Dynamic Transition from Î±-Helices to Î²-Sheets in ... - ACS Publications

Article Cite This: J. Am. Chem. Soc. 2017, 139, 16168-16177

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Dynamic Transition from α‑Helices to β‑Sheets in Polypeptide CoiledCoil Motifs Kirill A. Minin,† Artem Zhmurov,† Kenneth A. Marx,‡ Prashant K. Purohit,§ and Valeri Barsegov*,†,‡ †

Moscow Institute of Physics and Technology, Dolgoprudny 141701, Russia Department of Chemistry, University of Massachusetts, Lowell, Massachusetts 01854, United States § Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, Pennsylvania 19104, United States ‡

S Supporting Information *

ABSTRACT: We carried out dynamic force manipulations in silico on a variety of coiled-coil protein fragments from myosin, chemotaxis receptor, vimentin, fibrin, and phenylalanine zippers that vary in size and topology of their α-helical packing. When stretched along the superhelical axis, all superhelices show elastic, plastic, and inelastic elongation regimes and undergo a dynamic transition from the α-helices to the β-sheets, which marks the onset of plastic deformation. Using the Abeyaratne-Knowles formulation of phase transitions, we developed a new theoretical methodology to model mechanical and kinetic properties of protein coiled-coils under mechanical nonequilibrium conditions and to map out their energy landscapes. The theory was successfully validated by comparing the simulated and theoretical force-strain spectra. We derived the scaling laws for the elastic force and the force for α-to-β transition, which can be used to understand natural proteins’ properties as well as to rationally design novel biomaterials of required mechanical strength with desired balance between stiffness and plasticity.

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INTRODUCTION In 1953, coiled-coils were proposed independently by Crick and Pauling as protein motif structures comprised of supercoiled α-helical segments; at the time, Crick realized these structures would be stabilized by side chain interactions from the α-helices whose axes were twisted about 20 degrees with respect to each other, thereby repeating similar interactions every seven residues along each α-helix.1 In the intervening decades, structural biology studies have identified a wider range of coiled-coil motifs, including examples that deviate from Crick’s canonical heptad pattern. Currently, coiled-coils have been identified and characterized as ubiquitous, critically important, highly stable biomechanical structures, occurring either macroscopically at the tissue level (hair, nails, blood clots, etc.), microscopically within individual cellular structures (intracellular cytoskeleton, extracellular matrix, flagella, etc.), or as motif components of proteins involved in a variety of functions (membrane fusion, signal transduction, solute transport, etc.). Many well-studied proteins performing mechanical functions in biological processes, utilize the © 2017 American Chemical Society

superhelical coiled-coil architecture, including ones we have incorporated into the present study based upon their structural diversity and available atomic structures: muscle proteins (myosin), intermediate filaments (vimentin), whole blood clots and thrombi (fibrin), chemotaxis (chemotaxis receptors), cellular transport (kinesin), and bacterial adhesion (protein tetrabrachion).1 More recently, the unique superhelical symmetry and physical properties of coiled-coils has inspired the design of new materials,2,3 from short supercoils,4 to long and thick fibers,5 to nanotubes,6 spherical cages,7 and synthetic virions.8 In our previous work, we studied equilibrium mechanical properties and the α-to-β transition in fibrinogen coiled coils using a constant pulling force.9 In this study, we took a step further. We have explored the physicochemical properties of a wide range of polypeptide motifs formed by two-, three-, four-, and five-stranded coiled-coil protein supehelices, involving both Received: July 6, 2017 Published: October 18, 2017 16168

DOI: 10.1021/jacs.7b06883 J. Am. Chem. Soc. 2017, 139, 16168−16177

Article

Journal of the American Chemical Society

Figure 1. Structure of proteins containing the α-helical coiled-coil motifs. The protein fragments used in this work are boxed by dashed lines. (a) Myosin (shown in red, blue, and gray) is a motor protein moving actin filaments (shown in green and yellow). The myosin head domain contains the ATP binding site. When myosin moves along the actin filaments, it generates tension along the myosin tail, which forms a left-handed coiled-coil. (b) Bacterial chemotaxis receptor is a transmembrane protein with a long rodlike cytoplasmic domain, which hosts essential binding sites. The receptor contains two identical chains that make a “U-turn” at the end of the cytoplasmic domain. Each chain forms an antiparallel two-stranded lefthanded coiled-coil, and these wrap around one another forming a four-stranded coiled-coil. (c) Vimentin is a protein building block of intermediate filaments, an important component of the cytoskeleton of eukaryotic cells. The vimentin dimer contains the α-helical central domain made of twostranded coiled-coils. The tetramer forms when two dimers associate laterally. (d) Fibrin (top structure) is a building block of fibrin protofibrils (bottom structure) and fibrin fibers responsible for maintaining the stability of blood clots. Fibrin monomers contain the globular domains connected by two three-stranded parallel coiled-coils. (e) Phenylalanine zipper is an example of a de novo designed protein. Five identical chains with phenylalanine residues at position a and d (indicated by arrows) form a motif that folds into a five-stranded left-handed parallel coiled-coil. A single-point mutation Phe27 → Met27 (marked by asterisks) disrupts the five-stranded architecture thereby destabilizing the structure.

