Article pubs.acs.org/Macromolecules
Tensile Mechanics of α‑Helical Polypeptides Korosh Torabi* and George C. Schatz Department of Chemistry, Northwestern University, Evanston, Illinois 60208-3113, United States S Supporting Information *
ABSTRACT: We have developed a statistical mechanical model of the force−extension behavior of α-helical polypeptides, by coupling a random-coil polypeptide elastic model of an inhomogeneous partially freely rotating chain, with the latest version of the helix− coil transition model AGADIR. The model is capable of making quantitatively accurate predictions of force−extension behavior of a given polypeptide sequence including its dependence on pH, temperature and ionic strength. This makes the model a valuable tool for single-molecule protein unfolding experimental studies. Our model predicts the highly reversible unraveling of α-helical structures at small forces of about 20 pN, in good agreement with recent experimental studies.
20 and 25 pN.3 Such reversible elongation prevents a large amount of energy dissipation while work is being produced by the myosin motor proteins. Recent high resolution measurements reveal how α-helical structures function as ideal reversible springs which are essential components of the mechanically functioning protein myomesin.4 Recently αhelical polypeptides have also found application in synthetic molecular machine studies.6 Molecular dynamics (MD) simulation can be used to study the tensile mechanics of α-helices. However, in order to make quantitatively reliable predictions of polypeptide mechanical properties, improved force fields and solvation models still need to be developed.7 At the same time, equilibrium sampling using an MD simulation of a polypeptide pulling experiment is a computationally expensive task. The pulling velocity of a typical single-molecule experiment is about 10−7 to 10−5 Å/ns 3, which is too small to be accessible to MD simulation. Polypeptides that form helices in solution do not show a simple-two state equilibrium between fully helical and fully random-coil conformations. Instead, they sample numerous possible helical windows of different length formed at different locations along the polypeptide chain. It is known that the helix nucleation time scale is of order of 10−100 ns,8 therefore equilibrium sampling of all polypeptide configurations is impossible for pulling velocities larger than 10−2 Å/ns. Numerous MD simulation studies,9−11 with pulling velocities of several orders of magnitude larger than what is required for an equilibrium sampling of all available conformations, have led to rather unjustified conclusions on the mechanical behavior of α-helical polypeptides.
I. INTRODUCTION Proteins are large molecules consisting of one or more polypeptide chains folded into a biologically functional structure. Of special interest is to understand the mechanical properties of proteins that carry out their biological function under tension. Examples include the giant muscle protein titin responsible for passive elasticity of muscle,1 the cytoskeletal protein spectrin responsible for cell shape and flexibility,2 the myosin coiled-coil unit that undergoes a massive structural deformation facilitating the motor protein’s mechanical function,3 and fast-folding domains of the muscle protein myomesin which act as reversible strain absorbers.4 Singlemolecule experiments in which a protein is unfolded by application of mechanical force (via an atomic force microscope (AFM) probe or magnetic or optical tweezers) provide a unique opportunity to investigate the forced unfolding behavior of these mechanically functioning proteins. In this work we focus on the mechanical properties of the most abundant secondary structure of proteins namely the α-helix. The secondary structure (local structure) of a polypeptide is determined by a special arrangement of the backbone dihedral angles that allows for a certain hydrogen bonding pattern along the polypeptide chain. The α-helix is a secondary structure in the form of a right-handed helix which is stabilized by hydrogen bonding of the backbone CO group of every ith residue with the backbone NH group of (i + 4)th residue (residues are conventionally numbered from the N-terminus). A polypeptide chain void of any structure is said to be in a random-coil conformation. It is known that certain polypeptide sequences have a large tendency to fold into an α-helical structure, even in absence of any tertiary structure.5 The tensile mechanics of a polypeptide chain is affected by its tendency to form α-helical structures. It is has been experimentally shown that coiled-coil structures (consisting of two α-helical strands) undergo a drastic but highly reversible elongation at small forces between © 2013 American Chemical Society
Received: July 29, 2013 Revised: September 6, 2013 Published: September 26, 2013 7947
