The objectives of the thesis: The aim of the studies is to gain fundamental understanding of the role of hydrophobic and polar sequence on folding mechanism of proteins and aggregation of peptides
MINISTRY OF EDUCATION VIETNAM ACADEMY AND TRAINING OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY SCIENCE AND TECHNOLOGY ——————— NGUYEN BA HUNG THE ROLE OF HYDROPHOBIC AND POLAR SEQUENCE ON FOLDING MECHANISMS OF PROTEINS AND AGGREGATION OF PEPTIDES Major: Theoretical and computational physics Code: 44 01 03 SUMMARY OF PHYSICS DOCTORAL THESIS HANOI − 2018 INTRODUCTION The problem of protein folding has always been of prime concern in molecular biology Under normal physiological conditions, most proteins acquire well defined compact three dimensional shapes, known as the native conformations, at which they are biologically active When proteins are unfolding or misfolding, they not only lose their inherent biological activity but they can also aggregate into insoluble fibrils structures called amyloids which are known to be involved in many degenerative diseases like Alzheimer’s disease, Parkinson’s disease, type diabetes, cerebral palsy, mad cow disease etc Thus, determining the folded structure and clarifying the mechanism of folding of the protein plays an important role in our understanding of the living organism as well as the human health Protein aggregation and amyloid formation have also been studied extensively in recent years Studies have led to the hypothesis that amyloid is the general state of all proteins and is the fundamental state of the system when proteins can form intermolecular interactions Thus, the tendency for aggregation and formation amyloid persists for all proteins and is a trend towards competition with protein folding However, experiments have also shown that possibility of aggregation and aggregation rates depend on solvent conditions and on the amino acid sequence of proteins Some studies have shown that small amino acid sequences in the protein chain may have a significant effect on the aggregation ability As a result, knowledge about the link between amino acid sequence and possibility of aggregation is essential for understanding amyloid-related diseases as well as finding a way to treat them Although all-atom simulations are now widely used molecular biology, the application of these methods in the study of protein folding problem is not feasible due to the limits of computer speed A suitable approach to the protein folding problem is to use simple theoretical models There are quite a number of models with different ideas and levels of simplicity, but most notably the Go model and the HP network model and tube model Considerations of tubular polymer suggest that tubular symmetry is a fundamental feature of protein molecules which forms the secondary structures of proteins (α and β) Base on this idea, the tube model for the protein was developed by Hoang and Maritan’s team and proposed in 2004 The results of the tube model suggest that this is a simple model and can describes well many of the basic features of protein The tube model is also the only current model that can simultaneously be used for the study of both folding and aggregation processes In this thesis, we use a tube model to study the role of hydrophobic and polar sequence on folding mechanism of proteins and aggregation of peptides Spatial fill of the tubular polymer and hydrogen bonds in the model play the role of background interactions and are independent of the amino acid sequence The amino acid sequence we consider in the simplified model consists of two types of amino acids, hydrophobic (H) and polar (P) To study the effect of HP sequence on the folding process, we will compare the folding properties of the tube model using the hydrophobic interaction (HP tube model) with tube model using the pairing interaction which is similar to the Go model (Go tube model) This comparison helps to clarify the role of non-native interactions in non-native interactions To study the role of the HP sequence on aggregation of protein, we will compare the possibility of aggregation of peptide sequences with different HP sequences including the consideration of the shape of the aggregation structures and the properties of aggregation transition phase In addition, in the study of protein aggregation, we propose an improved model for hydrophobic interaction in the tube model by taking into account the orientation of the side chains of hydrophobic amino acids Our research shows that this improved model allows for obtaining highly ordered, long-chain aggregation structures like amyloid fibrils The objectives of the thesis: The aim of the studies is to gain fundamental understanding of the role of hydrophobic and polar sequence on folding mechanism of proteins and aggregation of peptides The main contents of the thesis: The general understanding of protein and protein folding, protein aggregation is introduced in chapters 1, of this thesis Chapter presents the methods used to simulate and analyze the data The obtained results of role of HP sequence for protein folding are presented in chapter The results of role of HP sequence for protein aggregation are presented in chapter Chapter Protein folding 1.1 Structural properties of proteins Proteins are macromolecules that are synthesized in the cell and responsible for the most basic and important aspects of life Proteins are polymers (polypeptides) formed