# Manual Fractals in Molecular Biophysics

Click a menu option to jump or scroll down for more information on biophysical research by Physics faculty. Hong's lab uses atomic force microscopy to manipulate biomolecules into functional nano-structures. Huan-Xiang Zhou 's lab uses computer modeling to study protein folding, stability, and protein-protein and protein-DNA interactions.

David van Winkle 's lab uses a wide range of experimental techniques to study biophysical problems in crystals, gels, and liquid crystals. In addition, Prof. Bernd Berg and Prof. Per Arne Rikvold , with backgrounds in high-energy and condensed-matter physics, are branching into biophysics and are using their expertise in developing Monte-Carlo algorithms and other statistical mechanics methods to study biological processes such as protein folding. There are many opportunities for graduate and undergraduate research in biophysics at FSU. Zhou's lab uses computer modeling to study protein folding, stability, and protein-protein and protein-DNA interactions.

Berg and Prof. Rikvold, with backgrounds in high-energy and condensed- matter physics, are branching into biophysics and are using their expertise in developing Monte-Carlo algorithms and other statistical mechanics methods to study biological processes such as protein folding. FSU has a rich biophysics research environment.

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Interactions between proteins are essential for biological functions. Our research aims to gain fundamental understanding of the determinants of binding affinity and binding rates. Based on statistical mechanics, we develop physical models that can be conveniently implemented on the computer to realistically predict binding properties.

An example is the transition-state theory we developed to predict the enhancement of protein association rates by electrostatic interactions. Other questions being addressed include:. Biomotors are protein molecules that convert chemical energy to mechanical motion. They generate basic motions for many moving biological systems including our muscles.

One example is actomyosin that is comprised two components: actin and myosin. Actin provides a track, on which myosin walks while consuming chemical energy from ATP hydrolysis. The biomotor is a fascinating masterpiece of the Nature. Moreover, the fuel efficiency of biomotors is higher than any existing man-made engines. Use our Program and contact us! Fractals in Molecular Biophysics. Previous Netter's Weiner and Leibel and Atlas of Radiology at Murray and Oxford Desk Oxford Case Textbook of The ESC Bernstein, Chemical Reactions in Clusters J.

Simons and J. Nichols, Quantum Mechanics in Chemistry G. Published by Oxford University Press, Inc. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press.

ISBN 1. Molecular biology-Mathematical models. BiophysicsMathematical models. Topics in physical chemistry series. D49 Preface Throughout its history, science has focused on a reductionist approach. A complicated phenomenon is studied by dissecting it into simpler components, ascertaining how these components work and then putting it all back together again. Most graduate students learn early in their careers to simplify. We are taught to control as many variables as possible, to change one variable at a time and that many simple experiments usually yield better results than one grand, but convoluted, experiment.

Indeed, this reductionist approach is largely responsible for the great success of science. However, there comes a time when the limits of reductionism are reached. As one colleague complained, "The easy experiments have all been done. This in part reflects the success in tackling simple problems.

It also reflects a deeper sense that perhaps reductionism does not always work, that perhaps there are some problems that cannot by understood from their parts. This change in attitude has been spurred by the development of the appropriate mathematical tools to handle complexity. Just as the advent of calculus stimulated the development of Newtonian mechanics, the mathematics of complexity is promoting interest in problems that just a few years ago where deemed too messy for proper consideration.

While "the calculus" provided a single unifying formalism for approaching a wealth of problems, the current era sees a plethora of mathematical techniques arising to comprehend complexity. There is now an arsenal of new approaches. These go by different names: chaos, fractals, nonlinear dynamics, and computational complexity are just a few of the subdisciplines that have arisen. Preface vii How do you measure complexity? In our current formative period, there is no single, easy answer to such a question. Most often, the mathematical techniques one uses are dictated by the problem at hand.

Fractal geometry offers a particularly appealing, visual approach to understanding complex and disordered systems. Fractal geometry is, amongst other things, a mathematical technique for handling nonanalytic functions: the jagged, nondifferentiable curves that, for instance, occur in the path of a lightning bolt. This new mathematics provides a new tool, the fractal dimension, that in a sense is the equivalent of the derivative of calculus. This tool can be used to describe those untidy, jagged structures that occur in nature. Although fractals have found their greatest applications in condensed matter physics, there have been a number of "little invasions" into other disciplines.

