membranes before we discuss biological
and artificial sources for active forces on membranes. We will then
introduce the experimental evidences for active membrane fluctuations
and review the theoretical approaches to describe them. The debate about
active and passive contribution for the RBC membrane fluctuations will
be focused on in an extra chapter, before we discuss the important
questions whether active forces can modulate the mechanical parameters
of membranes such as bending modulus and tension. Here it is important
to investigate with great care if such effective changes could be the
outcome of pure interpretation of complex data, or if they are obtained
by direct measurements. Finally, we discuss the current challenge posed
by both better theoretical as well as experimental approaches to
describe the composite of active bio-polymers and active membranes as
found at the cellular cortex.

~

Here we do not talk~ about activity that changes the membrane
organization~ or chemical composition, where reaction diffusion systems
are coupled to membrane mechanics via curvature or stretch dependent
diffusion or reaction rates.~

\section*{The mechanics of passive membranes (Figure
1)}

{\label{quick-introduction-of-passive-membrane-mechanics}}

The mechanical properties of phospholipid bilayer membranes has been
intensively studied with many excellent resources available summarizing
this complex field of physics~\cite{Seifert_1997,Nelson_2004}.~ Thanks to seminal
work by Nicholson and Singer,~ who established the fluid mosaic
model~\cite{Singer_1972} it is clear that biological membranes are
complex entities that have a self organized structure on the molecular
level. However, a much simple model introduced by Helfrich just one year
later is typically sufficient to describe membrane
mechanics~\cite{Helfrich_1973}. Helfrich suggested to view~ the membrane
as a 2D fluid (Fig 1), which allows to construct a global free energy
hamiltonian based on the local energies required for bending and
stretching the membrane. This is a beautiful example of coarse graining
a complex system to the relevant mechanical features thus allowing
construction of an effective mean field theory. The core of his approach
is found in the Helfrich hamiltoian:

\par\null

$$ F=\int_A \frac{1}{2} \kappa H(x,y)^2 dx dy + \int_A \sigma dA$$

here~\(F\)~is the free energy,~\(\kappa\)~the
bending modulus,~\(H\) is the local curvature
,~\(\sigma\) the membrane tension and~\(A\) the
area of the membrane. In a more general description, the spontaneous
membrane curvature~\(H_o\) is included by replacing the~
with~\(H\left(x,y\right)-H_0\). It is important to note here that
~\(\kappa\)~is a material properties of the membrane that
depends on its molecular composition while~\(\sigma\)~is a
situation dependent mechanical parameter, describing the energy required
to change the total apparent membrane surface. A more convenient
description is based on Monge gauge (Fig 1), where a flat orthogonal
coordinate system is defined parallel to the average membrane and the
membrane confirmation is captured as height~\(h\left(x,y\right)\) of the
membrane at position x,y (Fig1). It is important to realize that this
approximation only works for quasi-flat membranes with small deviations
of the membrane from the equilibrium flat position. Thanks to this
approximation the Helfrich Hamiltonian can be expressed in terms of
differential operators, where the curvature
becomes~~\(H=\nabla^{2\ }h\), and~\(dA=\frac{1}{2}\left(\nabla h\right)^2dx\ dy\ \), thus yielding the
commonly used expression:~

\par\null\par\null

$$ F=\int_A dx dy [ \frac{1}{2} \kappa (\nabla^2 h)^2 + \frac{1}{2} \sigma (\nabla h)^2]$$

The typical values for the bending modulus~\(\kappa\)~of
biological~and reconstituted membranes range from 5 to
50~\(k_BT\), while the tension depends largely on the actual
mechanical situation of the membrane. Hence any value smaller than the
critical tension~ of~\(\sigma_c\approx3\times10^{-3}\) N/m at which the membrane
ruptures~ is possible. Due to low bending modulus, typical membranes
show a significant deformation due to thermal excitation at ambient
temperature. In the purely passive, thermally excited situation,
equation (2) can be used to predict the amplitude of individual
wave-modes~\(h_q\) by simply expressing the spatial Fourier
transformation of~\(F\propto\int_{\ }^{\ }dq\ \frac{1}{2}\left(\kappa q^4+\sigma q^2\right)h\left(q\right)h^{\ast}\left(q\right)\), and using the equipartition
theorem to contribute the thermal energy~\(\frac{1}{2}k_BT\)~ to every q
mode immediately yields the static fluctuation spectrum of (which ref is
here best???):

$$ <h_q^2>=\frac{k_BT}{\kappa q^4+\sigma q^2}$$

To study the dynamics of the membrane fluctuations, the Navier Stokes
equation needs to be solved, where the~ pressure generated by a deformed
membrane is balanced by the fluid viscous forces. Since the systems of
interest here are in the low Raynolds numbers regime inertia can be
omitted, and the Navier Stokes equation is solvable. Each excited mode
decays with an relaxation time~\(\tau_r=\frac{\left(\kappa q^4+\sigma q^2\right)}{4q\eta}\),
where~\(\frac{1}{4q\eta}\) represents the Oseen tensor for a 2D membrane
submerged in a liquid of viscosity~\(\eta\). The dynamic
equation of motion for each q mode in a thermally excited environment
can be expressed as a Langevin equation:

\par\null

$$ \frac{dh_q}{dt}=\tau_r^{-1} h_q(t)+\zeta(t) $$

where~\(\zeta\left(t\right)\) represents the delta correlated thermal noise
that obeys the fluctuation dissipation theorem. Based on this Langevin
appraoch, the average membrane dynamics can be easily expressed by the
experimentally accessible auto-correlation function (ACF):

\par\null

$$<h_q(0)h_q(t)>=\frac{k_BT}{\kappa q^4+\sigma q^2}\times \exp(-\frac{t}{\tau_r})$$

Besides the ACF the membrane dynamics can be probed by the power
spectral density (PSD) of a small membrane patch.~ Due to the Wiener
Khinchin theorem, the PSD~~\(<h\left(\omega\right)^2>\) can be calculated as the
time dependent Fourier transformation of the autocorrelation function.
Since experimentally, often only a single point is measured, we need
here to sum over all wave-modes to obtain the measureable PSD:

\par\null

$$<h(\omega)^2>=\frac{4 \eta k_BT}{\pi} \int_{q_{min}}^{q_{max}} \frac{dq}{(\kappa q^3+\sigma q)^2 + (4\eta \omega)^2}$$

All of these analysis are based on a statistical mechanics approach that
assumes membrane fluctuations that are excited by pure thermal energy.
These fluctuations hence lead to a significant amount of membrane to be
hidden in the fluctuations if only the projected area of the membrane is
measured, as in most experimental situations (Fig 1) based on video
microscopy. Here fluctuations immediately average out because of the
limited time resolution of the camera and the limited spatial resolution
of optical microscopy.~ A beautiful experimental way to prove this idea
of excess area that is stored in fluctuations has been developed by
Evans and coworkers\cite{Evans_1987}. Here a micropipette applies a
suction pressure to part of the membrane of a giant unilamellar vesicle
(GUV), thus fixing the membrane tension that can be easily calculated
using Laplace's law. Via video microscopy, the apparent area
change~\(\alpha=\frac{\Delta A}{A}\) of the vesicle can be
obtained\cite{Evans_1990,Evans_1997,Rawicz_2000}. Theoretically, in the entropic limit where
the thermal fluctuations are pulled out of the membrane we expect:

\par\null

$$\alpha_1(\sigma)=\frac{k_BT}{8\pi\kappa}ln(\frac{\sigma}{\sigma_0}).$$

Once the amplitude of the fluctuations becomes negligible the actual
membrane is extended by increasing the distance between individual
phospholipid molecules, thus leading to a linear
relation~\(\alpha_2\left(\sigma\right)=\frac{\sigma}{K_A}\) that gives access to the stretching modulus
of the membrane, which is a material parameter directly related to the
bending modulus.

This classical Helfrich approach allows to define the passive mechanical
situation of a membrane essentially by only two parameters, the bending
modulus~\(\kappa\) and the tension~\(\sigma\). A special
role can be attributed to the tension, which is a key element for
biological membranes, since any shape change in a closed membrane leads
automatically to a variation of surface to volume ratio. Analyzing the
shape changes of GUVs, it is reasonable to fix the volume as the liquid
is in-compressible, hence it is necessary to adjust the area in a self
consist way, leaving the membrane tension as an effective Lagrange
multiplier\cite{Deuling_1976}. Using this self consist approach in a
closed membrane system leads to exactly the same interpretation of
tension as a definition based on the energy required to extend the
membrane. In a more intuitive picture, the membrane tension in
practically any experimentally relevant system can be seen as an
entropic spring like extension, where entropy tries to maximizes the
possible accessible states by exiting all modes, and thus reducing the
apparent, projected area (Fig1). Hence keeping the surface constant
requires a balancing force, which is effectively the membrane tension.~~

\par\null

\section*{What makes membranes active}

{\label{what-makes-membranes-active}}

In the thermodynamic equilibrium, and hence passive situation the
membrane fluctuations are exclusively excited by the stochastic
forces~\(\zeta\left(t\right)\) driven by thermal energy. Hence, we can use
the general assumption from statistical mechanics of random,
uncorrelated forces on the membrane. In contrasts, we call a membrane
`active'~if besides the purely thermal force, any additional force acts
on the membrane, thus exciting additional fluctuations that eventually
decay by dissipation via the fluid environment. This will drive the
system out of thermodynamic equilibrium, because the these forces
effectively drive additional fluctuations that, when dissipated heat the
system, and thus violate any equilibrium assumption.~ In the light of
the above introduced Langevin equation we can easily propose:~