from sequences of 20 diffirent types of amino acids, the monomers of the polymer The amino acids in the protein differ only in their side chains and are linked together through peptide bonds that form a linear sequence in a particular order Under normal physiological conditions, most proteins acquire well defined compact three dimensional shapes, knows as the native conformations, at which they are biologically active The amino acid sequence in the protein determines the structure and function of the protein Proteins has four types of structure Primary structure: It is just the chemical sequence of amino acids along the backbone of the protein These amino acid in chain linked together by peptide bonds Secondary structure is the spatial arrangement of amino acids There are two such types of structures: the α-helices and the β-sheets This kind of structure which maximize the number of hydrogen bonds (H-bonds) between the CO and the NH groups of the backbone Tertiary structure: A compact packing of the secondary structures comprises tertiary structures Usually, theses are the full three dimensional structures of proteins Tertiary structures of large proteins are usually composed of several domains Quaternary structure: Some proteins are composed of more than one polypeptide chain The polypeptide chains may have identical or different amino acid sequences depending on the protein Each peptide is called a subunit and has its own tertiary structure The spatial arrangement of these subunits in the protein is called quaternary structure There are a number of semi-empirical interactions that are introduced by chemists and physicists to describe interactions in proteins: disulfide bridges, Coulomb interactions, Hydrogen bonds, Van der Waals interactions, Hydrophobic interactions 1.2 Protein folding phenomenon Once translated by a ribosome, each polypeptide folds into its characteristic three-dimensional structure from a random coil Since the fold is maintained by a network of interactions between amino acids in the polypeptide, the native state of the protein chain is determined by the amino acid sequence (hypothesis of thermodynamics) 1.3 Paradox of Levinthal Levinthal paradox which addresses the question: how can proteins possibly find their native state if the number of possible conformations of a polypeptide chain is astronomically large? 1.4 Folding funnel Based on theoretical and empirical research findings, Onuchic and his colleagues have come up with the idea of the folding funnel as depicted in Figure 1.1 The folding process of the protein in the funnel is the simultaneous reduction of both energy and entropy As the protein begins to fold, the free energy decreases and the number of configurations decreases (characterized by reduced well width) entropy g energy folding N Figure 1.1: The diagram sketches of funnel describes the protein folding energy lanscape Figure 1.2: Free energy lanscape in the two-state model In this model, ∆F is the diference between the free energy of the folded and unfolded states ∆FN and , ∆FD , ∆F are the height of barrier from the unfolded and folded states and free energy difference between the N and U states , respectively In the canonical depiction of the folding funnel, the depth of the well represents the energetic stabilization of the native state versus the denatured state, and the width of the well represents the conformational entropy of the system The surface outside the well is shown as relatively flat to represent the heterogeneity of the random coil state 1.5 The minimum frustration principle The minimum frustration principle was introduced in 1989 by Bryngelson and Wolynes based on spin glass theory This principle holds that the amino acid sequence of proteins in nature is optimized through natural selection so that the frustrated caused by interaction in the natural state is minimal 1.6 Two-state model for protein folding Experimental observations suggest that the two-state model is a common mechanism used to characterize folding dynamics of the majority of small, globuar proteins In a two-state model of protein folding, the single domain protein can occupy only one of two states: the unfolded state (U) or the folded state (N) The free energy diagram for two-state model is characterized by a large barrier separating the folded state and the unfolded state corresponding minima of the free energy of a reaction coordinate The free energy difference between the N and U states (∆F ) characterize the degree of stability of the folding state called folding free energy Rates of folding kf and unfolding ku obey the law Vant Hoff5 Arrhennius: kf,u = ν0 exp − ∆FN,D kB T (1.1) For ν0 is constant, T is the temperature and kB is the Boltzmann constant The change of such as temperature, pressure, and concentration may affect on the ∆F 1.7 Cooperativity of protein folding Cooperativity is a phenomenon displayed by systems involving identical or near-identical elements, which act dependently of each other The folding of proteins is cooperative process In the protein, cooperativity is applied to the twostate process and is understood as the sharpness of thermodynamic transitions In practice, cooperativity is determined by the parameter measured by the ratio between the enthalpy van’t Hoff and the thermal enthalpy κ2 = ∆HvH /∆Hcal (1.2) High cooperativity means that the system satisfies the two-state standard and κ2 is closer to 1, the higher the