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Determination of entropy change, rate constant, and kinetic energy change during the process of clathrin-mediated endocytosis by computation modeling

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Determination of entropy change, rate constant, and kinetic energy change during the process of clathrin-mediated endocytosis by computation modeling
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Lowrance, Chanda M.
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Denver, Colo.
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Metropolitan State University of Denver
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Determination of Entropy Change, Rate Constant, and Kinetic Energy Change during the Process
of
Clathrin-Mediated Endocytosis by Computation Modeling By Chanda M. Lowrance
An undergraduate thesis submitted in partial completion of the Metropolitan State University of Denver Honors Program
May 5, 2017
Dr. Brendan Fry
Dr. Megan Filbin
Primary Advisor
Second Reader
Dr. Megan Hughes-Zarzo Honors Program Director


HONORS THESIS | SPRING 2017
DETERMINATION OF ENTROPY CHANGE, RATE CONSTANT, AND KINETIC ENERGY CHANGE DURING THE PROCESS OF CLATHRIN-MEDIATED ENDOCYTOSIS BY COMPUTATIONAL MODELING
CHANDA M. LOWRANCE
DEPARTMENT OF CHEMISTRY, METROPOLITAN STATE UNIVERSITY OF DENVER, CO,


Table of Contents
Abstract................................................................... 2
1. Introduction.............................................................. 3
1.1 Clathrin-Mediated Endocytosis........................................ 3
1.2Clathrin.............................................................. 4
1.3 Computational Modeling of Biological Processes....................... 5
2. Theory.................................................................... 7
2.1 Determination of Free Energy Change.................................. 7
2.2 Determination of Change in Enthalpy..................................... 11
2.3 Determination of Change in Entropy...................................... 12
2.4 Determination of Change in Kinetic Energy............................... 12
2.5 Clathrate Topographical Determinations.................................. 12
2.5.1 Spherical Approximation........................................... 12
2.5.2 Decreasing Volume Approximation................................... 13
2.6 Determination of the Rate Constant...................................... 13
2.7 Computational Modeling Methods for Determination of Entropy Change... 13
3 Experimental Procedure....................................................... 14
4 Results...................................................................... 14
5 Discussion................................................................... 14
6 Conclusion and Future Research............................................... 15
7 Acknowledgements............................................................. 16
8 References................................................................... 17
Appendix A: Mathematica Code.................................................... 19
Appendix B: Supplementary Figures and Diagrams............................... 20