parallel and antiparallel arrangements of their α-helices. Dynamic force manipulations in silico, which fully mimic single-protein forced unfolding experiments in vitro, have been combined with theoretical modeling of protein coiled-coils. Using this approach, we have found that all the coiled-coil protein fragment systems studied, ranging from two-to-five helices and forming parallel or antiparallel supercoil architecture, are characterized by three distinctly different regimes of forced extension and that they uniformly undergo a remarkable dynamic structural transition from all α-helices to all β-sheets. Experimental biophysical force extension studies have been carried out on a number of coiled coil containing protein fiber systems, such as keratin,10 marine snail egg case protein,11,12 and hagfish slime protein,13 (reviewed in ref 14). These systems all show dynamic structural transition results qualitatively similar to our simulations. However, these experimental studies are carried out on far longer time scales than our simulation results can achieve, as well as resulting from far more heterogeneous structures than our short-defined coiled coil motifs. In a theoretical study, two-stranded coiled coils were analyzed to determine the critical length (number of amino acids) for formation of the β-phase.15 It is worth noting that our study continues beyond a mere characterization of the coiled coils’ dynamic transitions, to provide a theoretical basis for understanding the different coiled-coils’ stability and their

nanomechanics when undergoing the pulling force induced elongation and transition from the α-helices to the β-sheets. To span the range of polypeptide coiled-coil motifs, here we used the coiled-coil fragments from the available atomic structure models of myosin, vimentin, fibrin, bacterial chemotaxis receptor and phenylalanine zippers all displayed in Figure 1; see also the Supporting Information for more details on architecture of polypeptide superhelices. (i) Myosin II is a muscle protein containing a double-stranded parallel coiledcoil.16 Upon muscle contraction, tension is transferred along the myosin tail16 (there is a body of experimental force data available for myosin17,18). (ii) Vimentin is an elementary building block of intermediate filaments in cells,19,20 helping to determine their resistance to mechanical factors.21 The structure of vimentin contains an α-helical rod domain, which can be divided into several double-helical parallel coiled-coil segments.21 (iii) Fibrin forms a fibrous network in blood,22 which acts as a scaffold to stem bleeding. The triple-helical parallel coiled-coils are responsible for the unique viscoelastic properties of fibrin.9,23 (iv) Bacterial chemotaxis receptor is responsible for signal transduction across cell membranes.24 The cytoplasmic domain contains two double-stranded antiparallel coiled-coils wrapped around each another forming a four-stranded superhelix. Lastly, (v) Phenylalanine zippers, 16169


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Figure 2. α-to-β transition in polypeptide coiled-coils. (a) Pulling setup with pulling plane and resting plane. The transition is shown for the (b and c) chemotaxis receptor and for (d and e) myosin. The (b and d) f ε-curves, showing elastic, plastic, and inelastic regimes of coiled-coils’ unfolding, are overlaid with the continuum theoretical curves. The insets in (b and d) are Ramachandran plots of dihedral (ϕ,ψ) angles in the two states. (c and e) Snapshots from similarly numbered regions in the f ε-curves show unfolding progress from the initial α-structure (0) to the mixed α/β-structures (1−3) and to the final β-structure (4). Snapshots (3−4) show exotic spirals formed by the β-sheets.

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artificial four-to-five-stranded parallel coiled-coils, are promising biomaterials with tunable materials properties.6,25 We mechanically tested the coiled-coil motifs by applying a time-dependent pulling force, which corresponds to the dynamic force-ramp mode used in single-molecule protein stretching experiments. Direct comparison of the results of dynamic force manipulations in silico and theoretical modeling of the force spectra have enabled us to develop a new theory to describe the nonequilibrium dynamic response of protein coiled-coils to external mechanical factors. The quantitative agreement between the theory and simulations of the systems studied allows for a new approach to rationally design coiledcoils with desired mechanical properties into novel biomaterials for specific technological applications. For example, experimentally manipulating protein systems to incorporate coiled coil segments with specific properties, such as the recombinant fibers containing vimentin fragments26 and hagfish slime protein27 coiled coil fragments, could take advantage of a rational design approach utilizing the theory of coiled coils’ functional dependencies and scaling laws we presented here to create better biomaterials, both faster and cheaper.

EXPERIMENTAL PROCEDURES

Structural Models of Polypeptide Coiled-Coils. We used the atomic structural models of myosin (PDB entry: 2FXO28), vimentin (1GK429), fibrin (3GHG30), bacterial chemotaxis receptor (1QU724), and phenylalanine zippers (2GUV and 2GUS25). All the models of polypeptide coiled-coils are displayed in Figure 1 (see the Supporting Information for more details). Force Spectroscopy in Silico. We employed the all-atom molecular dynamics (MD) simulations in implicit and explicit solvent accelerated on GPUs. Implicit solvent modeling: we used the solvent accessible surface area (SASA) model of implicit solvent solvation with CHARMM19 unified hydrogen force-field31 (see the Supporting Information).9 In silico models of proteins were constructed using the CHARMM program.32 We used a lower damping coefficient γ = 0.15 ps−1 versus γ = 50 ps−1 for ambient water at 300 K for more efficient sampling of the conformational space.33 To mimic the experimental force-ramp conditions and to enable rotational motions of coiled-coils around the pulling axis, we implemented the pulling plane with harmonically attached tagged residues at one end of the molecular system and the resting plane with constrained residues at the other (Figure 2a). This approach ensures even tension distribution in polypeptide chains and suppresses chain sliding past each other. The pulling plane was connected to a virtual cantilever moving with a velocity vf = 104−106 μm/s, thereby ramping up the applied force f = rf t with a loading rate rf = ksvf = 10−3−10−1 N/s (ks = 100 pN/nm is the cantilever spring constant). Explicit solvent modeling: we also 16170