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II. THEORY AND MODEL DEVELOPMENT A. Force−Extension Behavior of a Random-Coil Polypeptide. The elasticity of a random-coil polypeptide (a polypeptide chain without any tertiary or secondary structure) is mainly due to its backbone dihedral angle degrees of freedom. For experimentally and biologically relevant forces (below ∼1 nN) the changes in backbone bond lengths and bond angles are negligible. Therefore, a freely rotating chain (FRC) (chain of fixed bond length and angles with free dihedral angles) seems to be a rational choice to model the force−extension behavior of a random-coil polypeptide. In this approach, the average extension of a FRC under a given tensile force can be calculated using the transfer matrix method.20 To make the FRC model more realistic for a polypeptide chain, one needs to consider that one out of every three backbone dihedral angels of a polypeptide chain is constantly locked due to double bond formation.19 Furthermore, due to steric effects, the other two dihedral angles are not completely free to assume all values. Hanke et al. investigated the effect of dihedral angle constraints on the tensile mechanics of the FRC model.19 They showed that fixing one out of every three dihedral angles has a considerable effect on the force−extension curve. They demonstrated that the force−extension behavior of a random-coil polypeptide is best modeled by an inhomogeneous partially freely rotating chain (iPRFC) model (“inhomogeneous” refers to using different bond lengths to realistically describe the polypeptide chain, while “partially” refers to fixing one out of every three dihedral angles). On the basis of iPFRC model predictions, they suggested the following simple formula for the force−extension curve of a random-coil polypeptide
Several statistical-mechanical models have been developed to explain the equilibrium force−extension behavior of helical polypeptides.12−14 These models in general couple an elastic model for the polypeptide chain in a random-coil conformation with a helix−coil transition model that predicts the secondary structure propensity of the polypeptide. A typical helix−coil transition model predicts the equilibrium helical content of a polypeptide based on probabilities of all the possible helical windows formed along the polypeptide chain. All the statistical mechanical models cited above rely on basic helix−coil transition models which make them suitable for only qualitative predictions of force−extension behavior of homopolypeptides. The main purpose of the present work is to develop a statistical-mechanical model capable of making quantitative predictions of force−extension behavior of helical polypeptides comparable to data obtained from high resolution singlemolecule pulling experiments. The helix−coil transition algorithm incorporated into our model is the latest version of AGADIR.15 AGADIR is at present the only algorithm capable of making accurate predictions of helical propensity of a given polypeptide sequence for a large range of experimental conditions. As will be explained in more detail in the next section, AGADIR takes into account numerous energetic and entropic effects experimentally proved to influence the helical propensity of a given sequence of amino acids. AGADIR is parametrized based on a large amount of experimental data and its accuracy has been tested and improved by comparison to numerous circular dichroism (CD) and NMR spectroscopy measurements.5,15−18 For the tensile mechanics of the polypeptide segments in a random-coil conformation we have used the inhomogeneous partially freely rotating chain19 (iPFRC) model. The iPFRC model is shown to better represent the elastic behavior of a random-coil polypeptide in comparison to commonly used worm-like chain (WLC) or freely jointed chain (FJC) models.19−21 By coupling AGADIR with iPRFC, we have developed a computationally inexpensive statistical mechanical coarse grained model capable of making quantitatively reliable predictions on the tensile mechanics of α-helical polypeptides. The model further includes an algorithm for including an AFM tip (or optical tweezers potential) to describe a typical singlemolecule pulling experiment. This makes the model a valuable tool for single-molecule studies of proteins. Among all the previously developed models and MD simulation studies, to the best of our knowledge, our model is the only one capable of predicting the small unfolding force of 20−30pN recently measured in a high resolution AFM single-molecule experiment.4 Moreover, the potential of mean force (PMF) predicted by our model can explain the highly reversible and nondissipative elasticity of α-helical polypeptides (also experimentally observed3,4). The present article is organized as follows. In section II, after a brief review of the iPFRC and AGADIR models, we develop our model for force−extension predictions of an α-helical polypeptide within both a constant force ensemble (equivalent to a constant force magnetic tweezers experiment) and an AFM (or optical tweezers) pulling experiment. In this section we also show how the PMF can be calculated based on the force− extension curve of a stiff cantilever AFM experiment. In section III, we present our model’s predictions for several case studies on different helix forming polypeptides in different singlemolecule pulling experimental setups. Section IV includes a brief summary and discussion of the model and its predictions.