2
Determination of Entropy Change, Rate Constant, and Kinetic Energy Change During the Process of Clathrin-Mediated Endocytosis by
Computational Modeling
Chanda M. Lowrance
Department of Chemistry, Metropolitan State University of Denver, CO 80217 Clathrin-mediated endocytosis is a critical process that takes place in higher level organisms. The purpose of this research was to determine the entropy change, rate constant, and change in kinetic energy during the assembly stage of the process of clathrin-mediated endocytosis via computational modeling. Although there are many proteins involved in this process, this model focuses on the lattice formed by clathrin proteins. Using Wolfram Mathematica and Microsoft Excel, the change in entropy of assembly at 298 /fwas calculated to be 3746.56 kcal mol~1 K-1 while the experimental value is 3221.48 kcal mol~1 K~x. Thus, there is a 14.02% error in change of entropy. The values of the rate constant of assembly and the kinetic energy change are yet to be determined, but the process in which the two variables may be calculated are outlined.
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Notes
Clathrate is the term used for the cage-like structure, not to be confused with clathrin, which is the triskelion protein. Throughout the paper, clathrate may be referred to as the cage or basket. These terms are synonymous.
1. Introduction
1.1 Clathrin-Mediated Endocytosis
Clathrin-mediated endocytosis (also called receptor-mediated endocytosis) is an important but not well understood process. The mechanism was first discovered in the late 70s by scientists Roth and Porter. The discovery was made while investigating the uptake of yolk protein by insect oocytes.1 Further observation showed that the proteins involved in the process form a lattice that resembled the pattern of the seam of a soccer ball. This repeating pattern of pentagons and hexagons curved around to enclose a bilayer vesicle. The protein structure was later named a clathrate by Pearse for its cage-like structure (geometrically known as a truncated icosahedron).2
Figure 1 The structure of a clathrin-cage. Note that the blue highlighted portion is a single clathrin protein (triskelion).3
The uptake of small molecules by cells is critical for the survival of the cell. Clathrin-mediated endocytosis allows for proteins and lipids to be transported from plasma to the internal compartment of a cell. In all nucleated cells, this process occurs. There are five distinct steps of this process: (1) initiation of the coated pit formation (initiation), (2) the propagation of the clathrin lattice in addition to bilayer vesiculation and cargo recruitment (cargo propagation), (3) the completion of the lattice and the pinching off of the membrane vesicle (budding), (4) the transport or diffusion of the coated vesicle away from the membrane
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(traffic), and (5) the removal of the clathrin coat (uncoating).2 In this experiment, steps 1 and 2 (clathrate assembly) are modeled.
1.2 Clathrin
The molecular organization of clathrin is due to the lattice that is formed by triskelion. These triskelia contain a central hub from which three legs extend.
Figure 2 As shown, the triskelion is composed of smaller proteins. These proteins interact differently with adjacent triskelion to weave and form a lattice structure.
Each leg of the triskelion has a length of approximately 475 A (4.75xl0-8 m), and have a relatively uniform thickness of about 20 A (2.0xl0-9 m). The legs have
rotational symmetry and are made up of smaller proteins (Figure 2). The heavy chain proteins extend the full length of the leg, from the central hub to the spherical terminal domain at the distal end. Studies show that the terminal end is comprised of about 350 residues.2 The terminal end connects to the distal leg by a linker segment consisting of about 150 residues. The light chain proteins are tightly bound to the proximal leg in an extended conformation such that their C termini are close to the vertex of the triskelion.2 It was found that the removal of the light chain from the triskelion has little effect on the assembly properties of pure clathrin or of the clathrin with adaptor protein complexes in vitro.2 There is, however, a small inhibitory affect when light chain proteins are present with pure clathrin. Hence, it is
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It is important to keep track of the different effects that the proteins experience in order to make the model as realistic as possible.
1. Hydrophobic Effect
2. Hydrogen Bonds and
Conformational
Changes
3. Protein Immobilization Effect
4. Lipophobic Effect
probable that the light chains have a regulatory function in vivo.
Although the triskelion is somewhat stiff, there is a significant pucker. Recent studies show that if the triskelion were allowed to lie on a plane, with the terminal domains in contact with the surface, the vertex is approximately 200 A above the plane, and the terminal domains are about 400 A apart.
It is important to note that there are many proteins, other than clathrin, that are involved in the assembly and disassembly of the cage.4 However, it has been experimentally determined that clathrin can be prepared by gentle dissociation of coats from coated vesicles,5 and reassembled into coat-like lattices in the absence of other proteins.6'7 Thus, this experiment will focus solely on clathrin.
1.3 Computational Modeling of Biological Processes
A better understanding of the thermodynamic and kinetic properties of the process can be acquired through mathematical and computational modeling. This increased knowledge may permit for the manipulation of the clathrin assembly process, potentially enabling the use of clathrin for drug delivery. This will allow for a more efficient uptake of drugs, and could lead to the management of diseases and illnesses that are caused by toxins that function primarily by infiltrating the inner compartment of cells.
The technique of computational modeling may seem daunting at first, but it is actually quite simple. The diagram seen in Figure 3 shows the general path one can take to go from a
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Sixty (60) spheres are used to model the clathrin proteins. This allows for simpler mathematical calculations. Because spheres are used, the interactions between the spheres are not as accurate as they would be if triskelion are modeled. However, measures are taken to account for the differences in structure.
real system to a computational model or theory.
Figure 3 Starting from a real system, a model can be formed and perfected based on experimental and theoretical results.8
The real system being emulated in this experiment is the process of clathrin-mediated endocytosis. Experiments have been previously performed by other scientists and their results will be used as a basis for this model. Once the model system has been developed, simulations are run to imitate the experiments performed in the lab. Additionally, theoretical predictions are made to hypothesize how the simulation will compare to empirical data. Once the results are obtained from the simulation, percent errors and uncertainties can be calculated relative to the empirical data,
and the model and theory can be improved.
There are several classification criteria that can be used to create an appropriate model.9
(i) Linearity The linearity of a model applies to the operators in the mathematical model. For this experiment, the model is described as linear because the overall process is reversible.
(ii) Static vs. dynamic The model being examined is static (steady-state), a criterion that is applicable to systems in equilibrium, as opposed to dynamic (time-dependent).
(iii) Implicit vs. Explicit Because all the input parameters of the model are known (conditions of Drosophila melanogaster), and the output parameters are determined by a finite series of
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Although computational and mathematical modeling are different techniques, computational modeling is highly dependent on mathematical modeling. In Section 1.3, both computational modeling and mathematical modeling are defined. However, some of the criteria for mathematical modeling may be used to describe the computational model, based on the aforementioned dependency.
calculations, this model is explicit. The alternative would be an implicit model, in which the output parameters are known, and the input parameters are determined by iterative procedure.
(iv)Discrete vs. Continuous This research uses a discrete model, treating the objects being examined as particles. A continuous model
further knowledge. A floating model uses unfounded theories and observations to establish facts.
These criteria are fundamental to the development of an accurate model. It is important to note that this model is only an approximation, and not a literal representation of clathrin-
This research uses a discrete model instead of a continuous model. In future research, a continuous model may be used to better emulate the interactions between proteins involved during the process. However, a discrete model is sufficient in obtaining the desired data for this model.
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represents the objects as a continuous flow, such as a vector field.
(v) Deterministic vs Stochastic A deterministic model yields the same results for a given set of initial conditions. This model, however, uses a stochastic model to better simulate the randomness based on a probabilistic distribution.
