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Brownian Motion

Module by: Jason Holden, Kevin Kelly. E-mail the authors

Summary: This 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.

Note:

"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."

Figure 1: Clarkia pulcgella
Figure 1 (Graphic1.jpg)

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, 1828

Introduction

The 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.

Objectives

By 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

Figure 2: Robert Brown (1773 - 1858)
Figure 2 (Graphic3.jpg)
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 Theory

Figure 3: A grain of pollen colliding with water molecules moving randomly in all directions as a result of heat energy.
Figure 3 (Graphic4b.jpg)

The 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.

  1. Heat is a form of energy.
  2. Molecules carry two types of energy: potential and kinetic.
  3. Potential energy results from the electric force between molecules.
  4. Kinetic energy results from the motion of molecules.
  5. 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 Model

The 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.

Figure 4: A Brownian motion curve – time vs.x-coordinate of walk.
Figure 4 (Graphic5.png)
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

Figure 5: A random Brownian “walk” method fractal.
Figure 5 (Graphic6.jpg)
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 Nanotechnology

Figure 6: An artist’s rendition of a tourist submarine, shrunk to cellular size, in Asimov’s Fantastic Voyage.
Figure 6 (Graphic2.jpg)
If 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.

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