f=
kBT ⟨ξ⟩ ⎡ 1 ⟨ξ⟩ 3⎤ + ⎥ ⎢ Lc ⎣ cB Lc − ⟨ξ⟩ a⎦
(1)
where f is the tensile force exerted on the polypeptide molecule, ⟨ξ⟩ is the average extension of the molecule in the direction of pulling, kB is the Boltzmann constant, T is the temperature, and a and Lc are the Kuhn length and contour length of the molecule, respectively, and c is a fitting parameter that depends on the geometry (bond lengths and angles) of the iPFRC chain. The Kuhn length determines the slope of the linear section of the force−extension curve for small forces and the contour length is the length of the iPFRC chain under an infinitely large tensile force. Lc = NB, where N is the number of residues (amino acids) in the polypeptide and B is the contour length per residue, which is estimated (based on the backbone bond lengths of an amino acid) to be 3.654 Å.19 The following values for the Kuhn length and the fitting parameter are suggested based on the numerical solution of the iPFRC model for a polypeptide molecule, a = 7.45 Å and c = 0.807.19 The iPFRC formula (eq 1) ignores the effect of finite size on the force−extension curve of a short polypeptide molecule. However it has been shown20 that the finite size effect becomes irrelevant for forces larger than (kBTlp)/(Lc2), where lp = 3.18 Å is the persistence length of the iPFRC chain.19 This means that for any short polypeptide made up of more than two residues, size effects can be neglected for forces larger than about 3pN. Therefore, the force−extension formula in eq 1 can be safely used for polypeptide molecules without introducing any significant finite size error. Another model commonly used to describe the force− extension behavior of a flexible polymer molecule is the worm7948
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like chain (WLC) model.22 However, there are some drawbacks for using this model for a polypeptide molecule. It can be shown that when force exceeds kBTlp/B2 the asymptotic behavior of the continuous WLC model deviates from that of a discrete chain.20 For molecules with large persistence length such a double stranded DNA (lp ≈ 30 nm) such deviation occurs at very large forces above about 1 nN. However, for a polypeptide chain that has a small persistence length of about 4 Å20,23 the continuous assumption of the WLC model breaks down for tensions above 10pN. Figure 1 is a plot of the iPFRC
charged residues, electrostatic interactions of the helix macrodipole with the charged residues, side-chain/side-chain interactions, formation of local motifs, etc.24 Helix formation in polypeptides, as opposed to the protein’s tertiary structure, cannot be explained in terms of a two-state equilibrium between an entirely helical and an entirely randomcoil conformation. Instead numerous different helical segments (of different lengths at different positions) can form along a given polypeptide chain. There is an equilibrium probability associated with each helical segment. Therefore, in order to estimate the average helical content of a given polypeptide molecule, one needs to estimate the probability of formation of all the possible helical windows. To the best of our knowledge, the only model capable of making quantitatively accurate predictions for almost any polypeptide sequence is the helix− coil transition model developed by Serrano, Muñoz, and coworkers, AGADIR.5,15−18 The algorithm is parametrized based on a large amount of experimental data available for the different energetic effects that contribute to the stability of a given α-helical segment along a given polypeptide chain. AGADIR is formulated by writing the partition function of a polypeptide Q as a summation of different possible helical conformations along the chain,
Figure 1. iPFRC force−extension curve of eq 1 (solid red line) along with the Marko and Siggia22 force−extension curve of the WLC with lp = 3.18 Å (solid circles) for a polypeptide chain of 60 residues at a temperature of 273 K. The WLC force−extension curve reproduces the iPFRC results if an artificial contour length is used (here, 65 residues instead of 60).