One can see a smattering of applications in diverse fields ranging from astrophysics to organismal biology. This disordered advance of fractals into other fields has left its share of skeptics. It is the goal of this book to pull together diverse applications and to present a unified treatment of how fractals can be used in molecular biophysics.

This book is intended for two audiences: the biophysical chemist who is unfamiliar with fractals, and the "practitioner" of fractals who is unfamiliar with biophysical problems. It is my hope that this will stimulate both groups to work in this infant field. People not familiar with the mathematics of fractals often associate them with fantastic repetitive poster images or unusual scenes of landscapes. An obvious question then is: What do fractals have to do with molecular biophysics? There are a number of answers to this question. A theme that runs through this book is the close association of fractals and renormalization group theory.

Renomalization group theory is intimately associated with phase behavior of polymers and aggregates after all, much of biochemistry is polymer science. The renormalization group will appear in different guises throughout the book. Basic fractal concepts and their association with renormalization group theory are introduced in Chapter 1. This chapter is a sufficient introduction to the field to allow the book to be selfcontained. However, Chapter 1 should be used by the interested laymen as an entree into the fractal literature rather than as a comprehensive exposition. In the following chapters 2 and 3 the association between polymer statistics and fractal concepts is made in a pedagogical fashion.

This association is illustrated with examples describing the gross morphology of proteins as well as the loop structure of both proteins and nucleic acids. A number of other special topics are discussed as well.

## Fractals in molecular biophysics [1997]

These two chapters might well have been written from a polymer theory point of view without reference to fractals. What fractals do in these examples is to provide a unifying formalism or umbrella for discussing complex structures. This formalism will be put to good use in the later chapters that discuss dynamic phenomena. Chapter 4 discusses the multifractal formalism, an extension of fractal concepts to probability distributions. It describes the origin of multifractal behavior and discusses its appearance in proteins. It provides a formal, mathematical association between the statistical mechanics of order-disorder transitions in biopolymers and a generalization of fractal concepts multifractals.

In the following three chapters , the application of fractals to temporal phenomena is considered. Fractals provide a powerful connection between structure viii Preface and dynamics, and this connection is used to advantage throughout the three chapters. It is seen that fractal descriptions naturally lend themselves to the complicated, nonexponential rate processes that are so pervasive in biological systems.

Chapter 5 deals with the effects of dimensionality on diffusional processes. It considers chemical kinetics in restricted and fractal geometries. Applications to hydrogen isotope exchange in proteins and diffusion in biomembranes are considered. Chapter 6 discusses how and why protein dynamics might be fractal.

It examines specific results for ion channel gating kinetics. Vibrational relaxation processes in proteins are considered in Chapter 7. This area is of great historical value in the development of concepts such as the fracton. We revisit some of these early applications with the hindsight accrued by the passage of time. In a treatment close in spirit to the three chapters on dynamics, we consider the fractal analysis of sequence data in proteins and nucleic acid in Chapter 8.

This is an area of considerable recent interest, as well as controversy. The final two chapters 9 and 10 are reserved for processes that are perhaps the best examples. In Chapter 9, we deal with intrinsically fractal structures known as percolation clusters. It is seen that these arise in a number of biochemical settings: antibody receptor clustering, microdomains in biomembranes, and in the hydration of proteins. Chapter 10 presents a brief review of chaos in enzymatic systems.

Chaos is a companion discipline to fractals and utilizes fractal descriptions to describe complex dynamics. The "strange attractors" characteristic of chaotic dynamics are fractals. Examples of two enzyme systems that exhibit chaos are discussed. As in any new and rapidly developing field, it is extremely difficult to keep up with the literature. I apologize in advance for any oversight concerning references, old or new.

While I did intend to survey much of the field, I also wished to present a cohesive and somewhat pedagogical treatment of fractals in molecular biophysics. I have on occasion omitted results that were inconclusive or were tangential to the themes of the book. I have had valuable discussion with a number of people and wish to thank P.

Pfeifer, L. Liebovitch, M. Saxton, A. Grosberg, F. Family, and M. Shlesinger for inputs of various kinds. Lattice Diffusion and Fracton Dynamics 7.

