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We characterized the α-to-β transition in α-helical coiled-coil connectors of human fibrin(ogen) molecule using biomolecular simulations of their forced elongation, and theoretical modeling. The force (F) - extension (X) profiles show three distinct regimes: (1) the elastic regime, in which the coiled-coils act as entropic springs (F < 100–125 pN; X < 7–8 nm); (2) the constant-force plastic regime, characterized by a force-plateau (F≈150 pN; X≈10–35 nm); and (3) the non-linear regime (F >175–200 pN; X > 40–50 nm). In the plastic regime, the three-stranded α-helices undergo a non-cooperative phase transition to form parallel three-stranded β-sheets. The critical extension of α-helices is 0.25 nm, and the energy difference between the α-helices and β-sheets is 4.9 kcal/mol per helical pitch. The soft α-to-β phase transition in coiled-coils might be a universal mechanism underlying mechanical properties of filamentous α-helical proteins.
Large structural transitions comprise an essential mechanism of protein function and dysfunction, including folding, unfolding, and misfolding, conformational activation and inactivation, aggregation, protein-protein interactions, etc. The secondary structure alterations, including the α-helix to β-sheet transition, have been suggested as a universal mechanism of protein unfolding.1–4 It has been shown experimentally5–8 and theoretically9–11 that, upon mechanical unfolding, protein polymers containing two-stranded α-helical coiled-coils, such as α-keratin,7 α-keratin-like intermediate filaments,8 wool fibers,5 vimentin11 and desmin intermediate filaments,6 form structures with β-sheet architecture. In the α-helical coiled coil, two or more right-handed α-helices wind around each other to form a left-handed supercoil.12 Fibrin is another protein polymer containing relatively large but three-stranded α-helical coiled-coils that could undergo the α-to-β phase transition, directly related to fibrin deformability underlying its vitally important functions.
Fibrin polymers form in the blood at sites of vascular injury and provide the scaffold for hemostatic plugs and pathological thrombi that block blood vessels. Fibrin is formed from fibrinogen, a 340-kDa dimeric blood plasma protein, consisting of three pairs of Aα, Bβ, and γ chains linked by S-S bonds.13,14 Two distal and one central globular parts of fibrinogen are connected by two 17-nm-long three-stranded α-helical coiled-coils (Fig. 1). Fibrin formation is initiated by the enzymatic cleavage of the fibrinopeptides A and B from the N-termini of the Aα and Bβ chains, respectively, converting fibrinogen to fibrin monomer (containing α, β, and γ chains), which maintains major structural features of fibrinogen. Monomeric fibrin self-assembles, resulting in a filamentous network, stabilized by covalent cross-linking.15
Structural changes underlying the critically important mechanical properties of fibrin were found to occur at different spatial scales.16 At the molecular level, response to fibrin clot stretching represents forced unfolding of structural domains,16,17 but there is disagreement regarding the particular structures of fibrin(ogen) (Fg) involved.16–20 Our recent work, utilizing AFM-based single-molecule unfolding and pulling simulations, has shown that mechanical unraveling of Fg is determined by the coupled step-wise unfolding of the globular γ-nodules and reversible extension-contraction of the coiled-coil connectors.21
It has been hypothesized that molecular extension of fibrin is accompanied by the α-helix to β-strand conversion of the coiled-coils.18 Using Congo red staining, which is commonly used to identify stacks of β-sheets in amyloid proteins, formation of a congophilic material in stretched fibrin fibers, presumably new β-sheets, was revealed,17 although the specificity of this method is not fully justified. The direct experimental evidence for the α-helix to β-sheet transition in stretched and compressed hydrated fibrin clots has been recently obtained using FTIR-spectroscopy.22 Hence, given the immense biological importance of fibrin deformation and lack of information about underlying molecular structural changes, monomeric Fg is a prime system to study the mechanisms of the mechanical α-to-β transition in fibrin and other α-helical proteins.