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experimental data13,17,18,35−38 and MD studies, performed by us9 and by other researchers.15,33,39 Dynamics of Dihedral Angles and Hydrogen Bonds. Individual amino acid dihedral angles (ϕ,ψ) are well-known to be sensitive to dynamic changes in the protein secondary structure where they occur.40 This is reflected in migration of ϕ,ψ-angles from the α-region to the β-region in the Ramachandran plots (the insets to Figure 2, panels b and d). Further, in the α-state the H-bonds are all intramolecular. The β-sheets form due to intermolecular (interchain) hydrogen bonds (H-bonds) linking parallel or antiparallel β-strands. We profiled the probabilities of finding a system in the α-state pα and β-state pβ using dihedral angles and H-bonds. Specifically, we calculated the relative amount of intrachain bonds pα = Nintra/NH and interchain bonds pβ = Ninter/NH (NH-total number of H-bonds) and the relative amounts of residues in the α-region pα = Nα/Nϕψ and β-sheet region pβ = Nβ/Nϕψ (Nϕψ-total number of ϕ,ψ-angles) of the Ramachandran plot. The profiles of pα and pβ displayed in Figure 3 for bacterial

performed MD simulations in explicit solvent to validate the results of SASA implicit solvent-based modeling (see the Supporting Information for more detail).

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RESULTS α-to-β Transition. We mechanically tested myosin, vimentin, fibrin, bacterial chemotaxis receptor, and phenylalanine zippers (Figure 1) by employing Molecular Dynamics simulations of protein-forced unfolding accelerated on graphics processing units (GPUs). We implemented dynamic force ramp conditions of time-dependent pulling force as described in the Experimental Procedures. All the coiled-coils undergo the force-driven structural transition from the α-helices to the β-sheets regardless of the number of helices and parallel or antiparallel architecture. In Figure 2, we display the force-strain (fε) curves and structure snapshots for bacterial receptor and myosin. The fε-curves are reminiscent of experimental force extension profiles that can be found elsewhere.17,18 The plateau force for myosin f * ∼ 100 pN (Figure 2d) is higher than the experimental value of ∼40 pN18 due to 104−105-fold faster pulling speeds used in silico. Indeed, experimental pulling speeds are typically 101−103 nm/s;17 yet, our results in Figure 5a indicate that these give plateaus in the tens of pico-Newtons, which is in qualitative agreement with the experiments. Also, friction due to the surrounding fluid in our simulations is small compared to the experiments, which corresponds to a shorter relaxation time. Therefore, notwithstanding, the high pulling speeds used in simulations, mechanical tension propagates rapidly along the filament in a manner similar to that in the experiments. Results obtained for vimentin, fibrin, and phenylalanine zippers are shown in Figure S2. The f ε-curves reflect the three regimes of dynamic mechanical response of the coiled-coils to an applied pulling force: (i) the elastic regime I of coiled-coil elongation at low strain (f ∼ ε; ε < 0.2), characterized by a linear growth of f with ε; (ii) the plastic transition regime II of protein unfolding at intermediate strain (f = const; 0.2 < ε < 0.9), in which the α-to-β transition occurs; and (iii) the inelastic regime III of nonlinear elongation of the β-structure at high strain ( f ∼ ε2; ε > 0.9). In the α-to-β transition regime, the coiled-coils unwind and undergo a large 80−90% elongation. The α-to-β transition nucleates at both ends of the molecule, and two phase boundaries propagate toward the center (snapshots 1−2 in Figure 2, panels c and e and Figure S7). Nucleation occurs first at the end x = 0 because tension is largest at this end when the molecule immersed in fluid is pulled toward the right at the other end x = L,34 as in our simulations. Nucleation also occurs at the end x = L because bonding interactions are disrupted at both ends due to perturbations caused by the pulling planes. To avoid boundary effects due to the fact that terminal residues are not fully incorporated into the α-helices and hence might be less stable, we performed pilot simulations on myosin tail in which the residues one helical turn away from the terminals were constrained at one end (x = 0) and were tagged at the other end (x = L). In this trajectory, the nucleation also started from the tagged residues (Figure S7a, right). We also witnessed the formation of exotic, spiral segments of β-sheets (snapshots 3−4 in Figure 2, panels c and e). Importantly, all of these features were observed irrespective of the number of α-helices (from two- to five-stranded) and architecture of α-helical packing (parallel and antiparallel) comprising the coiled-coil. Overall, the transition process is in a very good agreement with available

Figure 3. Dynamics of populations pα and pβ for chemotaxis (a) receptor and (b) myosin tail, described using the dihedral (ϕ,ψ) angles and hydrogen bonds (H-bonds). Compared are curves from MD simulations with green curves from continuum theory based modeling. We used the intervals in the Ramachandran plot −80° < ϕ < − 48° and −59° < ψ < − 27° for the α-phase and −150° < ϕ < − 90° and 90° < ψ < 150° for the β-phase.40 For H-bonds, we used the 3 Å cutoff distance between hydrogen donor (D) and acceptor (A) atoms and the 20° cutoff for the D−H···A bond angle.