(vi) Deductive, inductive, or floating -This model is both deductive and inductive. A deductive model uses theory to form a logical structure. An inductive model uses empirical data and from them generalizes the data to
mediated endocytosis. However, the criteria previously outlined will allow for the model to be relatively accurate.
2. Theory
2.1 Determination of Free Energy Change
The computational model for clathrin-
mediated endocytosis is based on an
in vivo analysis. The cellular
conditions of internal body
temperature is based on the conditions
found in Drosophila melanogaster,
which has been widely studied.
Additionally, as previously
mentioned, the analysis is of pure
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clathrin, so all accessory proteins are ignored.
The driving force for the spontaneous incorporation of proteins into membranes is widely accepted to be the hydrophobic effect between the proteins and the solvent, water.10 However, there are other contributing factors that either support or act against clathrate assembly. These factors include, but are not limited to changes in protein structure, protein state, lipid state, and attractive interactions between protein and water of lipid molecules.10
The reduction of the mobility of the water molecules surrounding the protein are due to the hydrophobic effect. External translational and rotational degrees of freedom account for the change of the protein state. These degrees of freedom become immobilized upon incorporation of the protein and the internal degrees of freedom (electronic and vibrational)
whose changes involve
conformational and internal bond changes.10 Changes in the lipid state account for the reduction of the mobility of the lipid molecules in a fluid cell membrane. Polar interactions, such as hydrogen bonds and dipole interactions, are taken into account by the changes in bonds at the proteins surface. These interactions are relatively small. Because the energetics of the proteins in the membrane are well studied, this research focuses on the initiation and propagation.1516 The effect of protein immobilization was studied by Jahnig,10 and was determined to lead to a considerable reduction of the binding energy available from the hydrophobic effect.17'18
The free energy change gained from the hydrophobic effect, denoted AGw, can be determined from the change of
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the protein-water interfacial area and a value for the free energy change per unit area.12 The free energy per area was determined to be 2.0xl021 2.5x 1021 cal mol^m-2, for hydrocarbons and hydrophobic amino acids.12'22 This value will be used for the hydrophobic effect contribution of clathrin.
Additionally, each clathrin will be modeled as a sphere. The surface area will be multiplied by a factor of 1.7 to account for the surface area of the actual protein.12 Each sphere will be assigned a radius of 4.75xl0_8m, in accordance with the length of each leg of the triskelion.4 Therefore, the surface area of the clathrin sphere is 47t(4.75x10-8 )2 x
1.7 m2. We will assume that the free energy change per area for this model is
the lower limit, 2.Ox
1021 cal mol^m-2. Thus, the
theoretical total free energy change upon
incorporation is A Gw = 9.6x 105 kcal/mol.
Since the clathrin will be modeled as spheres, we can neglect the conformational change contribution to free energy. For the breakage of hydrogen bonds between the protein and water upon incorporation of the protein (protein is not restored in the membrane), an energy of 5.8 kcal/ mol per hydrogen bond is lost.23 This decrease in hydrogen bond energy is proportional with the decrease in surface area of the clathrin sphere. Because this energy is negligible when compared to that of the total free energy change, it will be ignored in this model.24
The immobilization of external degrees of freedom contribution to the free energy can be determined by the equation
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AGt=NkTln(^ (1)
where AGt is the change in translation free energy, N is the number of proteins, k is the Boltzmann constant, T is temperature, and Vfree is the volume in which the protein resides. Because the proteins are assembling on the cell membrane, the clathrin essentially occupy the entire hemolymph volume of the fruit fly. Accordingly, Vfree = 1 L is a suitable volume size to employ in this model. The variable A is the de Broglie wavelength, which can be calculated using the equation
A = <2>
where h is Plancks constant and m is the proteins mass, in this case ~651 kDa.25 Applying the same assumptions to rotational degrees of freedom we get
A Gt = A Gr (3)
A Gt = 2A Gt (4)
where AGr is the change in free energy due to rotational degrees of freedom and AGt is the change in free energy due to the immobilization effect. Again, because the clathrin are model as spheres, and they are confined to a small box, we will ignore these contributions to free energy change.
The lipophobic effect is the final effect to be considered. The free energy of the lipid agitation caused by proteins upon incorporation is derived within the framework of a continuum model for lipid order.26'27 The orientation order of the hydrocarbon structures, described by the order parameter that is the average of the segmental order parameters of the hydrocarbon, dictate the lipid order. The average segmental order will be symbolized by 8. The perturbation of the lipid order by the protein molecule is done by the imposition of a
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boundary order parameter, S0, at the protein surface. As the distance from the protein surface increases, the perturbation decreases exponentially. The lipid coherence length, -ft, terminates when the unperturbed order parameter, 8U, is reached. The lipids are ordered by the proteins above the lipid phase transition (80 > 8U) and are disordered by the proteins below the lipid phase transition (80 < 8U). For one protein molecule at low protein concentration, where the overlap of perturbations from neighboring proteins are neglected,14 the free energy of pure clathrin does not include lipids, so the contribution to free energy by the lipophobic effect is zero.
2.2 Determination of Change in Enthalpy
The free energy change due to the hydrophobic effect is largely entropic in nature.10 Thus, the enthalpy change due to this effect, AHw, is relatively zero. This
facet of clathrate assembly is also applicable to the effects of hydrogen bonding and conformational changes. Consequently, the change in enthalpy due to hydrogen bonding and conformational change, AHc ~ 0. Likewise, AHt and AHh the respective translation and immobilization enthalpies, are also essentially zero. This is due to the entropic nature of translation and immobilization. Finally, the enthalpies of lipid-protein association due to the lipid perturbation effect were determined by calorimetric measurements.19'21 Enthalpies up to hundreds of kcal/ mol were determined experimentally. However, since our free energy is on the order of 105, this enthalpy will be disregarded, as well.
Considering that enthalpy is virtually zero, enthalpy is not included in any of the thermodynamic calculations.
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Recall that entropy is the unavailability of a system's thermal energy for conversion into mechanical work. Entropy is often referred to as the arrow of time because its asymmetry can be used to determine the direction of a reaction or event.
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2.3 Determination of Change in Entropy
Taking what is known about the free energy and enthalpy changes, the following equation is established.
A G = -TAS (5)
f = -AS (6)
The computational determination of the change in entropy will act as a standard of which the accuracy of the other computations will be compared.
2.4 Determination of Change in Kinetic Energy
From the determination of enthalpy and entropy change, we can assume that the internal energy of the system is by and large kinetic in nature. Staying true to the first law of thermodynamics, the change in total energy must equal the sum of all energies. The second law tells us that entropy is a measurement of the unavailability of a systems thermal energy for conversion into mechanical
work. Using this information, in addition to applying more complex equations of thermodynamics, the change in kinetic energy of clathrate assembly can be approximated.
2.5 Clathrate Topographical Determinations
Two methods of approximation are created to optimize accuracy of the determination of surface area of the clathrate cage.
2.5.1 Spherical Approximation
The surface area of the clathrate can be determined by simply integrating the surface of the cage as clathrin are added to the structure. Using Mathematica, a mesh can be designed to wrap around the entire structure, and the area of the mesh would be equal to the overall surface area of the clathrate.
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In this experiment, the change in entropy was determined using the decreasing volume approximation to determine surface area. It is predicted, however, that the spherical
approximation will yield a more accurate approximation.
2.5.2 Decreasing Volume Approximation
Another approximation of surface area is performed by starting with the clathrate fully assembled, and then deducting the surface area of each clathrin sphere for 60 time steps (representative of the 60 spheres).
2.6 Determination of the Rate Constant