We present our results of theoretical and computational exploration of the mechanism of forced elongation of the Fg molecule and its portions - two symmetrical coiled-coils connected by the central nodule (each coiled-coil contains residues Aα27-200, Bβ58-197, γ14-139) and a single coiled-coil portion (residues Aα45-200, Bβ76-197, γ19-139). Biomolecular simulations provide a detailed nano-scale picture of structural transitions in proteins.23,24 Yet, due to the very large system size (from 500 amino acid residues for a single coiled-coil portion to 2,868 residues for the full-length Fg molecule), theoretical modeling of fibrin(ogen) remains challenging. We utilized all-atom Molecular Dynamics (MD) simulations implemented on Graphics Processing Units (GPUs) to speed up the simulations.25,26
We have shown that the force-induced molecular elongation of Fg is determined by the dynamic interplay between unwinding/extension of the α-helical coiled-coils and unfolding of the γ-chain nodules. The mechanical response of the coiled-coil connectors shows remarkable α-helix to β-structure transition to form elongated three-stranded parallel β-sheets. The force-induced structural rearrangements described at the nano-scale provide a molecular basis for understanding the unique physical properties of fibrin fibers, including their extensibility, viscoelasticity, and strain hardening.27
We carried out pulling simulations for the single three-stranded coiled-coil portion of Fg (SCC), the double three-stranded coiled-coil portions connected by the central nodule (DCC), and the full-length Fg molecule without flexible αC regions and N-termini of the Aα and Bβ chains unresolved in the crystal structure (Fig. 1; see, also, Supplementary Information (SI)). We employed a time-dependent (ramped) force f(t)=rf t (rf - force loading rate) to study the dependence of unfolding force F (mechanical response) on the molecular extension X, and a force-clamp protocol to resolve the average extension <X> as a function of constant tensile force f (see Methods). The former characteristic can be compared with experimental force-extension curves of protein mechanical unfolding. The latter is a phase diagram to describe different regimes of the mechanical elongation. Because of the high complexity of the Fg molecule, we employed Molecular Dynamics (MD) simulations in implicit water using the Solvent Accessible Surface Area (SASA) and Generalized Born (GB) models.28–31
We first studied forced elongation of the full-length Fg. The sawtooth-like force-extension profiles for Fg capture multistep sequential unfolding transitions, which occur in the symmetric left and right γ-nodules. These transitions give rise to distinct ~75–150 pN force peaks separated by ~20–30 nm peak-to peak distance (Fig. 2). These findings agree well with our previous results from coarse-grained modeling of forced Fg unfolding.21 The first two force signals are due to the β-strand (residues γ379-395) pull-out (structures 2R and 2L). The next three force signals correspond to unfolding transitions in the left γ-nodule (structures 3L–5L): first, unfolding of the C-terminal part of the γ-nodule (residues γ309-326 and γ339-379) occurs (structure 3L; transition of type 2 described previously21); second, the central domain of the γ-nodule (residues γ219-309) unfolds (structure 4L; transition of type 121); third, unfolding of the N-terminal domain of the γ-nodule (residues γ139-153 and γ182-219) occurs (structure 5L, transition of type 321). The remaining three force peaks correspond to the same unfolding transitions in the right γ-nodule (structures 5R, 3R, and 4R in Supplementary Figure 1). The mechanical elongation of the coiled-coils does not result in detectable force signals.