receptor and myosin show the following: (i) the α-helical (βstrand) content decreases (increases) with time t (and force f = rf t), (ii) the transformation from α-helices to β-sheets is a twostate transition, and (iii) this transition is accompanied by redistribution of ϕ,ψ-angles (α → β; Figure 2, panels b and d insets) and reconfiguration of H-bonds (from intra- to interchain H-bonds; see snapshots in Figure 3). We obtained 16171


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Figure 4. (a) Free-energy landscape ΔG(x) for myosin coiled-coil at different pulling forces. (b) Histogram-based estimate of the distribution of projected lengths for myosin residues P(x) for f = 90 pN and the energy landscape reconstructed using the model parameters (see Table 1).

similar results for the vimentin, fibrin, and Phe-zippers (Figure S3). These results demonstrate that the hydrogen bonds and dihedral angles can be used as molecular signatures to detect the α-to-β transition in protein motifs with superhelical symmetry. Theoretical Methodology. Two-State Model. In our previous study,9 we showed that under the near-equilibrium constant-force conditions (force-clamp) the α-to-β transition in protein coiled-coils can be described using a two-state model. A similar two-state-type model was used to describe the α-to-β transition in a protein from whelk egg capsule.3,12 The sigmoidal extension-force phase diagram for myosin and fibrinogen coiled-coils in Figure S5 is divided into the αphase and the β-phase.9 The α-state can be modeled as an entropic spring with energy Gα = kαΔX2α/2, where kα is the spring constant and ΔXα = f/kα is the extension. The β-state can be described by a wormlike chain41 with energy Gβ = (3kBT/2lβ)∫ [∂n(s)/∂s]2 ds, where lβ is the persistence length and T is the temperature. An applied pulling force stretches the α-helices by a fractional extension yα( f) = ΔXα( f)/Lα, where Lα is the maximal extension in the α-state, lowers the free energy difference ΔG(f), and increases the transition probability pβ(f)/pα(f) = exp[−ΔG(f)/kBT]. The force induced elongation in the β-states is given by yβ(f) = ΔXβ( f)/Lβ, where Lβ is the maximal extension in the β-states. For the wormlike chain, yβ( f) = 1 − {ξ(q)1/3 + [4q/3−1]/ξ(q)1/3}−1, where ξ(q) = 2 + {4 − [(4/3) q − 1]3}1/2 is a function of q = f lβ/kBT (rescaled energy in units of kBT). The total extension can be calculated as ΔX( f) = pα yα Lα + pβ yβ Lβ. This expression can be used to describe the dependence of total extension on pulling force.9 The results of two-state modeling for myosin and fibrin coiled coils are displayed in Figure S5. The model assumes that the two states (α and β) can coexist. Thus, some parts of the molecule can be in the α-phase, whereas other parts of the molecule can be in the β-sheet phase with a few interfaces separating them. The interfaces move to accommodate the kinematic boundary conditions applied to the molecule, and the transition is “continuous”. If force boundary conditions are applied (e.g., constant tension through the molecule) then either the whole molecule is in the α-phase or in the β-phase and the transition sets in within a short force range (the width of transition range depends on molecular cooperativity). Continuum Theory. To describe the α-to-β transition in the nonequilibrium regime of time-dependent mechanical perturbation, we used the Abeyaratne-Knowles formulation of phase transitions in continua.42,43 In a one-dimensional continuum,

the displacement of a material point at reference position x at time t is given by u(x,t) = X(x,t) − x. The end at x = 0 is fixed, and u(0,t) = 0 for all t. At the other end, x = L, u(L,t) = ΔX(t), where ΔX(t) = vf t > 0 ( fαβ > f 0, where f 0 is the Maxwell force at which the free energies per residue for the two phases become equal. The β-phase is described by a stretch Γβ(f) = Xβ(f)/L, f*βα < f < ∞, where Xβ(f) and f*βα are the total deformed length in the β-phase and critical force in the β-phase. A transformation strain is defined as γ(f) = Γβ( f) − Γα( f), f*αβ ≥ f ≥ f*βα. The equation of motion for the 1D continuum is ∂f/∂x = 0, so that f(x,t) is constant for 0 < x < L. If fαβ * < f < fβα * , there is a mixture of the α- and β-phases, and the total extension is X (L , t ) = L + ΔX (t ) = L[Γα(f (t ))pα (t ) + Γβ(f (t ))pβ (t )] (1)

We assumed that the strain in the α-phase/β-phase is Γα( f)/ Γβ( f) throughout the molecule (force is uniform over the entire molecule). A kinetic relation expressed in terms of the pulling force f(t) describes the evolution of pβ, i.e., pβ̇ = Φ[f (t )]

(2)

where Φ(f) is a material property. We can rewrite eq 2 as pβ =

⎤ ΔX 1 ⎡ − Γα(f )⎥ ⎢⎣1 + ⎦ γ (f ) L

(3)

