Free-Energy Landscape for Peptide Amphiphile Self-Assembly

Oct 22, 2013 - The mechanism of self-assembly of 140 peptide amphiphiles (PAs) to ... A 2D free-energy landscape with respect to the fraction of nativ...
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Free-Energy Landscape for Peptide Amphiphile Self-Assembly: Stepwise versus Continuous Assembly Mechanisms Tao Yu and George C. Schatz* Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3113, United States S Supporting Information *

ABSTRACT: The mechanism of self-assembly of 140 peptide amphiphiles (PAs) to give nanofiber structures was investigated using a coarse-grained method to quantitatively determine whether the assembly process involves discrete intermediates or is a continuous process. Two novel concepts are introduced for this analysis, a cluster analysis of the time dependence of PA assembly and use of the fraction of native contacts as reaction coordinates for characterizing thermodynamic functions during assembly. The cluster analysis of the assembly kinetics demonstrates that a pillar-like intermediate state is formed before the final cylindrical semifiber structure. We also find that head group assembly occurs on a much shorter time scale than tail group assembly. A 2D free-energy landscape with respect to the fraction of native contacts was calculated, and the pillar-like intermediate structure was also found, with free energies about 1.2 kcal/mol higher than the final state. Although this intermediate state exists for only hundreds of nanoseconds, the PA self-assembly process can be recognized as involving two steps, (a) transition from the disordered state to the noncylindrical pillar-like intermediate and (b) pillar-like to final semifiber transition. These results are important to the further design of PAs as functional nanostructures.

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these questions can provide a more quantitative physical picture of self-assembly that could be important to the design of PAs to make specific functional structures. In the present work, we address the above questions using several long-time-scale (around 22 microseconds) unconstrained coarse-grained MD simulations. This unbiased simulation strategy provides a more adequate sampling of the assembly process, which should be more accurate than is possible using constrained molecular dynamics. Possible intermediate states and their kinetic and thermodynamic properties are identified by a cluster analysis, and free-energy profile calculations are used to characterize relative stability. In analogy to what is done in protein folding studies,24−27 we consider the fraction of native contacts to define the collective variables (CVs) as these provide a straightforward physical picture to describe the assembly dynamics. The new results give a more detailed picture concerning PA self-assembly than has been provided previously in which the spherical micelle intermediates previously identified do not play an important role, but instead, there are intermediates close to the final state involving the linking of large-scale structures.

eptide amphiphiles (PAs) form self-assembled supramolecular structures that are promising soft gel materials for applications in biomedicine,1−6 as templates for nanomaterials synthesis,7 as light-harvesting materials,8 and as nanoreactors.9 In particular, PAs that form cylindrical micelles have been reported by Stupp and co-workers1,2,10 for use in promoting the growth of blood vessels, healing broken bones, and stimulating nerve growth. Many theoretical studies11−18 have been done to understand the PA self-assembly processes, including recent studies revealing structural details of the micelles at an atomistic level.19,20 In addition, the assembly of micelles starting from homogeneous initial conditions has been simulated using a coarse-grained model.21 Most recently, we have studied the thermodynamic properties and driving forces for the PA selfassembly process at the atomistic level.22,23 One issue that was unclear in the earlier work was whether the assembly process involves stable distinct intermediate states or is a continuous process in which PAs collectively aggregate in one step. The previous studies23 using biased MD simulation calculated the free-energy, enthalpy, and entropy changes between the initial and final states, without any observed intermediate states. However, a mechanism involving a stepwise process with spherical micelle intermediate states was speculated21 based on unbiased coarse-grained simulations. However, this earlier coarse-grained modeling did not provide sufficient detail to clarify this stepwise assembly mechanism; therefore, there remains the following questions: (1) Are the spherical-micelle-like structures transient or metastable states? (2) Does the free-energy landscape possess local minima, thus supporting a stepwise mechanism? (3) Are there transition energy barriers between the steps? Finding out the answers for © 2013 American Chemical Society

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METHODS Coarse-Grained Simulations. We have chosen a PA with the peptide sequence Ser-Leu-Ser-Leu-Ala-Ala-Ala-Glu-Ile-LysVal-Ala-Val for all simulations. Figure 1a displays the setup of the the CG model using the MARTINI force field.28,29 Further

