Refolding of homopolymer under quenched force - Maksim Kouza

Vietnam Journal of Science and Technology 55 (6A) (2017) 1-8 REFOLDING OF HOMOPOLYMER UNDER QUENCHED FORCE Maksim Kouza 1, 2, * , Andrzej Kloczkowski 2 , Pham Dang Lan 3 , Mai Suan Li 4, * 1 Faculty of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warsaw, Poland 2 Nationwide Childrens Hospital, Battelle Center for Mathematical Medicine, Columbus, OH 43215 USA 3 Institute for Computational Science and Technology, Tan Chanh Hiep Ward, District 12, Ho Chi Minh City, Viet Nam 4 Institute of Physics, Polish Academy of Science, Al. Lotnikow 32/46 02-668 Warsaw, Poland * Email: mkouza@chem.uw.edu.pl; masli@ifpan.edu.pl Received: 15 July 2017; Accepted for publication: 15 December 2017 ABSTRACT Recently single molecule force spectroscopy has become an useful tool to study protein, DNA and RNA. However, very little attention was paid to homopolymer which plays an important role in many domains of science. In this paper we make the first attempt to decipher the free energy landscape of homopolymer using the external force as reaction coordinate. The impact of the quenched force on the free energy landscape was studied using simplified coarse- grain Go model. Similar to protein, we have obtained a clear switch from the thermal regime to force-driven regime. The distance between the denatured state and transition state in the temperature-driven regime is smaller than in the force-driven one. Having a rugged free energy landscape without a pronounced funnel the homopolymer folding is much slower than that of protein making study of homopolymer very time consuming. Keywords: protein refolding, folding pathways, free energy landscape, distance between denaturate and transition states, Go model. 1. INTRODUCTION Recently great advance in understanding free energy landscape (FEL) of biomolecules has been achieved [1 - 3], but very little was known about FEL of homopolymer which is important not only for basic research but also for application. In single molecule force spectroscopy (SMFS) experiment, which is capable to distinguish the fluctuations of individual molecules from the ensemble average behaviour, the external mechanical force f is used as an additional parameter to probe FEL. At sufficiently low f one can assume that the external force modulate the free energy barriers leaving the distance between the native state (NS) and transition state (TS), , and the distance between the denatured state (DS) and TS, , unchanged (Fig. 1). Because the force applied to chain terminus facilitates unfolding the unfolding barrier is reduced by amount of . In contrast, the folding barrier is elevated by Maksim Kouza, Andrzej Kloczkowski, Pham Dang Lan and Mai Suan Li 2 as the force obstacles refolding. This linear dependence of the barrier shift on , first proposed by Bell [4], leads to the following force dependence of unfolding ( ) and folding time ( ): , (1) here , , and are Boltzmann’s constant, temperature, the unfolding and folding time in the absence of external force, respectively. In this paper we restrict ourselves to the case of refolding under external force. At high forces one has to go beyond the Bell theory by two possible ways. The first one is to apply non-linear theories developed by different groups [5 - 9]. Despite clear improvement of the Bell approximation these new theories are valid for not high enough forces. The second way to deal with arbitrary forces is to apply the Bell formula to force regions with different values of (and ). Because the Bell theory is widely used in interpretation of experimental data, here we follow this way to decipher FEL. The unfolding FEL of proteins have been thoroughly investigated [2, 11 - 12]. The mechanical unfolding of homopolymer was studied [13] but the impact of the force on its FEL has not been addressed. Motivated by experimental works [14 - 18], the refolding of proteins under quenched force has been considered by simulation [19, 20]. In particular, recently we have disclosed [21] that the switch from the thermal to force-driven regime occurs at the critical force separating the folded state from the unfolded one. At low forces folding pathways coincide with those of folding in the absence of the external force. In addition, we have demonstrated that experimentally measured in the middle force regime which is just above the temperature- driven or low force regime. The refolding of homopolymer was not probed by either theory or experiment. Figure 1. Schematic plot for FEL without (blue) and with (red) the external force. Refolding of Homopolymer under Quenched Force 3 The major goal of this paper is to study the refolding FEL of homopolymer using the external force as an useful reaction coordinate. The most complete picture of FEL may be obtained using using all-atom models as it has been done for short peptides [22, 23]. However under external force the refolding time exponentially increases with (Eq. 1) making the problem too hard to deal with even on fast computers [24]. Therefore, we used the coarse- grained model [13, 25] to study refolding of homopolymer. We have shown that, similar to the protein case [21], there exists the switch from the low- force to the high-force regime where changes. The difference in folding pathways was demonstrated through different FELs for these regimes. The value of seems to be small compared to that of protein. 2. MATERIALS AND METHODS In order to investigate the effect of external force on refolding of homopolymer we use one of the previously proposed models [13, 25]. The energy of the model is as follows: (2) The first harmonic term accounts for chain connectivity. The second term represents a non- local interaction energy between homopolymer residues which is given by 12-6 Lennard-Jones potential. The last term accounts for the force applied to C and N termini along the end-to-end vector . We choose , and . The fraction of native conformation is defined as a fraction of conformations that have the root mean square deviation, relative to the native state, less than . 3. RESULTS "Native state" and folding temperature of homopolymer Taking a clue from the previous studies [13. 