By differentiating eq 1 with respect to time, and eliminating ṗβ using eq 2 and pβ using eq 3, we obtain [g (f ) − γ ′(f )(1 + ΔX /L)]f ̇ + γ(f )vf /L = γ 2(f )Φ(f ) (4)

where g(f) = Γα( f) Γ′β(f) − Γ′α( f) Γβ(f) . From eq 4, a force plateau ( f ̇ = 0) forms at f = f * if either γ( f *) = Γβ(f *) − Γα( f *) = 0 or vf/L = γ(f *) Φ(f *) . Because in experiments, the height of the force plateau depends on rf and, hence, is strainrate (vf/L) dependent, vf/L = γ(f *) Φ(f *) defines the force plateau height (Figure 2). Application to α-to-β Transition. Calculating the f εcurves requires a kinetic relation and a nucleation criterion. By 16172


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Table 1. Mechanical, Thermodynamic, and Kinetic Parameters for Protein Coiled-Coils from Myosin, Chemotaxis Receptor, Vimentin, Fibrin, and Phenylalanine Zippers Differing in the Number of α-Helices Nh: Lα, κα, and zαβ (α-Phase), Lβ, lβ, and zβα (β-Phase), f 0, ϵαβ, ϵ0, and A (All These Model Parameters Are Defined in the Main Text)a superhelical protein

Nh

Lα (nm)

κα (pN)

lβ (nm)

Lβ (nm)

myosin

2

18.1[18.1]

2169[2170]

[0.38]

46[43]

chemotaxis receptor vimentin

4

13.6[13.6]

2367[2370]

[0.33]

34[32]

2

11.5[11.5]

1375[1375]

[0.36]

28[27]

fibrin

3

16.2[16.2]

1135[1135]

[0.35]

41[38]

4-strand Phezipper Phe-zipper

4

6.2[6.2]

2644[2645]

[0.36]

15[14]

5

7.8[7.8]

1793[1795]

[0.35]

20[19]

zαβ (nm) (0.13) [0.13] (0.11) [0.11] (0.14) [0.14] (0.11) [0.11] (0.11) [0.11] (0.11) [0.11]

zβα (nm) (0.07) [0.07] (0.08) [0.08] (0.06) [0.06] (0.09) [0.09] (0.08) [0.08] (0.09) [0.09]

f 0 (pN)

ϵαβ (kBT)

ϵ0 (kBT)

A (μs−1)

103(90) [90]

(4.4) [4.4]

(3.3) [3.3]

[0.5]

232[176]

(8.3) [8.3]

(5.3) [5.3]

[0.2]

103[90]

(4.3) [4.3]

(3.6) [3.6]

[1.0]

187(160) [160] 268[200]

(7.8) [7.8]

(4.6) [4.6]

[0.4]

(9.3) [9.3]

(5.8) [5.8]

[0.2]

383[350]

(17) [17]

(9.4) [9.4]

[0.1]

Standard deviations (not shown) are ≤3% of the average parameter values obtained directly from MD simulations or by using the energy landscapes (in parentheses) or by performing a fit of eqs 7−9 to the simulated f ε-curves in Figures 2 and Figure S2 (in squared brackets). a

treating the α-phase as an entropic spring and the β-phase as a wormlike chain, we obtain Γα( f) = f/κα + 1 and Γ′α(f) = 1/κα for the α-phase, where κα = kα/Lα, and Γβ(f ) = (Lβ /Lα)(1 − kBT /4lβf ) a n d Γ β′ ( f ) =

⎛ zαβ ⎞−1⎡ vf ϵ ⎤ Lα ⎟⎟ ⎢ln − + 0 ⎥ f * = ⎜⎜ kBT ⎥⎦ κα(Lβ − Lα) ⎠ ⎢⎣ A(Lβ − Lα) ⎝ kBT (8)

Eq 8 captures the loading rate and temperature dependence of plateau force (note that the attempt frequency A should also depend on temperature). The temperature dependence of plateau force in quasistatic stretching can be computed using the Clausius−Clapeyron equation as has been shown for coiled coils by Miserez et al.11 and for DNA by other authors.34,44,45 However, our high force-loading rates cause the structural transition to occur away from the equilibrium, and so the calculation of plateau force should account for the kinetics of the α-to-β transition (see the Supporting Information). To address the question of dependence of the critical force f * on temperature T, we performed pulling simulations for myosin tail and chemotaxis receptor coiled coils in the 280−340 K temperature range. The results displayed in Figure 5c show that f * decreases with T as expected for the transitions driven by enthalpy changes (ΔH). From our simulations, ΔH = 0.2−0.3 kcal/mol per residue, which also agrees with the experimental data.11,14 After the β-phase has formed and stretching of the β-phase continues, the mechanical response in the inelastic regime III (Figure 2, panels b and d) is given by the wormlike chain formula:

(Lβ /4Lα) kBT /lβf 3 for the β-phase. The α-to-β transformation strain becomes γ(f ) = (Lβ /Lα)(1 −

kBT /4lβf ) − f /κα − 1

(5)

and γ′( f) = (Lβ/4Lα) kBT /lβf 3 − 1/κα. To obtain the kinetic relation (eq 2), we use the Arrhenius rates for forward and backward transitions kαβ and kβα,34 pβ̇ = Φ(f ) = A[e−ΔGαβ(f )/ kBT − e−ΔGβα(f )/ kBT ]

