Heitjans P., Karger J. (Eds.). Diffusion in Condensed Matter: Methods, Materials, Models
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466 Roger Biel et al.
at least 1 kHz can usually be obtained, leading to a temporal resolution of
at least 1 ms. More than two paths can be used with the aid of electronic
switches.
11.3 Observation of Diffusion of Heavy Water in Gels
and Living Cells by Scanning Acoustic Microscopy with
Phase Contrast
Scanning acoustic microscopy (SAM) [4] in the reflection mode is a confo-
cal version of microscopy similar to confocal laser microscopy but based on
acoustic waves. In commercially available equipments frequencies up to 2 GHz
are used. With water as a coupling fluid a lateral resolution down to 0.6 µm
can be achieved.
Recently, high resolution phase contrast has been developed for this ver-
sion of scanning microscopy [5]. For demonstration of a typical image obtain-
able by scanning acoustic microscopy with phase contrast (PSAM), Fig. 11.5
shows an image of a cell.
The phase contrast can be employed to detect local changes of the trans-
port properties of ultrasound induced by diffusion processes [6]. To establish
this method droplets of gel deposited on a glass substrate have been used.
The gel is manufactured from household gelatin dissolved in ordinary water.
Heavy water (D
2
O) is used in exchange for ordinary water as a coupling fluid.
This leads to time-dependent changes in the phase images. A set of images
Amplitude contrast Phase contrast
Fig. 11.5. Images of an XTH2-cell on glass. The grey scale in the image in phase
contrast is proportional to the phase of the reflected ultrasonic waves. The width
of the images is about 100 µm. The images were obtained at a frequency of 1 GHz
with water as a coupling fluid. The maximum thickness of the cell somewhat below
the center of the images is about 2 µm.
11 Diffusion Measurements by Ultrasonics 467
Fig. 11.6. Time dependence of the phase of the reflected ultrasonic waves (1.2 GHz)
observed with an ultrasonic microscope for a position on a sample of gelatin de-
posited on glass. The coupling fluid has been changed from normal water to heavy
water at time equal zero.
Fig. 11.7. Time dependence of the phase of the reflected ultrasonic waves (1.2 GHz)
observed with an ultrasonic microscope for a fixed position on a living cell on glass.
The coupling fluid has been changed from normal water to heavy water at time
equal zero.
was taken at constant time intervals. A reference area close to the deposited
droplet is used as a reference to detect changes of the phases in the image of
the droplet. Figure 11.6 shows the result for a fixed position in the image of
a droplet. The area at this position is only limited by the lateral resolution
of the acoustic microscope of about 1 µm. The temporal dependence of the
phase signal exhibits a distinct change following the contact to heavy water
over a time span of about 100 s. This change is attributed to the exchange
468 Roger Biel et al.
of normal water by heavy water in the gel by diffusion. Variations at later
times are in part caused by solution processes of the deposited material.
Fig. 11.7 shows the response of a living cell under similar conditions. A
relaxation time of about 200 s can be derived from the early part of the
response. The distinct change at time 1800 s represents the exitus of the cell
due to an overdose of heavy water.
These observations represent only a first step to demonstrate the pos-
sibility to observe transport processes including diffusion with acoustic mi-
croscopy. We restrain therefore from a quantitative analysis of the data. The
transport properties in the cell are expected to be dominated by diffusion
through the cell membrane. The observation is at least sufficient to deter-
mine the typical time scales for transport processes even in living objects.
For determining quantitatively the diffusivities, the thickness of the ob-
jects is needed. The technique is therefore especially suitable for coatings of
constant thickness. For irregularly shaped objects of limited size on planar
substrates (as employed here), the thickness can be monitored by atomic
force microscopy (AFM), available in the developed microscope [5]. Homoge-
neous materials can also be characterized by phase sensitive scanning acoustic
microscopy (PSAM) and model calculations fitted to the data [7].
In addition to the techniques demonstrated here microscopic holographic
detection schemes have been developed [5, 8], which can be employed for
the spatially resolved detection of the variation of the elastic properties in
extended objects.
11.4 Conclusion
The novel detection schemes based on ultrasonic interferometers can be used
to determine the diffusion coefficient following a non-equilibrium distribution
of the diffusing species – in the demonstrated application the diffusing hydro-
gen. The rather high sensitivity allows the observation of slow processes, re-
spectively small deviations from equilibrated distributions. Scanning acoustic
microscopy with phase contrast representing also an interferometric method
can be used in a similar way to observe relaxation processes with spatial
resolution down to about 1 µm (lateral). Due to the rather short interaction
length in the observed objects which is typically in the range of the lateral
resolution or below, the sensitivity is significantly reduced with respect to
interferometric detection schemes employed in macroscopic objects.