Received: September 17, 2013 Revised: October 14, 2013 Published: October 22, 2013 14059

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Figure 1. (a) Schematic structures of three different types of PAs, PA1, PA2, and PA3, used in the CG model. The details of each bead are described in ref 21. (b) The coarse-grained model of 140 PAs in the left initial homogeneous mixture state and right final noncontinuous semifiber state. Tail and head segments of the PAs are color coded as red and lime green, respectively.

variables in calculating the free energy. The definitions of Qt and Qh are described in the SI. By counting the relative probability of the configurations with different Qt and Qh in the three 21.6 μs unbiased coarse-grained simulations, the free energy was calculated using

details are presented in ref 21. Note that the atomistic simulation results demonstrated that the PA nanofiber displays an inhomogeneous secondary structure distribution. Because the MARTINI CG model cannot introduce the secondary structure relaxation in the MD simulation, we have chosen to predefine the secondary structure distribution of the PAs when building up the simulation model to be consistent with the previous work. As shown in Figure 1a, three different types of PAs were defined, as labeled PA1, PA2, and PA3. Detailed information to construct these PAs is provided in Table S1 (Supporting Information (SI)). Here, 140 PAs were used in the assembly simulations, composed of 35 PA1 (25%), 21 PA2 (15%), and 84 PA3 (60%) to match the results of atomistic simulations. To begin the MD simulations, these 140 PAs with 17385 CG waters, 182 sodium ions, and 42 chloride ions were randomly distributed in a box with dimensions 16 × 16 × 16 nm 3 . The GROMACS-4.5.1 simulation package 30 was employed to carry out three independent unconstrained NPT simulations31−34 with periodic boundary conditions. The setup of simulation parameters and details can be found in the SI. Free-Energy Calculations. Two choices of the fraction of native contacts, Qt and Qh, were employed as collective

ΔGα = −kT[ln(P(qα)) − ln(Pmax(qα))]

(1)

where k is the Boltzmann constant, P(qα) is an estimate of the probability density function obtained from a histogram analysis in the MD simulation, and qα is the predefined reaction coordinate, that is, Qt or Qh.

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RESULTS AND DISCUSSION Kinetics of Clustering. Figure 1b displays the initial disordered homogeneous mixture state and final ordered nanofiber-like state, which is called the “semifiber” state. The peptide head and tail segments are color coded as red and lime green, respectively. Note that the semifiber state is not a real nanofiber structure as it displays a discontinuous feature for the tail portion even though the head portion displays one continuous fiber. This phenomenon indicates that the tails are not completely packed with each other after more than 20 14060

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The tail cluster number has average values of ∼6, 4, and 5 for the three trajectories during the time intervals 0.2−0.6, 0.3−0.7, and 0.2−0.5 μs. During these intervals, the tails show a small pillar-like structure, that is, a fragment of a fiber, in contact with a bent fiber-like structure. Subsequently, the pillar-like shape transitions to a loose semifiber structure whose tail cluster number is ∼3 for the time intervals of 0.6−1.2 and 0.5−1.3 μs in the red and blue curves, followed by the final semifiber structure at longer times. Meanwhile, the head cluster number in Figure 2 shows only one flat stage prior to the final semifiber cluster that forms at 0.1−0.45 μs (red curve). The green/blue curves suggest an even shorter flat stage between 0.05 and 0.1 μs. The intermediate flat stage (0.1−0.45 μs) is associated with cluster numbers of 2 (red) or 1 (green and blue), both corresponding to the pillar-like structure, where there is better contact in the pillar-like structure in the green and blue simulations. Overall, we see that kinetic factors lead to rapid aggregation of heads, creating an outside shell to the final fiber structure inside of which occurs the slower aggregation of the tails. An example of the tail fusion is is shown in Figure S1 (SI). The significant differences between the tail and head assembly can be attributed to the fact that heads are more exposed to solvent than tails; thus, the heads in the clusters have more opportunity to fuse into larger structures. The flat regions in Figure 2 show that the pillar-like intermediate only lasts for a few tenths of a microsecond, with the tail and head cluster measures of the existence of this structure not being entirely synchronized. Note that the pillar-like structure is distinguished from the final semifiber in that the tail clusters form a linear chain in the semifiber, whereas this is not present in the pillar structure. However, it should be noted that it is the heads that provide the primary driving force for making cylindrical (as opposed to spherical for flattened) structures; therefore, the instability of the piller structure is primarily due to the head organization. Thermodynamic Analysis. As a second method of analysis, we use the contact numbers (associated with the units T1 and Ab7) to determine the fractions of native contacts, Qt and Qh, as collective variables representing the different configurations between the initial and final states. We then calculate free-energy profiles as a function of these fractions. The scale of these two collective variables is set between 0 and 1. In the initial state, Qt and Qh are close to zero, and in the final state, they are close to 1. When the PAs transition from disordered into ordered states, both Qt and Qh increase. By counting the relative probability of all-collective configurations (45000 × 3) with different Qt and Qh values, the free-energy landscape with respect to Qt and Qh is calculated using eq 1. Figure 3 displays the 2D free-energy landscape with respect to Qt and Qh. The dark blue ranges correspond to several shallow minima viewed as stable or metastable configurations, while the light orange ranges correspond to structures with higher free energy. The multiple-minimum character of the free-energy landscape demonstrates that there exist metastable intermediates in assembly; however, none of them are very deep; therefore, they have short lifetimes. Several selected ranges are labeled, and detailed information and representative configurations are listed Figure 4. It is found that the most stable configuration (labeled e) is the semifiber. The second most stable one (labeled f) also displays a semifiber structure. These two share the same Qt but different Qh values, indicating different semifiber structures with slightly different local tail structures but similar global features in which the heads form a