To characterize unfolding of the coiled-coil connectors, we carried out pulling simulations for the double three-stranded α-helical coiled-coil portion of the Fg molecule (DCC, see Methods) without globular β- and γ-nodules. The force-extension profiles have three distinct regimes (Fig. 3; Table I). At lower forces (<100–125 pN), the unfolding force (F) grows linearly with the extension (X), i.e. F~X (Fig. 3). In this linear regime, starting from the twisted state (structure 1 in Fig. 3), the coiled-coils unwind by a large angle θ = 2π (Table I) while responding elastically to the applied force (structure 2). We observed formation of transient short β-strands (“β-seeds”) comprising 4–6 amino acid residues and random coils in the least structured parts of the Aα chain (Aα88-102), Bβ-chain (Bβ133-142), and γ-chain (γ66-84). We calculated the apparent spring constant for the DCC construct in the linear regime by taking the derivative of unfolding force (F) with respect to extension (X) and found that dF/dX≈33 pN/nm. To study reversibility of the elastic transitions, we performed the refolding simulations for the DCC construct using the structures stretched by 3 nm and 7 nm as the initial states (see Methods). All the simulation runs showed full refolding; the force-retraction curves follow closely the force-extension curves, and the hysteresis is small (Fig. 3).
The linear regime persists until X = 7–8 nm. Beyond this point, the unfolding force levels off reaching a long plateau at F = 150 pN. In this constant-force regime, the coiled-coils extend to ~35–40 nm (Fig. 3), i.e., behave as a plastic material. The Aα, Bβ, and γ-chains quickly rewind back undergoing the reverse -2π rotation around the pulling axis to form the initial twisted state. Significantly, the three-stranded α-helices gradually transform to three-stranded parallel β-sheets (Fig. 3). The dynamics of the structural transitions in the DCC construct in terms of the propensity to form α-helices or β-strands, is presented in Supplementary Figure 2. The Ramachandran plots of the dihedral angles and ψ, which are sensitive to changes in the secondary structure of proteins (Fig. SI3), show that and ψ move from the α-helical region (−90°< < −30° and −60°< ψ < 0°) to the β-structure region (−180°< < −60° and 90°< ψ < 180°) through the transition region with mixed α+β character (−100°< < −75° and 0°< ψ < 90°).
When the α-to-β transition sets in, we witness the formation of exotic, spiral segments of the three-stranded parallel β-sheets (Fig. 3, structure 4), which unfold and unwind simultaneously (compare, e.g., structures 4 and 5). The coupled unfolding-unwinding transition starts from the least structured central part of the coiled-coils and involve ~90% of the residues in the Aα, Bβ, and γ chains (structure 3 in Fig. 3). Due to tension fluctuations, there is formation of transient π-helical and 310-helical structures (not shown). At X ≈ 40 nm extension, almost all the α-helical segments had converted into the β-sheets (structure 5 in Fig. 3). Because upon unwinding the internal energy of the chain does not change, +2π unwinding (−2π rewinding) in the linear (constant-force) regime corresponds to entropic contributions to the molecular extension (contraction). To examine the reversibility of unfolding in the constant-force regime, we carried out the refolding simulations for the DCC construct using partially unfolded structures stretched by 13 nm and 21 nm. We found that the force-retraction curves notably deviate from the force-extension curves, and the hysteresis is large (Fig. 3). None of the simulation runs showed full refolding to the initial folded state in the 2-μs timescale.
At a 35–40 nm extension, the unfolding force F starts to increase again, marking the beginning of the non-linear regime (Fig. 3). Here, an elongation of the coiled-coils to X≈55-60 nm is accompanied by a parabolic increase in the unfolding force, i.e., F~X2 (structure 6 in Fig. 3). This regime corresponds to excitation of the bond angles (straightening of the protein backbone), which is reflected in an unusually steep rise of F to 700 pN combined with a moderate 10-nm increase in X to 60 nm (Fig. 3). The unwinding angle is close to zero (Table I) since the Aα, Bβ, and γ chains are fully extended (structure 6 in Fig. 3). A small variation in F at X≈50-55 nm is due to unfolding of the central nodule (structure 6 in Fig. 3), which occurs only after all the coiled-coils have unwound, fully stretched and converted to β-sheets. Refolding simulations for the DCC construct in the non-linear regime from the stretched structures by 40 nm and 60 nm did not show any refolding (Fig. 3).
