Brownian Motion1.32006/01/10 13:04:36 US/Central2007/05/29 16:55:39.867 GMT-5JasonChristopherHoldenjch3921@rice.eduKevinF.Kellykkelly@rice.eduJasonChristopherHoldenjch3921@rice.eduKevinF.Kellykkelly@rice.eduTishMarieStringertish@rice.edubrowniankinetic-molecular theorymotionnanonanotechnanotechnologypollenRobert BrownThis module presents an overview of Brownian motion, with a focus on how it relates to nanotechnology. It looks at Brownian motion from a historical, physical, mathematical, and biological perspective in order to give the reader a complete view of this phenomenon."This module was developed as part of a Rice University Class called "Nanotechnology: Content and Context" initially funded by the National Science Foundation under Grant No. EEC-0407237. It was conceived, researched, written and edited by students in the Fall 2005 version of the class, and reviewed by participating professors."
This plant was
Clarkia pulchella, of which the grains of pollen, taken from
antherae full grown, but before bursting, were filled with
particles or granules of unusually large size, varying from nearly
1/4000th to 1/5000th of an inch in length, and of a figure between
cylindrical and oblong, perhaps slightly flattened, and having
rounded and equal extremities. While examining the form of these
particles immersed in water, I observed many of them very evidently
in motion; their motion consisting not only of a change of place in
the fluid, manifested by alterations in their relative positions,
but also not unfrequently of a change of form in the particle
itself; a contraction or curvature taking place repeatedly about
the middle of one side, accompanied by a corresponding swelling or
convexity on the opposite side of the particle. In a few instances
the particle was seen to turn on its longer axis. These motions
were such as to satisfy me, after frequently repeated observation,
that they arose neither from currents in the fluid, nor from its
gradual evaporation, but belonged to the particle itself.-Robert Brown, 1828IntroductionThe physical phenomena described in the
excerpt above by Robert Brown, the nineteenth-century British
botanist and surgeon, have come collectively to be known in his
honor by the term Brownian motion.Brownian motion, a simple stochastic process,
can be modeled to mathematically characterize the random movements
of minute particles upon immersion in fluids. As Brown once noted
in his observations under a microscope, particulate matter such as,
for example, pollen granules, appear to be in a constant state of
agitation and also seem to demonstrate a vivid, oscillatory motion
when suspended in a solution such as water.We now know that Brownian motion takes place
as a result of thermal energy and that it is governed by the
kinetic-molecular theory of heat, the properties of which have been
found to be applicable to all diffusion phenomena.But how are the random movement of flower
gametes and a British plant enthusiast who has been dead for a
hundred and fifty years relevant to the study and to the practice
of nanotechnology? This is the main question that this module aims
to address. In order to arrive at an adequate answer, we must first
examine the concept of Brownian motion from a number of different
perspectives, among them the historical, physical, mathematical,
and biological.ObjectivesBy the end of this module, the student should
be able to address the following critical questions.- Robert Brown is generally credited to have
discovered Brownian motion, but a number of individuals were
involved in the actual development of a theory to explain the
phenomenon. Who were these individuals, and how are their
contributions to the theory of Brownian motion important to the
history of science?- Mathematically, what is Brownian motion? Can
it be described by means of a mathematical model? Can the
mathematical theory of Brownian motion be applied in a context
broader than that of simply the movement of particles in
fluid?- What is kinetic-molecular theory, and how is
it related to Brownian motion? Physically, what does Brownian
motion tell us about atoms?- How is Brownian motion involved in cellular
activity, and what are the biological implications of Brownian
motion theory?- What is the significance of Brownian motion
in nanotechnology? What are the challenges posed by Brownian
motion, and can properties of Brownian motion be harnessed in a way
such as to advance research in nanotechnology?A Brief History of Brownian Motion
The phenomenon that
is known today as Brownian motion was actually first recorded by
the Dutch physiologist and botanist Jan Ingenhousz. Ingenhousz is
most famous for his discovery that light is essential to plant
respiration, but he also noted the irregular movement exhibited by
motes of carbon dust in ethanol in 1784.Adolphe Brongniart made similar observations
in 1827, but the discovery of Brownian motion is generally
accredited to Scottish-born botanist Robert Brown, even though the
manuscript regarding his aforementioned experiment with primrose
pollen was not published until nearly thirty years after
Ingenhousz’ death.At first,
he attributed the movement of pollen granules in water to the fact
that the pollen was “alive.” However, he soon observed the same
results when he repeated his experiment with tiny shards of window
glass and again with crystals of quartz. Thus, he was forced to
conclude that these properties were independent of vitality.