Q= +
∫ dr e−u(r)/k T = ∫rc dr e−u(r)/k T B
∑∫ i,j
B
dr e−u(r )/ kBT
ij
(2)
where r is the configuration space vector and u(r) is potential energy as a function of the molecule’s configuration (since we are dealing with a constant temperature system we can leave the momentum space out of the partition function). The first integral on the right-hand side of eq 2 is the partition function of the entirely random-coil conformation with the subscript rc indicating that the configuration space is limited to that of the entirely random-coil conformation and the second term is the summation over partition functions of all possible helical windows. Subscripts ij on the corresponding integral terms represent additional limits on the configuration space r for the chain that includes a j-residue-long helical segment that begins at ith residue. Dividing the partition function by that of the entirely random-coil conformation (which is equivalent to shifting the free energy, by a constant term), we can write the partition function as
formula of eq 1 (solid red line) along with the Marko and Siggia22 formula for the WLC with lp = 3.18 Å (solid circles) for a polypeptide chain of 60 residues at a temperature of 273 K. Apparent deviation of the WLC model cannot be fixed for both small and large force regimes by only adjusting the persistence length of the WLC. However, by using an artificial contour length Lc, the WLC model reproduces the iPFRC result. The open circles in Figure 1 show the WLC formula with 65 residues. Therefore, using the WLC formula to fit the force− extension data for a random-coil polypeptide molecule is likely to result in about 10% overestimation of the contour length (or number of residues) of the polypeptide molecule. Of course this assumes that the iPFRC model, which captures the molecular construction of the random-coil polypeptide chain with more detail, predicts a more realistic force−extension curve. B. AGADIR: A Helix−Coil Transition Model. Within an α-helical structure the backbone dihedral angles maintain certain values in a way that position the backbone carbonyl group of the ith residue close to the backbone amide group of the (i + 4)th residue. Therefore, a random-coil structure, in which the backbone dihedral angles are free to explore a larger region of configuration space, is entropically more favorable. Different amino acids (due to differences in their side-chains) have different sterically hindered dihedral angles and therefore different entropic aversion to form helical structures. An αhelical structure is mainly stabilized by (i, i + 4)NH−CO hydrogen bonding which is enthalpically more favorable than hydrogen bonds formed between the same groups and the water molecules (by almost 1 kcal/mol5). Besides the backbone hydrogen bonding, other interactions that determine the tendency of a given sequence of amino acids to form an αhelical structure are electrostatic interactions among the
Q=1+
∑ K ij (3)
i,j
where Kij is the probability of formation of the ij helical window relative to an entirely random-coil conformation:
K ij =
∫ij dr e−u(r)/ kBT ∫rc dr e−u(r)/ kBT
(4)
Therefore, the free energy cost of forming a given helical window relative to the entirely random-coil conformation would be:
ΔGij = −kBT log K ij
(5) 15
In the latest version of AGADIR, the free energy of each possible helical window is estimated as the summation of terms related to “intrinsic helical propensities” (entropy loss) of all 7949
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residues within the helical window, energy associated with all the backbone hydrogen bonds formed, side-chain interaction of all the residues within the helical window, energy contribution of all the residues outside the helical segment (including capping box motifs, hydrophobic staple motifs, and ...), electrostatic interactions among all the charged residues, and the electrostatic interactions of the helix macrodipole with the charged residues. The effect of N and C termini protection is also included in the model (It is known that acetylation or succinylation of the N-terminus or amidation of the C-terminus tends to increase the helicity of a polypeptide by removing the charges that interact unfavorably with the helix macrodipole and by providing an extra hydrogen bonding site with the helix backbone24). All the above entropic and energetic terms are parametrized based on an