## Fractals in Molecular Biophysics

River basins are fractal. So is the vasculation of the cornea. A lightning bolt shows a certain hierarchical jaggedness characteristic of fractals. Fractals are self-similar or scale-invariant objects. Regardless of the magnification with respect to a given variable, the structure remains statistically invariant. Often the fractal nature of such macroscopic phenomena, especially growth processes, is readily visualized. It is not so easy to visualize the fractality of a microscopic object such as a protein or a biomembrane.

Temporal processes can also exhibit fractal time behavior. These too are not always easily visualized. A fractal is said to have no characteristic scale, whether it be spatial or temporal. We are used to thinking in a "building block" fashion where smaller units are assembled into larger ones. At each level of construction, one has a characteristic length that appears in a natural way.

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An apartment building serves as a good example of an assemblage with several scales. Starting with bricks and mortar, one has, within an order of magnitude, a single size. Likewise, as a practical matter, most rooms in apartment buildings are more or less the same size. While the size of the building itself may vary considerably, usually these variations are not greater than an order of magnitude.

The apartment building has at least three characteristic sizes: bricks, rooms, and the building itself. One can construct characteristic scales for many such familiar objects. It may, in fact, be difficult to think of something without characteristic sizes. Consider a molecular liquid and the variety of characteristic lengths and volumes associated with it. A few examples are the size of the atoms, the bond lengths within the molecule, the specific volume of the molecule, and the nearest 3 4 Fractals in Molecular Biophysics neighbor distance between molecules.

If the liquid has long-range order, like water, then there are additional, more complicated spatial scales. For every spatial scale, there often is a temporal counterpart. Liquids have a complicated array of vibrational, collisional, and diffusive modes. Yet these modes would all appear to have their own characteristic time scale. If the pressure and temperature are adjusted so the liquid approaches its critical point, it begins to show large-scale fluctuations.

The fluid is caught between a liquid and gaseous state and fluctuates between the two states. The fluctuations at one level of magnification look essentially the same as those at another level. At the critical point, fluctuations occur at all length scales. This gives rise to a light-scattering phenomenon known as critical opalescence. The details of the various interactions in the liquid become unimportant and the fluctuations are governed by certain "universal" behavior.

The "universality" is a result of the fluctuations between two phases, and the microscopic details of the system are washed out. This phase transition phenomenon is in a broad sense a fractal phenomenon. It has been successfully treated theoretically using renormalization group theory Wilson, Renormalization group theory appears again and again in physics and materials science.

It reveals a plethora of phenomena that are scale invariant. One of the goals of renormalization group approaches is to establish the universality class of a set of phenomena. Once this is done, seemingly different phenomena can be related to each other and the role of dimensionality in the system is more clearly revealed. A number of biochemical systems exhibit structures that are fractals. These include percolation clusters in biomembranes, antibody aggregates, and the "strange attractor" of chaotic enzyme reactions.

In addition to these examples, we consider biophysical phenomena that are fractal in the sense of renormalization group theory. In the following chapters, scale invariance will appear in a range of settings that include the polymer statistics of proteins, the symbolic dynamics of DNA sequences, and the kinetics of the opening and closing of ion channels. It is the aim of this book to provide a unified treatment of such phenomena. In this introductory chapter the basic concepts of fractals and multifractals are presented.

This chapter is not meant to be a comprehensive discussion of the field but rather is intended to provide sufficient background for the following chapters. It will provide the interested reader with an entree into the subject. Indeed, entire books are devoted to this material, and the interested reader is referred to the excellent introductory texts by Feder , Vicsek , and Falconer Of course, the very colorful The Fractal Geometry of Nature by Mandelbrot is also strongly recommended. The chapter begins with an introduction to basic ideas that will be used throughout the book. Chapter 2 provides a reinforcement of this introduction by showing a direct application to polymer configurational statistics, in general, and to the specific details of protein structure.

An elementary discussion of fractals and set theory is also provided. Often, simple set theoretical arguments can be of great benefit when developing scaling laws. Such scaling arguments are employed in most of the following chapters. This section is followed by a discussion of self-affme fractals. Self-affine fractals are fractals that do not scale identically along each coordinate. These occur in growth and dynamical What Are Fractals? Finally, a discussion of fractals and renormalization group theory is presented. Renormalization group methods are used throughout the book and, as discussed above, provide a physical underpinning for a number of examples.