\par\null

$$ \frac{dh_q}{dt}=\tau_r^{-1} h_q(t)+\zeta^{\rm{a}}(t,q)+\zeta^{\rm{th}}(t) $$

where an additional active force~\(\zeta\left(t,q\right)_a\)~has been introduced.
The statistics of this active force determine the effect of activity on
the membrane.~In principle, any additional force will generate such
active, non-equilibrium states. Although, up to now only a certain set
of biological and artificial sources have been considered, obtaining an
active membrane in the sense of this article can be achieved by any
additional force that is superimposed on the thermal excitation. Please
note that additional processes, such as chemical reaction on or in the
membrane that will not generate a force on the membrane but potentially
lead to changes in membrane organisation or phase transition are beyond
the classical mean field description covered here and will hence not be
discussed in this review.~ It is also instructive to point out here that
any active force that has the same statistics like zero mean value and
delta correlation will automatically lead to a system that can be
experimentally described by a passive model if only the spontaneous
fluctuations are measured. In this case, the activity would be disguised
as an increased thermal energy, which is often used to define an
effective temperature.~ Hence, finding that a passive model can explain
the found fluctuations should not used as argument that a membrane is
indeed passive. Independent measurements of the mechanical propertis
would be required to support this claim.~~

\subsection*{Biological sources (Figure
2)}

{\label{biological-sources}}

\textbf{Ion pumps and channels} are key components of biological
membranes. Pumps allow to generate concentration gradients across the
membrane by pumping ions,~ even against a chemical potential difference.
In contrast, ion channels allow the passive flow of ions across the
membrane, for example to balance chemical or electrical potentials. In
both cases ions are transported across the membrane which requires a
mechanical momentum. To obey Newton's laws, this momentum is balanced by
the membrane, which effectively results in a force kick experience by
the membrane. Additionally, the ion transport will locally change ion
concentration and hence the osmotic pressure, which in turn will force
some water molecules to also pass the membrane, thus increasing the
force on the membrane ~\cite{Prost_1996}. It should be noted that the
energy source for the pumps is typically provided as a chemical energy
by the dissociation of ATP or GTP, while the ion flux through channels
is driven by chemical potential differences due to concentration
gradients across the membrane. In consequence these point forces will
lead to a deformation of the membrane that is pushing against the
surrounding liquid, ultimately resulting to frictional forces applied on
the membrane that are effective over the length-scale of the deformation
and balancing the initial point forces. This opposing force is not a
point force, because it couples to the fluid environment via the
membrane and is hence `smeared out'. As pointed out by several reports,
in total the resulting forces should be described as force
dipoles~\cite{Chen_2004,Gov_2004,Lacoste_2005}.~

While channels and pumps are located in the membrane, also external
forces can act on the membrane via
the~\textbf{cytoskeleton}~\cite{Bezanilla_2015}. Virtually every biological
cell has a polymer based cytoskeleton that is mechanically coupled to
the plasma membrane. In the case of red blood cells, this is primarily a
spectrin network\cite{Lux_2015}, while in more complex eukariotic
cells a thin meshwork of actin filaments is mechanically anchored to the
membrane\cite{Salbreux_2012}. The actin cytoskeleton is inherently active,
since it continuously reorganizes, either by
polymerization/depolymerization or by a highly dynamic de- and
attachment\cite{Blanchoin_2014}. Additionally,~ motor proteins, such as
myosins can directly pull the membrane or deform the cytoskeleton, thus
effectively pulling on the membrane (Fig2). Such reorganization forces
may lead to both normal and tangential forces on the membrane. Again,
the resulting membrane deformation will immediately balanced by the
viscous forces actin on the membrane, hence generating effective force
dipoles~\cite{Shlomovitz2007}.

The biological membranes are in continuous reorganization by the
addition and removal of small vesicles from the membrane. This
\textbf{endo- and exocitosis} requires well defined membrane deformation
and fission/fusion, both process require protein generated forces that
actively interact with the membrane thus driving it out of equilibrium.~
Such protein driven local curvatures are not only a key step for
endocytosis, but can be used by cells to regulate the total membrane
area, thus fixing membrane tension~\cite{Sinha2011}. This is highly
important in muscle cell and other highly deformable cell
types~\cite{Ramaswamy_2001}.~

Besides such direct forces applied on the membrane,
also~\textbf{rotating motors} embedded in the bilayer can drive
non-equilibrium behaviour. A highly relevant example is the
transmembrane protein~F0F1-ATP synthase, responsible for the synthesis
of ATP~\cite{Junge2015}.~ This impressive protein machinery is
powered by the chemical potentialdifference of a proton gradient across
the mitochondrial membrane. The ion flow drives rotational steps that
lead to the~ generation of ATP. Hence, the chemical potential energy is
not only translated into the chemical energy of the ATP molecule, but
also generates a mechanical forces acting on the membrane. Theoretically
already described in 2003~\cite{Lenz2003,Lenz2004} this form of active
membranes was only recently observed experimentally~\cite{Almendro-Vedia2017}.
The important feature of rotational motors is expected to be the
hydrodynamic interaction potentially leading to self-organized
arrangement of the motors~\cite{Lenz2004}. Besides these
experimentally verified processes driving active membrane fluctuations,
another process has been proposed to drive the membrane out of
equilibrium by inducing local jumps of the spontaneous membrane
curvature via flipping of phospholipids from one leaflet to the other.
Such processes are highly relevant for the membrane organization
~\cite{Montigny_2016} , since the asymmetry of the different membrane
compartments is vital for the physical properties of the membrane and
controls local phase transition and small domains. Flipping
phospholipids against a gradient requires chemical energy in the form of
ATP and is accomplished by a protein class called flippases and
floppases~\cite{Holthuis2005}. Furthermore, scamblases are independent
of an external energy source and allow to equilibrate the composition
gradients of well defined lipids by a bidirectional switching processes
~\cite{Hankins2015}. The mechanical forces are here induced by a local
change of spontaneous curvature since either a whole lipid is moved from
one leaflet to the other or by exchanging lipids with different
properties. The rapid and local change of spontaneous curvature will
effectively lead to a local deformation of the membrane thus generating
force. Although these forces have been already integrated into the
proposed theoretical descriptions, so far only static shape changes of
GUVs have been studied experimentaly~\cite{Papadopulos_2007}.~