empirical analysis of a large amount of helical content experimental data for different polypeptides in aqueous solutions. Several thermodynamic models incorporated into the algorithm improve the model predictions for a large range of temperature, pH and ionic strength.17 The formulation presented in eq 2 is a single-segment approximation of the algorithm (AGADIR-1s). This formulation can be easily generalized to allow for conformations that include more than one helical segment simultaneously formed along the polypeptide chain (AGADIR-ms).18 The effect of the single-segment approximation on the force−extension predictions of the model is studied in Supporting Information. C. Tensile Mechanics of an α-Helical Polypeptide: Constant Force Ensemble Model. In the previous section we discussed how the AGADIR algorithm estimates the probabilities of different helical windows along a polypeptide chain (Kijs in eq 4). To calculate the equilibrium force− extension curve of a polypeptide chain or to estimate its helical tendency when it is being pulled, we need to modify those probabilities to include the effect of the tensile force. Here we start with the fact that the partition function used in AGADIR (eq 2) is merely the partition function of the molecule in a constant force ensemble where the value of force is zero, i.e., Q = Q( f = 0). The partition function for any given value of tensile force f would then be defined as: Q (f ) = +
∫
dr e−[u(r ) − fξ(r )]/ kBT =
∑∫ i,j
∫rc
K ij(f ) = e
×
∫rc dr e−[u(r ) − fξ(r )]/ kBT
(8)
∫0
f
df ′ ⟨ξ⟩ij , f ′ +
∫0
f
df ′ ⟨ξ⟩rc , f ′ (9)
where ⟨ξ⟩ij,f and ⟨ξ⟩rc,f are the average extensions in the direction of the applied tension f, for the polypeptide chain with the ij helical window and for the one that is in an entirely random-coil conformation, respectively. Modeling a randomcoil polypeptide chain with iPFRC, the integral for the entirely random-coil conformation (third term on the right-hand side of eq 9) could be easily calculated via eq 1. However the integral for the polypeptide that includes a helical segment (second term on the right-hand side of eq 9) needs further consideration. If we assume there is only one helical segment along the polypeptide chain, then the molecule is composed of three distinct segments: one helical segment and two randomcoil segments at both ends (Figure 2). Assuming the three
Figure 2. Schematic of polypeptide containing a j residue long helical segment starting at the ith residue under constant tension f.
dr e−[u(r ) − fξ(r )]/ kBT
segments are freely jointed to each other (no coupled energy terms), the average extension of the molecule containing an ij helical segment could be written as
(6)
⟨ξ⟩ij , f = ⟨ξrc , N − j⟩f + ⟨ξhel , j⟩f
(10)
where ⟨ξrc,N‑j⟩f is the average extension of a N−j residue long random-coil polypeptide chain (where N is the total number of residues in the polypeptide molecule) and ⟨ξhel,j⟩f is the average extension of a j residue long helical segment in the direction of pulling. The assumption of freely jointed segments is only made at a few points along the chain and therefore does not affect the force extension behavior of a long polypeptide chain. Furthermore, since AGADIR treats different conformational segments as energetically independent and within iPFRC the finite size effect is shown to be negligible, this assumption is not
∫ij dr e−[u(r ) − fξ(r )]/ kBT ∫rc dr e−[u(r ) − fξ(r )]/ kBT
∫ij dr e−u(r )/ kBT
∫rc dr e−u(r )/ kBT
ΔGij(f ) = ΔGij(f = 0) −
where ξ(r) is the extension of the molecule in the direction of the force. According to the above partition function, the probability of forming an ij helical window relative to the entirely random-coil conformation (when both are subject to fixed tension f) is: K ij(f ) ≡
×
∫ij dr e−[u(r ) − fξ(r )]/ kBT
By defining ΔGij(f) ≡ −kBT log Kij(f), it is straightforward to show that (see the Supporting Information for more details)
dr e−[u(r ) − fξ(r )]/ kBT
ij
−ΔGij(f = 0)/ kBT
(7)
Using the zero force helical window free energies ΔGij(f = 0) (which is AGADIR’s ΔGij defined in eq 5), we can write 7950
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include additional terms for all the possible combinations of nonoverlapping helical windows. For AGADIR-2s, eq 3 should be modified to