This chapter begins with a particularly simple example of such methods. Throughout this chapter, the motivation for using fractals as physical models is emphasized. This chapter focuses more on the underlying physics rather than on mathematical rigor. Benoit Mandelbrot had the great insight to realize that complex forms, as often arise in nature, follow more general scaling laws.

The fractal is created by an iterative process of scaling and replacement of units from the previous iteration. Each step is known as a prefractal and the fractal is formed after an infinite number of iterations. The triangular carpet in the bottom of the figure has concentric circles superimposed on it and allows the mass fractal dimension to be calculated. Thus, the iterative scheme gives the expected result that the triangles cover the plane and, therefore, have a fractal dimension of 2. While the Sierpinski carpet is a very specific type of fractal, it does show properties common to all fractals.

It is constructed by two operations: scaling and replacement. The initial triangle top of Figure 1. The central triangle is then replaced with an open region. This second structure is then scaled to the size of the third triangle, and again all the central triangles are removed. The structures formed by these iterations are called prefractals. Technically, a fractal is the infinite iteration of this process. Figures 1. They are the triadic Koch curve and the middle-third Cantor set, respectively. Again, in these cases the initial structure is scaled reduced in this instance and line segments in the original structure are replaced by the scaled structure.

All these examples are deterministic fractals. In this case, as the magnification is changed the structure is not identical but does have a similar jaggedness. The jaggedness reflects the randomness of the process used to form the fractal and the fractal dimension can be used as a measure of that jaggedness.

Lightning or dielectric breakdown is an example of such a random fractal, with statistical similarity on many different scales. An example of such a fractal is illustrated in Figure 1. Fractals form "naturally" in a number of different settings. Some physiological structures are almost deterministic fractals. The mammalian lung is an example of one of these Shlesinger and West, Often, a growth phenomenon, whether it is biological or chemical, leads to a fractal What Are Fractals? Each line segment is replaced by a scaled version of the previous iteration.

A common growth process that forms a fractal is diffusion-limited aggregation. A diffusion-limited aggregate is shown in Figure 1. These structures have been observed in electrodeposition experiments and it has been possible to simulate them on the computer. How and why these fractals form under otherwise homogeneous conditions is one of the unsolved problems of fractal physics.

Although these physical systems may appear self-similar over a range of magnifications, ultimately there will be lower and upper length cutoffs at which fractal behavior cannot exist. Physical structures will not be infinitely repetitive. At first appearance this would seem to limit greatly the ability of fractals to model a physical phenomenon.

In actuality this is not a serious constraint. A unit line segment has the middle third interval removed. An example of a random fractal. Structure is formed by aggregation of diffusing particles. Diffusing particles perform a random walk, and when they hit the aggregate they stick. This simple physical process creates a fractal, and has been observed both experimentally and in computer simulations.

What Are Fractals? The examples shown in Figures 1. Technically, it is only when the iteration goes to infinity that a fractal is formed. Although the limit of infinity rarely appears in physical applications, this turns out not to be a serious constraint. Properties of the finite iteration so closely capture the true fractal that the distinction between these two structures is often unimportant. Typically, fractal models are used to predict scaling laws over several orders of magnitude of an experimental parameter. The models will break down when the "length" scale is as small as the fundamental unit or as large as the entire structure.

Thus, there should be well-defined limits for any fractal model. If enough physical insight exists on a given problem, then the "crossover" points between regimes can be predicted. This can greatly assist any effort to assess whether a limited scaling regime is due to fractal behavior. Another important consequence of the scale-invariant property of fractals is that the functions describing them can never be smooth witness the Koch curve, Figure 1. A common first approach to a problem in mathematical physics is to expand a function to first order and solve the linearized problem.

Because fractals are scale invariant, one cannot find a scale on which they are smooth enough to linearize. Any expansion will not generate a smooth, local region, but rather will generate an identical structure, a consequence of self-similarity. Fractals intrinsically are nonanalytic functions, i.

This makes them particularly useful in describing irregular shapes found in a range of natural phenomena. Before the advent of fractal geometry, such functions were considered to be mathematically "pathological. In their place is the concept of the fractal dimension. The irregular shape of a fractal is characterized by the fractal dimension s associated with it. There are a number of different ways of defining this parameter and, depending on the type of fractal one is dealing with, these definitions are not always equivalent.