co-operation and vice versa 1.8 Hydrophobic interaction The hydrophobic effect is the observed tendency of nonpolar substances (such as oil, fat) to aggregate in an aqueous solution and exclude water molecule The tendency of nonpolar molecules in a polar solvent (usually water) to interact with one another is called the hydrophobic effect In the case of protein folding, the hydrophobic effect is important to understanding the structure of proteins The hydrophobic effect is considered to be the major driving force for the folding of globular proteins It results in the burial of the hydrophobic residues in the core of the protein 1.9 HP lattice model In the HP lattice model, there are two types of amino acids with respect to their hydrophobicity: polar (P), which tend to be exposed to the solvent on the protein surface, and hydrophobic (H), which tend to be buried inside the globule protein The folding of the protein is defined as a random step in a 2D or 3D network Using this model, Dill had design some HP sequence that the minimal energy state in the tight packet configurations was unique The phase transition of the sequences is designed to be well cooperative Research shows that aggregate due to hydrophobic interaction is the main driving force for folding 1.10 Go model The Go model ignores the specificity of amino acid sequences in the protein chain and interaction potential is build based on the structure of the folded state The basis of the Go model is the maximum consistent principle of protein interactions in the folded state The results of the study show that the Go model for the folding mechanism is quite good with the experiment, especially in determining the contribution of amino acid positions in the polypeptide chain to the transition state during protein folding Because the model is based on a native state structure, the Go model can not predict the protein structure from the amino acid sequence that is only used to study the folding process of a known structure 1.11 Tube model Considerations of symmetry and geometry lead to a description of the protein backbone as a thick polymer or a tube At low temperatures, a homopolymer model as a short tube exhibits two conventional phases: a swollen essentially featureless phase and and a conventional compact phase, along with a novel marginally compact phase in between with relatively few optimal structures made up of α-helices and β-sheets The tube model predicts the existence of a fixed menu of folds determined by geometry, clarifies the role of the amino acid sequence in selecting the native-state structure from this menu, and explains the propensity for amyloid formation Chapter Amyloid Formation 2.1 The structure of amyloid fibril (a) (b) Figure 2.1: 3D structure of the Alzheimer’s amyloid-β (1-42)fibrils has a PDB code of 2BEG (a) view along the direction of fibril axis (b) view perpendicular to the direction of fibril axis Amyloid fibrils possess a cross-β structure, in which β-strands are oriented perpendicularly to the fibril axis and are assembled into β-sheets that run the length of the fibrils (Figure 2.1) They generally comprise 24 protofilaments, that often twist around each other Repeated interactions between hydrophobic and polar groups run along the fibril axis 2.2 Mechanism of amyloid aggregation The formation of amyloid can be considered to involve at least three steps and are generally referred to as lag phase, growth phase (or elongation) phase and an equilibration phase Seeding involves the addition of a preformed fibrils to a monomer solution thus increasing the rate of conversion to amyloid fibrils Addition of seeds decreases the lag phase by eliminating the slow nucleation phase Chapter Methods and Models for simulations 3.1 HP tube model The backbone of the protein is models as a string of Cα atoms separated by an interval of 3.8˚ A, forming a flexible tube of 2.5˚ A also has a constraint with both the tube’s three radii (local and non-local) Potential objects describing this condition are given in figure 3.1) Vtube (i, j, k) = ∞ if Rijk < ∆ if Rijk ≥ ∆ ∀ i, j, k (3.1) The bending potential in the tube model is related to the spatial constraints of the polypeptide chain The bending potential at position i given by (Figure 3.1) ∞ Vbend (i) = eR 0 if Ri−1,i,i+1 < ∆ if ∆ ≤ Ri−1,i,i+1 < 3.2 ˚ A if Ri−1,i,i+1 ≥ 3.2 ˚ A (3.2) eR = 0.3 > and the unit corresponds to the energy of a local hydrogen bond In the tube model, local hydrogen bonds are made up of atoms i and i+3 and assigned to energy equal to − Non-local hydrogen bonds are formed between the atoms i and j > i + and have the energy of −0.7 The energy and geometric constraints of a local hydrogen bond between the atom i and the atom j are defined as follows: j =i+3 ehbond = − A ≤ rij ≤ 5.6 ˚ A 4.7 ˚ |bi · bj | > 0.8 |bj · cij | > 0.94 |bi · cij | > 0.94 (r i,i+1 × ri+1,i+2 ) · ri+2,i+3 > The same for a non-local hydrogen bond: (3.3) εHH=-0.70 εHH=-0.50 20 εHH=-0.30 εHH=-0.21 εHH=-0.20 εHH=-0.19 (b) Rg (units of A0) 18 (c) 16 14 12 10 (a) (d) 0.2 (e) Figure 4.9: Ground state conformations obtained by the simulations for 3HB protein with varying hydrophobic interaction intensities The display structure corresponds to eHH = −0.2 (a), eHH = −0.21 (b), eHH = −0.3 (c), eHH = −0.5 (d), eHH = −0.7 (e) 0.3 0.4 0.5 T (units of ε/kB) 0.6 0.7 Figure 4.10: Temperature dependence of the specific heat of 3HP protein in the HP tube model with different hydrophobic