To model the α-to-β phase transition in the coiled-coils, we carried out force-clamp simulations (see Methods) for the single coiled-coil portion of Fg (SCC). The profiles of the average total extension of the SCC construct <X> as a function of tensile force f are sigmoidal (S-shaped; Fig. 4). In the low-force regime (<100 pN) and high-force regime (>200 pN), which correspond mostly to pure α-helices and β-structures, respectively, the molecular extension is very small. In the 100–200 pN transition range, the SCC construct undergoes a remarkable 15 nm elongation from ~3 nm to ~18 nm (Fig. 4). Here, the α-helical coiled-coils unwind and transform into three-stranded parallel β-sheets (Fig. 4). The α-helical content decreases while the share of β-strands increases with increasing pulling force.
The mechanisms of α-to-β transition in the SCC and DCC constructs are similar (Table I). The low-force regime, transition regime, and high-force regime observed in the force-clamp simulations (f – const.; Fig. 4) correspond to the linear, constant-force, and non-linear regimes of mechanical elongation in the ramp-force simulations (f(t)=rft; Fig. 3). We analyzed the propensities of the coiled-coils to form the α-helices and β-strands as a function of pulling force (Fig. SI3). These can be viewed as the force-dependent populations of the α-helices pα(f) and β-structures pβ(f). At low forces, ~90% of the DCC (or SCC) structure is α-helical, whereas the percentage of β-strands is zero. Hence, in the folded state ~10% of the coiled-coiled connectors contain random coils. When f > 200-300 pN, the amount of α-helices drops to zero, while the amount of β-strands rises up to 90%. In the fully unfolded state the same residues remain in the random coil conformation (~10% of the chains). Although the number of hydrogen bonds in the DCC construct decreases from 375 to 200 in the 0–300 pN force range (Fig. SI3), it does not reach zero even at the highest f = 500-600 pN. This is because the nascent β-sheet structure is stabilized by the transverse (inter-chain) hydrogen bonds.
We analyzed the phase diagram (Fig. 4) using a two-state model (Methods), in which the SCC is partitioned into the “α-state” (for α-helical coiled coil) and “β-state” (for three-stranded β-sheets). Because of the linear regime (F~X) observed at low forces (Fig. 3), we modeled the α-state as an entropic spring. The β-state was described by a worm-like chain.32,33 Force application induces the molecular elongation in the α-state and β-state, and lowers the energy barrier for the α-to-β transition (Methods). The total extension X(f) = pα(f)yα(f)Lα + pβ(f)yβ(f)Lβ is the superposition of extensions, where pα and pβ are populations, yα and yβ are fractional extensions, and Lα and Lβ are total extensions for the α-state and β-state, respectively. We performed a fit of the simulated data points for the SCC construct using the theoretical curve of X(f), and obtained a very good match (Fig. 4). We found that the spring constant is kα = 35.1 pN/nm and the maximum extension is Lα = 4.4 nm for the α-state. For the β-state, the persistence length is lβ = 1.5 nm, and maximum extension (contour length) is Lβ = 19.7 nm. The equilibrium distance between the α-state and transition state is zαβ = 0.25 nm, and the Gibbs energy difference separating the α-state and β-state is ΔG(0) = 4.9 kcal/mol (for a helical pitch).
We performed an in-depth computational and theoretical study of the fibrin(ogen) nanomechanics. To separate contributions to molecular elongation from the coiled-coil connectors of Fg, we performed pulling simulations for the double three-stranded coiled-coil (DCC) and single three-stranded coiled-coil (SCC) portions of Fg. These mimic the ramp-force and force-clamp modes used in experimental forced protein unfolding assays. Due to the very large numbers of degrees of freedom, we used implicit solvation models. To exclude possible model-dependent artifacts, we adopted the SASA and GB models, which are based on different approaches to solvation effects.