μs of simulation. Similar results were observed in ref 21, where it was indicated that a much longer simulation time was required to remove this discontinuity. In the present work, we employed the semifiber structure as the reference to assess the relative stability intermediate configurations in the simulation. We first examine the time evolution or dynamical history of the different intermediate configurations by doing a cluster analysis (counting the number of clusters at different times). This analysis will reveal important information about the kinetics of self-assembly and, in particular, whether there are well-defined intermediates. Our cluster analysis is based on the proximity of tails and heads (specifically the units T1 and Ab7) of the PAs, with the three independent simulations used for the calculations. A cluster is defined as two or more tails or heads contacting within a 7 Å radius. The results, expressed as cluster number as a function of time, are presented in Figure 2. This

Figure 2. The time evolution of the number of clusters defined by (a) the tails and (b) the heads of the PAs. Results for three trajectories are shown in red, green, and blue.

shows that the number decreases with time, corresponding to the fact that the semifiber is forming while other small clusters are disappearing. In addition, the decrease with time shows well-defined stepwise features. The counting of cluster number is initiated at 2.8 ns, where the tails form 57 clusters and the heads form 14 clusters on average. After 1.3 μs, the average number of tail and head clusters is 2.5 and 1, corresponding to the semifiber structure where all of the heads form a single cluster but the tail clusters are not continuous. This stepwise dependence on time is connected to the formation of specific intermediate configurations, in which the first stage of selfassembly involves a disorder (homogeneous mixture structure) to meta-order (pillar-like structure that we define below) transition and the second stage is from meta-order (pillar-like structure) to full-order (semifiber structure). More interesting, it is found that the tail and head steps occur at different times. 14061

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Figure 3. The free-energy landscape associated with the self-assembly of 140 PAs as a function of the fractions of native contacts of the tails and heads. The definitions of Qt and Qh refer to eqs S1 and S2 (Supporting Information, respectively. Detailed information about the free energy and configurations associated with the highlighted regions a−f is displayed in Figure 4.

Figure 5. The free-energy profile as a function of (a) Qt and (b) Qh.