interaction intensities eHH = −0.2 , −0.3 , −0.5 v −0.7 the phase transition type From eHH = −0.3 to eHH = −0.7 graph has a small shoulder, it expands when eHH increases At the values |eHH | < 0.3 epsilon the shoulder does not exist or very small to be recognized on the graph 4.11 depicts the dependence of the average energy E and the radius of gyration Rg on the temperature Average energy changes at the folding transition temperature Tf When |eHH | > 0.2 , then the change of Rg by the temperature is monotonous The change of Rg by temperature occurs more slowly and the inflection point of the graph occurs at higher temperatures as |eHH | increases This proves that as |eHH | increases, the collapse phase occurs at higher temperatures For |eHH | ≤ 0.2 , the radius of the radius depends on temperature in the form of non-monotonous: at low temperature Rg has a large value corresponding to the basic state is single-α; as the temperature rises, the single helix becomes unstable due to thermal oscillations and therefore Rg decreases; As temperatures continue to rise, the hydrogen bonds break down and the protein configuration is folded in size increasing lead the Rg increase The cooperativity depend on the hydrophobic force intensity is determined by the ratio between the enthalpy van’t Hoff and the thermal enthalpy κ2 = ∆H vH /∆Hcal The value κ2 equal to 0, 5975 ± 0, 0166; 0, 6181 ± 0, 0116; 0, 7267 ± 0, 0206; 0, 7475 ± 0, 0256 for HH = 0, 2; 0, 3; 0, 5; 0, The results show that when the hydrophobic interaction is stronger, the cooperation also becomes stronger show by the increasing of the value of κ2 18 (units of ε) 20 -20 -40 (a) -80 -100 0.2 0.3 22 0.4 0.5 eHH=-0.19 eHH=-0.20 20 (Angstroms) eHH=-0.19 eHH=-0.20 eHH=-0.21 eHH=-0.30 eHH=-0.50 eHH=-0.70 -60 0.6 0.7 eHH=-0.21 eHH=-0.30 0.8 0.9 eHH=-0.50 eHH=-0.70 18 16 (b) 14 12 10 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 T (units of ε/k B) Figure 4.11: Temperature dependence of the average energy E (a), the averaged radius of gyration, Rg (b) of 3HP protein in the HP tube model with different hydrophobic interaction intensities eHH = −0.2 , −0.3 , −0.5 v −0.7 19 Chapter the role of hydrophobic and polar sequence on aggregation of peptides This chapter studies the aggregation of the short peptide in the tube model with correlated side chain orientations We study the role of the HP sequence on protein aggregation and formation of amyloid fibrils We consider 12 HP sequences of length N = as given in table 5.1 with number of peptide in each systems changing from m = to m = 20 The sequences, denoted as S1 through S12, are selected in such a way that they contain only or hydrophobic (H) residues, corresponding to hydrophobic fraction of 25% and 37.5%, respectively Figure 5.1 shows that the lowest energy conformation obtained in the simulations,supposed to be the ground state of a given system, strongly depends on the sequence 5.1 Sequence dependence of aggregate structures Fig 5.1 shows that the lowest energy conformation obtained in the simulations Two sequences, S2 and S11, form a double layer β-sheet structure with characteristics similar to that of a cross-β structure A similar structure but less fibril-like is also found for sequence S12 with some parts that are non-β-sheet Both sequences S3 and S4 form a α-helix bundle The helix bundle of sequence S4 however is more ordered and has an approximate cylinder shape, in which the α-helices are almost parallel to each other The role of hydrophobic residues in aggregation can be figured out from the structures of the aggregates The packing of hydrophobic side chains is best Table 5.1: HP sequences of amino acids of peptides considered in present study (H – hydrophobic, P – polar) The parameter s denotes the minimal sequence separation between two consecutive H amino acids Sequence name S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 Sequence PPPHHPPP PPHPHPPP PPHPPHPP PHPPPHPP PHPPPPHP HPPPPPHP HPPPPPPH PPHHHPPP PPHPHHPP PHPPHHPP PHPHPHPP PHPPHPHP 20 s 1 2 S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 Figure 5.1: Ground state conformations obtained by the simulations for systems of M = 10 peptides for 10 HP sequences (S1–S10) as given in Table 5.1 observed for sequences S2 and S11, for which the hydrophobic residues are aligned within each β-sheet and the hydrophobic side chains from the two β-sheets are facing each other This packing is possible due to the HPH pattern in these sequences which position the hydrophobic side chains on one side of each β-sheet An alignment of hydrophobic residues is also seen for sequence S12 due to the HPH segment of this sequence In the aggregate of sequences S4, which is a helix bundle, the hydrophobic side chains are gathered along the bundle axis, thanks to to the alignment of hydrophobic side chains along one side of each α-helix This alignment is due to the HPPPH pattern in the S4 sequence On the other hand, the S3 sequence with the HPPH pattern also forms a helix but the hydrophobic side chains are not well aligned in the helix, leading to a less ordered aggregate 5.2 Thermodynamics of aggregation We find that the specific heat strongly depends on both the sequence and the system size Fig 5.2 and Fig 5.3 show the temperature dependence of the specific heat per molecule for various system sizes for sequences S2 and S4, respectively For sequence S2, it is shown that as M increases the specific heat’s peak shifts toward higher