Entropy Protein Folding

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The Effect of Entropy on Protein Folding

L.Strube

Abstract

This article concerns the modern challenge of determining a pro-

teins tertiary structure and folding mechanism based on its primaryamino acid sequence. We explore the concept of entropy as an influ-

ential factor in this process by describing a mathematical model for

the attractive force of two large macromolecules, modeled as hard

spheres, immersed in a low-concentration solution of particles, repre-

sented as small hard spheres. Using this model we demonstrate that

the greatest entropy of the small particles occurs when the surfaces of

the two large macromolecules are in contact. In addition,we describe

a recent extension of this model to solutions of high small-particle

concentration. Finally we discuss a current application of this model

to protein research.

1 Introduction

In 2003 the Human Genome project was completed, having determined thenucleotide sequence of the last gene in the last molecule of human DNA [14].As each gene within a molecule of DNA codes for the amino acid sequence ofa specific protein, this event marked the beginning of a new era in molecularbiology,an era in which the quantity of known protein sequences far exceedsthe quantity of known protein structures [5].

1.1 Protein Structure Determination

The gap between the volume of known protein sequences and the volume ofknown protein structures is due both to the complexity of protein moleculesand the technologies currently available to scientists [5]. A proteins complexstructural organization consists of four distinct levels: primary, secondary,

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tertiary and quaternary (Figure 1). The primary level consists of the amino

acid sequence coded for by individual genes and can be visualized as a lineararrangement of amino acid beads. The twisting and folding of this stringof beads into -helices and -pleated sheets forms the proteins secondarystructure which then folds in on itself to form the proteins tertiary struc-ture. Each of these first three levels of organization consists of the twistsand folds of a single amino acid sequence known as a peptide. While manyproteins contain only one peptide, some are formed via the aggregation ofmultiple strands. Such proteins are described as having a quaternary struc-ture which forms when multiple peptides, of tertiary structure, bind into asingle, functioning aggregate [16].

Figure 1: The Four Levels of Protein Structure [19]

Without a knowledge of the gene sequence that codes for a specific pro-tein, both the determination of its structure and the determination of itssequence requires involved chemical protocols. [5]. However, if the gene

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sequence code for a protein is known, then the gene sequence can be

translated into the corresponding protein sequence in a fairly straightfor-ward manner. Thus, while protein structure determination continues to bea tedious process, the rate of protein sequence determination has increasedrapidly since the completion of the Human Genome Project [5].

As a proteins function is tied tightly to its overall structure, this floodof new sequences represents a gold-mine of scientific breakthroughs, if thecorresponding structures can be determined [21]. Thus, the volume of knownprotein sequences is serving as a catalyst for the development of new predic-tive structure determination methods. These methods aim to use a proteinsamino acid sequence as a bottom-up predictor of its final structure. This

is in contrast to the top-down classical methods such as x-ray crystallogra-phy and NMR spectroscopy which determine protein structure via protocolswhich analyze the chemical characteristics of the structures as a whole [5].

1.1.1 Modern Sequence-Based Methods

Sequence Comparison

Current prediction methods are divided into two general categories based onthe strategy used [5]. The first category utilizes the fact that a proteins struc-ture is primarily determined by the identity of the amino acids it contains. Inother words, proteins with similar sequences typically produce similar over-

all structures and thus have similar functions [15] [5]. Scientists using thismethod compare new protein sequences, those with unknown structure, tothe sequences of proteins whose structures have already been determined byclassical methods. This method has proved to be relatively accurate, exhibit-ing a success rate similar to that of low resolution crystallography and NMR[20].

The ab initio Method

The second strategy of prediction attempts to determine a proteins structure

based on the energetics involved without any comparison to known structures[5][20]. It involves the development of both an energy function and a searchprotocol by which all possible configurations of a given sequence can be eval-uated. This method is based on the thermodynamic hypothesis which statesthat the protein structure with the lowest free energy is the native (natural)state structure [5] [18]. This category of structure prediction is often referred