(6)

where A is an attempt frequency and ΔGαβ(f) = ϵ0 − fzαβ and ΔGβα( f) = ϵ0 − ϵαβ + fzβα are the energy barriers; ϵ0 and ϵαβ are the force-free energy barrier and energy difference (per residue length) between the α- and β-states, respectively, and zαβ and zβα are transition distances (Figure 4b). Then, the energy difference ΔG( f) = ΔGαβ(f) − ΔGβα( f) provides the detailed balance pβ/pα = exp[−ΔG( f) /kBT] = exp[−(ϵαβ−f(zαβ+zβα))/ kBT]. Initially, the continuum is in the α-phase. Before the nucleation of the phase boundary, the force in the linear elastic regime I (Figure 2, panels b and d) is given by f (t ) = καvf t /Lα = καΔXα(t )/Lα

f (t ) = (kBT /4lβ)[1 − (Lα /Lβ )(1 + vf t /L)]−2

(7)

= (kBT /4lβ)[1 − (Lα /Lβ )(1 + ΔXβ(t )/L)]−2

The nucleation criterion requires that when f = fαβ and ΔX = ΔXαβ = fαβL/κα, a phase boundary nucleates at x = 0 and x = L. Note that it is assumed that the force is uniform throught the molecule, but in reality, nucleation occurs first at the ends (x = 0 and x = L) since tension is higher there due to fluid drag on the molecule in our simulations (see Figure S7). Nucleation may also occur at “defect sites” (unstructured regions) or points of stress concentration as in “stutters”15 along the coiledcoil. The force is governed by eq 4 with the initial condition f = fαβ. The phase boundaries move through the continuum and convert all the residues into the β-phase. This marks the onset of the plastic transition regime II, which corresponds to the force plateau in the f ε-curves (Figure 2, panels b and d). The plateau force is given by (see the Supporting Information for the derivation of eq 8):

(9)

We used eqs 7−9 to theoretically model the simulated f εcurves for all the protein coiled-coil superhelices at all pulling speeds. The results of fitting of simulated f ε-curves are presented in Figure 2 and Figure S2. Numerical values of model parameters are accumulated in Table 1.

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DISCUSSION Physical Chemistry of Protein Coiled-Coil Superhelices. We carried out the first, to our knowledge, comprehensive study of the biomechanics of the full range of polypeptide fragment types with coiled-coil motif architecture, which vary in the number of strands (from two to five) and topology (parallel and antiparallel) of α-helical packing. We 16173


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the difficulty of comparing macroscopic fiber pulling behavior to that of our coiled-coil motifs, there are some general agreements in the behavior of these systems. Bendit46,47 noted in an early keratin fiber study that β-sheet structure content scaled linearly with strain. Our coiled coils motif fragment systems exhibit similar behavior (see Figures 3 and Figure S3) following an initial elastic behavior region up to 0.2 strain. Interestingly, this behavior in our simulations is in agreement with more recent careful studies of pulled keratin fibers10 where the coiled coils begin to unravel α-helices below this same level of strain, and then begin to convert the α-helices to β-sheets above this level of strain until a strain level ∼0.5, during which both α-helices and β-sheets coexist as the former is converted to the latter (see Figure 2c and 2e). Another point of agreement with the previous studies15 is that the smallest number of amino acids per chain in phenylalanine zippers exhibiting the α-to-β transition is 42, which is larger than the critical value of amino acids (40) necessary for formation of βsheet structure.15 Force Spectra and Energy Landscape. Parameters that can be accessed directly using the MD simulation output are Lα, κα, Lβ, and f 0: Lα measures the end-to-end distances in the αphase; κα is the slope of the elastic portion of an f ε-curve; Lβ is the maximum extension in the β-phase; and f 0 ≈ f* (lβ is obtained using a wormlike chain fit to the inelastic part of a f εcurve; see Figure 2). The other parameters, zαβ, zβα, ϵαβ, and ϵ0, can be estimated by mapping the free-energy landscape. We employed a mean-field approach,48,49 ΔG(x) = −kBT ln P(x), to profile the free energy ΔG as a function of projection of the average residue length (unit length) along the superhelical symmetry axis x (reaction coordinate). An example of P(x) for myosin is in Figure 4b (see Figure S4 for P(x) and ΔG(x) for other superhelices). For ΔG(x) sampled at a small force f ≈ 0, zαβ and zβα can be readily estimated. Since zαβ and zβα barely change with f (Figure 4), ϵαβ is obtained using the Maxwell force [f 0 = ϵαβ/(zαβ + zβα)], and ϵ0 is obtained using the expression for ΔGαβ or ΔGβα. The attempt frequency A can be estimated by fitting the profiles of pα and pβ = 1 − pα with eq 6 as shown in Figure 3. The values of all these parameters for all coiled-coil systems are accumulated in Table 1. It is worth noting that for the free energy landscapes, we calculated for the 6 coiled-coil systems presented in Figure S4, there are quite similar free energy values calculated for the αstates (native states) and their corresponding α + β-states (intermediate states), as well as there being significant overlaps in the density distributions for these two sets of states. This is certainly true for the two coiled-coil containing Phe-zipper systems (see panels e and f in Figure S4). These observations are in good agreement with an interesting feature of coiled-coil systems pointed out by other authors,1 namely the existence of a flat energy landscape with a large number of conformational states being nearly isoenergetic. The isoenergetic states are perhaps best expressed in the GCN4 leucine zipper, similar to the two Phe-zipper systems (tetramer and pentamer) we studied here, that allows this coiled-coil containing regulatory protein to thereby exist in many variant coiled-coil forms, dimers, trimers, tetramers, and heptamers, in both parallel and antiparallel orientations, as a result of only minor changes in sequence and environmental conditions.1 Therefore, the significant functional diversity that can result from nearly isoenergetic state structural variants, evident in our energy landscapes’ reconstruction and from experimental studies relying upon minor changes in sequence and environments,