temperature and its height increases (Fig 5.2) This result indicates that the aggregate becomes increasingly stable and the transition becomes more cooperative as the system size increases For sequence S4, for which the aggregates are helix bundles, the height of the main peak increases with M but the position of the peak varies non-monotonically (Fig 5.3) Note that the aggregation transition for sequences S4 is always found at a slightly lower temperature than the folding transition of individual chain This is in contrast 21 M=1 M=2 C/M (kB) 1000 M=4 M=5 M=1 1000 M=1 M=2 M=3 M=4 M=5 M=6 M=8 M=10 S2 M=6 M=2 M=6 M=1 M=2 M=4 M=6 M=10 S4 100 100 10 M=8 10 T* T* 0.14 M=4 C/M (kB) 10000 M=3 M=10 0.16 0.18 0.2 0.22 0.24 0.26 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 T (ε/kB) T (ε/kB) M=10 Figure 5.2: Temperature dependence of the specific Figure 5.3: Same as Fig 5.2 but for sequence S4 heat C per molecule for sequence S2 systems with the systems at mM concentration For clarity, the system number of chains M equal to 1, 2, 3, 4, 5, 6, and 10 sizes shown are fewer than for sequences S2 as indicated The position of a putative physiological temperature, T ∗ , is indicated with sequence S2, whose aggregation transition temperature is always higher than the folding temperature of a single chain In Fig 5.4, the results of the maximum specific heat per molecule, Cpeak /M , and the temperature of the peak, Tpeak , are combined for all sequences considered and for several values of M It is shown that the variation of both Cpeak /M and Tpeak increases with M Note that for M = 10, the highest specific heat maxima correspond to sequences S2 and S11 whose aggregates are fibril-like (see Fig 5.1) For sequences S2 and S11, Cpeak /M is not only the highest among all sequences but also increases with M much faster than other sequences Our results indicates that the propensity of forming fibril-like aggregates is associated with the cooperativity of the aggregation transition The wide variation in the transition temperatures Tpeak among sequences suggests another interesting aspect of aggregation Suppose that we consider the systems at the physiological temperature, T ∗ In our model, a rough estimate of T ∗ could be 0.2 /kB , which corresponds to a local hydrogen bond energy of kB T ∗ For M = 1, one finds that all sequences but S10 has Tpeak < T ∗ suggesting that the peptides are substantially unstructured at T ∗ as a single chain For M = and M = 10, only three sequences, S3, S4 and S5, have Tpeak < T ∗ , while the other have Tpeak > T ∗ Thus, sequences S3, S4 and S5 not aggregate at T ∗ while other sequences This result indicates that the variation of aggregation transition temperatures among sequences is also a reason why protein sequences behave differently towards aggregation at the physiological temperature Some se22 quences not aggregate because aggregation is thermodynamically unfavorable at this temperature Note that the ability of forming fibril-like aggregates is not necessarily associated with a high aggregation transition temperature In fact, Fig 5.4b shows that sequences S2 and S11 have only a medium value of Tpeak among all sequences, for both M = and M = 10 Some sequences with a higher Tpeak , such as S8, S9 and S10, form disordered aggregates (a) Cpeak/M (units of kB) 2500 M=10 M=6 M=1 2000 1500 1000 500 0 (b) 10 11 12 E (units of ε) Tpeak (units of ε/kB) 0.4 0.36 0.32 0.28 0.24 -5 -10 S2, M=4 -15 T=0.2 -20 T* 0.2 -25 0.16 10 11 12 100 200 300 400 500 600 MC steps (x106) sequence # Figure 5.4: Dependence of the maximum of the spe- Figure 5.5: Energy as function of Monte Carlo steps in cific heat Cpeak per molecule (a) and its temperature a trajectory at T = 0.2 for the sequence S2 system with Tpeak (b) on the sequence for systems of M = 10 (solid), M = at mM concentration The conformation shown M = (dashed) and M = (dotted) peptides at mM is a metastable state with a 3-peptide β-sheet in contact concentration The horizontal line in (b) indicates a pu- with a disordered helix formed by the 4th peptide tative physiological temperature T ∗ Fig 5.2 shows that for sequence S2, systems of M ≤ have the specific heat peaked at a lower temperature than T ∗ = 0.2 /kB , which means that these systems not aggregate at T ∗ Only for M > 4, the specific heat peak temperature is higher than T ∗ indicating that the fibril-like aggregates formed by this sequence are stable at T ∗ Thus, a sufficient number of peptides is needed for the aggregation to happen at a given temperature We also find that the lower peak in the specific heat of the system of M = (Fig 5.2) corresponds to a transition from metastable aggregates at intermediate temperature to the ground state at low temperature Fig 5.5 shows the trajectory of an equilibrium simulation at T = 0.2 /kB for sequences S2 with M = The time dependence of the system’s energy in this trajectory indicates that the peptides not aggregate most of the time, so that the energy is relatively high, but for some short periods they can spontaneously 23 form a metastable aggregate of a much lower energy This metastable aggregate has a three-stranded β-sheet (Fig 5.5, inset) and could act as a template for fibril growth in systems of more peptides 5.3 Kinetics of fibril formation First, we consider a system of M = 10 peptides with concentration c = mM under equilibrium condition Fig 5.6 shows the dependence of the total free energy of the system on the size of