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to as the ab initio (lit. in the beginning) or de novo (lit. anew) method

in reference to its focus bottom up structure prediction [20] [6] [1] [11].Currently the ab initio method is, in theory, the most valuable of the twostructure prediction categories in that it is not limited by the diversity (orlack thereof) of classically determined protein structures. Further, by pre-dicting a proteins structure from its sequence alone, scientists could detectstructural, and thus functional, similarities between proteins with very dif-ferent sequences. However, at this time the success rate ofab initio structureprediction is quite low, a fact which appears to be the result of our limitedunderstanding of electrostatics and solvent/protein interactions [20]. Thusthe improvement of this valuable method of requires further research into

each of these factors.It is in the search to understand these factors that two fields of studymeet: the field of protein structure determination we have been discussing,and the field of protein folding research. While to the casual observer thesetwo fields appear to be one and the same, they are in fact at their core quitedistinct in their focus. The former determination field is primarily focusedon the end result of protein folding while the later folding field is primarilyinterested in the process by which a protein folds. Thus while the same forcesand environmental influences affect both a proteins final structure and themanner by which it folds, these two topics represent branches of two distinctfields: molecular biology and the study of self-assembly.

1.2 The study of Self-Assembly

1.2.1 Nanotechnology

The study of self-assembly is a fairly new research area spanning the last twoto three decades. However, it is perhaps more accurately understood as anatural outgrowth of the development of nanotechnology which was inspiredby living systems and first introduced in the 1950s [18]. In an era of room-sized computers, when the structure of DNA had just been elucidated, the

concept of nanotechnology held an almost science fiction quality. RichardFenymann, one of the foremost visionaries in the field of nanotechnologymarveled in his speach Theres Plenty of Room at the Bottom at thecapabilities of the living cell. He exhorted scientists to search for ways tomimic the cell, saying: Many of the cells are very tiny, but they are veryactive; they manufacture various substances; they walk around; they wiggle;

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and they do all kinds of marvelous things-all on a very small scale. Also,

they store information. Consider the possibility that we too can make athing very small, which does what we want-that we can manufacture anobject that maneuvers at that level! [12] [18].

As Dr. Feynmann envisioned, technology has in fact become smaller andsmaller; calculations once performed by room-sized computers can now becompleted on hand-held calculators. However, while smaller size does implygreater efficiency and lower cost of materials, in reality the tools required toconstruct such technology have become quite expensive and is now a limitingfactor in the progress of this field [21]. Thus the need for cost-effective waysof producing nanotechnology serves as one of the driving forces of the study

of self-assembly. It is hoped that by studying the process by which naturalsystems, such as proteins, self-assemble we can artificially create our ownself-assembling nanoscale machines [21].

1.2.2 The Turing Machine

If nanotechnology is the driving force behind the study of self assembly,then it is the concept of a Turing machine that is the foundation for thefield. The concept of a Turing machine was developed in the early 1900sby Alan Turing [18]. His idea was that any function or operation could bebroken down into a sequence of discrete actions and that by following such

a sequence of actions the end result would always be the same [7]. His ideasmarked the birth of algorithmic thinking [18]. Thus, a greater understandingof solvent interactions is necessary for both protein stucture determinationand for the development of the discrete steps in the Turing Machine ofprotein folding.

1.3 Entropy

It has been known for many years that the chemical characteristics of theamino acids which make up a protein play a significant role in its overall

structure[18]. This influence is due both to interactions between individualamino acids and to interactions between a given amino acid and its environ-ment. For example, each amino acid can be classified as charged (positively,or negatively), polar, or non-polar. If a protein sequence is composed entirelyof negatively charged residues it will not form a helix, the residues repellingone another like matching ends of a magnet. However, appropriate spacing

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negatively and positively charged amino acids can lead to the formation of

a helix [16]. Similarly water, being itself a polar molecule, is attracted toother polar/charged molecules and repelled by non-polar molecules. Thusnon-polar amino acids are likely to orient themselves inside the overall struc-ture of a protein away from water whereas charged or polar amino acids arefound on the exterior surface of the protein in contact with water [16].

Only in recent years have non-specific interactions between a protein andits environment been considered. One such interaction now being researchedis the role of entropy in determining a proteins native (naturally occuring)structure [22] [13] [17]. Entropy is a measure of disorder, or randomness, ina system [9]. For example, imagine a flat tray containing a set of checkers

(Figure 2). If you confine these checkers to one corner of the tray theirorganization is high, implying a low level of entropy. On the other hand, ifyou shake the tray allowing the checkers to disperse where they will, theirorganization becomes low, implying a high degree of entropy. The second lawof thermodynamics states that the the universe is tending towards disorder[9]. Thus reactions which increase entropy are spontaneous and an energyinput is required to prevent an increase in entropy.