Transition theory can be used to determine the rate constant for the assembly stage of clathrin-mediate endocytosis. As previously mentioned, the entire process is assumed to occurs in less than a minute. Thus, the rate constant should indicate that the assembly stage of endocytosis should occur in approximately 10 seconds, assuming that all stages occur on a similar timescale. The transition state will be the point of clathrate assembly in which the first pentagon appears in the structure. This is the point in which there is no clathrate disassembly, and there is a large free
energy loss. This is also the point in which the clathrate cage develops curvature. Taking into account the difference between the potential energy of the transition state and the potential energy of the individual clathrin spheres, the rate constant of the reaction can be determined.
2.7 Computational Modeling Methods for Determination of Entropy Change
Using Mathematica, the clathrin are modeled as spheres. In each trial run, the spheres will begin in a randomly disbursed arrangement. In each time step, the clathrin will move to one of the vertices of a truncated icosahedron structure, similar to the clathrin being incorporated into the clathrate.
The changes in surface area of the clathrate were determined using method 2.5.2, and the values were used to determine the change in free
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Wolfram Mathematica and Microsoft Excel were both used to make the final determinations in free energy and entropy change. Although Mathematica is capable of advanced calculations, Excel allows for easy editing and displays value in an easy-to-read tabular format.
energy change. Finally, the change in entropy is determined.
3. Experimental Procedure
The process of clathrin-mediated endocytosis is modeled using the mathematical program Wolfram Mathematica. Using the parameters outlined in the theory section of this article, the thermodynamic and kinetic properties of clathrin are applied to the clathrin spheres (refer to section 2.1). Furthermore, a space-filling model of a truncated icosahedron is used to simulate the bonds between the clathrin (Figure B.2). Using method 2.5.2, the change in surface area of the clathrate is calculated and used to determine the change in Gibbs free energy. Then using the information about free energy in Section 2.1, and Equation 6, the free energy change was determined. Using Microsoft Excel, the change in entropy was calculated. The calculated change in
entropy was compared to the theoretical change in entropy, and the percent error was determined. Using Equation 6, a command is written to calculate the entropy at 298 K (room temperature) (Appendix A).
4. Results
The experimental value of entropy was found to be
3221.48 kcal mol-1 K-1, while the computational value of entropy was 3746.56 kcal mol-1 K-1. Thus, the computational model gave a 14.02% error. The model will continue to be refined in order to accurately predict the change in kinetic energy and the rate constant of assembly.
5. Discussion
There are some obvious shortcomings of this model. For instance, there is a dynamic opening and closing of the clathrin-cage as it assembles that is
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Consumers of prescription medications are aware of the high cost of medications. A large component to the price of medications is the cost of research and development. Being able to model drug-interactions on a computer will help reduce or eliminate the need to perform hundreds of tests in the lab.
not represented in this model. The contribution of the accessory proteins was not considered. Additionally, only the first two stages of the process were accounted for. There were many other factors that were ignored for this model, and inclusion of these factors will provide a more complete analysis. In light of this, this model relatively accurate.
6. Conclusion and Future Research
A more detailed model of the process of clathrin-mediated endocytosis can be constructed to better resemble the process in vivo. Creating a more
detailed, realistic representation of the clathrin triskelions will allow for the modeling of the interactions between adjacent triskelion legs. This will also ensure that the predictive computations are more reliable.
Streamlining this process may potentially lead to the use of the model as a means of constructing a drug-delivery system in
which the synthetic clathrates are customized to bind to only specific drugs. Additionally, determining the rate constants of the overall reaction and the changes in kinetic energy of the full process will allow for the times the drug is delivered, absorbed, and becomes no longer effective. Such an advancement of this model for use in the pharmaceutical industry will decrease the amount of money spent on research and development of medications. This could lead to a dramatic decrease in the price of drugs for consumers. Therefore, there is a critical need for the use of computational modeling in chemistry and biochemistry.
Despite the simplistic nature of the model, the determination of the change in entropy was relatively accurate. Thus, computational modeling is an accurate way of
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determining the thermodynamic and kinetic properties of biological processes.
16
7. Acknowledgements
I would like to acknowledge and thank Dr. Brendan Fry, who helped advise me on this project. I would also like to thank Dr. Joshua Martin and Dr. Megan Filbin-Wong for their respective physical chemistry and biochemistry knowledge.
Additionally, thank you to Dr. Megan Hughes-Zarzo for her guidance in the Metropolitan State University of Denver Honors Program.
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8. References
(1) Roth, T. F.; Porter, K. R. Electron Microscopy 1962.
(2) Kirchhausen, T. Annual Review of Biochemistry 2000, 69 (1), 699-727.
(3) Clathrin
https://en.wikipedia.org/wiki/Clathrin (accessed Apr 2, 2017).
(4) Clathrin uncoating: Auxilin comes to life. http://www.cell.com/current-biology/fulltext/S0960-9822(01 )00010-0?_returnURL=http%3A%2F%2Flinkinghu b. el sevier. com%2F retrieve%2F pii%2F S096 0982201000100%3Fshowall%3Dtrue (accessed May 7, 2017).
(5) Keen, J. FL; Willingham, M. C.; Pastan, I.
H. Cell 1979, 16 (2), 303-312.
(6) Kirchhausen, T.; Harrison, S. C. Cell 1981, 23 (3), 755-761.
(7) Crowther, R.; Pinch, J.; Pearse, B. Journal of Molecular Biology 1976, 103 (4), 785-798.
(8) Computer simulation. https://en.wikipedia.org/wiki/Computer_sim ulation (accessed May 7, 2017).
(9) Mathematical model. https://en.wikipedia.org/wiki/Mathematical_ model (accessed May 7, 2017).
(10) Jahnig, F. Proceedings of the National Academy of Sciences 1983, 80 (12), 3691 3695.
(11) Tanford, C. Science 1978, 200 (4345), 1012-1018.
(12) Richards, F. M. Annual Review of Biophysics and Bioengineering 1977, 6 (1), 151-176.
(13) Heijne, G.; Blomberg, C. European Journal of Biochemistry 1979, 97 (1), 175-181.
(14) Engelman, D.; Steitz, T. Cell 1981, 23 (2), 411-422.
(15) Tanford, C. Journal of the American Chemical Society 1962, 84 (22), 4240-4247.
(16) Hermans, J. The Journal of Physical Chemistry 1966, 70 (2), 510-515.
(17) Page, M. I.; Jencks, W. P. Proceedings of the National Academy of Sciences
1971, 68 (8), 1678-1683.
(18) Janin, J.; Chothia, C. Biochemistry 1978, 17 (15), 2943-2948.
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(19) Rosseneu, M.; Soetewey, F.; Middelhoff, G.; Peeters, H.; Brown, W. Biochimica et Biophysica Acta (BBA) Lipids and Lipid Metabolism 1976, 441 (1), 68-80.
(20) Massey, J. B.; Gotto, A. M.; Pownall, H.
J. Biochemistry 1981, 20 (6), 1575-1584.
(21) Epand, R. M.; Sturtevant, J.
M. Biochemistry 1981, 20 (16), 4603-4606.
(22) Reynolds, J. A.; Gilbert, D. B.; Tanford,
C. Proceedings of the National Academy of Sciences 1974, 71 (8), 2925-2927.
(23) Encyclopedia of Genetics, Genomics, Proteomics and Informatics 2008, 1186 1187.
(24) Nossal, R. Traffic 2001, 2 (2), 138-147.
(25) Ferguson, M. L.; Prasad, K.; Boukari, FL; Sackett, D. L.; Krueger, S.; Lafer, E. M.; Nossal, R. Biophysical Journal 2008, 95 (4), 1945-1955.
(26) Owicki, J. C.; Mcconnell, H.
M. Proceedings of the National Academy of Sciences 1979, 76 (10), 4750-4754.
(27) Jahnig, F. Biophysical Journal 1981, 36(2), 329-345.
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Appendix A: Mathematica Code
(*
Chanda Lowrance, Dr. Brendan Fry HON 4950, CHE 4950 Updated May 4, 2017
This program is a computational model of the assembly of the clathin cage that is formed during the process of clathrin mediated endocytosis (receptor-mediated endocytosis). To make the model more simple the AP complexes and other proteins are disregarded in the model. Therefore only the triskelion(light and heavy chains) are included and represented by a sphere. Each vertex represents the point in which the central hub of the triskelion overlaps with the knee of the adjacent triskelion.
*)
(Location of clathrin in clathrate*)
clathrateVertices = PolyhedronData["TruncatedIcosahedron", "VertexCoordinates"];
(Randomization of clathrin starting position*) RandLoc = Table[RandomReal[{-10, 10}, 3], {60}];
sATI = PolyhedronData["TruncatedIcosahedron", "SurfaceArea"] sAClathrin =4* Pi; temp = 298;
(Move clathrin spheres for clathrate vertices*)
Do [
RandLoc[[i]] = clathrateVertices[[i]];
(Parameterization of clathrin spheres*) clathrinSphere = Sphere[#, 1] & /@ RandLoc; sABound [i] = sATI + sAClathrin (60 i);
Graphl[i] = Graphics3D [clathrinSphere PlotRange {{-12, 12}, {-12, 12}, {-12, 12}}, Prolog -* Text [i 1 "s"] ], {i, 60}];
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Appendix B: Supplementary Figures and Diagrams
Coat assembly
Bud formation
Fission
Clathrin release
Hsc70
Auxilin
Synaptojanin
Endophilin