the largest aggregate, m, formed at three temperatures slightly below Tpeak including T = T ∗ = 0.2 /kB It is shown that for all these temperatures the free energy has a maximum at m = 3, suggesting that m = could be the size of the critical nucleus for fibril formation The free energy barrier for aggregation in Fig 5.6 is found to increase with T and is about of kB T to kB T This barrier is not large and is consistent with the fact that the sequence considered is highly aggregation-prone For m > 3, Fig 5.6 shows that the free energy decreases almost linearly with n, which is consistent with the fact that the growth of the aggregate in size is essentially one-dimensional We then considered a larger system of M = 20 peptides and studied the time evolutions from random configurations of dispersed monomers Up to 100 independent trajectories are carried out to determine the statistics We first consider the system at concentration c = mM and T = 0.2 /kB Fig.5.7 (a and b) shows three typical trajectories with the total energy E and the size of the largest aggregate m as functions of time Interestingly, these trajectories show clear evidence of an initial lag time, during which m fluctuates but remains small (m ≤ 3) before a rapid and almost monotonic growth (Fig 5.7 b) They also shows that nucleation is complete for m = A peptide configuration at a nucleation event is shown on Fig 5.7d indicating that a possible nucleus is a three-stranded β-sheet formed by three peptides (Fig 5.7e) Fig 5.7c shows that the system can form multiple aggregates of various sizes The distribution of the aggregate size obtained after a sufficient long time is bimodal reflecting the fact that the system size is finite and clusters of less than peptides are unstable Thus, one either observes one large cluster with size close to the system size or several smaller clusters The largest aggregates of m = 20 peptides have the form of an elongated double β-sheet strongly resemble a cross-β-structure (Fig 5.7f) It is shown in Fig 5.8 (a and b) that for T = 0.2 /kB , the time dependence of nβ can be fitted well to the exponential relaxation function of M (1 − e−t/t0 ), where t0 is the characteristic time of aggregation This time dependence also depends strongly on the concentration c with t0 increases more than times by 24 S2, M=10 F (units of ε) -1 -2 m* -3 -4 -5 T=0.20 T=0.21 T=0.22 -6 -7 10 m Figure 5.6: Dependence of total free energy, F , on the size of the largest aggregate, m, for the sequence S2 system of M = 10 peptides at mM concentration and at three different temperatures, T = 0.2, 0.21 and 0.22 /kB , as indicated A barrier with the maximum located at m = is indicated (b) (c) 20 T=0.2 T=0.2 mM mM m 15 10 nucleation 500 1000 1500 0 500 1000 10 number of aggregates 20 -20 -40 -60 -80 -100 -120 -140 -160 E (units of ε) (a) 1500 t (×10 MC steps) (d) t=1.5×10 6 t (×10 MC steps) T=0.2 mM 10 15 20 aggregate size (f) (e) Figure 5.7: Kinetics of fibril formation for sequence S2 with M = 20 peptides at concentration mM and temperature T = 0.2 /kB (a) Dependence of the energy, E, on time, t, measured in MC steps for three different trajectories (b) Time dependence of the maximum aggregate size m for the same three trajectories as shown in (a) Arrows indicate nucleation event for each trajectory (c) Histogram of the aggregate size given by the number of peptides obtained at a large time of t = 1.5 × 109 MC steps (d) Snapshot of peptide configuration at a nucleation event (e) Conformation of the nucleated cluster formed by three peptides taken from the configuration shown in (d) (f) Conformation of an elongated fibril-like structure formed by 20 peptides 25 changing c from mM to 0.5 mM There seems to be no evidence of a lag phase at T = 0.2 /kB as nβ increases linearly with t for small t (Fig 5.8b) This lack of evidence, however,may be due to the fact that the deviation from the exponential growth is too small to be observed Indeed, we find that if the temperature is increased a little to T = 0.21 /kB , the lag phase can be observed Fig.5.8c shows that the growth of nβ in time is significantly deviated from the exponential relaxation function at small time This growth when plotted in a log-log scale (Fig 5.8c) shows that at small time nβ ∝ tα with α ≈ 1.25 The exponent α > indicates that the time dependence of nβ behaves like a convex function, which proves the existence of the lag phase at small time The stronger evidence of the lag phase at T = 0.21 /kB compared to that at T = 0.2 /kB is consistent with the higher free energy barrier for nucleation at the former temperature previously shown in Fig 5.6 5.4 Aggregation of mixed sequences Finally, we study the aggregation for a binary mixture of two sequences, S2 and S4 It was shown that in homogeneous systems, the first sequence is strongly fibril-prone, whereas the second one forms only α-helices Furthermore, the sequence S4 has the aggregation transition temperature lower than T ∗ , so the its aggregate is not stable at T ∗ Strikingly, our simulations at T ∗ show that in a binary system of equally 10 chains of each sequence, after a sufficiently long time, a fraction of the S4 chains aggregate and convert into β-sheet conformation on an existing aggregate formed by the S2 chains (see Fig 5.9) Though this fraction is only about 10% on average, this observation shows that the template-based mechanism for fibril formation can be effective for polypeptides of very different