Figure 2: Entropy

Using this analogy a cell is essentially a tray of checkers. However, unlike

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the example above, a cell contains many different objects of different sizes.

Some are relatively large like the proteins we have been discussing, otherssuch as sodium and potassium ions or free amino acids are quite small. Inaddition the cell is an aqueous environment and its contents being immersedin solution of high water concentration [10]. Recently it has been proposedthat an increase in the entropy of the small particles within a cell may be adriving force favoring the organized folding of protein structures [22] [13] aswell as the aggregation of protein/DNA complexes [17].

2 Interaction between Particles Suspended in

Solutions of MacromoleculesAs cells are essentially fluid filled compartments, any revelent model of inter-actions between particles within a cell must involve solution/solute scenario.The fundamental model for non-specific interactions between particles insolution was produced by Asakura and Oosawa in the 1950s [3][4] and isthe basis for multiple modern models involving protiens [17][13][22]. In theAsakura and Oosawa (AO) model, the authors focused primarily on the effectof entropy on macromolecule aggregation. In doing so they chose to ingnoreinteractions between solutes due to charge or polarity.

2.1 Osmotic Pressure as an Attractive Force

To understand the AO model, imagine two metal plates immersed in a solu-tion containing only small particles (diameter = d) (Figure 3). Observe thatif the plates are oriented such that the distance between the plates is lessthan d, the small particles are excluded from the space between the plates.This produces a concentration gradient from the region of pure solvent be-tween the plates, to the region of higher particle concentration surroundingthe plates. As a result, an osmotic force acts on the system causing puresolvent to rush out of the interior space, diluting the exterior solution, and

pulling the plates together [23]. Referring back the the tray of checkersanalogy above (Figure 2), this scenario causes the metal plates to becomemore organized, having been pulled together, while the small particles be-come less organized, having been given a greater free volume in which todisperse.

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Figure 3: Two metal plates suspended in a solution with particles of diameterd

This idea can then be generalized to a system containing two large macro-molecules (diameter = D) suspended in a similar solution of small particles

(diameter = d), such that the distance between the large macromoleculesis less than d. As in the case of the metal plates, the small particles areexcluded from the region between the macromolecules and a force equivalentto osmotic pressure acts on the outer edges of the macromolecules drivingthem together and increasing the entropy of the small particles (Figure 4).

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Figure 4: Two hard spheres of diameter D suspended in a solution of particleswith diameter d

The goal of the AO model was to quantify this attractive force acting

on the surface of a macromolecule. Once quantified, this attractive forcecan then be integrated to determine the change in free energy resulting froma change in the macromolecules orientation relative to one another. Asenergy is required to prevent an increase in entropy, energy and entropy areessentially two sides of the same coin: an increase in free energy correspondsto a decrease in entropy and an increase in entropy corresponds to a decreasein free energy. Thus determining the change in free energy is mathematicallyequivalent to determining the change in entropy. If we let a represent thedistance between the centers of the two large macromolecules, minimizingthe free-energy function for this system in terms of a yields the separation

distance at which the system exhibits the greatest small-particle entropy.

2.2 Quantifying the Attractive Force

The first step in determining the value for a which produces the greatestentropy, is to quantify the force acting on single large macromolecule. This

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pressure can be expressed as:

P =

V

(x)w(x, a)dx (1)

where x is a position vector from the center of the large macromolecule tothe center of the small particle concentration(Figure 5), (x) is the numericaldensity of the small particles at the position x, a is the distance between thetwo large macromolecules andw(x, a) represents the gradient of the averagepotential energy of a small particle at x which results from the forces actingon it.[3] [23].

Figure 5: Position Vector

2.2.1 Statistical Mechanics: Approximating (x)

Several challenges arise when using this equation. The first is determining(x), the density of the small particles. To find (x), Asakura and Oosawaassumed that the concentration of the small particles in solution was low,

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and used a first-order approximation of a statistical mechanics equation to

determine their density [8]. This first order approximation is equivalent toBoltzmans equation and is expressed as:

(x) = New(x,a)KT (2)

Substituting this into equation (1) for P yields:

P =

V

New(x,a)KT 1w(x, a)dx

where N represents the concentration of particles, K is the Boltzman con-stant, and T is absolute temperature. As before, w(x, a) represents the av-

erage potential energy of a small particle at x which results from the forcesacting on it when the centers of the two large macromolecules are separatedby a distance a.