Notation
c hydrogen concentration in atomic percent
c
11
,c
44
stiffness constants
D diffusion coefficient
11 Diffusion Measurements by Ultrasonics 469
D
0
diffusion coefficient at T →∞
E
a
activation energy
k
B
Boltzmann constant
t time
T temperature
v
t
velocity of transverse ultrasonic waves
x position
ρ density
References
1. K.U. W¨urz: Ein Ultraschall-Zweistrahlinterferometer zur Messung des Diffu-
sionskoeffizienten von H in Ta. Diploma thesis, Frankfurt am Main (1987)
2. A. Magerl, B. Berre, G. Alefeld: Phys. Stat. Sol. (a) 36, 161 (1976)
3. J. V¨olkl, G. Alefeld: Diffusion of Hydrogen in Metals. In: Hydrogen in Metals
I (Springer, Berlin Heidelberg New York 1978) p 332
4. R.A. Lemons, C.F. Quate: Appl. Phys. Lett. 24, 163 (1974)
5. W. Grill, K. Hillmann, K.U. W¨urz, J. Wesner: Scanning Ultrasonic Microscopy
with Phase Contrast. In: Advances in Acoustic Microscopy, vol 2, ed by A.
Briggs, W. Arnold (Plenum Press, New York 1996)
6. K. Hillmann, W. Grill, J. Bereiter-Hahn: J. of Alloys and Compounds 211/212,
625 (1994)
7. R. Biel: Ultraschallrastermikroskopie mit Phasenkontrast. Diploma thesis,
Frankfurt am Main (1995)
8. K.U. W¨urz, J. Wesner, K. Hillmann, W. Grill: Z. Phys. B 97 (1995)
12 Diffusion in Membranes
Ilpo Vattulainen and Ole G. Mouritsen
12.1 Introduction
Nature is always in motion. As simple as it is, this statement is true in the
sense that numerous phenomena in living systems are characterized by non-
equilibrium. Our muscles are in constant need of energy provided by the
metabolic pathway, the blood flows in our veins as long as the heart keeps
going, and yet old cells are constantly being replaced by fresh ones as a
typical life time of a cell is on the order of one day. The state of living beings
can therefore only rarely be described by equilibrium. However, even if a
true equilibrium were possible in living systems, we would find spontaneous
thermal fluctuations to occur around the equilibrium state, again implying
that the matter were moving in time.
This book is devoted to the idea of particles moving in condensed matter
systems. Our aim in this chapter is to discuss the role of diffusion in soft
biological interfaces known as membranes.
In biological systems, diffusion is one of the most intriguing processes that
arise from the dynamics of biological molecules. It may sound surprising but
many of the diffusion processes in living systems are based on a very simple
idea – a random walk. Microtubules – long filaments of proteins that search
for other molecules inside cells and serve as highways for transporting them
inside the cell – grow and shrink in a manner that reminds us of a random
walk [1]. Small bacteria, in turn, swim in a fashion involving a long ballistic
movement followed by a period during which the particle “tumbles” in a
highly erratic manner, thus choosing a new direction for its motion, again
followed by another long jump [2]. Interestingly, this joint motion of rushing
and tumbling allows the bacterium to move efficiently around its environment
as it searches for the best food markets. Then, if we look at the motion of
molecules in cell membranes that act as a permeable barrier between the
inside and the outside of the cell, we again may notice that the motion of
molecules in the plane of the membrane can essentially be described as a
random walk [3].
All these examples are kind of confusing since one might expect that
Nature works in a manner that is more orderly and deterministic and less
random than finance or lottery, in which the random walk plays a major
role [4]. Yet we have plenty of evidence that the diffusion in living systems
472 Ilpo Vattulainen and Ole G. Mouritsen
is often governed by this simple principle. To the best of our knowledge,
the first (reported) observations of the random walk were made in 1828 by
the botanist Robert Brown [5], who studied the pollen of different plants
and observed that, when placed in water, the pollen particles were in unin-
terrupted and irregular “swarming” motion. Today, we understand that the
Brownian motion (as it is called to honor the person who discovered it) con-
cerns the motion of a colloidal particle in a liquid and results from random
molecular collisions with the liquid molecules, leading to motion, which can
be described in terms of the random walk picture. In biological systems, in
turn, we are dealing with molecules or clusters of molecules whose size is
far greater than the size of surrounding solvent particles. Therefore there is
an analogy from colloidal particles to biological molecules, thus providing us
with some grounds to use the random walk concept to describe diffusion in
living systems.