Clathrin
triskelion
, Clathrin lattice
AP
complex
o Hsc70 Auxilin O [j,geacne(§tor/
Figure B.l There are proteins other than clathrin involved in clathrin-mediated endocytosis.
Figure B.2 A space-filling model of a truncated icosahedron accounts for the bonds between the clathrin. Note that the clathrin are modeled as spheres, as opposed to triskelion.
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Full Text

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Determination of Entropy Change, Rate Constant, and Kinetic Energy Change during the Process of Clathrin Mediated Endocytosis by Computation Modeling By Chanda M. Lowrance An undergraduate thesis submitted in partial completion of the M etropolitan State University of D enver Honors Program May 5, 2017 Dr. Brendan Fry Dr. Megan Filbin Dr. Megan Hughes Zarzo Primary Advisor Second Reader Honors Program Director

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Table of Contents Abstract .................. 2 1. Introduction 3 1 .1 Clathrin Mediated Endocytosis ... 3 1.2 Clathrin 4 1 .3 Computational Modeling of Bio logical Processes .... ... 5 2. Theory ... 7 2.1 Determination of Free Energy Change ... 7 2.2 Determination of Change in Enthalp y 11 2.3 Determination of Change in Entropy ... 12 2.4 Determination of Change in Kinetic Energy 12 2.5 Clathrate Topographical Determinations ... 12 2 .5.1 Spherical Approximation ......... 12 2 .5.2 Decreasing Volume Approximation 13 2 .6 Determination of the Rate Constant 13 2.7 Computational Modeling Methods f or Determination of Entropy Change 13 3 Experimental Procedure 14 4 Results 14 5 Discussion 14 6 Conclusion and Future Research 15 7 Acknowledgements 16 8 Reference s 17 Appendix A: Mathematica Code 19 Appendix B: Supplementary Figures and Diagrams 20

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Lowrance HON 4950 | Spring 2017 | MSUD 2 D eterminati on of Entropy Change Rate Constant, and Kinetic Energy Change During the Process of Clathrin Mediated Endocytosis by Computational Modeling Chanda M. Lowrance Department of Chemistry, Metropolitan State University of Denv er, CO 80217 Clathrin mediated endocytosis is a critical process that takes place in higher level organisms. The purpose of this research was to determine the entropy change rate constant, and change in kinetic energy during the assembly stage of the proc ess of clathrin mediated endocytosis via computational modeling. Although there are many protei ns involved in this process, this model focuses on the lattice formed by clathrin proteins. Using Wolfram Mathematica and Microsoft Excel the change in entropy of assembly at !"# $ % was calculated to be &'() +) $ ,-./ $ 01/ 2 3 $ % 2 3 while the experimental value is &!!4 (# $ ,-./ $ 01/ 2 3 $ % 2 3 Thus, there is a 14.02% error in change of entropy. The values of the rate constant of assembly and the k i netic energy change are yet to be determined, but the process in which the two variables may be calculated are outlined.

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Lowrance HON 4950 | Spring 2017 | MSUD 3 Notes Clathrate is the term used for the cage like structure, not to be confused with clathrin, which is the triskelion protein. Thro ughout the paper, clathrate may be referred to as the "cage" or "basket". These terms are synonymous. 1. Introduction 1.1 Clathrin Mediated Endocytosis Clathrin mediated endocytosis (also called rec eptor mediated endo cytosis) is an important but not well un derstood process The mechanism was first discovered in the late 70's by scientists Roth and Porter. The discovery was made while investigating the uptake of yolk protein by insect oocytes. 1 Further observation showed that the proteins involved in the proc ess form a lattice that resembled the pattern of the seam of a soccer ball. This repeating pattern of pentagons and hexagons curved around to enclose a bilayer vesicle. The protein structure was later named a clathrate by Pearse for its cage like structure (geometrically known as a truncated icosahedron) 2 Figure 1 The structure of a clathrin cage. Note that the blue highlighted portion is a single clathrin protein (triskelion). 3 The uptake of small molecules by cells is critical for the survival of the cell Clathrin mediated endocytosis allows for proteins and lipids to be transported from plasma to the internal compartment of a cell. In all nucleated cells, this process occurs There are five distinct steps of this process: (1) initiation of the coated pit formation (initiation), (2) the propagation of the clathrin lattice in addition to bilayer vesiculation and cargo recruitment (cargo propagation), (3) the completion of the lattice and the pinching off of the membrane vesicle (budding), (4) the transport or diffusion of the coated vesicle away from the membrane

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Lowrance HON 4950 | Spring 2017 | MSUD 4 (traffic), and (5) the removal of the clathrin coat (uncoating). 2 In this experiment, steps 1 and 2 (clathrate assembly) are modeled. 1.2 Clathrin The molecular organiz ation of clathrin is due to the lattice that is formed by triskelion. These triskelia contain a central hub from which three "legs" extend. Figure 2 As shown the triskelion is composed of smaller proteins These p roteins interact differently with adjacent triskelion to weave and form a lattice structure. Each leg of the triskelion ha s a length of approximately ('+ $ 5 ( ( '+ 6 47 2 8 $ 0 9 and have a relatively uniform thickness of about !7 $ 5 ( 7 6 47 2 : $ 0 9 The legs have rotational symmetry and are made up of smaller proteins (Figure 2) The heavy chain protein s extend the full length of the leg, from the central hub to the spherical terminal domain at the distal end. Studies show that the terminal end is comprised of about 350 residues. 2 The terminal end connects to the distal leg by a linker segment consisting of about 150 residues. The light chain proteins are tightly bound to the proximal leg in an extended conformation such that their C termini are close to the vertex of the triskelion. 2 It was found that the removal of the light chain from the triskelion has little effect on the a ssembly properties of "pure clathrin" or of the clathrin with adaptor protein complexes in vitro. 2 There is, however, a small inhibitory affect when light chain proteins are present with "pure clathrin". Hence, it is

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Lowrance HON 4950 | Spring 2017 | MSUD 5 probable that the light chains have a r egulatory function in vivo Although the triskelion is somewhat stiff, there is a significant pucker. Recent studies show that if the triskelion were allowed to lie on a plane, with the terminal domains in contact with the surface, the vertex is approximately !77 $ 5 above the plane, and the terminal domains are about (77 $ 5 apart. It is important to note that there are many proteins, other than clathrin, that are involved in the assembly and disassembly of the cage. 4 How ever, it has been experimentally determined that clathrin can be prepared by gentle dissociation of coats from coated vesicles, 5 and reassembled into coat like lattices in the absence of other proteins. 6,7 Thus, this experiment will focus solely on clathri n. 1.3 Computational Modeling of Biological Processes A better understanding of the thermodynamic and kinetic properties of the process can be acquired through mathematical and computational modeling. This increased knowledge may permit for the manipulation of the clathrin assembly process, potentially enabling the use of clathrin for drug delivery. This will allow for a more efficient uptake of drugs and could lead to the management of diseases and illnesses that are caused by toxins that function primarily by infiltrating the inner compartment of cells. The technique of computational modeling may seem daunting at first, but it is actually quite simple. The diagram seen in Figure 3 shows the general path one can take to go from a It is important to keep track of the different effects that the proteins experience in order to make the model as realistic as possible. 1. Hydrophobic Effect 2. Hydrogen Bonds and Conformational Changes 3. Protein Immobilization Effect 4. Lipophobic Effect