In the specific scenario we are considering this equation further simplifiesto the following:

P = kT N

a(ln Q) (3)

Q =

V

ew(x,a)

kT dx. (4)

2.2.2 Q as the Free Volume

The second challenge in using this equation is determining the energy, w,of the small particles; however, in this scenario Q is equivalent to the freevolume available to the centers of the small particles [3] [4]. To determine thevalue for Q, note that surrounding each of the large macromolecule spheresthere exists a shell of depth d

2into which the centers of the small particles

cannot enter (Figure 6). Also note that if the spheres are at a distance greaterthan d apart, that is, if the small particles can invade the space between themacromolecules, the space excluded from the center of the small particles has

a volume equal to that of the two spheres and their shells. If on the otherhand, the shells overlap, the volume of the excluded space is reduced by thevolume of the overlapping region.

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Figure 6: The Overlap Region

To determine the Q, the free volume available to the small particles, webegin by calculating volume of the overlapping region of the shells(Figure 7).First let a equal the distance between the centers of the spherical macro-molecules and fix the centers of the two macromolecules at (0, 0, a

2) and

(0, 0,a2). Observe that if the overlapped region is divided in two longi-tudinally that each half of this region is a portion of a sphere of radius D+d

2

(Figure 4) and has a depth of a2

(Figure 7). Thus to find the volume of theoverlapping region we find the volume of one sphere portion and multiplyby two. To simplify the computation let R = D+d

2, h = a

2. Note that the

upper bound of the volume to be calculated is z = 0 and the lower bound isdescribed by the following equation:

x2 + y2 + (z h)2 = R2

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Figure 7: Half the Volume of the Overlap Region

Using polar coordinates (r2 = x2 + y2), the following integral can be usedto determine the volume of this half region:

Vhalf =

20

R2h20

R2 r2 h

rdrd

with the entire volume of the overlapping region equaling:

Voverlap = 2

20

R2h20

R2 r2 h

rdrd.

Next we integrate:

4R2h20

R2 r2 h rdr = 4 R2h20

R2 r2rdr R2h20

hrdr

.

Integrating again gives:

Voverlap = 4

(h

2)3/2

3+

(R2)3/2

3 R

2h

2+

h2h

2

=

4

3

h3 + R3 3

2R2h +

3

2h3

.

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Therefore the total volume of the excluded space can be calculated as follows:

Vexcluded = 2

4

3R3

4

3

R3 3

2R2h +

1

2h3

=4

3R3 +

4

3R3 4

3

R3 3

2R2h +

1

2h3

=4

3R3 4

3

3

2R2h +

1

2h3

=4

3

R3 +

3

2R2h 1

2h3

.

Substituting R = D+d2 and h = a2 back into our solution yields:

Vexcluded =4

3

D + d

2

3+

3

2

D + d

2

2 a2

1

2

a2

3.

Thus the free volume Q can be described by:

Q = Vtotal Vexcluded = Vtotal 43

D + d

2

3+

3

2

D + d

2

2 a2

1

2

a2

3

= Vtotal

42

3(D + d)2 + (D + d)2a

1

3a3

.

Note: Based on these calculations and our specific definition ofQ, when thereis no overlap, i.e. a D + d, we have:

Q = Vtotal 83

D + d

2

3.

We can now use this form of Q to get an expression for P, namely,

P =

poS

where po =kTNV

.

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2.2.3 Assumption: V

the Excluded Space

Recalling our original formula (Equation 3) for P we have:

P = kT N

a(ln Q)

= kT NV

S

= poS

Thus our equation for S can be written as follows:

S = V

a (ln Q)

= V Q

Q

=

4

[(D + d)2 a2]

1 ExcludedSpaceV

If we assume that the small particle concentration is low and the volumeV of our region is significantly larger than the excluded volume we can makethe following approximation:

S

4

(D + d)2 a2 (5)

This in turn produces the following tractable approximation for P whena (D + d):

P

4

(D + d)2 a2

Note that based on this approximation and our definition of Q above, ifa (D + d) then P = 0.