Puzzled, Brown was in the end never able to adequately explain the
nature of his findings.The first person to put forward an actual
theory behind Brownian motion was Louis Bachelier, a French
mathematician who proposed a model for Brownian motion as part of
his PhD thesis in 1900.Five years later in 1905, Albert Einstein
completed his doctoral thesis on osmotic pressure, in which he
discussed a statistical theory of liquid behavior based on the
existence of molecules. He later applied his liquid
kinetic-molecular theory of heat to explain the same phenomenon
observed by Brown in his paper Investigations on the Theory of the
Brownian Movement. In particular, Einstein suggested that the
random movements of particles suspended in liquid could be
explained as being a result of the random thermal agitation of the
molecules that compose the surrounding liquid.The subsequent observations of Theodor
Svedberg and Felix Ehrenhaft on Brownian motion in colloids and on
particles of silver in air, respectively, helped to support
Einstein’s theory, but much of the experimental work to actually
test Einstein’s predictions was carried out by French physicist
Jean Perrin, who eventually won the Nobel Prize in physics in 1926.
Perrin’s published results of his empirical verification of
Einstein’s model of Brownian motion are widely credited for finally
settling the century-long dispute about John Dalton’s theory for
the existence of atoms.Brownian Motion and Kinetic TheoryThe kinetic theory of matter states that all
matter is made up of atoms and molecules, that these atoms and
molecules are in constant motion, and that collisions between these
atoms and molecules are completely elastic.The kinetic-molecular theory of heat involves
the idea that heat as an entity is manifested simply in the form of
these moving atoms and molecules. This theory is comprised of the
following five postulates.Heat is a form of energy.Molecules carry two types of energy: potential and
kinetic.Potential energy results from the electric force between
molecules.Kinetic energy results from the motion of molecules.Energy converts continuously between potential energy and
kinetic energy.Einstein used the postulates of both theories
to develop a model in order to provide an explanation of the
properties of Brownian motion.Brownian motion is characterized by the
constant and erratic movement of minute particles in a liquid or a
gas. The molecules that make up the fluid in which the particles
are suspended, as a result of the inherently random nature of their
motions, collide with the larger suspended particles at random,
making them move, in turn, also randomly. Because of kinetics,
molecules of water, given any length of time, would move at random
so that a small particle such as Brown’s pollen would be subject to
a random number of collisions of random strength and from random
directions.Described by Einstein as the “white noise” of
random molecular movements due to heat, Brownian motion arises from
the agitation of individual molecules by thermal energy. The
collective impact of these molecules against the suspended particle
yields enough momentum to create movement of the particle in spite
of its sometimes exponentially larger size.According to kinetic theory, the temperature
at which there is no movement of individual atoms or molecules is
absolute zero (-273 K). As long as a body retains the ability to
transfer further heat to another body – that is, at any temperature
above absolute zero – Brownian motion is not only possible but also
inevitable.Brownian Motion as a Mathematical ModelThe Brownian motion curve is considered to be
the simplest of all random motion curves. In Brownian motion, a
particle at time t and position p will make a random displacement r
from its previous point with regard to time and position. The
resulting distribution of r is expected to be Gaussian (normal with
a mean of zero and a standard deviation of one) and to be
independent in both its x and y coordinates.Thus, in summary a Brownian motion curve can
be defined to be a set of random variables in a probability space
that is characterized by the following three properties.For all time h > 0, the displacements X(t+h) –
X(t) have Gaussian distribution.The displacements X(t+h) – X(t), 0 < t1
< t2 < … tn, are independent of previous
distributions.The mean displacement is zero.