What verifies the simulation approach used in this study is that the mechanism of unfolding of the γ-nodules observed in all-atomic detail is quite similar to the one described in our coarse-grained modeling study.21 A minor difference was that in the all-atomic modeling the C-terminal β-strand “pull-out” from the γ-nodule did not always lead to its immediate dissociation into the C-terminal and N-terminal domains. The binding contacts at the interface of C- and N-terminal domains in the γ-nodule stabilize the structure but for a short period of time. Hence, the C-terminal β-strand “pull-out” and unraveling of the central domain (transition of type 1)21 might occur in two steps. Yet, due to the stochastic nature of the protein mechanical unfolding, both scenarios are possible. Our data are consistent with the calorimetric study, which showed that the removal of the C-terminal β-strand renders the γ-nodule less stable.34
Previous studies23 have attempted to resolve the molecular mechanism of Fg unfolding. Because of the large number of degrees of freedom (1,018,000 atoms), these authors used a 25-times faster pulling speed than the one used here. We were able to utilize a slower pulling speed and follow fewer degrees of freedom because we utilized implicit water models, which have been used in the theoretical studies of forced protein unfolding,9,11 and GPU computing. The point of force application used in Ref. 23 was different from ours; we applied pulling force to the last crystallographically resolved residue in the γ-nodule, which corresponds to the covalent γ-γ crosslinking site. In spite of the differences in methodology, the force-extension profiles for the Fg coiled-coils presented here and in Ref. 23 are very similar.
The force-extension profiles for a portion of Fg containing two coiled-coil connectors (DCC) show three distinct regimes (Fig. 3). In the linear regime, the system responds elastically to pulling force, whereas in the constant-force regime (with force plateau) the molecule behaves as a plastic material. Molecular elongation of the coiled-coils continues in the non-linear regime, where force performs work against the bond angles and dihedral angles. A similar shape of the force-extension profiles with three distinct regimes was observed for the myosin double-helical coiled-coils albeit at lower 30–40 pN forces.35 Remarkably, when forced unfolding of coiled-coils occurs concomitantly with unfolding of the globular domains as in the case of Fg monomer, the force peaks from abrupt unraveling of the globular structures overlap and mask the force plateau from unfolding of the coiled-coils (Fig. 2).
In the linear (elastic) regime, elongation of the coiled-coils is due to their unwinding around the uniaxial direction of pulling force (coiled-coils extends as a whole but Aα-, Bβ-, and γ-chains do not elongate). The internal energy of the coiled-coil connectors remains constant (ΔHel ≈ 0; data not shown) and the Gibbs energy change (ΔGel) is due to the entropy change, i.e. ΔGel ≈ -TΔSel. Hence, in the elastic regime the Fg coiled-coils act as an entropic spring. For the SCC construct, TΔSel≈ −90 kJ/mol. This is about a half of TΔS=−201 kJ/mol for unfolding of the two coiled-coils from our earlier study,21 which shows good agreement between these results and our previous estimates. In the linear regime, the coupled elongation and unwinding are fully reversible (Fig. 3). In the constant-force (plastic) regime, the forced elongation of the coiled-coils is driven by the change in the internal energy (enthalpy). For the DCC construct, ΔHpl≈ 330 kJ/mol, which is comparable to ΔH=285 kJ/mol for complete unfolding of two coiled-coils from our earlier study.21 In the constant-force regime, the coiled-coils contract and rewind but only partially refold. In the non-linear regime when the pulling force is quenched, the coiled-coils contract but remain in the random coil configuration. Hence, the three regimes are different in the unfolding reversibility, i.e., fully reversible in the linear elastic regime, partially reversible in the constant-force plastic regime, and irreversible in the non-linear regime (in microsecond simulation timescale).