2.3 Determining the Free Energy EquationRecall that Force = Energy [2]. To determine the potential energy of ourscenario we compute the following:

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U(a) =

P da

=

4

(D + d)2 a2

= 4

(D + d)2a + a

3

3+ c

To determine c we set U(D + d) = 0 and solve:

U(D + d) = (D + d)3 +(D + d)3

3+ c = 0

=

2(D + d)

2

3+ c

= 0

Thus we have c = 2 (D+d)2

3which results in the following free-energy equation:

U(a) =

4

(D + d)2a + a3

3+ 2(D+d)

2

3

: a (D + d)

0 : a (D + d)

If we assume that D d, this equation simplifies to:

U(a) =

3

2kT

(D+da)

d

2: a (D + d)

0 : a (D + d)(6)

where is the volume concentration of the macromolecules and = Dd

.

2.4 Minimizing Free-Energy Maximizing EntropyFrom this approximation it is clear that the free-energy U(a) is minimizedwhen a=(D+d). In other words, the free-energy of the system reaches a

minimum when the surfaces of the macromolecules are in contact. As afree-energy minimum represents an entropy maximum we may state thatentropy reaches a maximum when the macromolecules are in contact. Whilethis result was achieved by assuming a low concentration of small particlesAsakura and Oosawa hypothesized that it would also be applicable at highconcentrations as well [4].

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3 Relaxing an Assumption: A High Concen-

tration Scenario

It is important to note that the development of the free energy approxi-mation U(a) required multiple assumptions which must be evaluated beforethese results can be applied to the problem of protein folding and structuredetermination. The first of these assumptions was that the concentration ofsmall particles was low (Equation 2), the second, that the total volume toexcluded space ratio is high (Equation 5), and the third that the diameterD of the large macromolecules is significantly larger than the diameter d ofthe small particles (Equation 6). In addition the macromolecules were mod-

eled as spheres whereas proteins themselves are far more complex as notedpreviously.

The third of these assumptions, D d is fairly reasonable as proteinsare significantly larger than the free amino acids, ions, and water moleculessurrounding them; however, the first and second assumptions both depend ona low concentration of small particles. If we let these small particles representwater molecules these two assumptions are clearly violated, the cell being anaqueous environment.

In 1994, Walz and Sharma revisited the AO model recalculating the at-tractive force for higher concentrations of small particles [23]. They did this

by returning to the statistical mechanics equation used to approximate (x),using a second-order approximation of the density of the small particles in-stead of the first-order approximation described in the AO model. In doingso they found that while an increase in entropy certainly acts as an attractiveforce at separations D a (D + d), it also serves as a repulsive force atseparations (D + d) a 2(D + d) (Figure 8).

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Figure 8: Increase in Entropy as a Repulsive Force

On the surface the presence of a repulsive barrier to the aggregation of

the large macromolecules appears to limit the applicability of the AO modelwith regards to living systems. However, osmotic pressure as it relates tothis scenario, is not the only force acting on proteins within cells, nor is itthe strongest. In fact Walz and Sharma noted in their article Effect of LongRange Interactions on Depletion Force between Colloidal Particles that theweakness of this force has proved to be an obstacle in the direct experimentalstudy of the scenario described by the AO model [23]. Thus in the crowded,energy-rich environment of the cell, the likelihood that two portions of aprotein molecule will enter the attractive range is high [17] and the AOmodel can be used as a basic model for proteins in solution.

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4 Discussion: The AO Model Applied to Pro-

tein Folding

In recent years the AO model has been applied to the problem of proteinfolding in hopes that doing so will improve the accuracy of folding models[13]. In 2005, Harano and Kinoshita noted that the problem of protein foldingis one of the most difficult problems facing molecular biology today. Theyalso noted that the isolated folding of a protein results in a decrease of theproteins entropy. Thus for such a reaction to occur spontaneously it mustbe coupled to an energetically favorable reaction [13]. Prior to their work,hydrogen bonding, hydrophobic effects and the formation of salt bridges had

been studied as potential favorable reactions. In each case the results wereinconclusive, thus Harano and Kinoshita chose to explore possibility that anincrease in the entropy of water could serve as the energetically favorablereaction coupled to protein folding [13].

They found that due to the small size and hence high numerical concen-tration of water in aqueous living systems, the entropic gain of water wassufficient to counterbalance even the largest entropy losses due to proteinfolding. They also noted that if cells contained a solvent with a larger molec-ular size, such a counterbalance would not occur as the increase in solvententropy would not be sufficient to counter balance protein folding [13].