From a resulting
curve, it is evident that Brownian motion fulfills the conditions
of the Markov property and can therefore be regarded as Markovian.
In the field of theoretical probability, a stochastic process is
Markovian if the conditional distribution of future states of the
process is conditionally independent of that of its past states. In
other words, given X(t), the values of X before time t are
irrelevant in predicting the future behavior of X.Moreover, the trajectory of X is continuous,
and it is also recurrent, returning periodically to its origin at
0. Because of these properties, the mathematical model for Brownian
motion can serve as a sophisticated random number generator.
Therefore, Brownian motion as a mathematical model is not exclusive
to the context of random movement of small particles suspended in
fluid; it can be used to describe a number of phenomena such as
fluctuations in the stock market and the evolution of physical
traits as preserved in fossil records.When the simulated Brownian trajectory of a
particle is plotted onto an x-y plane, the resulting curve can be
said to be self-similar, a term that is often used to describe
fractals. The idea of self-similarity means that for every segment
of a given curve, there is either a smaller segment or a larger
segment of the same curve that is similar to it. Likewise, a
fractal is defined to be a geometric pattern that is repeated at
indefinitely smaller scales to produce irregular shapes and
surfaces that are impossible to derive by means of classical
geometry.
Figure 5. The
simulated trajectory of a particle in Brownian motion beginning at
the origin (0,0) on an x-y plane after 1 second, 3 seconds, and 10
seconds.
Because of the
fractal nature of Brownian motion curves, the properties of
Brownian motion can be applied to a wide variety of fields through
the process of fractal analysis. Many methods for generating
fractal shapes have been suggested in computer graphics, but some
of the most successful have been expansions of the random
displacement method, which generates a pattern derived from
properties of the fractional Brownian motion model. Algorithms and
distribution functions that are based upon the Brownian motion
model have been used to develop applications in medical imaging and
in robotics as well as to make predictions in market analysis, in
manufacturing, and in decision making at large.Rectified Brownian Motion In recent years, biomedical research has shown that
Brownian motion may play a critical role in the transport of
enzymes and chemicals both into and out of cells in the human
body.Within the cells of the body, intracellular
microtubule-based movement is directed by the proteins kinesin and
dynein. The long-accepted explanation for this transport action is
that the kinesins, fueled by energy provided by ATP, use their two
appendage-like globular heads to “walk” deliberately along the
lengths of the microtubule paths to which they are attached.
Kinesin, as a motor protein, has conventionally been understood to
allow for the movement of objects within cells by harnessing the
energy released from either the breaking of chemical bonds or the
energy released across a membrane in an electrochemical gradient.
The kinesin proteins thus were believed to function as cellular
“tow trucks” that pull chemicals or enzymes along these microtubule
pathways.New research, however, posits that what
appeared to be a deliberate towing action along the microtubules is
actually a result of random motion controlled by ATP-directed
chemical switching commands. It is now argued that kinesins utilize
rectified Brownian motion (converting this random motion into a purposeful unidirectional one).We begin with a kinesin protein with both of
its globular heads chemically bound to a microtubule path.
According to the traditional power stroke model for motor proteins,
the energy from ATP hydrolysis provides the impetus to trigger a
chemo-mechanical energy conversion, but according to the rectified
Brownian motion model, the energy released by ATP hydrolysis causes
an irreversible conformational switch in the ATP binding protein,
which in turn results in the release of one of the motor protein
heads from its microtubule track. Microtubules are composed of
fibrous proteins and include sites approximately 8 nm apart where
kinesin heads can bind chemically. This new model suggests that the
unbound kinesin head, which is usually 5-7 nm in diameter, is moved
about randomly because of Brownian motion in the cellular fluid
until it by chance encounters a new site to which it can bind.