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Lowrance HON 4950 | Spring 2017 | MSUD 6 real system to a computation al model or theory. Figure 3 Starting from a real system, a model can be formed and perfected based on experimental and theoretical results. 8 The real system being emulated in this experiment is the process of clathrin mediated endocyto sis. Experiments have been previously performed by other scientists and their results will be used as a basis for this model. Once the model system has been developed, simulations are run to imitate the experiments performed in the lab. Additionally, theor etical predictions are made to hypothesize how the simulation will compare to empirical data. Once the results are obtained from the simulation, percent errors and uncertainties can be calculated relative to the empirical data, and the model and theory can be improved. There are several classification criteria that can be used to create an appropriate model. 9 (i) Linearity The linearity of a model applies to the operators in the mathematical model. For this experiment, the model is described as linear be cause the overall process is reversible. (ii) Static vs. dynamic The model being examined is static (steady state), a criterion that is applicable to systems in equilibrium, as opposed to dynamic (time dependent). (iii) Implicit vs. Explicit Because all the input parameters of the model are known (conditions of Drosophila melanogaster ), and the output parameters are determined by a finite series of Sixty (60) s pheres are used to model the clathrin proteins. This allows for simpler mathematical calculations. Because spheres are used, the interactions between the spheres are not as accurate as they would be if triskelion are modeled. However, m easures are taken to account for the differences in structure.

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Lowrance HON 4950 | Spring 2017 | MSUD 7 calculations, this model is explicit The alternative would be an implicit model, in which the output parameters are known, and the input parameters are determined by iterative procedure. (iv) Discrete vs. Continuous This research uses a discrete model, treating the objects being examined as particles. A continuous model represents the objects as a continuous flow, such as a vector field. (v) Deterministic vs Stochastic A deterministic model yields the same results for a given set of initial conditions. This model, however, uses a stochastic model to better simulate the randomness based on a probabilistic distribution. (vi) Deduc tive, inductive, or floating This model is both deductive and inductive A deductive model uses theory to form a logical structure. An inductive model uses empirical data and from them generalizes the data to further knowledge. A floating model uses unfo unded theories and observations to establish facts. These criteria are fundamental to the development of an accurate model. It is important to note that this model is only an approximation, and not a literal representation of clathrin mediated endocytosis. However, the criteria previously outlined will allow for the model to be relatively accurate. 2. Theory 2.1 Determination of Free Energy Change The computational model for clathrin mediated endocytosis is based on an in vivo analysis. The cellular conditions of internal body temperature is based on the conditions found in Drosophila melanogaster, which has been widely studied. Additionally, as previously mentioned, the analysis is of "pure Although computational and mathematical modeling are different techniques, computational modeling is highly depend ent on mathematical modeling. In Section 1.3, both computational modeling and mathematical modeling are defined. However, some of the criteria for mathematical modeling may be used to describe the computational model, based on the aforementioned dependency This research uses a discrete model instead of a continuous model. In future research, a continuous model may be used to better emulate the interactions between proteins involved during the process. However, a discrete model is sufficient in obtaining t he desired data for this model.

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Lowrance HON 4950 | Spring 2017 | MSUD 8 clathrin", so all accessory proteins are ignored. The driving forc e for the spontaneous incorporation of proteins into membranes is widely accepted to be the hydrophobic effect between the proteins and the solvent, water 10 However, there are other contributing factors that ei ther support or act against clathrate assembly These factors include, but are not limited to changes in protein structure, protein state, lipid state, and attractive interactions between protein and water of lipid molecules. 10 The reduction of the mobility of the water molecules surrounding the prote in are due to the hydrophobic effect. External translational and rotational degrees of freedom account for the change of the protein state. These degrees of freedom become immobilized upon incorporation of the protein and the internal degrees of freedom (e lectronic and vibrational) whose changes involve conformational and internal bond changes. 10 Changes in the lipid state account for the reduction of the mobility of the lipid molecules in a fluid cell membrane. Polar interactions, such as hydrogen bonds and dipole interactions, are taken into account by the changes in bonds at the proteins surface. These interactions are relatively small. Because the energetics of the proteins in the membrane are well studied, this research focuses on the initiation and prop agation 15 ,1 6 The effect of protein immobilization was studied by Jahnig, 10 and was determined to lead to a considerable reduction of the binding energy available from the hydrophobic effect. 17 ,18 The free energy change gained from the hydrophobic effect, d enoted ; < = can be determined from the change of

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Lowrance HON 4950 | Spring 2017 | MSUD 9 the protein water interfacial area and a value for the free energy change per unit area. 1 2 The free energy per area was determined to be 7 6 47 >3 ? + 6 47 >3 $ -./ $ 01/ 2 3 0 2 > for hydroca rbons and hydrophobic amino acids. 12 2 2 This value will be used for the hydrophobic effect contribution of clathrin. Additionally, each clathrin will be modeled as a sphere. The surface area will be multiplied by a factor of 4 to account for the surfac e area of the actual protein. 1 2 Each sphere will be assigned a radius of ( '+ 6 47 2 8 $ 0 in accordance with the length of each leg of the triskelion. 4 Therefore, the surface area of the "clathrin sphere" is ( @ A ( '+ 6 47 2 8 $ 9 > $ 6 4 $ 0 > We will assume that the free energy change per area for this model is the lower limit, 7 6 4 7 >3 $ ./ $ 01/ 2 3 0 2 > Thus, the theoretical total free energy change upon incorporation is ; < = B ? ) 6 47 C $ ,-./ 01/ Since the clathrin will be modeled as spheres, we can neglect the conformational change contribution to free energ y. For the breakage of hydrogen bonds between the protein and water upon incorporation of the protein (protein is not restored in the membrane) an energy of + # $ ,-./ D 01/ per hydrogen bond is lost. 2 3 This decrease in hydrogen bond energy is proporti onal with the decrease in surface area of the clathrin sphere. Because this energy is negligible when compared to that of the total free energy change, it will be ignored in this model. 2 4 The immobilization of external degrees of freedom contribution to th e free energy can be determined by the equation