utilized an in silico analogue of dynamic force spectroscopy, which utilizes tensile pulling force to mechanically test proteins. The main result of our study is that, regardless of the strand number, size, and topology of their α-helical packing, all polypeptide superhelices uniformly undergo a dynamic transition from the α-helix state to the β-sheet state. All the force spectra reveal very similar force-strain profiles which differ only in the length of elastic regime (regimes I in Figure 2), and in the length of plastic transition regime (regimes II in Figure 2), which sets in at different specific values of characteristic force f * (force plateau in Figure 2) for different coiled coils. These measures depend on the size Lα and number Nh of αhelices and topology of their packing. Another important result is our finding that the characteristic force f * is a linear function of ln vf and Nh (see Figure 5). To arrive at these results, we employed the all-atom MD simulations in implicit solvent accelerated on a GPU, which has enabled us to span the simulation time scale of a few tens of microseconds in reasonable time. To confirm that the results obtained are not an artifact of the force field used in simulations, we performed a pilot study of myosin forced unfolding using the all-atom MD simulations in explicit solvent (see the Supporting Information). We observed similar dynamic behavior of the myosin coiled-coil superhelix under linearly increasing tension with the elastic regime I, plastic transition regome II, and the inelastic regime III (see Figure S6). Here too, the elastic regime continues up until the strain ε < ∼ 0.2, and the transition regime sets in as soon as the characteristic force reaches the constant value f * ∼ 150 pN (see the plateau force in Figure S6). This is slightly higher than f * = 139 pN detected in implicit solvent modeling, which can be attributed to the low-friction limit used. Nevertheless, the plasticity sets in due to the α-to-β transition (see snapshots in Figure S6). Also, in good agreement with results from implicit solvent modeling, the nonlinear regime III sets in at ε ≈ 0.9. Having confirmed that implicit solvation models of protein coiled-coils provide a good description of their properties and using the structures generated for the mixed α + β-phase, we carried out long 10 μs equilibrium simulations for myosin and fibrin, in which the force was gradually quenched to zero (with the same rate rf). The reverse β → α transition was not observed (i.e., coiled-coils extended and β-sheets formed permanently). Hence, on the 10 μs time scale of computational experiments, the α-to-β-transition marks the onset of plasticity in protein coiled-coils. Experimental biophysical studies have been carried out by many investigators on coiled-coil containing protein systems such as fibers of keratin,10 marine egg snail case protein,11,12 and hagfish slime protein,13 subjected to forced extension, while wide-angle X-ray scattering measurements determined the degree of conversion of the α-to-β transition. It is difficult to compare these results with ours directly in much detail. For example, experimentally the marine egg snail case protein fiber was observed to possess a fully reversible α-to-β transition,11 while the hagfish slime protein’s transition was irreversible13 and exhibited plastic deformation behavior as did our simulations of the coiled-coil motif fragment systems. However, it should be noted that the issue of reversible vs irreversible transitions is not addressable definitively in our computational studies given the modest time scales achievable in our current simulations (tens of microseconds), compared to those achievable experimentally (minutes), especially where transition reversibility may be a kinetically slow process. Notwithstanding 16174


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Figure 5. (a) Plot of f * vs ln vf (data points from simulations) and theoretical curves of f * vs ln vf predicted using eq 8 for chemotaxis receptor and myosin (error bars show standard deviations). The slopes and y intercepts were calculated using parameters taken from Table 1. (b) Plot of f * vs Nh (number of α-helices) and a linear fit with equation y = aNh (a = 88 and 103 for vf = 104 − 106 μm/s; error bars are standard deviations). (c) Temperature (T)-dependence of the average plateau force f * (with standard deviations) for the α-to-β transition in myosin tail and chemotaxis receptor. Red lines are a linear fit f *= 617.4 − 1.5T for myosin data and f *= 1271.5 − 3.2T for chemotaxis receptor data (υf = 105 μm/s).

predicted and simulated values of f * extracted from the f εspectra (Figures 2 and Figure S2), which further validates our theory. We also found a linear scaling of f *∼ Nh with the number of α-helices forming a coiled-coil for Nh = 2−5 (Table 1). The profiles of f * versus Nh in Figure 5b show a roughly additive contribution to the mechanical strength from the αhelix number, which weakly cooperates to sustain the stress. Indeed, for vf = 104 μm/s, f * increases from 103 pN for doublehelical myosin tail and vimentin (∼52 pN per helix) to 187 pN for the triple-helical fibrin coiled-coil (∼62 pN per helix), to 268 pN for the four-stranded Phe-zipper (∼67 pN per helix), and to 383 pN for Phe-zipper with five α-helices (∼77 pN per helix). Also, a lower f*= 232 pN for the four-stranded chemotaxis receptor (∼58 pN per helix) is due to the antiparallel arrangement, which goes to show that the parallel arrangement of α-helices provides higher mechanical strength.