located near Qt = 0.42, 0.48, 0.52, 0.60, 0.62, 0.70, and 0.8, where Qt = 0.8 corresponds to the most stable semifiber structure. At around Qt = 0.62 and 0.70, the very shallow minima correspond to pillar-like structures, as shown in Figure 4c and 4d. For Qt between 0.8 and 1, the free energy continuously increases. On the other hand, Figure 5b shows a rougher landscape for the free-energy profile as a function of the reaction coordinate Qh. Like ΔG(Qt), ΔG(Qh) also shows a slowly decreasing profile first and then quickly increasing after the semifiber structure is formed. However, it is clear that ΔG(Qh) shows many more local minimum than are found for Qt. This bumpy character indicates many metastable configurations along the Qh coordinate, but with small barriers between them so that the PA head segments are highly dynamic during the self-assembly process. Note that the number of local minima along Qh before the semifiber is formed (i.e., Qh < 0.7) is much larger than that after. Similar results are also found for Qt, suggesting that head and tails segments become rigid once the final semifiber structure is generated. Also, heads are more exposed to the solvent environment; therefore, heads are more dynamic compared with tails. Note that we are missing free-energy information near the very beginning of the self-assembly process, where Qt is less than 0.3. A lot of spherical-micelle-like configurations were obtained in this regime; therefore, we refined this part of the analysis by collecting 12500 extra frames. It is found that no obvious stable minima exist in that range of Qt in Figure S2 (SI). Recalling that there is a rapid decay of tail cluster number before the pillar-like structure is formed, it may be concluded that the spherical micelle structures are transient states. In addition, we also calculated a one-dimensional free-energy profile based on the tail segment cluster number; however, this did not provide further information compared with the present thermodynamic analysis using Qt.

Figure 4. Snapshots of representative structures and free-energy information corresponding to the labeled ranges a−f in Figure 3.

continuous network. The states labeled c and d correspond to the pillar-like intermediate states that were seen earlier in the cluster analysis. Note that the two fragments connect with each other only through the head regions, with the tail parts still spatially separated. This explains why their Qt values are smaller than those in states e and f. Therefore, to insert the pillar-like fragment into the larger fragment and to form the semifiber structure requires a locally significant reorganization to make further contact of both heads and tails, thus increasing both Qt and Qh. From the 2D free-energy plot, there is almost no barrier for this transition to occur. This is consistent with the few hundred nanosecond time scales for this transition to occur, but the cluster analysis shows that the pillar-like structure exhibits metastable intermediate stepwise assembly behavior. By projecting the free-energy landscape onto Qt and Qh separately, one-dimensional free-energy profiles as a function of Qt and Qh have also been generated. The results are displayed in Figure 5. In Figure 5a, the free-energy profile along Qt is a relatively smooth function with several shallow minimum 14062

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CONCLUSION

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ASSOCIATED CONTENT

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S Supporting Information *

Details of the MD simulations, definitions of the fraction of native contacts, the mechanism of transformation of the pillarlike structure to semifiber as shown in Figure S1, the freeenergy profile along Qt in the range between 0.15 and 0.35 as shown in Figure S2, and information about secondary structure chosen for the coarse-grained PAs in Table S1. This material is available free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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In conclusion, we found that the self-assembly of PAs evolving into a semifiber final state involves many intermediate configurations that have only low barriers between them (a few kT at most). Therefore, there are only weak bottlenecks to self-assembly on a time scale of a few microseconds. By examining native contacts of tail and head groups during unbiased simulations of self-assembly, a time-dependent evaluation of cluster number and a 2D free-energy landscape were used to provide more detailed kinetic and thermodynamic information than we were able to generate previously using atomistic simulations. In particular, the results demonstrated that (1) there exists pillar-like metastable states with a hundreds of nanoseonds lifetime, prior to the formation of the final semifiber structure; this divides the self-assembly mechanism into two primary steps; (2) the spherical micelle structures previously noted near the beginning stage of self-assembly are just unstable transient states; (3) the head segments of the PAs are much more dynamical than the tail segments during the self-assembly process, even in the final state; and (4) head segments aggregate faster than tail segments in the assembly process. While these details of the self-assembly process have not yet been studied experimentally, we hope that this work will stimulate interest in such studies. The mechanism of selfassembly is expected to influence the functional nanostructure that results from assembly. Also, it might be possible to study such mechanisms experimentally through studies of the time evolution of fluorescence (and other optical properties) associated with chromophores that are attached to the PAs. More importantly, in the present work, we proposed new data analysis methods based on unbiased MD simulations, that is, using a cluster analysis and the fraction of native contacts to understand the complex PA self-assembly process from both a kinetics and thermodynamics point of view. The methods proposed here can be easily extended to study other selfassembly processes involving soft materials.

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ACKNOWLEDGMENTS

This research was supported by the National Science Foundation (Grant CHE-1147335) (for methods development) and by the DOE NERC EFRC (DE-SC0000989) (for applications). We thank Dr. One-Sun Lee and Marty Blaber for useful comments. 14063

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