adding to the assumptions made in the constituent models (AGADIR and iPFRC) of our model. ⟨ξrc,N‑j⟩f is calculated via eq 1 of the iPFRC model. Considering the fact that the persistence length of the α-helical segment is about 15 nm25 (which is equivalent to 100 residues in the helical conformation), we assume the average extension of the helical segment is equal to the average projected length of a rigid rod in the direction of pulling (see the Supporting Information for more details): ⎡ ⎛ flhel , j ⎞ ⎛ flhel , j ⎞−1⎤ ⟨ξhel , j⟩f = lhel , j⎢coth⎜ ⎟ ⎥ ⎟−⎜ ⎢⎣ ⎝ kBT ⎠ ⎝ kBT ⎠ ⎥⎦
Q=1+
i,j
⟨ξ⟩f =
N−j
(11)
N
N−j
⟨ξ⟩ij , i ′ j ′ = ⟨ξrc , N − (j + j ′)⟩ + ⟨ξhel , j⟩ + ⟨ξhel , j ′⟩
(12)
As is apparent in eq 12, in AGADIR the minimum length of a helical segment (or cooperative length) is taken to be six residues, four residues being in the helical conformation plus two nonhelical caps.18 On the basis of the above formulation, the probability of finding the kth residue in a helical conformation when molecule is under tension f would be N
Phel , k(f ) =
(16)
Here the first term on the right-hand side is the extension of a N − (j + j′) residue long random-coil polypeptide chain, which is calculated via eq 1, and ⟨ξhel,j⟩f and ⟨ξhel,j′⟩f are average extensions of the two helical segments in the direction of pulling that are calculated by eq 11. D. Tensile Mechanics of an α-Helical Polypeptide: AFM Experiment Model. In the previous section, we formulated our model in the constant force ensemble. The experimental equivalent of the constant force model would be attaching one end of the polypeptide molecule to a fixed surface and the other end to a magnetic bead that is placed in the uniform force field of magnetic tweezers.26,27 Single molecule experiments are also done with AFM or optical tweezers where the free end of the molecule is trapped in a parabolic potential well (see Figure 3). In this case, the force exerted on the
⟨ξ⟩rc , f + ∑ j = 6 ∑i = 1 K ij(f )⟨ξ⟩ij , f 1 + ∑ j = 6 ∑i = 1 K ij(f )
(15)
i ,j i′,j′
where the second summation runs over all the possible (nonoverlapping) helical windows ij and i′j′, simultaneously formed along the chain. Since AGADIR treats different segments of the chain as being independent of each other, the probability of spontaneously forming two helical windows is simply the product of probabilities of the two windows formed individually. To incorporate the two segment approximation within the force−extension model, instead of eq 10, we use the following to calculate the average extension of a chain containing ij and i′j′ helical windows:
where lhel,j is the length of the helical segment,lhel,j = j × 1.5 Å (each amino acid contributes 1.5 Å to the length of the αhelical segment24). On the basis of the above model one can calculate ΔGij(f) in eq 9 for any given value of the force f. Consequently, the probability of all the possible ij helical windows, for any given tension, can be calculated as Kij( f) = e−βΔGij(f). Having calculated Kij( f) one can estimate the average extension of the molecule as: N
∑ K ij + ∑ ∑ K ijK i′ j ′
k−1
∑ j = 6 ∑i = k − j + 1 K ij(f ) N
N−j
1 + ∑ j = 6 ∑i = 1 K ij(f )
(13)
And the overall helical content of the polypeptide molecule under tension f is N
H (f ) =
∑k = 1 Phel , k(f ) N
(14)
In the model developed above we have invoked a single helical segment approximation. Thus, the probability of having more than one helical window at the same time is assumed to be negligible. Because of the free energy barrier of nucleating each separate helical window, in general, the above assumption would not introduce a large error. For each helical segment, initially three residues need to be fixed into helical dihedral angles (against the forces of entropy) before the first enthalpically favorable (i, i + 4) hydrogen bond can form. Once the initial nucleus is formed, for any additional residue that gives up its dihedral angle freedom, one additional hydrogen bond is formed. Therefore, in most cases, the formation of more than one helical segment along the same chain is highly unlikely in comparison to conformations including one helical segment with frayed ends. It has been shown that results based on AGADIR-1s, for polypeptides not longer than 56 residues, coincide with those from AGADIR-ms within 0.3% error.18 However, it is straightforward to modify the above model to include multiple helical segments. For example, to allow up to two helical windows, we just need to
Figure 3. Schematic of single-molecule pulling of a helical polypeptide with AFM.