A broad 100–200 pN transition range in the phase diagram (Fig. 4) reveals that the α-to-β transition is non-cooperative (α-helical segments unravel independently). The estimated spring constant in the α-helical state kα = 35.1 pN/nm is close to the linear slope of 33 pN/nm from the force-extension profiles (Fig. 3). The maximal extension of the α-helices, Lα = 4.4 nm, is close to the 3–4 nm extension for the “α-state” from the phase diagram (Fig. 4). The long persistence length lβ = 1.5 nm (~4–5 residues) in the “β-state” is reasonable given planar extended structure of β-sheets. The unfolding forces for the DCC and SCC portions are considerably larger than ~50 pN forces for unfolding for the double-helical myosin coiled-coils (35). We ascribe this difference to the more mechanically resilient three-stranded coiled-coils of the Fg molecule. The maximal extension of the β-sheets obtained Lβ = 19.7 nm corresponds to the total extension Lα + Lβ ≈ 24 nm observed at larger forces > 300 pN (data not shown).
The distance to the transition state for a helical pitch zαβ = 0.25 nm is close to the published estimate (0.12 nm),9 but shorter than the same quantity for the globular D regions (~1 nm).21 This indicates that α-helices readily transform into β-strands when the pulling force is high enough to change the dihedral angles. The α-to-β transition sets in at f*≈ 150 pN force (Fig. 4), and the energy barrier is ΔG ≠ ≈ f*zαβ /kBT = 5.4 kcal/mol. This is only slightly larger than the energy difference between the α-state and β-state at equilibrium, ΔG(0) ≈ 4.9 kcal/mol, which accounts for the disruption of one N-H…O hydrogen bond (~3 kcal/mol)37 and changing the dihedral angles (~2 kcal/mol).38 By taking the difference (Δxαβ) between the length of the α-helical pitch xα ≈ 0.54 nm (α-state), and the contour length for ~3.6 residues forming the pitch xβ≈1.3 nm (β-state), Δxαβ=xβ - xα ≈ 0.75 nm, we find that the distance between the β-state and the transition state for the reverse β-to-α transition is zβα=Δxαβ - zαβ ≈ 0.5 nm.
The mechanical and rheological behavior of fibrin clots can only be understood by integration of their materials properties at the bulk, network, fiber and molecular levels.16 The mechanism of α-to-β transition in fibrin upon forced elongation described here can explain some of the most unique features of fibrin polymers, such as strain hardening, viscoelasticity, and negative compressibility.27 The hypothetical α-to-β transition preceding the unfolding of globular γ-nodules was estimated to account for up to 100% strain of fibrin fibers.18 According to our new data (Fig. 2 and Ref. 21), unfolding of the globular γ-nodules occurs before or concurrently with the α-to-β transition, but the contribution of the coiled-coils to the overall extensibility should not depend on the order of unfolding events. This remarkable contribution is based on elongation per residue from 0.15 nm in the α-helix to 0.32–0.34 nm in the β-strand. If the fraction of unfolded fibrin molecules reaches the maximum at a whole clot macroscopic strain of ≈1.2,17 then at this strain additional unfolding mechanisms must play a role. The first structure to unfold in Fg is the globular γ-nodule,21 and unfolding of one γ-nodule with 80-nm contour length may account for about 2-fold extension of a 45-nm-long folded fibrin molecule. The additional unfolding mechanism at the strains ≥1.2 could be unwinding/unfolding and the α-to-β transition in the coiled-coils. Remarkably, this strain corresponds to a non-linear increase of the clot stiffness (strain-hardening), which can be accounted for by formation of the more mechanically resistant β-sheets.9
Both unwinding and unfolding of the coiled-coils are necessary to account for the elastic and plastic properties of the fibrin clots, inasmuch as these structural transitions have both elastic (fully reversible) and inelastic (partially reversible) regimes as the strain increases (Fig. 3). In agreement with these data, it has been shown that force–stretch response of the network17 and individual fibers19 is partially reversible with some hysteresis, which increases with strain. Also, the α-to-β transition is followed by protein aggregation within and between fibrin fibers, which can be driven by newly formed intermolecular β-sheets, shown earlier to promote formation of protein aggregates.40 In turn, protein aggregation is followed by reduction of water accessible surface area and water expulsion, which can underlie volume shrinkage of fibrin clots in response to extension (negative compressibility).16,17
Our results offer new mechanistic, structural, and thermodynamic insights into the complex dynamic behavior of stretched fibrin polymers and understanding of the unique role of coiled coils in fibrous proteins. The dynamic transition from the elastic regime to plastic regime of mechanical elongation of Fg coiled-coils followed by the remarkable α-to-β phase transition might provide a molecular basis underlying unique physical properties of fibrin network. The characteristic plateau regions in the force spectra of filamentous proteins with high α-helical content, such as fibrin(ogen), might serve as dynamic signatures for the α-to-β transition. The two-state model used here to map the free energy landscape provides a theoretical framework to describe the α-to-β transition in proteins. The unwinding of the Fg coiled coils indicates that the physical properties of proteins with α-helical coiled-coil structures are best understood in terms of the multi-dimensional energy landscape, where multiple coupled reaction coordinates characterize their unfolding nanomechanics.