It should be noted that potentially any protein located at the membrane
and undergoing transformational changes by using either chemical energy,
or a chemical potential maintained by concentration gradients across the
membrane may lead to active forces on the membrane. So the here
presented summary might be far from complete.

\subsection*{Artificial sources}

{\label{artificial-sources}}

Besides these biological relevant active forces driving mechanical
membrane fluctuations, activity can also be induced by artificial, force
generating processes. A beautiful example of this are active particles
that can be embedded in the membrane or in the liquid around the
membranes. Especially in the case of active microswimmers encapsulated
in a membrane, the particles will not only generate hydrodynamic forces
but can also directly bump into the membrane and thus transmit direct
pushing forces. So far, mainly theoretical studies and simulations have
been published in this respect\cite{Paoluzzi_2016}. If embedded in the
membrane, active molecules can be undergo confirmation changes for
example by the illumination with light of a specific wavelength, or by a
chemical reaction as well as simple fluorescence excitation happening at
the membrane\cite{Veatch_2008,Reusch_2013}. Typically such sources are random in
nature, and hence similar to the classes of active biological sources.~

A further well studied source of activity is membrane adhesion to
surfaces, here the pure presence of a close wall was proposed
theoretically to greatly change the fluctuations~\cite{Prost_1998}.
Furthermore, the adhesion is typically a non-equilibrium processes as it
required adhesion energy to be transformed in membrane deformations.
This subject has been in detailed studied by reconstituted liposomes
attaching to patterned surfaces \cite{Fenz_2017,Schmidt_2014,Bihr_2012}.~

A very attractive approach to systematically study active forces on the
membrane is by using optical tweezers to drive membrane fluctuations.
Here, random forces can be applied, but also controlled pulling forces
are used \cite{Bar_Ziv_1998}. An exceptionally nice example was
developed recently by multiplexed optical tweezers that could excite
individual bending modes, and thus directly test the mechanical
properties of GUVs in a mode dependent way~\cite{Brown2011}. Finally,
hydrodynamic flow gradients will also apply a force on the membrane and
thus lead to active fluctuations. Such hydrodynamic interactions and
activities are especially important when active swimmers are moving in
close vicinity to the membrane.~~

\par\null

\section*{Direct observation of activity (Experiments, Fig
3)}

{\label{308954}}

Although the first clear hints for an active contribution to membrane
fluctuations of the red blood cell membrane where given in the middle of
the 20th century, the mixture of both active and pure thermal
contributions led to the generally accepted interpretation that~ RBC
flickering is a purely Brownian motion process~\cite{PARPART1956,Brochard1975,Strey1995}. The
reason for this long and sometime even emotional pursued scientific
discussion can be found in the fundamental resemblance of actively
driven fluctuations with purely thermal fluctuations if activity is
injected by randomly distributed and uncorrelated force centers. Since
one fingerprint of thermal motion is exactly this randomness, it is not
surprising that all experimentally recorded data that is not actively
deforming the system can indeed be fitted by passive theories. It is
very important to stress here that the pure fact that a passive model
can fit the experimental data is a not sufficient to claim that the
fluctuations are purely passive.~