molecule is f = k(L − ξ), where k the spring constant or stiffness of the cantilever, L is the displacement of the cantilever base or center of the optical trap with respect to the fixed surface and ξ is the extension of the molecule in the direction of pulling. For such a setup, we need to modify our model to include the potential energy of the cantilever in the partition function of the combined system of the molecule plus cantilever tip. For this system, the partition function and the free energy as a function of L are Q (L ) = 7951
k
2
∫ dr e−β[u(r)+ 2 (L−ξ(r)) ]
(17)
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and
k k (L − ξij , L)2 − (L − ξrc , L)2 2 2
ΔGij(L) = ΔGij +
G(L) = −kBT log Q (L)
(18)
+ fij , L ξij , L −
The derivative of the free energy with respect to cantilever base displacement L is the average force measured by the AFM:
∂G = k⟨(L − ξ)⟩ ∂L
+
2
(20)
ξij , L = ⟨ξrc , N − j⟩ f
ij , L
The molecule’s extension and therefore measured force fluctuate about their average values. Therefore, the average extension vs force measured in a constant force setup is not exactly the same as the average extension vs average force measured with an AFM setup. The force−extension curve measured with an AFM setup generally depends on the cantilever stiffness or spring constant k. To be able to calculate the Kij values using the iPFRC model (which is originally developed within a constant force ensemble20), we assume the extension fluctuation of the molecule with a given ij helical window is negligible relative to fluctuations due to transitions between conformations of different helical windows. Our model thus only includes the extension fluctuations due to helix−coil transitions occurring within the polypeptide molecule. Therefore, the extension of the molecule of a given helical conformation is equal to the value corresponding to the mechanical equilibrium between the cantilever and the polypeptide molecule (simultaneously satisfying eqs 23 and 24). This assumption becomes more realistic when a relatively stiff cantilever is used. For a soft cantilever, even for a fixed helical conformation, the molecule’s extension fluctuates considerably. Nevertheless, except for a very soft cantilever (k < 1 pN Å) and large forces (extensions above 80% of the contour length), the summation of the iPFRC potential of mean force and the cantilever potential would still be reasonably symmetric about its minimum and this assumption would not result in a substantial error. On the basis of the above assumption, we can rewrite eq 20 as 2
K ij(L) ≡
/ kBT
∫ij dr e−u(r)/ kBT δ(ξ(r ) − ξij , L)
2
/ kBT
∫rc dr e−u(r)/ kBT δ(ξ(r ) − ξrc , L)
e−(k /2)(L − ξrc ,L)
≡ e−ΔGij(L)/ kBT
df ′ ⟨ξ⟩rc , f ′
(22)
(23)
To calculate f ij,L and ξij,L, eq 23 (representing the mechanical equilibrium) needs to be solved simultaneously with the following equation:
∫ij dr e−[u(r ) + (k /2)(L − ξ(r )) ]/ kBT
e−(k /2)(L − ξij ,L)
df ′ ⟨ξ⟩ij , f ′ − frc , L ξrc , L
fij , L = k(L − ξij , L)
2
∫rc dr e−[u(r ) + (k /2)(L − ξ(r )) ]/ kBT
frc , L
fij , L
where ΔGij(defined in eq 5) is determined by the AGADIR algorithm and f ij,L is the average tensile force for a given displacement L which is:
(19)
At a given value of L, the probability of any possible ij helical window relative an entirely random-coil conformation is:
K ij(L) ≡
∫0
∫0
+ ⟨ξhel , j⟩ f
(24)
ij , L
⟨ξrc,N−j⟩f ij,L and ⟨ξhel,j⟩f ij,L are calculated via eq 1 and eq 11, respectively. Once we have solved for f ij,L and ξij,L at a given value of L, we can calculate ΔGij(L) via eq 22 and then the helical window probabilities as Kij(L) = e−(ΔGij(L)/(kBT). For a given cantilever base displacement L, the average extension of the molecule, the residue level helicity and the overall helical content of the molecule are calculated via eq 12, eq 13 and eq 14, respectively. For the present work, we have only developed a single-segment approximation for the AFM model. E. Potential of Mean Force. The molecule plus cantilever model developed in the previous section can be used to calculated the potential of mean force (PMF) of a given polypeptide molecule. The PMF as a function of the molecular extension ξ is defined as ϕ(ξ) ≡ −kBT log
∫ dr e−u(r)/k T δ(ξ(r ) − ξ) B
(25)
−(u(r))/(kBT)
It can be shown that ∫ dr e δ(ξ(r) − ξ) is proportional to the partition function of the molecule plus cantilever system (eq 17) in limit of a very large spring constant k → ∞.28 Therefore, the PMF can be calculated as the integral of the force−extension curve at the stiff cantilever limit. The PMF can be considered as the free energy surface of the molecule with the extension ξ as the reaction coordinate. The probability density of finding the molecule at a given extension ξ in the direction of a constant tensile force f is proportional to e−(ϕ(ξ)−fξ)/(kBT). The PMF predicts bistability if ϕ(ξ) − fξ displays two distinct minima for a given value of force. Bistability of a molecule with a large free energy barrier between the folded and unfolded states leads to a less reversible response (larger dissipation) for a pulling and contraction cycle. As will be discussed in more detail in the Results, a typical αhelical peptide reveals the opposite behavior. For forces that unfold the helical segment, the mostly helical and mostly random-coil states are mediated by numerous transition states of partially helical conformations with free energies only slightly above the minima.