We performed all-atom Molecular Dynamics (MD) simulations implemented on a GPU using the structural models of the whole human fibrinogen molecule, the double three-stranded coiled-coil (DCC) portion, and the single coiled-coil (SCC) portion described in the Supplementary Information (SI). We employed the Solvent Accessible Solvent Area (SASA) model and Generalized Born (GB) model39,28–31 of implicit solvation (see SI). Ramp-force simulations: To imitate the experimental ramp-force measurements for the Fg and DCC systems, we constrained one end of the molecule and applied a time-dependent force f(t) = f(t)n to the other in the direction n parallel to the end-to-end vector. The constrained and pulled residues were γ394 and symmetrical γ395 (for Fg), and γ139 and symmetrical γ139 (for DCC), respectively. The force amplitude f(t)=rf t was increased linearly with the loading rate rf = kspvf, where ksp = 100 pN/nm is the spring constant. The slowest pulling speed vf we could utilize to complete 2-μs runs in reasonable “wall-clock” time (~48 days per trajectory) on a GPU (GeForce GTX 580) was vf = 105 μm/s. In the simulations of force-quenched refolding, we used structures generated in unfolding runs as initial states. The force f(t)= f0-rf t was decreased linearly with the same rate rf starting from some initial value f0. Force-clamp simulations: We carried out pulling simulations at a constant force f = fn for the SCC portion. The constrained and pulled positions were γ19 and symmetrical γ139. We generated five 2-μs runs for each force value to resolve the average extension (<X>) and fluctuations (ΔX).
The “α-state” is modeled as a harmonic spring with energy Eα = kαXα2/2, where kα is the spring constant, Xα = f/kα is the average extension, and with the maximum extension Lα. The “β-state” is modeled by a worm-like chain32,33 with bending energy , where lβ is the persistence length, T is the absolute temperature (kB – Boltzmann’s constant), and with the contour length (maximum extension) Lβ. Because at f=0, the two states differ in their free energy by an amount ΔE0=Eβ - Eα, which is large compared to the thermal fluctuations (ΔE0kBT), and the molecule is in the α-state. Force application stretches the molecule, which can be quantified by the fractional extension yα(f)= Xα(f)/Lα, and lowers the energy barrier by an amount f·zαβ, i. e., ΔE(f) = ΔE0 – f·zαβ, where zαβ is the distance from the minimum (α-state) to the transition state, which for sufficiently high values of force leads to the α-to-β transition. The transition probability is given by the Boltzmann’s factor, exp[− ΔE(f) / kBT] = pβ(f)/pα(f) - the ratio of populations pα and pβ of the two states (pα+pβ = 1). In the β-state, the molecule can be extend by an amount yβ(f)=Xβ(f)/Lβ = 1-(χ(t)1/3 + ((4/3)t – 1) / χ(t)1/3)−1, where and. t = flβ / kBT.
This work was supported by the American Heart Association grant 09SDG2460023 (to VB), the National Institutes of Health grants HL030954/HL090774 (to JWW), and the National Science Foundation grant MCB-0845002 (to RID).