\subsection*{}

{\label{120770}}

\subsection*{Activity and excess area
increase}

{\label{120770}}

The first experimental proof that active forces have a measurable impact
on the bilayer membranes was given in 1999 ~\cite{Manneville_1999}~3 years
after the first theoretical prediction that activity should modify
membrane fluctuations by Prost and Bruinsma~\cite{Prost_1996}. The
experimental system was based on the micropipette aspiration approach,
where the excess area is measured as a function of membrane tension.
Active forces are applied on the GUV membrane by embedding the light
activated proton pump bacteriorhodopsin (BR)~\cite{Lanyi_2004}. BR is
a well characterized photon driven ion pump~ pump optimally operating
when illuminated with wavelength around 566nm. The measurement of
membrane excess area strikingly showed that in the active case more
membrane area was stored in the fluctuations if compared to the case
where the GUV was illuminated with green light, where no active pumping
is expected. As pointed out, it came with no surprise that the passive
fluctuation model could very well fit the active case since the
logarithmic dependence of the excess area on the applied tension was
nicely valid even when activating the BR. However, between the active
and the passive case, the experimentally obtained~ prefactor of the
expected logarithmic dependence~(see Eq. 7) even doubled in many
experiments. Since the bending modulus was independent of the included
BR concentration~\cite{Manneville1999} the authors argued that the model
can be used to determine the activity by introducing an effective
temperature. The concept of effective temperature is very appealing
here, as it suggests that the active forces have a statistical~signature
similar to thermal forces, such as zero mean and a uncorrelated action.
In a follow up approach the same technique was applied to ATP driven
Ca2+ pumps embedded in SOPC~ membranes\cite{Girard_2005}, that showed a
clear dependence of bending modulus on the amount of in-cooperated
proteins. In a more detailed analysis, again a strong increase of the
fluctuations was measured and interpreted in the light of an effective
temperature\cite{Gov_2004}.~However, slowly models were proposed
that took the finite correlation time of any active process into
account~\cite{Lacoste2005}. Here the interpretation of activity by an
effective temperature is questionable, since a new timescale in the
activity needs to be introduced, which conflicts with the statistical
requirement on thermal forces to be delta correlated.~~

\par\null

\subsection*{Flickering analysis}

{\label{855677}}

To get more insights into the actual membrane fluctuations in both the
active and the passive case, several experimental approaches have been
developed in the past decades. For the interested reader, a detailed
review about these methods has been recently
published~\cite{Monzel_2016}. After the first quantitative measurements
by Brochard and Lennon\cite{Brochard1975}~more detailed video microscopy
analysis was pursued. Initial theoretical work~\cite{Schneider_1984}
extended later to tensed vesicles by Safran and
Milner~\cite{Milner_1987} paved the way for to describe and analyze the
mode dependent fluctuations. Initially, the shape analysis using video
microscopy has been applied on flaccid vesicles~\cite{Engelhardt_1985} and
was soon extended to RBCs~\cite{Duwe}. Applying these contour
detection methods to analyze shape fluctuations of active membranes
driven by light induced BR suggested that activity can effectively
reduce membrane tension~\cite{Faris_2009}. A big strength of the
contour analysis is access to both, the spatial modes and the temporal
relaxation times, because both carry information about the mechanics and
the~ energy injected in the system by activity . this was exploited more
recently to analyze active membranes driven by the pumps Na+, K+
ATPase~\cite{Bouvrais_2012}, where slight effects of the activity on the
bending modulus were reported. Strikingly, in this later study,two
relaxation timescales were identified. Besides the expected, mode
dependent relaxation an additional slow relaxation timescale of about
0.5 sec was identified that was consistent with the cycling time of the
pumps used in these experiments. In a more theoretical approach the
flickering has also been simulated~\cite{Loubet_2013}, which confirm
that membrane mechanics can be varied by the addition of active forces
on the membrane. Very recently, the first experimental evidence of
active membrane fluctuations induced by the rotary motor
F0F1-ATPsynthase has been reported in GUVs, where the increased
fluctuations where detected using video microscopy based mode analysis.
These experiments showed that the activity was highly localized to a
few, but well defined positions on the GUV
membrane\cite{Almendro-Vedia2017}.~~

{[}MOVE TO RBC DISCUSSION??{]}In the context of cellular systems, video
microscopy based mode decomposition has been used to study the active
fluctuations in RBCs, with partially conflicting results.
While~\cite{Evans2008} found no change in fluctuations upon ATP
depletion, a later paper that also used contour analysis did see drastic
changes upon ATP depletion~\selectlanguage{ngerman}\cite{Rodríguez-García2016}. It should however be
noted here that the pure change of fluctuation amplitudes under ATP
depletion~ in complex systems like RBCs is not a clean proof that active
forces are the reason for the increased fluctuations, as the abundance
of ATP has a drastic influence on the cell mechanics, and thus recorded
fluctuation changes can be easily explained~ by stiffening of the cells
upon ATP depletion.~

Besides these video microscopy based analysis, a number of additional
techniques have been developed in the past ten years. Namely reflection
interference contrast microscopy (RICM) and total internal reflection
microscopy (TIRF) allow to precisely monitor membrane fluctuations close
to the glass surface of common microscopy setups. These methods give
access to nm precise height fluctuation in the substrate plane, and
hence allow to test simultaneously temporal and spatial correlations.
Both methods have been used to to study active membrane fluctuations in
cells~\cite{Fenz_2017,Biswas_2017}. While these methods work only very close to
the glass substrate, diffraction phase microscopy (DPM) allows to image
sample thickness by a highly sensitive measurement of the phase-shift
experienced by photons passing through an object with different
refractive index as the environment. If the refractive index of the
object is known, the absolute thickness can be readily calculated with
nm precision~\cite{Popescu2006,Bhaduri_2014}, and hence the height fluctuations are
accessible. This method was used to investigate the question of active
contributions to the red blood cell membrane, and indeed signs of
non-equilibrium processes, such as non-Gaussian fluctuation
distributions measured by the kurtosis~\cite{Park2011,Ben-Isaac2011}~~were
observed.~ It was also used to confirm previous measurements of ATP
dependent fluctuation amplitudes~\cite{Park_2010}. In a more recent
experiments, the same technique was used to show that a purely thermal
model can explain the measured fluctuations~\cite{Evans_2017}. It
should be emphasized again, that due to the randomness of the active
forces, it is expected that passive models can explain the fluctuation
data, however that the mechanical properties extracted from such passive
theories might be corrupted, and need to be crosschecked by an
independent measurement before taken serious.~