Indeed, even the definition of a fractal itself has often been rather loose. Falconer makes the analogy that a mathematician defining fractals is akin to a biologist defining life. While no simple or general definition has come forth, workers in the field still have a good sense of the attributes of the respective phenomenon. Thus, the nebulous status of the definition of a fractal should not be misleading. In most applications a very specific class of fractals is considered, which can be rigorously defined.

This lack of a general definition is also a reflection of the openness of the ongoing research in this area. In the present treatment, a pragmatic approach is taken and simple operational definitions are used to define the fractal dimension. Initially, the fractal dimension, DM, associated with "mass fractals" is considered. To measure a mass fractal dimension, the total mass within concentric circles of radius, R, is determined. For the Sierpinski carpet, in which the darkened triangles have uniform mass, the mass dimension is readily determined see Figure 1.

Placing the origin at one of the vertices of the carpet, concentric circles can be drawn which include an integer number of triangles. For the carpet shown in Figure 1. Using Eq. Note that the density, p R , of a mass fractal embedded in a Eiuclidean space of dimension E is a function of the radius.

It scales as: Density distribution as in Eq. There are situations where one has an image or set of points and does not know the mass of the objects. For instance, an electron micrograph of the clustering of proteins in a biological membrane gives information on point distributions but not on the mass content. A similar situation arises when observing the distributions of stars in a galaxy. In such cases, it is more convenient to define a different fractal dimension known as the correlation dimension.

This correlation dimension is closely related to the mass dimension. To determine the correlation dimension, the number of pairs of points, C r , that lie within a radius, r, of each other is counted Grassberger and Procaccia, This quantity scales as: and the correlation dimension is determined from the slope of a log-log plot of C r versus r. This definition has found popular application in determining the fractal dimensions of the "strange attractors" that characterize a chaotic system.

Chaos in enzymatic systems is discussed in Chapter The fractal dimension may be defined in other ways. One such definition that is computationally very useful is the "box-counting" dimension. Consider a set of points as in Figures 1. The covering of a smooth curve is illustrated in Figure 1.

From a more technical point of view, hypercubes are used to encompass the structure. For a structure embedded in a Euclidean space of dimension E, the hypercube will have E orthogonal edges. Now, the box size is changed to determine how N depends on the length, 8. For a fractal structure this dependence will be described by an inverse power law such that: where DB is known as the "box-counting" dimension.

The slope of a plot of log A' versus log 8 provides this fractal dimension. However, for self-similar, deterministic fractals both procedures give the same dimension. As a simple example of the determination of a fractal dimension by box counting, the triadic Koch curve shown in Figure 1. Let the end-to-end length of the curve be of unit length.

Note that we require the boxes to be centered on the line segments. From Eq. Consider now the Cantor set shown in Figure 1. The Cantor set or "dust" has a dimension less than 1 because it has fewer points than a line.

With the relationship given in Eq. For example, the apparent length of a curve, L 8 , will depend on the size of the ruler. This length is simply given by the ruler size, 8, multiplied by the number of steps, N S. From our definition in Eq. For smooth curves, as the ruler size approaches zero the length converges to a finite limit. For fractals, no such convergence occurs, rather it diverges as the inverse power law of Eq. This is the famous conundrum of the "coastline" problem Mandelbrot, , which shows that the length of a coastline or boundary depends on the size of the measuring device.

By measuring the length of a curve with different step sizes, the fractal dimension is readily determined from a log-log plot. Similarly, a surface area, 5 8 , can be measured by covering the region of interest with squares the edges of which have length 8. This gives: Likewise a volume, V 8 , is measured with cubes of edge length 8, giving: The relationships shown in Eqs 1. This is a direct consequence of self-similarity see Pietronero, Scaling relationships as in Eq. The most elementary example is the scaling of state functions, such as the free energy, with extensive variables of the system.

A more interesting situation occurs for systems near their critical point. Renormalization group theory leads to the following general expression for the free fnnrow F of the system where G is a regular function of a scaling field, x, of the system. Inhomogeneous functional equations of the form of Eq. These What Are Fractals? As will be discussed in the section on renormalization group approaches, the solution to the homogeneous equation equations of the form of Eq.

The homogeneous solution is readily obtained by differentiating Eq. From relationships such as Eq. Power laws do not have a characteristic scale. This might not be obvious, but can be readily seen by comparing a power law with an exponential function that does have a "characteristic length. Therefore, a scaling relation as in Eq. So why isn't L the characteristic length of the system? Now, consider the next decade, The exponential is greatly diminished over the second decade. On the other hand, the power law behaves identically over the second decade as over the first.