natures Here, the fibril-like aggregate formed by the aggregation-prone peptides acts as the template for the aggregation of non-aggregation-prone peptides Note that due to the mismatch of different hydrophobic patterns in the two sequences, the aggregates formed by the two sequences are more disordered than the homogeneous ones (Fig 5.9b) It is also shown in Fig.5.9c that the growth of this mixed aggregate at the given temperature remains exponential but the characteristic time for aggregation is larger than in corresponding homogeneous system of sequence S2 26 (a) (b) 20 T=0.2 T=0.2 10 mM 10 0.5 mM mM 15 0.5 mM 0.25 mM 0.25 mM 0 500 1000 1500 100 t (×106 MC steps) (c) T=0.21 12 10 T=0.21 15 total 10 mM (c) (d) 20 10 mM 0.5 mM 1000 t (×106 MC steps) (a) (b) 0.5 mM 1000 2000 t (×10 MC steps) 3000 100 1000 sequence S4 t (×10 MC steps) 500 1000 1500 2000 t (x106 MC steps) Figure 5.8: Time dependence of the average number of peptides in β-sheet conformation, nβ , in the aggregation of sequence S2 with M = 20 The system is considered at temperatures T = 0.2 /kB (a,b) and 0.21 /kB (c,d) and at several concentrations, c = mM (squares), 0.5 mM (circles) and 0.25 mM (triangles), as indicated The average of nβ for each concentration is taken over 100 independent trajectories Right figures (b and d) plot the same data as in the left figures (a and c), respectively, except that in loglog scale Data points are fitted to an exponential relaxation function of M (1 − e−t/t0 ) for c = mM (solid) with t0 = 570 × 106 for c = mM in (a) and t0 = 1850 × 106 for c = 0.5 mM in (a), and t0 = 109 for c = mM in (c) The log-log plots shows that the growth of nβ at small times follows a power law, nβ ∝ tα , with α = in (b) and α = 1.25 in (d) for both concentrations of mM and 0.5 mM 5.5 Figure 5.9: (a) Snapshot of a conformation obtained in a simulation of the binary mixture of 10 chains of sequence S2 and 10 chains of sequence S4 at concentration c = mM and temperature T = 0.2 /kB H residues are shown in dark green P residues are in light green and pink colors for the S2 and S4 chains, respectively (b) Zoom-in side and top views of the aggregate shown in a Note that six S4 chains are present in the aggregate, and five of them are in the β-sheet configuration (c) Time dependence of the average number of peptides in β-sheet conformation, nβ , obtained from 100 independent simulations, for both sequences together (squares) and for sequences S4 only (circles) A fit to the exponential relaxation function as given in the caption of Fig 5.8 with t0 = 832 × 106 (solid line) is shown for the case of both sequences Discussion Previous study of the tube model has shown that hydrophobic-polar sequence can select protein’s secondary and tertiary structures In particular, the HPPH and HPPPH patterns have been identified as strong α-formers, whereas the HPH pattern is a β-former Strikingly, exactly the same binary patterns have been used in experiments that allow the successful design of de novo proteins In the present study, we find that these simple selection rules still hold for the peptides in aggregates, even though the model has been changed by considering the orientations of side chains The present study shows that the binary pattern also determines the orderness of the aggregate In particular, there should be some compatibility between the alignment of hydrophobic side chains and the overall 27 symmetry of the aggregate Interestingly, the HPH pattern appears to be both a strong β-former and a highly aggregation-prone sequence Our finding is in a full agreement with experimental design of amyloids, which shows that segments of alternating hydrophobic and polar pattern (such as PHPHPHP) can direct protein sequences to form amyloid-like fibrils The role of side-chains in amyloid fibril formation has been stressed in early all-atom simulations of short peptides In the tube model without consideration of side chain orientations, amyloid aggregation may also be obtained However, these aggregation are sometimes disorganized in arranging the β-sheet Here, we show that the correlated orientations of hydrophobic side-chains are important for both the ordered packing of β-strands within a β-sheet and the stacking of β-sheets in the fibril In particular, the alternating hydrophobic polar pattern leads to β-sheets of hydrophobic side chains oriented on one side of the β-sheet This one-sided orientation stabilizes the two-layered β-sheet aggregate, which is the system’s ground state and can grow into a long fibril, as shown for the case of sequence S2 Our thermodynamics calculations show that the formation of fibril-like aggregates is much more cooperative than that of non-fibril-like aggregates This cooperativity was indicated by both the height of the specific heat peak and the increase of the maximum specific heat per molecule with the system size The high cooperativity of fibril formation can be understood as due to the highly ordered nature of fibril structures and the dominating contribution of intermolecular interactions in these structures We also find that thermodynamic stability is not a distinguished feature of fibril-like aggregates In particular, sequences associated with very high aggregation transition temperature not necessarily form fibrillike aggregates The increased overall hydrophobicity of the sequence is shown to enhance the stability of the aggregates without impact on their fibril characteristics