Recently, an elegant methods called dynamic optical displacement
spectroscopy (DODS)~\cite{Monzel_2015} was introduced to determine
membrane height fluctuations using a fluorescence correlation
spectroscopy related approach. Here, the fluorescence intensity of a
confocal volume intersecting the membrane is recorded over time. In a
calibration step the fluorescence intensity is assigned to the axial
position of the membrane with respect to the excitation laser. This
allow to determine the membrane height fluctuations with 20 nm spatial,
and \selectlanguage{greek}µ\selectlanguage{english}s temporal resolution. Applying this technique to GUVs, RBCs and
human macrophages showed clear signs of non-gaussian behaviour in the
cells, but not in the passive GUVs.~ This technique also confirmed in
RBCs the excitance of ATP dependence of the fluctuations, with a
decrease of the non-gaussian contribution is the ATP depleted case,
again supporting that active fluctuations might be relevant in RBC
flickering.~

Another point detection of the membrane fluctuations called point dark
field microscopy (PDM) was used by Korenstein and coworkers in a series
of papers investigating the red blood cell
flickering~\cite{Levin1991,Tuvia1998,Tuvia1997,Tuvia1992}. Here the light barely touching the edge
of the RBC is refracted, and hence can pass the blocking filter that
removes all unscattered light in a conventional microscope. These papers
were the first known quantitative measurement of ATP dependent
flickering after the initial phenomenological description by
Blowers~\cite{BLOWERS1951}, and can be seen as the trigger for the
scientific discussion related to the origin of RBC flickering. Besides
the observation that the flickering was ATP dependent, it was also
reported that the flickering amplitude depends on the viscosity of the
surrounding fluid~\cite{Tuvia1997}, which was considered a clear sign
that the flickering is active, as in a thermal equilibrium, a dynamical
variable such as viscosity cannot influence time averaged values like
the mean fluctuation amplitude.~

Similar to point dark field microscopy is a more recent laser based
technique is time resolved membrane fluctuation spectroscopy
(TRMFS)~\cite{G_gler_2007,Betz_2009,Betz_2012}, which uses a quadrant photodiode to measure
the asymmetric interference pattern of a highly focused infrared laser
that is partially interacting with the membrane. This allows for very
high spatial (nm) and temporal (\selectlanguage{greek}µ\selectlanguage{english}s) resolution. In contrast to PDM it is
possible to calibrate the signal, and thus gain a high resolution of the
membrane fluctuations. Using this method, a clear dependence of red
blood cell membrane fluctuation on ATP was confirmed~\cite{Betz_2009},
but the previously reported dependence of viscosity~\cite{Tuvia_1997}
was not observed. It is possible that the integration time used in PDM
was not adjusted in the high viscosity regime, so that the statistics
were not sufficient to fully explore the possible membrane states. This
would be consistent with the values reported by TRMFS and PDM.~

In sum, all of these flickering analysis experiments developed in the
past 3 decades are based on the pioneering work of Brochard and
Lennon~\cite{Brochard1975}~who basically founded the field of
quantitative flickering analysis. It is remarkable that these initial
experiments and theoretical analysis already covered many aspects of our
current understanding of membrane fluctuations and red blood cell
flickering, although the model proposed was purely passive.~

Similar to the experimental obtained by micropipette aspiration, the
pure measurement of fluctuations is commonly in well accordance with
passive model, even in the case of active membranes. This finding that
fluctuation can be modeled by a passive theory is also one of the
arguments used to propose that the red blood cell flickering is a
passive process, simply because it can be described so. But this
argument does not hold if the active forces are random, as in this case
the effect of activity will have the same phenotype as pure thermal
forces. Even worse, the passive model will give a false sense of
correctness and lead to the notion that the extracted mechanical
parameters are correct, although they might be systematically wrong
because the energy used in the model is underestimated. Unfortunately,
even if activity is included in the model, it is still unclear if the
extracted parameters are reliable, since the characteristics of the
included noise might be inaccurate.~

\subsection*{}

{\label{247382}}

\subsection*{Check of equilibrium
thermodynamics:~~}

{\label{247382}}

As the expected active forces acting on the membrane are inherently
random as they do indeed largely resemble a thermal contribution, which
easily explains why equilibrium models can also model the fluctuations
of active membranes. The error of the passive model would only become
obvious when comparing the extracted mechanical membrane properties with
independent mechanical measurements. In principle, in any purely passive
system there is a deep relation between mechanical properties and the
thermal fluctuations, expressed by the fluctuation dissipation theorem
(FDT). Simply speaking, the FDT connects the dissipation of fluctuation
to the thermal excitation of exactly the same fluctuations. The membrane
`borrows' energy from the thermal bath to excite a fluctuation, but
exactly the same energy is returned by dissipating these fluctuation.
Any systematic difference between fluctuations and dissipation would
lead to a direct violation of the second law of thermodynamics.
Formally, the FDT can be expressed as~\(<\tilde{x}\left(\omega\right)^2>\ =\frac{2k_BT}{\omega}\chi''\left(\omega\right)\), where~~