Random diffusion is a non-specific process, which invariably leads to dis-
order in a system. Although random diffusion over time may provide for
essential reaction pathways in living matter, it is too disordered and disor-
ganized to be relied on in delicate life processes. Therefore, in order to take
advantage of the omnipresence and robustness of random diffusion, Nature
has over evolutionary time scales developed strategies to compartmentalize
and structure living matter on scales from nanometers to the size of whole
cells and organisms. Within these structures, macromolecules, macromolec-
ular assemblies, as well as sub-cellular entities perform random as well as
directed diffusion. The compartmentalization and structure are provided by
biological soft interfaces, so-called membranes, as well as a host of fibers
and scaffolding structures. It is noteworthy that the typical length scales of
these structures are in the 1 – 1000 nm range, i.e. on a scale where diffusional
processes can be effective on the time scales that are relevant in biology. Due
to the structuring, one can anticipate that the various diffusional processes
take place in highly heterogeneous matter and that deviations from normal
diffusion under isotropic conditions are likely to occur.
An understanding of the nature of diffusional processes in living matter,
and in particular on the level of the individual cell and its various sub-cellular
components, is one of the grand challenges in the so-called post-genomic era.
Within the last few years, the complete genome of whole organisms, rang-
ing from bacteria, yeast, worms, insects, to that of man, has been mapped
out. The genome provides the information about which macromolecules (e.g.
proteins) the organism can produce. However, the genome contains seem-
ingly no information on how these macromolecules are organized in space
and time and which molecular mechanisms are controlling the organization.
Physical principles are here called for. At almost every stage of these con-
trolling mechanisms, elements of diffusional processes are involved: in the
molecular self-assembly of fibers and membranes, in the transport and traf-
ficking of RNA, DNA, sugars, fats, and metabolites, in biochemical signaling
12 Diffusion in Membranes 473
cascades as well as intra- and inter-cellular communication, in cell growth,
and ultimately in cell death.
In this short review, our aim is to illustrate some of the charming aspects
of diffusion in model membranes. We feel that the topic is of a fundamental
nature as it is related to the dynamics of cellular functioning. Many present-
day medical applications such as drug delivery and gene therapy [6] are es-
sentially dynamic processes that involve the transport of molecules through
membranes. Diffusive transport in membranes furthermore plays an impor-
tant role in processes such as membrane fusion [7], charge transfer [8], and
intracellular signaling [9]. It is rather surprising, though, how little is known
about the microscopic mechanisms of these dynamic processes in membrane
systems. Thus we hope that you will enjoy this little journey, and especially
we wish that you will find this topic exciting and let the spirit guide your way
to work on these problems. To ease your way on this road, we would like to
direct your attention to a number of comprehensive review articles about the
overall properties of membrane systems [9–16] as well as about the dynamic
processes in membranes and other biological systems [2, 15–17]. It is not an
easy road, but it is worth it.
12.2 Short Overview of Biological Membranes
Nature is full of examples of surfaces or interfaces that separate one phase
from another and have some important function. For instance, our skin pro-
tects our body from hazardous chemicals of the environment and from dry-
ing out, and soap bubbles are fragile fluctuating interfaces [18] that are both
charming to wonder about and important as an example of the behavior of
the detergents we use in our daily life. Further, soap bubbles remind us of
biological membranes that surround our cells and serve as an example of
soft condensed matter surfaces whose properties can be tuned by weak in-
teractions on the order of thermal energy. Yet there is no doubt that the
complexity and biological relevance of membranes are far greater than those
of soap bubbles.
Biological membranes are an intriguing example of Nature’s talent to
make complex systems with a huge number of degrees of freedom but with
a well-defined function. The complexity, however, makes our study difficult
as we try to understand how membranes work under a variety of physiolog-
ically relevant conditions. To ease this burden, one often starts from simple
descriptions of such complex systems, and then, after having understood how
they work, it is natural to proceed to a higher level of complexity.
Using this idea, we may first think of cell membranes as thin elastic sheets,
whose total thickness is typically about 5 nm and depends only weakly on
thermodynamic conditions such as the temperature of the system. The mem-
brane is comprised of two monolayers, each of which is made up of lipid mole-
474 Ilpo Vattulainen and Ole G. Mouritsen
Fig. 12.1. Representation of a typical phosphatidylcholine (PC) lipid molecule on
an atomic level (on the left). Additionally, as physicists, we feel obliged to present
a simplified description of a lipid (in the middle), where the aim is to grasp the
essential features of both the hydrophilic and hydrophobic nature of lipid molecules.