25] we obtained the "native state" of homopolymer (or the lowest energy state) by performing system annealing at different temperatures. An example of the obtained homopolymer native state for N = 16 is shown in Fig. 2(a). One can show that due to a very rugged FEL homopolymers fold much slower than their protein counter parts. Therefore, we confined our study to the short chain with 16 beads. First, we collected data for 13 temperatures at zero force. We generated 50 trajectories at a given value of temperature. At folding temperature the fraction = 1/2, e.g. half of the trajectories are folded and the another half unfolded (Fig. 2 (b)). Note, that shown in Fig. 2(b) is the temperature at which the folding is fastest in the absence of force. Then, we performed constant force simulations at different values forces at to study the homopolymer refolding process kinetics starting from fully extended conformations. Trajectories were considered to reach the native basin if RMSD relative to the native state is less than the cutoff . The refolding time is defined as an average of the first passage times to reach the native basin. Maksim Kouza, Andrzej Kloczkowski, Pham Dang Lan and Mai Suan Li 4 Figure 2. (a) An example of the homopolymer native state for N = 16. Snapshot was created with VMD software [26]. (b) The temperature dependence of (open circles) defined as the normalized number of RMSD (relative to the native conformation) values within . Figure 3. Dependence of refolding times on the external force for a N = 16 polymer (upper panel). Free energy surfaces obtained at f = 0, 0.3, 0.5 and 0.7 (a, b, c, and d). E2E refers to the end-to-end distance. Results were obtained at . Refolding of Homopolymer under Quenched Force 5 Existence of different force regimes and change in refolding pathways of homopolymer Figure 3 (upper panel) shows the semi-log plot for the force dependence of refolding times at . As in the protein case, we can clearly distinguish different scenarios. The Bell formula works good within the two force intervals [0:0.5] and [0.5:0.7], corresponding distances from the unfolded state to transition states are nm and nm, respectively. Figures 3a-d show the free energy surfaces as a function of the end-to-end distance, R, and RMSD, at different values of force. In the first force regime (at f = 0.0, 0.3 and 0.5) no obvious barriers are seen as the free energy changes continuously between various states. In contrast, in the second force regime (Fig. 3d, f = 0.7) there is a clearly defined barrier located at large values of R and RMSD ( , ) not populated at zero or low values of force. Force-induced extended conformations represents the local minimum at high values of R and RMSD (43Å, 15Å). Thus, the external force, which is greater than , introduces significant conformations changes leading to the force-induced intermediate state. Homopolymer traps in force-induced conformation making its folding pathway more complex compared to zero and low force cases. Comparison with a N=16 hairpin The distance between TS and DS of a N = 16 homopolymer is lower than that of proteins with more than 16 residues [21]. In order to make a more precise comparison we considered a β-hairpin which also has 16 beads. The native structure of β-hairpin was retrieved from the protein data bank with code 2GB1 (Fig. 4a). To study folding kinetics of this peptide we have used the Go model developed by Clementi and Onuchic [27] with the set of parameters given in our earlier work [21]. Figure 4. (a) Native structure of β-hairpin, the C-terminus of protein G (residues 41-56, PDB code 2GB1). (b) Dependence of refolding times on the external force a for β-hairpin. One can show that in the absence of external force the folding is fastest at . As in the homopolymer case our study was conducted at this temperature. The clear crossover from the low to high force regime appears at (Fig. 4). For these two regimes we obtain nm and nm which are substantially higher than those of homopolymer. The difference might come from different criteria for folding. In both cases the folding time is defined as the median first passage time to reach NS. However, NS is very well-defined for Maksim Kouza, Andrzej Kloczkowski, Pham Dang Lan and Mai Suan Li 6 protein (when all native contacts are formed), while for homopolymer we use much broader definition (if RMSD ≤ ). 4. DISCUSSION AND CONCLUSIONS In this manuscript we have explored the influence of the external mechanical force on FEL of homopolymer using several concepts from protein physics like native state and folding. However, one has to bear in mind that there is no a full analogy between protein and homopolymer. If the native state of protein corresponds to minimum of well-defined funnel on FEL then the "native state" of homopolymer is a degenerate ground state. Because the degeneracy level increases with chain length the concepts of native state as well as folding become ambiguous for a long homopolymer. Determination of the critical length above which concepts adapted from protein folding theory cease to be valid is a highly non-trival task. Nevertheless we believe that this theory is still applicable for short homopolymers like the N=16 chain studied in this paper. The simple structure-based model has been already achieved surprising success in modeling protein’s mechanical behaviour [2, 3, 28, 29, 11]. Many studies have demonstrated that Go model - although having its approximations which include coarse-grained representation, neglecting explicit solvent and non-native interactions [30, 31] - is capable to reproduce the complex details of protein unfolding and (re)folding dynamics including free energy landscape parameters, pathways and mechanical stability [10, 19, 20, 32 - 34]. In the present paper we have tested a simple coarse-grained model [13, 25] in investigation of the influence of external force on the refolding of homopolymer. Using this CG model we have observed a switch from thermal to force-driven pathways of homopolymer refolding. Similar to the protein case [21] these two regimes are characterized by different values of measuring the distance from the transition state to the denaturate one. It would be interesting to test the prediction followed from our homopolymer unfolding simulation by experiment. Based on the results obtained for protein [21] and homopolymer one can anticipate that the crossover from the temperature-driven to force-driven regime takes place also for other biomolecules like DNA and RNA. Our work in this direction is in progress. Acknowledgement. This work was supported by Department of Science and Technology at Ho Chi Minh city, Vietnam and the Polish NCN Grant 2015/19/B/ST4/02721. M.K. acknowledges the Polish Ministry of Science and Higher Education for financial support through Mobilnosc Plus Program No. 1287/MOB/IV/2015/0. This research was also supported in part by the High Performance Computing Facility at The Research Institute at Nationwide Children’s Hospital. REFERENCES 1. Onuchic J. N. and Wolynes P. G. – Theory of protein folding, Curr. Opin. Struct. Biol. 14 (2004) 70. 2. Kumar S. and Li M. S. – Biomolecules under mechanical force, Physics Reports 486 (2010) 1. 3. Sun L., Noel J. K., Sulkowska J. I., Levine H., and Onuchic J. 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