5 Conclusion

We see that the problem of protein folding and structure determination is asignificant open problem today, so significant that it has been described bysome as the holy grail of molecular biology [15]. Due to the vital role of pro-teins within living organisms and their ability to self assemble, the solutionto this problem has far reaching ramifications for both molecular biology andnanotechnology. Currently, self-assembling drug carriers are being used todeliver drugs to the cancerous tissue of patients and an increased understand-

ing of self-assembly is expected to result in further medical breakthroughs aswell as to aid the development of microcomputers and other nanotechnologies[21].

Scientists, mathematicians and physicists are approaching this problemfrom a variety of perspectives including classical methods such as x-ray crys-tallography and NMR, comparative methods by which new protein structures

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are approximated via comparison to know structures, and ab initio structure

prediction which attempts to predict protein structure using energy func-tions and search protocols. Despite the extensive work being done, a correctelucidation of the protein folding process has yet to be determined and pre-diction models continue to exhibit limited accuracy [13] [20]. As a result,study is now focusing on individual forces and environmental characteristicsthat influence protein folding [20] [13].

One such influence, currently being researched is the favorable increase inwater entropy which occurs when the amino acids of protein molecules comeinto close proximity with one another. The mathematical model developedby Asakura and Oosawa in the 1950s for non-interacting hard particles in

solution forms the foundation of this work. A recent extension of this modelby Harano and Kinoshita suggests that the favorable increase in entropy ofwater molecules within a cell is sufficient to overcome the entropic loss of theprotein molecule itself as it folds. Harano and Kinoshita further note thatsuch a counterbalance of entropic gain and loss is dependent on the smallsize of water molecules and suggest that favorable protein folding would notoccur in other solvents [13].

From this work it is apparent that a further understanding of the pro-tein folding and structure determination is dependent on the integration ofmultiple fields of study including chemistry, physics, biology and mathe-matics. Such an integration requires scientists and mathematicians to bewell-rounded in their knowledge of these subjects as well as to engage incollaborative work with individuals outside their immediate field. It is onlythrough this collaboration that further breakthroughs in our understandingof these complex biological structures is possible.

Acknowledgements I would like to thank Dr. Beckham for his enthusi-astic and untiring support in what turned out to be a very challenging topic,Dr. Kreft for her willingness to explain basic physics assumptions to a

couple of literal mathematicians, and my roommate Carrie for all the latenight entropy conversations.

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6 Epilogue

When I began senior seminar in August I had conflicting expectations. Onthe one hand I had completed two other seminar-like math experiences in thepast and so I felt equipped for the challenge. On the other hand I suspectedthat choosing my own topic would prove more than I had bargained for.As it turned out, there was an element of truth in each of these. The skillsI learned in my previous projects were helpful in this class, but I had muchmore to learn and while this class was more than I bargained for, I learnedthat research is not a solo endeavor and more than I bargained for is justabout right if Im willing to ask for help.

Of all of the aspects of this seminar, two of my most significant learningexperiences were the process of researching an interdisciplinary topic andthe opportunity to discuss the concepts I studied with a mathematician,chemist, two physicists and a biochemist. In terms of research I learned thatit is important to read multiple papers addressing the same general topicrather than sequentially beating my way through one difficult paper. I alsodiscovered that sometimes a follow-up paper describes the ideas at hands ina more through and approachable manner than the original. Finally, learnedthat one way to analyze the validity and relevance of a paper is to determinehow many times it has been cited in other articles.

In terms of discussing my research with others, I learned that discussion

is one of my favorite parts of the process of learning a new topic. I havealways loved meeting with professors to talk with them about class material,but it was a completely different experience to discuss material that wasnew to both myself and the professor. In addition it was fascinating, albeitfrustrating at times, to see how multiple fields approach the same problemfrom very different perspectives. It was also encouraging to watch otherpeople get excited and interested in what I was studying.

Overall I would describe this experience as one of the most importantof my college career. As much as it freaked me out at times not to seewhere I was headed, looking back I realize that this was probably one of the

most important parts of the experience. Without the ability to learn newunplanned ideas I would never be able to mature past the comparativelypassive stage of student into an active researcher. Given my experience Iwish that undergraduates were required to participate in multiple seminarsover the course of their college career.

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