Because of the structural limits in the kinesin and because of the
spacing of the binding sites on the microtubules, the moving head
can only reach one possible binding site – that which is located 8
nm beyond the bound head that is still attached to the microtubule.
Thus, rectified Brownian motion can only result in moving the
kinesin and its cargo 8 nm in one direction along the length of the
microtubule. Once the floating head binds to the new site, the
process begins again with the original two heads in interchanged
positions. The mechanism by which random Brownian motion results in
movement in only one pre-determined direction is commonly referred
to a Brownian ratchet.Ordinarily, Brownian motion is not considered
to be purposeful or directional on account of its sheer randomness.
Randomness is generally inefficient, and though in this case only
one binding site is possible, the kinesin head can be likened to
encounter that binding site by “trial and error.” For this reason,
Brownian motion is normally thought of as a fairly slow process;
however, on the nanometer scale, Brownian motion appears to be
carried out at a very rapid rate. In spite of its randomness,
Brownian motion at the nanometer scale allows for rapid exploration
of all possible outcomes.Brownian Motion and NanotechnologyIf
one were to assume that Brownian motion does not exercise a
significant effect on his or her day-to-day existence, he or she,
for all practical purposes, would be correct. After all, Brownian
motion is much too weak and much too slow to have major (if any)
consequences in the macro world. Unlike the fundamental forces of,
for instance, gravity or electromagnetism, the properties of
Brownian motion govern the interactions of particles on a minute
level and are therefore virtually undetectable to humans without
the aid of a microscope. How, then, can Brownian motion be of such
importance?
As things turn out, Brownian motion is one of
the main controlling factors in the realm of nanotechnology. When one hears about the concept of
nanotechnology, tiny robots resembling scaled down R2D2-style
miniatures of the larger ones most likely come to mind.
Unfortunately, creating nano-scale machines will not be this easy.
The nano-ships that are shrunk down to carry passengers through the
human bloodstream in Asimov’s Fantastic Voyage, for example, would
due to Brownian motion be tumultuously bumped around and flexed by
the molecules in the liquid component of blood. If, miraculously,
the forces of Brownian motion did not break the Van der Waals bonds
maintaining the structure of the vessel to begin with, they would
certainly make for a bumpy voyage, at the least.Eric Drexler’s vision of rigid nano-factories
creating nano-scale machines atom by atom seems amazing. While it
may eventually be possible, these rigid, scaled-down versions of
macro factories are currently up against two problems: surface
forces, which cause the individual parts to bind up and stick
together, and Brownian motion, which causes the machines to be
jostled randomly and uncontrollably like the nano-ships of science
fiction.As a consequence, it would seem that a basic
scaling down of the machines and robots of the macro world will not
suffice in the nano world. Does this spell the end for
nanotechnology? Of course not. Nature has already proven that this
realm can be conquered. Many organisms rely on some of the
properties of the nano world to perform necessary tasks, as many
scientists now believe that motor proteins such as kinesins in
cells rely on rectified Brownian motion for propulsion by means of
a Brownian ratchet. The Brownian ratchet model proves that there
are ways of using Brownian motion to our advantage.Brownian motion is not only be used for
productive motion; it can also be harnessed to aid biomolecular
self-assembly, also referred to as Brownian assembly. The
fundamental advantage of Brownian assembly is that motion is
provided in essence for free. No motors or external conveyance are
required to move parts because they are moved spontaneously by
thermal agitation. Ribosomes are an example of a self-assembling
entity in the natural biological world. Another example of Brownian
assembly occurs when two single strands of DNA self-assemble into
their characteristic double helix. Provided simply that the
required molecular building blocks such as nucleic acids, proteins,
and phospholipids are present in a given environment, Brownian
assembly will eventually take care of the rest. All of the
components fit together like a lock and key, so with Brownian
motion, each piece will randomly but predictably match up with
another until self-assembly is complete.Brownian assembly is already being used to
create nano-particles, such as buckyballs. Most scientists view
this type of assembly to be the most promising for future
nano-scale creations.