In some cases, this structure can be coarse grained even further to obtain a minimal
description of a lipid in terms of a two-dimensional disk (shown on the right).
cules. The lipids
1
(see Fig. 12.1) can be thought of as surfactants with some
biologically relevant function, and they typically include two non-polar and
hydrophobic (water hating) acyl chains connected close to the head group,
which in turn is usually polar and hydrophilic (water loving) and therefore
capable of forming hydrogen bonds with neighboring water molecules. This
“schizophrenic” nature of lipid molecules is the underlying reason why they
self-assemble as closed objects such as micelles and liposomes, such that the
head groups face water molecules while the acyl chains are as far from the
water phase as possible. More precisely speaking, the question is about min-
imizing the free energy, which contains a substantial entropic component as
in most other strongly fluctuating soft matter systems [19]. Often the hy-
drogen bonds play the most important role for the balance between energy
and entropy since hydrogen bonds are significantly stronger than many other
non-bonded interactions, such as van der Waals interactions in soft matter
systems. Consequently, the lipids form phases of various kinds [19, 20] de-
pending on, e.g., the effective shape of the lipid molecules, their interactions
with other molecules in a system, and the relative concentrations of lipids and
the solvent. As a simple way of understanding the structure of membranes,
we may consider the single-component lipid bilayer shown in Fig. 12.2. Al-
though this example is a highly simplified description of an actual biological
membrane, it readily demonstrates the basic structure of membranes. As a
matter of fact, due to its “simplicity”, this model is often used as a starting
point when one aims to understand the complicated properties of biological
membranes.
1
We use the following short-hand notation for di-acyl glycero phospholipids:
PC, phosphatidylcholine; PLPC, palmitoyl-laureoyl PC; DMPC, dimyristoyl PC;
DPPC, dipalmitoyl PC; DSPC, distearoyl PC; POPC, palmitoyl-oleoyl PC.
12 Diffusion in Membranes 475
Fig. 12.2. A PLPC lipid bilayer based on molecular dynamics simulations [21]
as a representation of the basic model for biological membranes. Water molecules
appear at the top and bottom of the picture. Illustration by Marja Hyv¨onen. A
schematic view of the bilayer is shown on the right.
Turning to native biological membranes [9, 11] (see Fig. 12.3) found in
living systems, they contain a bilayer of lipids as discussed above. Yet this
is just a small part of the story, since biological membranes are not like
single-component lipid bilayers but rather they are mixtures of various types
Fig. 12.3. A cartoon of an actual biological membrane including the lipid bilayer,
integral proteins, the cytoskeleton (below the bilayer), and glycocalyx carbohy-
drates (above the bilayer). Illustration by Ove Broo Sørensen.)
476 Ilpo Vattulainen and Ole G. Mouritsen
of lipids that differ in a number of ways such as with respect to the size,
the chemical composition in the polar head group region, and the rigidity
of the molecular core [11]. Fats, oils, and certain vitamins and hormones
are all lipids. It is fascinating that there are typically more than a hundred
different lipid species in a given type of biological membrane, all assumed to
have some particular purpose, and many of them are unsaturated or poly-
unsaturated, increasing the complexity of their nature even further. The lipid
bilayer is then complemented by numerous kinds of integral and peripheral
proteins that are embedded or attached peripherally to the membrane. The
proteins are also related to a dynamic rubber-like network known as the
cytoskeleton [11], which is attached to the inner surface of the membrane.
The cytoskeleton moves the cell, gives further rigidity to the membrane, and
also allows the membrane to adjust its shape to varying non-spherical shapes
as is the case, e.g., when red blood cells travel through narrow capillaries.
Finally, to make things even more difficult, Nature has chosen that the outer
leaflet of the membrane is also covered by a network, which in this case is
made of glycocalyx carbohydrates. It has been suggested that this network
serves the purpose of cell-cell recognition and adhesion to other cells.
As biological membranes are highly complex objects, the overall picture
of their structural and dynamic properties has been rather shady for quite
some time. Some 30 years ago, membranes were thought of as fluid-like mat-
tresses of lipid molecules whose main purpose was to provide an environment
in which membrane proteins were able to function. This “fluid mosaic model
of membranes” [22] has had an enormous impact on the manner in which
membranes have been investigated, and it has served its purpose as a good
starting point for further work as the picture of membranes and their role
have been refined. Thanks to significant activities on the field based on novel
experimental techniques and theoretical modeling, a detailed understand-
ing of membranes and their biological relevance has emerged. Rather than
being static, membranes are nowadays known to be highly dynamic in na-
ture [23, 24]. Both in-plane and out-of-plane fluctuations are playing a role,
although their relative magnitudes are very different. While membranes can
be twisted and bent rather easily [12, 24, 25], they cannot be stretched by
more than about 2 – 5 % of their average area without rupturing [12]. Thus,
although there is a danger of misunderstanding here, one can roughly de-
scribe membranes as sheets of thin and flexible polymer networks. Further
characteristics of membranes include their fluid-like nature in the sense that
lipid molecules are able to pass one another relatively easily, their heterogene-
ity [26] in the sense that there are distinct different small-scale phases and
internal structures in membranes, and their mode of compartmentalization
in the sense that cells are divided in compartments with specialized functions
and membranes thus act as permeable barriers for molecules that are trying
to diffuse from the outside to the inside of the cell (or vice versa).