Our work shows that the HP pattern is a determinant of both amyloid aggregation and its thermodynamic stability, rather than all hydrophobicity of the sequence The sequence S2 in our study shows the one-step nucleation The impact of the HP sequence on nucleation is also associated with a small nucleation barrier and the rapid nucleation with almost invisible lag phase observed for this sequence For this fibril-prone sequence, it is found that the non-equilibrium behavior of a larger system is consistent with equilibrium properties of smaller systems at the same peptide concentration In particular, the frequent formation and dissolving of the aggregates before nucleation and the growth of the aggregates after nucleation are in accord with their thermodynamic stabilities as isolated systems 28 Interestingly, the small size of the critical nucleus found in our study agrees with those obtained in homopolymer studies as well as in lattice heteropolymer and all-atom simulations of short peptides Our simulation result on the peptide binary mixture is fully consistent with experiment of Ridgley and shows that a cross-β-sheet can be heterogeneous in its peptide composition It is possible that naturally occurring amyloid fibrils can possess this heterogeneity due to the templated self-assembly process 29 Conclusion Our studies in this thesis have led to the following conclusions about the role of hydrophobic and polar sequence on folding mechanism of proteins and aggregation of peptides: The hydrophobic and polar (HP) sequence has a strong influence on the folding mechanism of proteins The folding process of a protein with a defined HP sequence is characterized by two separate transitions: the collapse transition from a swollen state to a compact but disordered state occurring at a higher temperature and the folding transition from the compact disordered state to the folded state occurring at a lower temperature The disordered state is stabilized primarily by hydrophobic interaction The intensity of the hydrophobic interaction has a great affect on the temperature of the collapse transition and the temperature of the folding transition of the protein, but within a fairly wide range the intensity of interaction does not change the structure of the folded state of the protein The folding cooperativity and thermal stability of proteins in the tube model with HP sequence is considerably lower than in the tube Go model This shows that in the free energy landscape have been shaped by the geometric and symmetric elements of the protein, the conflict between the interactions (leading to frustration) persists for a considerable time with specific amino acid The conflict among interactions are almost completely eliminated in the Go tube model, a model with optimized interactions for protein folded structure The aggregate structure of short peptides and the thermodynamic properties of the aggregation transition strongly depend on the HP sequence In the HP sequences, there exist patterns of high aggregation propensities forming structures that are rich in beta-sheet or alpha-helix In addition to the HP sequence, the orientation of hydrophobic side chains also has a great influence on the order and the symmetry of aggregate structures Our simulation showed that the peptides contained sequences with the HPH pattern (two H amino acid separated by one P), the aggregate is a two-layer β-sheet structure similar to that found in amyloid fibrils The highest peak in the specific heat belongs to sequences which forms the double layer β-sheet structure This result suggests that the propensity to form amyloid may be linked to the cooperativity of the aggregation transition The formation of amyloid fibrils follows the nucleation and growth mechanism with the existence of a 30 lag phase The template structure plays an important role in fibrils formation Amyloid fibrils can be formed by a mixture of peptides of non-homogeneous amino acid sequences Our simulations also show another feature of amyloid formation, that is considerably non-specific to a sequence, namely the fibril induced aggregation of a non-aggregation-prone sequence This templating property certainly complicates the problem of amyloid formation as it suggests that the cross-β structure can be heterogeneous in their sequence or peptide composition 31 List of works has been published [1] Nguyen Ba Hung, Trinh Xuan Hoang, “Folding of proteins in presculpted free energy landscape”, Conmunications in Physics 23, 313–320 (2013) [2] Nguyen Ba Hung, Trinh Xuan Hoang, “Aggregation of peptides in the tube model with correlated sidechain orientations”, Journal of Physics: Conference Series 627, 012028 (2015) [3] Nguyen Ba Hung, Duy Manh Le, Trinh X Hoang, “Sequence dependent aggregation of peptides and fibril formation”, Journal of Chemical Physics 147, 105102 (2017) 32 ... conclusions about the role of hydrophobic and polar sequence on folding mechanism of proteins and aggregation of peptides: The hydrophobic and polar (HP) sequence has a strong influence on the. .. clarify the role of non-native interactions in non-native interactions To study the role of the HP sequence on aggregation of protein, we will compare the possibility of aggregation of peptide sequences... for the study of both folding and aggregation processes In this thesis, we use a tube model to study the role of hydrophobic and polar sequence on folding mechanism of proteins and aggregation of