$$\chi(\omega)=\tilde{x}(\omega)/\tilde{f}(\omega)$$

is the complex, frequency dependent mechanical response function that
can be accessed experimentally. A simple way of measuring the response
function is by applying a well defined time dependent force while
measuring the response in form of a movement. Dividing the time
dependent Fourier transformation of the membrane position by the Fourier
transformation of the applied force gives the complex response function.
In Fourier space, the complex part of the response
function~\(\chi''\left(\omega\right)\)~describes energy dissipation.~

By comparing now the power spectral density~\(<h\left(\omega\right)^2>\)~ of the
free fluctuations (thermal plus possible active contributions) with the
dissipative response function, the validity of the FDT can be directly
checked, and any difference where the spontaneous fluctuations are
larger than expected must be attributed to non-thermal active forces
driving at least partially the membrane. Recently, the validity of the
FDT was checked for RBC flickering and it became obvious that there is a
significant active contribution that leads to violation of the FDT at
timescales slower that about 100ms, while the FDT is perfectly valid for
faster timescales. This is the first direct evidence that at low
timescales the red blood cell membrane is actively driven. The model
explaining both the fluctuations and the response function is an
extension of the classical passive model, and as expected, it turns out
that the free, and active~fluctuations can even be explained by a
passive model, at the price of extracting largely underestimated values
for membrane tension.~

\par\null

\subsection*{Scattering and memrbane tether pulling
experiments:~}

{\label{521442}}

Besides these main experimental approaches to study membrane activity,
also scattering methods and membrane tether pulling experiments have
been used to study active fluctuations in membranes. In a first approach
using x-ray scattering, it was not possible to directly observe an
effect of the BR proteins on a model membrane, which might be due to the
highly packed layering structures required for x-ray
scattering~\cite{Giahi2007}. However, when optically pumping
fluorescent molecules directly attached to the membranes, it was
possible to directly observe changes in relaxation timescales of the
thus excited fluctuations using time resolved diffuse X-ray scattering.
The observed timescales are in the order of \selectlanguage{greek}µ\selectlanguage{english}s, and are hence only
detectable with such scattering approaches. Because of the short
timescales involved, the study concludes that although in conventional
video fluorescence microscopy these effect can be neglected, they might
play a role in scanning techniques like confocal microscopy or STED
microscopy that take very rapid local measurements, while depositing
high laser powers very locally~\cite{Reusch2013}. Beides these two
studies, the experimental investigation of active membranes by
scattering techniques remains poor.~

A further experimental approach that is frequently used to measure
membrane mechanics are controlled membrane tethers pulled by optical
tweezers~\cite{Dai_1995},~ AFM\cite{Diz_Mu_oz_2010}, magnetic
beads~\cite{Heinrich_1996} or hydrodynamic flows \cite{Borghi_2007}. The
pulling force~\(f_p\) required to maintain a membrane tether
can be precisely linked to the membrane mechanics by~\(f_p=2\pi\sqrt{2\sigma\kappa}\)
if any contribution for the cytoskeleton can be neglected. By observing
changes in the forces when electrically activating a membrane, rapid
changes in membrane tension can be observed~\cite{Brownell2010}, and
based on similar measurements the importance of membrane tension and the
active regulation of this key mechanical parameter have been in detailed
studied, as recently reviewed~\selectlanguage{ngerman}\cite{Diz-Muñoz2013}.

\section*{}

{\label{can-activity-affect-mechanical-properties}}\par\null

\section*{Theoretical approaches to model activity in membrane
mechanics~}

{\label{228665}}

1) initial ideas by Jacques and Bruinsma

2) Expansion~ of these approaches by including diffusion of active
proteins, activity couplding to the memrbane curvature, curvature
dependent dissusion, hydrodynanimics.~

3) Addition of active sticking and detaching from substrates

4) Incorporation of interaction with active actin cytoskeleton.~

5) new approaches by Lomholt via Gibbs

6) Investigation of broken detailed ballance and possibilities of
instabilities.~

7) Active membrane of the red blood cells.~

IMPORTANT: Can we give a recipie of how to analyse RBC fluctuations.
People are still using the passive model, because our description is so
complex that no-one understands how to integrate it. Even~ Ulrich
Schwarz was asking me what he should do to analyse the memrbane
flcutautions they have.~ Can we give any predictions about the
q-dependence of the activity?~

\par\null

FLipflop: Simulations (~Lipid flip-flop driven mechanical and
morphological changes in model membranes)

\par\null

\section*{The controverse discussion about RBC
flickering}

{\label{575553}}

Quickly review the discussion. This is somewhat what was already done in
the~ intro\ldots{}. I should substancially shorten the intro and expand
here a bit. Also remove any of the discussion in the previous part.

