# Folding of Proteins - Simulation using Monte Carlo Approach

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1 A Report On Folding of Proteins - Simulation using Monte Carlo Approach Prepared By Ramji T. Venkatasubramanian In Partial fulfillment of course Computational Nanomechanics ME 8253 Spring Semester, May 2006 University Of Minnesota TC

2 INTRODUCTION What are proteins? Several macromolecules like proteins, polysaccharides, lipids and nucleic acids are the important constituents of biological organisms. Amongst these, protein is the most important molecule in the class of biological macromolecules. These complex macromolecules (polypeptides) that play important roles as single molecules (drugs, enzymes), cellular constituents (membrane and cellular organelle components) as well as in tissues such as extracellular matrix components most notably collagenous tissues. Proteins are essentially polymers of amino acids that are linked to each other through amide bonds. There are a total of 20 amino acids and all proteins are chains the different probabilistic combinations. Different structures of proteins Proteins have complex structures which are important to their function. This structure has numerous levels including primary, secondary, tertiary and quaternary based on the amount of cross-linking involved. The different structures of proteins are: Primary structure: This is the simplest structure with minimal cross linking and consists of linear chains of amino acids referred to as polypeptides. The primary structure is associated with the covalent bonds between the atoms making up the protein molecule. Fig.1. Tertiary structure of protein Secondary structure: When two or more polypeptide chains are linked primarily through hydrogen bonds between atoms, it results in secondary structure. Structures such as alpha helix and beta sheet are secondary structures. A secondary structure may also involve disulfide bonding in some cases. Tertiary structure: These structures are formed when secondary structures fold to form a three-dimensional complex structure primarily through hydrophobic interactions, but hydrogen bonds, ionic interactions, and disulfide bonds are also involved. This is the ultimate 3D folded structure of a whole globular protein and is important to protein function (see Fig. 1).

3 Quaternary structure: This usually involves the conformational fitting of two proteins together associated with specific function. In addition to these levels of structure, proteins may shift between several similar structures in performing their biological function. These structures are usually referred to as "conformations," and transitions between them are called conformational changes. Protein Folding The process by which the higher structures are formed is called protein folding and is a consequence of the primary structure. The mechanism of protein folding is not entirely understood. Although any unique polypeptide may have more than one stable folded conformation, each conformation has its own biological activity and only one conformation is considered to be the active one. Function is associated with the native or higher structured state of the protein as shown for a simple idealized case in Fig. 2. When this structure is interrupted, the protein is unable to carry out its specific function. This may involve either partially or totally unraveling of the protein as shown in the figure, or re-organizing the hydrogen bonding which gives the protein its native higher level of structure. This process is called denaturation. It can be either partial or total, and it can also be reversible or irreversible. It does not involve breaking the individual covalent bonds between the atoms of the protein molecule. Also the structure of protein is dependent on the temperature and the protein denatures when exposed to higher temperature. This is again due to change in the protein structure from a native (folded) to a denatured (unfolded) state during heating. Increase in Temperature Fig.2. Denaturation or Unfolding of protein [2] Protein stability is extremely critical to the outcome of all thermally mediated applications to biomaterials such as thermal therapies and burn injury. It is therefore imperative to understand as much as possible about how a protein looses stability and to what extent we can control this through the thermal environment as well as through chemical or mechanical modification. For a review on protein, stability refer to Bischof et. al [2].

4 MONTE CARLO APPROACH TO PROTEIN FOLDING Monte Carlo simulation is commonly used to compute several pathways in understanding thermodynamic mechanisms. Denaturation of protein or unfolding of proteins can be viewed analogously as a phase change problem from the thermodynamic point of view. A simulation run is a series of random steps in conformation space, each perturbing some degrees of freedom of the molecule. A step is accepted with a probability that depends on the change in value of an energy function. Fig.3. 2D Lattice structure of protein In this project, the metropolis algorithm was used in the Monte Carlo simulations. All the simulations were performed in MATLAB v7.0. Proteins are assumed to be two dimensional structures in a lattice. The amino acids occupy the lattice points and the covalent amide bonds the lattice edge (see Fig. 3). Each run at a particular temperature consisted of 50,000 steps. In the first step a linear chain was assumed. In the subsequent steps, structure was chosen by picking a particular link randomly and giving it a rotation (clockwise or anticlockwise) again randomly. The new structure was accepted after checking it for steric hindrance. Assuming the energy of the new and old structures as E 1 and E 2 respectively, the probability condition parameter R is given by ( E2 E1) R = exp( ) (1) k T B where k B and T are the Boltzmann s constant and the temperature respectively. The new step was accepted for folding if R was greater than 1. If R was less than 1, the step was accepted only if a random number, generated from a uniform distribution between the interval 0 and 1, was less than R. The energy at each step is calculated based on interaction between the nearest neighbors that are not covalently linked. The energy of the structure was the sum of all the interactions. The interactions between any two amino acids were randomly assigned a value between -4 and -2 from a uniform distribution. In these simulations, the energy is given in Boltzmann s constant units and the temperature is dimensionless. Therefore, caution must be taken while interpreting results in absolute