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Lowrance HON 4950 | Spring 2017 | MSUD 10 ; < E B F,G/H I JKLL M N O (1) where ; < E is the change in translation free energy, F is the number of proteins, is the Boltzmann constant, G is temperature, and P QRSS is the volume in which the protein resides. Because the proteins are assembling on the cell membrane, the clathrin essentially occupy the entire hemolymph volume of the fruit fly. Accordingly, P QRSS B 4 $ T is a suitable volume size to employ in this model. The variable U is the de Broglie wa velength, which can be calculated using the equation U B V > WXYZ (2) where [ is Planck's constant and 0 is the proteins mass, in this case \ )+4 $ ,]. 2 5 Applying the same assumptions to rotational degrees of freedom we get ; < E B ; < R (3) ; < ^ B ; < E (4) where ; < R is the change in free energy due to rotational degrees of freedom and ; < ^ is the change in free energy due to the immobilization effect. Again, because the clathrin are model as spheres, and they are confi ned to a small box, we will ignore these contributions to free energy change. The lipophobic effect is the final effect to be considered. The free energy of the lipid agitation caused by proteins upon incorporation is derived within the framework of a cont inuum model for lipid order. 26 ,2 7 The orientation order of the hydrocarbon structures, described by the order parameter that is the average of the segmental order parameters of the hydrocarbon, dictate the lipid order. The average segmental order will be s ymbolized by The perturbation of the lipid order by the protein molecule is done by the imposition of a

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Lowrance HON 4950 | Spring 2017 | MSUD 11 boundary order parameter, ` at the protein surface. As the distance from the protein surface increases, the perturbation decreases exponentia lly. The lipid coherence length, a terminates when the unperturbed order parameter, b is reached. The lipids are ordered by the proteins above the lipid phase transition ( ` c b ) and are disordered by the proteins below the lipid phase tra nsition ( ` d b ). For one protein molecule at low protein concentration, where the overlap of perturbations from neighboring proteins are neglected, 1 4 the free energy of "pure clathrin" does not include lipids, so the contribution to free energy b y the lipophobic effect is zero. 2.2 Determination of Change in Enthalpy The free energy change due to the hydrophobic effect is largely entropic in nature. 10 Thus, the enthalpy change due to this effect, ; e = is relatively zero. This facet of clathr ate assembly is also applicable to the effects of hydrogen bonding and conformational changes. Consequently, the change in enthalpy due to hydrogen bonding and conformational change, ; e f g 7 Likewise, ; e E and ; e ^ the respective translation and immobilization enthalpies, are also essentially zero. This is due to the entropic nature of translation and immobilization. Finally, the enthalpies of lipid protein association due to the lipid perturbation effect were determined by calorimetric measur ements. 19 21 Enthalpies up to hundreds of ,-./ D 01/ were determined experimentally. However, since our free energy is on the order of 47 C this enthalpy will be disregarded, as well. Considering that enthalpy is virtually zero, enthalpy is not included in any of the thermodynamic calculat ions.

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Lowrance HON 4950 | Spring 2017 | MSUD 12 2.3 Determination of Change in Entropy Taking what is known about the free energy and enthalpy changes, the following equation is established. ; < B ? G ; h (5 ) ; i Z B ? ; h (6 ) The computational determination of the change in entropy will act as a standard of which the accuracy of the other computations will be compared. 2.4 Determination of Change in Kinetic Energy From the determination of enthalpy and entropy change, we can assume that the internal energy of the system is by and large ki netic in nature. Staying true to the first law of thermodynamics, the change in total energy must equal the sum of all energies. The second law tells us that entropy is a measurement of the unavailability of a system's thermal energy for conversion into me chanical work. Using this information, in addition to applying more complex equations of thermodynamics, the change in kinetic energy of clathrate assembly can be approximated. 2.5 Clathrate Topographical Determinations Two methods of approximation are created to optimize accuracy of the determination of surface area of the clathrate cage. 2.5.1 Spherical Approximation The surface area of the clathrate can be determined by simply integrating the surface of the cage as clathrin are added to the structure. Using Mathematica, a "mesh" can be designed to wrap around the entire structure, and the area of the mesh would be equal to the overall surface area of the clathrate. Recall that entropy is the unavailability of a system's thermal energy for conversion into mechanical work. Entropy is often referred to as "the arrow of time" because its asymmetry can be used to determine the direction of a reaction or event.

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Lowrance HON 4950 | Spring 2017 | MSUD 13 2.5.2 Decreasing Volume Approximation Another approximation of surface area is performe d by starting with the clathrate fully assembled, and then deducting the surface area of each clathrin sphere for 60 time steps (representative of the 60 spheres). 2.6 Determination of the Rate Constant Transition theory can be used to determine the rate c onstant for the assembly stage of clathrin mediate endocytosis. As previously mentioned, the entire process is assumed to occurs in less than a minute. Thus, the rate constant should indicate that the assembly stage of endocytosis should occur in approxima tely 10 seconds, assuming that all stages occur on a similar timescale. The transition state will be the point of clathrate assembly in which the first pentagon appears in the structure. This is the point in which there is no clathrate disassembly, and the re is a large free energy loss. This is also the point in which the clathrate cage develops curvature. Taking into account the difference between the potential energy of the transition state and the potential energy of the individual clathrin spheres, t he rate constant of the reaction can be determined. 2.7 Computational Modeling Methods f or Determination of Entropy Change Using Mathematica, the clathrin are modeled as spheres. In each trial run, the spheres will begin in a randomly disbursed arrangem ent. In each time step, the clathrin will move to one of the vertices of a truncated icosahedron structure, similar to the clathrin being incorporated into the clathrate. The changes in surface area of the clathrate were determined using method 2.5.2, an d the values were used to determine the change in free In this experiment, the change in entropy was determined using the decreasing volume approxim ati on to determine surface area. It is p redicted, however, that the spherical approximation will yield a more accurate approximation.

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Lowrance HON 4950 | Spring 2017 | MSUD 14 energy change. Finally, the change in entropy is determined. 3. Experimental Procedure The process of clathrin mediated endocytosis is modeled using the mathematical program Wolfram Mathematica. Using th e parameters outlined in the theory section of this article, the thermodynamic and kinetic properties of clathrin are applied to the clathrin spheres (refer to section 2.1). Furthermore, a space filling model of a truncated icosahedron is used to simulat e the bonds between the clathrin (Figure B.2). Using method 2.5.2, the change in surface area of the clathrate is calculated and used to determine the change in Gibb's free energy. Then using the information about free energy in Section 2.1, and Equation 6 the free energy change was determined. Using Microsoft Excel, the change in entropy was calculated. The calculated change in entropy was compared to the theoretical change in entropy, and the percent error was determined Using Equation 6, a command is w ritten to calculate the entropy at !"# $ % (room temperature) (Appendix A). 4. Results The experimental v alue of entropy was found to be &!!4 (# $ ,-./ $ 01/ 2 3 $ % 2 3 while the computational value of entropy was &'() +) $ ,-./ $ 01/ 2 3 $ % 2 3 Thus, the computational model gave a 14.02% error. The model will continue to be refined in order to accurately predict the change in kinetic energy and the rate constant of assembly. 5. Discussion There are some obvious shortcomings of this model. For instance, there is a dynamic opening and cl osing of the clathrin cage as it assembles that is Wolfram Mathematica and Microsoft Exc e l were both used to make the final determin ations in free energy and entrop y change Although Mathematica is capable of advanced calculations Excel allows for easy editing and displays value in an easy to read tabular format.