It is said to have no characteristic length. Equations 1. There is a wealth of natural phenomena that can be described by power laws or by exponentials of power laws. This has provided strong motivation for the adoption of fractal models. However, the converse of this proposition, i. In this case the inverse power law does not imply a selfsimilarity.

If the center of the distribution is moved from the origin to another point, the distribution for all other points changes accordingly. A self-similar fractal, on the other hand, is independent of the origin. Self-similarity means that the same structure will be generated regardless of location or magnification. Thus, our density distribution clearly does not behave as a fractal and yet it has a power law associated with it. The translational invariance property of self-similar fractals can be a serious restriction when considering real physical models.

This is especially true for finite systems in which edge effects are important. For a system to be considered fractal, power law behavior is not sufficient; rather, some degree of self-similarity must be 14 Fractals in Molecular Biophysics established. A rigorous demonstration of fractality involves demonstrating scale invariance in operations of dilation, translation, and rotation.

Returning to Eq. Often solutions to functional equations have a "unit periodic" term associated with them. This solution shows logarithmic oscillations in x. If the parameter X is not uniquely determined by the problem, it can take any value. In these cases, there is no physical reason for choosing a given Q and Eq. However, in a range of problems X is restricted. For instance, in the box covering that was used for the Koch curve and the Cantor set see Figures 1. These specific values fit with the self-similar structure of the specific fractals.

Logarithmic oscillations will occur in the mass covered as the radius is smoothly varied over the fractal. Equation 1. This shows that the fractal dimension need not be restricted to real numbers, but can be complex as well. Even Eq. The general solution is: where A m are the Fourier components that define the detailed shape of the oscillations.

As seen in Eqs 1. Have such oscillations ever been observed? Numerical simulations of certain random walk problems display this behavior. Oscillations have also been demonstrated experimentally for measurements of the magnetoresistance of fabricated, submicron Sierpinski gaskets that are made of aluminum Doucot et al. Perhaps the most striking example of logarithmic oscillations appears in the structure of the human lung.

The human lung is a fractal structure that has 23 generations of branches. This is an unusually high number of branching generations. Most deciduous trees have only 7 or 8 generations! Although the lung is asymmetric, it does show fractal behaviour. If the diameter of the bronchial tube is plotted versus branch or generation number, logarithmic oscillations are seen, as shown in Figure 1.

Shlesinger and West have analyzed this with a model that predicts a fractal dimension that is a complex number as in Eq. A number of protein dynamical phenomena show "wiggles" in log-log plots, and these have also been analyzed with a generalized noise model that yields complex fractal dimensions Dewey and Bann, In this section, we have discussed how to determine the fractal dimension of objects.

This is the basic tool of fractal geometry. However, the mathematics of What Are Fractals?

The log of the average bronchial diameter versus log of the branch order or generation number. Overall dependence is an inverse power law -- ; an oscillatory component - is superimposed on this. Figure from Shlesingen and West, For instance, the theory of fractal sets provides mathematical relationships that would be extremely difficult to derive by more conventional approaches. These relationships add to the predictive power of a given model and are often easily implemented. This is the subject of the next section.

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In both instances, new fractal structures are formed and, often, the fractal dimension of the new structure is readily determined if the dimensions of the initial fractals are known. As a word of caution, theorems involving fractal sets are often given as inequalities but frequently the limiting equality is used.

It is sometimes possible to find a counterexample that is not consistent with the equality. The interested reader is referred to Falconer's book for a rigorous approach to these problems Falconer, As will be seen in Chapters 2 and 5, simple set theoretical arguments can be used to advantage when considering surface properties of proteins. It is often easier to determine the fractal dimension of the curve B rather than the area, A X B. Bottom The product of the unit interval, A, with the triadic Cantor set, B. First, the product of two fractal sets is treated. The product of two sets, A and B, is defined as the set of points in which the first coordinate is any point in A and the second coordinate is any point in B.

The dimensionality of the product is given by: For cases in which the curves are "smooth," the equality holds in Eq. A simple example is a line segment along the x-axis set A and a second line segment set B along the y-axis. The product, A X B, is a rectangle in the xy plane.