\ldots{}.Tuvia reported that the static, time averaged fluctuation
amplitude decreases when increasing the medium viscosity was interpreted
as strong sign for an active process, since a purely passive origin of
the fluctuations would not change any static observable, such as the
mean fluctuation amplitude.~ Follow up experiments using laser
interferometry to precisely quantify the static fluctuations amplitudes
in different medium viscosity did not confirm these early experiments.
Possibly the acquisition time window used to get the time average was
not sufficiently long, so that the system could not explore all
positions in the cases of increased medium viscosity. Further doubts
about the active interpretation came from a series of papers that could
not confirm the reduction in fluctuation amplitude, and based on all
these controversial observations a highly controversial discussion
emerged in the field. Starting with more precises measurements of
membrane position, however, additional experimental results were
obtained that gave further hints towards an active contribution to the
flickering, such as Non-Gaussian contributions to the probability
distribution of the membrane position, or anomalous lateral correlation
functions. These can not be explained in the picture of a equilibrium
system that is operating in a linear regime with regard to the
force-distance situations. Although very intriguing, the asymmetric
nature of the RBC cell membrane can easily explain that the non-Gaussian
fluctuations could come from nonlinear contributions due to the
architecture of the membrane. Eventually, a direct comparison between
the fluctuations and an independent measurement of the spontaneous
fluctuations was required to give a direct test of fundamental
equilibrium physics, such as the fluctuation dissipation relation. Using
optical tweezers we could recently show that the question of active or
passive membrane fluctuations effectively boils down to a question of
timescales.~ For timescales slower that 0.1 sec, the effect of active
metabolic forces on the red blood cell membrane becomes relevant, while
for measurements on timescale faster than 100ms, the thermal
contribution dominates.~

\par\null\par\null

\section*{The effect of activity on membrane mechanics (tension, kappa
and effective
temperature).~}

{\label{300800}}

As already introduces, the effect that active forces may have on the
mechanical properties of a membrane

The proposition of renormalized bending modulus and tension, as well as
the concept of effective temperature. Mikhailov proposes even an
effective negative bending rigiity, leading to instabilities. Active
tension in the Lomhold Seifert paper depends on viscosity!

\subsection*{}

{\label{500281}}

\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%REMOVE??

\section*{The risk of
over-interpretation}

{\label{the-risk-of-over-interpretation}}

depending on the analysis tension is higher or lower\ldots{} but what
happens here!!! It needs to be carefully differenciated if we gain
tension by applying equilibrium model on active system (this leads to a
reduction of apparent tension because we observe larger fluctuations),
however, when we apply the simple definition of tension as entropic
effect, the actual tension is increase in the system, because I need to
apply larger forces to pull out the area stored in fluctuations.~

If the `real' tension is integrated correctly in the model, it remains
to be checked if the model is actually the correct description. We still
lack a fully and convincingly developped model that include the effects
on tension and bending~ modulus.~

For the effective Energy, the situation is similar. Introduced to
discribe the static experiments of micropieptte aspiration, well
described by a renormalization of the energy exciting the fluctuations,
it became obvious that in this static and lengthscale independent
measurement we effective integrate over time and lengthscale, thus
resulting an renormalization that simpliy adds up all the different
sources of active energies injected in the system. However, when
investigating the dynamics and/or the different wavelength involved in
the movement, both the theoretical and the experimental results clearly
show that the active energies coupling to the fluctuations are both time
and lengthscale dependent. Hence the interpretation as effective
temperature is missleading because a hallmark of thermal noise is that
it is a constant that is independent of time and lengthscales.~

\par\null

Any process that is actively applying forces on the membrane, but shows
the same, or similar statistics as thermal excitation, namely zero mean
and delta correlated autocorrelation, while violating the
fluctuation-dissipation theorem will ultimately lead to fluctuations
that are impossible to distinguish from thermal excitation unless a
clear measurements of the energy exciting these motions is possible.
Intrinsically, the mechanical properties also enter the amplitudes of
thermal fluctuations so that any additional energy besides the purely
thermal energy would be misinterpreted as changes in the mechanical
properties, for examples as an reduction of mechanical tension.~

The obvious~

\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%
END REMOVE



\section*{The challenge: Understanding the mechanics of the active cortex
as a membrane-cytoskeleton composite
material}

{\label{the-relevant-system-active-cortex-as-membrane-cytoskeleton-composite}}

Here we can expant in the RBCs, and in the experiments and theories that
couple the cytoskeleton to the membrane (Nir, Granek), leading to
travelling waves and pattern formation on the the membrane. impotance of
non-equilibrium `stickers' by Lipowsky. General model by sriram
(activating memrbanes 2014)

IMPORTANT, don't forget the recent paper by Darius:~Actomyosin dynamics
drive local membrane component organization in an in vitro active
composite layer

Actually, we need to more experiments I think. E.g. predictions of
increased diffusion, of increased fluctuations clcose to walls, and to
changes in the adhesion properties have never (I could not find
anything) been tested.~

On the theory side, a clear definition of tension and the confusion with
effective tension would be highly desirable.~

\par\null\par\null\par\null\par\null

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