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Lowrance HON 4950 | Spring 2017 | MSUD 15 not represented in this model. The contribution of the accessory proteins was not considered. Additionally, only the first two stages of the process were accounted for. There were many other factors that w ere ignored for this model, and inclusion of these factors will provide a more complete analysis. In light of this, this model relatively accurate. 6. Conclusion and Future Research A more detailed model of the process of clathrin mediated endocytosis can b e constructed to better resemble the process in vivo Creating a more detailed, realistic representation of the clathrin triskelions will allow for the modeling of the interactions between adjacent triskelion legs. This will also ensure that the predictiv e computations are more reliable. Streamlining this process may potentially lead to the use of the model as a means of constructing a drug delivery system in which the synthetic clathrates are customized to bind to only specific drugs. Additionally, determ ining the rate constants of the overall reaction and the changes in kinetic energy of the full process will allow for the times the drug is delivered, absorbed, and becomes no longer effective. Such an advancement of this model for use in the pharmaceutica l industry will decrease the amount of money spent on research and development of medications. This could lead to a dramatic decrease in the price of drugs for consumers. Therefore, there is a critical need for the use of computational modeling in chemistr y and biochemistry. Despite the simplistic nature of the model, the determination of the change in entropy was relatively accurate. Th us computati onal modeling is an accurate way of Consumers of prescription medications are aware of the high cost of medications. A large component to the price of medications i s the cost of research and development. Being able to model drug interactions on a computer will help reduce or eliminate the ne ed to perform hundreds of t ests in the lab.

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Lowrance HON 4950 | Spring 2017 | MSUD 16 determining the thermodynamic and kinetic properties of biological proces ses. 7. Acknowledgements I would like to acknowledge and thank Dr. Brendan Fry, who helped advise me on this project. I would also like to thank Dr. Joshua Martin and Dr. Megan Filbin Wong for their respective physical chemistry and biochemistry knowledge. A dditionally, thank you to Dr. Megan Hughes Zarzo for her guidance in the Metropolitan State University of Denver Honors Program.

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Lowrance HON 4950 | Spring 2017 | MSUD 17 8. Reference s (1) R oth, T. F.; Porter, K. R. Electron Microscopy 1962 (2) Kirchhausen, T. Annual Review of Biochemistry 2000 69 (1), 699 727. (3) Clathrin https://en.wikipedia.org/wiki/Clathrin (accessed Apr 2, 2017). (4) Clathrin uncoating: Auxilin comes to life http://www.cell.com/current biology/fulltext/S0960 9822(01)00010 0?_returnURL=http%3A%2F%2Flinkinghu b.elsevier.com%2Fretrieve%2Fpii%2FS09 6 0982201000100%3Fshowall%3Dtrue (accessed May 7, 2017) (5) K een, J. H. ; Willingham, M. C.; Pastan, I. H. Cell 1979 16 (2), 303 312. (6) Kirchhausen, T.; Harrison, S. C. Cell 1981 23 (3), 755 761. (7) Crowther, R.; Pinch, J.; Pearse, B. Journal of Molecular Biology 1976 103 (4), 785 798. (8) Computer simulation. h ttps://en.wikipedia.org/wiki/Computer_sim ulation (accessed May 7, 2017). (9) Mathematical model. https://en.wikipedia.org/wiki/Mathematical_ model (accessed May 7, 2017). (10) J ahnig, F. Proceedings of the National Acade my of Sciences 1983 80 (12), 3691 3695. (11) Tanford, C. Science 1978 200 (4345), 1012 1018. (12) Richards, F. M. Annual Review of Biophysics and Bioengineering 1977 6 (1), 151 176. (13) Heijne, G.; Blomberg, C. European Journal of Biochemistry 1979 97 (1), 175 181. (14) Engelman, D.; Steitz, T. Cell 1981 23 (2), 411 422. (15) Tanford, C. Journal of the American Chemical Society 1962 84 (22), 4240 4247. (16) Hermans, J. The Journal of Physical Chemistry 1966 70 (2), 510 515. (17) Page, M. I.; Jencks, W. P. Proceedings of the National Academy of Sciences 1971 68 (8), 1678 1683. (18) Janin, J.; Chothia, C. Biochemistry 1978 17 (15), 2943 2948.

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Lowrance HON 4950 | Spring 2017 | MSUD 18 (19) Rosseneu, M.; Soetewey, F.; Middelhoff, G.; Peeters, H.; Brown, W. Biochimica et Biophysica Acta (BBA) Lipids and Lipid Metabolism 1976 441 (1), 6 8 80. (20) Massey, J. B.; Gotto, A. M.; Pownall, H. J. Biochemistry 1981 20 (6), 1575 1584. (21) Epand, R. M.; Sturtevant, J. M. Biochemistry 1981 20 (16), 4603 4606. (22) Reynolds, J. A.; Gilbert, D. B.; Tanford, C. Proceedings of the National Academy of Sciences 1974 71 (8), 2925 2927. (23) Encyclopedia of Genetics, Genomics, Proteomics and Informatics 2008 1186 1187. (24) Nossal, R. Traffic 2001 2 (2), 138 147. (25) Ferguson, M. L.; Prasad, K.; Boukari, H.; Sackett, D. L.; Krueger, S.; Lafer, E. M.; Nossal, R. Biophysical Journa l 2008 95 (4), 1945 1955. (26) Owicki, J. C.; Mcconnell, H. M. Proceedings of the National Academy of Sciences 1979 76 (10), 4750 4754. (27) JŠhnig, F. Biophysical Journal 1981 36 (2), 329 345.

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Lowrance HON 4950 | Spring 2017 | MSUD 19 Appendix A: Mathematica Code

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Lowrance HON 4950 | Spring 2017 | MSUD 20 Appendix B: Supplementary Figures and Diagrams Figure B.1 There are proteins other than clathrin involved in cla thrin mediated endocytosis. Figure B.2 A space filling model of a truncated icosahedron accounts for the bonds between the clathrin. Note that the clathrin are modeled as spheres, as opposed to triskelion.