suggests a compelling rationale for the utility and widespread occurrence of coiled-coil systems in nature. We used the values of Lα, κα, Lβ, zαβ, ϵ0, and A from Table 1 as the initial input and eqs 7−9 to fit the simulated f ε-curves for all the protein coiled-coils. In eqs 7 and 9, ε(t) = ΔX(t)/L is the time-dependent strain. This enabled us to refine the parameter values and to estimate lβ (all shown in squared brackets in Table 1). The results of fitting for chemotaxis receptor and myosin presented in Figure 2 (panels b and d) and in Figure S2 for the other coiled-coils show excellent agreement between the simulated and theoretical f ε-curves, which validates our theory. The refined final values of model parameters from the fit do not differ much from the initial input, which points to the internal consistency between the results of theory and simulations. We used these parameters and eq 6 to reconstruct the profiles pα and pβ. Figure 3 shows that, although both H-bond redistribution and dihedral angles’ migration capture the α-to-β transition, the H-bond based estimation of pα and pβ results in a better agreement with the simulations. Scaling Laws. The theory predicts that the stiffness of coiled-coils κα/Lα is inversely proportional to their length Lα (eqs 7), and so shorter (longer) coiled-coils are stiffer (less stiff) and have smaller (larger) extensibility. Comparison of the κα values for different model systems reveals a nontrivial dependence of κα on a number of α-helices (Nh) and parallel versus antiparallel topology of coiled coils (see Table 1 and Table S1). For instance, κα = 2169 pN for myosin and 1375 pN for vimentin, although both are double-stranded parallel coiled coils. The value of κα for the three-stranded parallel coiled coils from fibrinogen is even lower (1135 pN). It should be expected that the initial elastic component of coiled coils could be also affected by the irregularities in heptad repeats, such are stutter in vimentin19 and plasmin binding site in fibrinogen.22 The theory also predicts a logarithmic scaling of the plateau force (critical force) f*, which marks the onset of transition from the α-phase to the β-phase, with the pulling speed vf and, hence, with the loading rate rf = ksvf (eq 8). We tested this prediction by performing a fit of the profiles of f * versus ln vf for coiled-coils from chemotaxis receptor and myosin using eq 8 and by borrowing parameter values from Table 1. The results shown in Figure 5a show excellent agreement between the

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CONCLUSIONS We developed a new theory, which can be used to accurately describe dynamic transitions in wild-type and synthetic coiledcoil superhelical polypeptides under mechanical nonequilibrium conditions and to model their force-strain spectra available from dynamic force manipulations in vitro and in silico. The theory can be used to probe the mechanical and kinetic properties of any coiled-coil superhelical polypeptide and to map out their free-energy landscapes. For example, the slope and y intercept of the line of critical force for the α-to-β transition to plastic deformation f * versus ln vf (Figure 5a) can be used to estimate the critical distance per unit length zαβ and the force-free barrier height ϵ0 (Supporting Information). The scaling laws for the elastic force f versus length L, f ∼ L−1, and f * versus loading rate rf, f *∼ ln rf, and the dependence of f * on L and number of helices Nh provide a new methodology to understand naturally occurring coiled-coil proteins’ mechanical behavior, as well as to rationally design coiled-coil motifs into novel synthetic biomaterials and nanomaterials with the required mechanical strength and desired balance between stiffness and plasticity. The pulling speeds used in our study (thousands of micrometers per second) are faster than those used in singleprotein stretching experiments (hundreds-to-thousands of nanometers per second), and biological sheer rates correspond 16175


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to much slower force-loading conditions than those used here. However, our force-loading rates are thousand times slower than those used in earlier steered MD simulation studies. This has been achieved by utilizing the computational acceleration on GPUs. This is in addition to our using a very low damping coefficient for faster equilibration and conformational relaxation. Our results indicate that notwithstanding fast force loading, the unfolding dynamics we have explored here for the coiled-coil containing motifs are near-equilibrium rather than far-from-equilibrium. The theory we have developed involves the kinematic boundary conditions which correspond to the time-dependent force application (dynamic force-ramp), and so both simulations and theory model the same near-equilibrium conditions of protein forced unfolding. Eqs 7−9 are the central theoretical results of our study. When viewed as functions of the strain (ΔX/L), the force for linear elastic regime I (eq 7) and inelastic regime III (eq 7) are independent of the pulling speed (vf), which means that these results can be directly applied to protein coiled coils in a biological setting. Eq 8 for the characteristic force for α-to-β transition (f *) does involve the dependence on vf, but when fast force loading is implemented, f * can be estimated by extrapolating to slower values of vf. For example, Figure 5a shows that f * for slow force loading should be in the tens of pico-Newtons, which agrees with experiments on myosin.17

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/jacs.7b06883. More information on model systems, computational methods, derivation of model equations, data analyses as well as figures that include data on all protein systems used in this work (PDF)

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AUTHOR INFORMATION

Corresponding Author

*[email protected] ORCID

Artem Zhmurov: 0000-0002-4414-8352 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by NSF (Grant DMR1505662 to V.B. and P.K.P.), American Heart Association (grant-in-aid 13GRNT16960013 to V.B.), and Russian Foundation for Basic Research (Grants 15-37-21027 and 15-01-06721 to A.Z.).

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