5 scale. Energy at each temperature is calculated as an average over a period of 50,000 MC steps. While it is important to understand the stability of proteins with temperature, it is also important to control the denaturation process using chemical or mechanical modification if heating is to be used in several therapies. To include the effect of mechanical or chemical modification in folding of proteins, MC simulations were carried out in the presence of some boundary conditions. In the current case, the folding of proteins was restricted to a box of 16 units (in the x-axis) and 6 units (in the y-axis). For a sense of dimension of the applied boundary, it should be remembered that the minimum distance between any two amino acids that are covalently linked is 1 unit. RESULTS AND DISCUSSION Any thermodynamic system without any constraint always has the drive to reach the minimum energy state. As expected, in all the runs the protein structure reached a stable low energy state at all temperatures, however it varied with temperature. The amount of folding increased with the number of MC steps as shown in Fig. 4. As the purpose of this simulation was to study the mechanism of denaturation using Monte Carlo simulations, MC steps were performed at each temperature and it this thus assumed that at the average energy would represent the favorable energy state at that particular temperature. Figure 5 shows the variation of energy over the 50,000 MC steps at T = a b c d Fig. 4. Folding of protein with increasing MC steps (a to d)

6 . Fig. 4. Energy versus Monte Carlo steps at T = It was observed that higher temperatures favored a higher energy state. This satisfies with the fact that denaturation of proteins occurs at higher temperatures. Denatured proteins can be termed as protein structures with high energy state due to absence of hydrogen bonds and other Van der Waal s forces as compared to the native or folded state. With increase in temperatures, these bonds that preserve the folded structure of protein are broken resulting in an unfolded high energy structure (see Fig. 6a for a kinetic model). The temperature where this unfolding takes place is referred to as the denaturation temperature and it has been reported to be over a range depending on the protein of interest. For instance, the reported denaturation temperature for collagen is about o C [1]. Energy -20 Temperature without boundary constraint -50 Fig. 6. a) Kinetic model for denaturation of protein b) Denaturation S-shaped curve

7 The trend of the variation of the energy of preferred protein structure, i.e. the average energy over MC steps, was similar to some experiment results. The energy of the protein structure increased gradually from T=1.7 to T=5.7 following an S-shaped curve and varied very little outside this temperature regime (shown in Fig. 6b). This behavior has also been observed in the experimental results from the FTIR (Fourier Transformed Infrared Spectroscopy) as shown in Fig. 7a. The FTIR is useful in determining the molecular structure of protein. The area between wave numbers cm -1 is referred to as amide-iii band and the increase in area suggests increase in β structures or the unfolded structure. A reduced area is reflective of an increased folded structure. a b Fig. 7. Comparison of MC simulation results with experimental results The control in Fig. 7a refers to protein without any cross-linking. It is observed from the FTIR studies that protein denaturation is delayed due to cross-linking. It has also been shown that mechanical stress during heating can delay denaturation and the effect of cross-linking can be compared to the effect of chemical modification or cross linking [3, 4]. The red and green circles in Fig. 7a are the denaturation curve on proteins with crosslinking. Having stated that, the purpose of Fig. 7a is therefore to show that application of mechanical stress will also produce similar results. Figure 7b represents the results from MC simulation and they seem to follow a similar trend. The plus in Fig. 7b are MC simulations that were performed without any boundary constraint and the red circles with a boundary constraint. It can be seen that application of a boundary constraint results in a similar shift of S-curve towards the right which might represent increase in denaturation temperature as expected from the experimental data. This can be imagined as a condition of application of mechanical stretch to a protein during heating which would increase its denaturation temperature. As seen in the Fig. 7b, application of boundary conditions also increases the vibration between energy states and variation is no longer a smooth curve as seen in the case of no boundary constraints.

8 REFERENCE: 1. Aksan, A. and J.J. McGrath, Thermomechanical Analysis of Soft-tissue Thermotherapy. Journal of Biomechanical Engineering, : p Bischof, J.C. and X. He, Thermal stability of proteins. Annals of the New York Academy of Sciences, : p Chen, S.S. and J.D. Humphrey, Heat-induced changes in the mechanics of a collagenous tissue: pseudoelastic behavior at 37 degrees C. Journal of Biomechanics, (3): p Chen, S.S., N.T. Wright, and J.D. Humphrey, Heat induced changes in mechanics of a collagenous tissue: Isothermal isotonic-shrinkage. ASME Journal of Biomechanical Engineering, : p

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