(21)
where ξij,L and ξrc,L are the average extensions (corresponding to mechanical equilibrium) of the molecule in the ij helical conformation and in the entirely random-coil conformation, respectively. δ is the delta function that limits the partition function integrals to the average extension values. Using eq 19, we can calculate ΔGij(L) as (see the Supporting Information for more details):
III. RESULTS The model developed in the previous section is capable of predicting the force−extension behavior of a helix-forming polypeptide of a given sequence, for a given temperature, pH, ionic strength and N and C termini protection states. In what follows, we present our model’s predictions for several 7952
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conformation has a larger extension in the direction of pulling and therefore is energetically more favorable than a random-coil conformation (see eq 6). When the force is increased to about 10 pN the (E)60 chain is mostly helical and the helical segment is almost aligned with the tensile force. Further increasing the force starts unraveling the helical segment and a distinct force plateau appears in the force−extension curve. As shown in Figure 5, increasing the force from 10 to 30 pN unravels the polypeptide from 80% to 0% helicity. For forces above 30 pN, the polypeptide is completely in a random-coil conformation and the force−extension curve follows that of the iPFRC. B. Residue Level Helicity Predictions. Our model calculates the probability of all the possible ij helical windows for a given polypeptide chain under tension. Therefore, the model is able to predict residue level helicities along the polypeptide molecule that is being pulled. Helical tendencies of individual residues can be experimentally measured by NMR analysis. Figure 6 shows the model predictions of residue level
polypeptide pulling case studies. Results reported for the constant force model (sections A and B) are based on the twosegments approximation model and those including AFM (sectionsD and C) are based on single-segment approximation. Please see the Supporting Information for a more detailed comparison of 1s vs 2s model results. A. Force−Extension Behavior of a helical vs a Random-Coil Polypeptide. In a neutral solution, the glutamic acid (E) side-chain is charged (pKa = 4.15). Because of unfavorable electrostatic interactions, a long chain of glutamic acids is a poor helix former at pH = 7.0. However glutamic acid has a high helical propensity at low a pH. Figure 4
Figure 4. Plot of average extension as a function of applied tensile force for (E)60 at pH = 2.0 (mostly helical) and pH = 7.0 (mostly random-coil). The red line represents the iPFRC force−extension curve of eq 1. Figure 6. Residue level helicity prediction for (AEAAKA)10 with protected termini for several different tensions.
shows the force−extension curve of (E)60 with unprotected termini at 273 K and ionic strength of 0.1 M at two different pH values, 2.0 and 7.0 (reported results are the average extensions in a constant force ensemble). At pH = 7.0 the force−extension curve closely follows that of an iPFRC model of a completely random-coil polypeptide. However, at pH = 2.0 the force−extension curve clearly deviates from that of a random-coil. As shown in Figure 5, a free (E)60 molecule is on
helicities of (AEAAKA)10 with protected termini calculated with a two-segment approximation at 295 K, pH = 7.0 and ionic strength of 0.1M. As shown in this figure, even for this highly helical polypeptide sequence the two tails of the molecule are mainly in the random-coil conformation. Figure 7 reveals that
Figure 7. Residue level helicity prediction for (AEAAKA)10 with unprotected termini for several different tensions. Figure 5. Average helical content of helical (E)60 pH = 2.0 as a function of applied tensile force. The dashed line represents the corresponding average extension of the chain.
the unprotected polypeptide has even longer frayed ends, in a way that the last residues at the two ends of the molecule hardly ever assume a helical conformation. Applying a small tensile force of up to 10 pN tends to induce more helicity by making the central helical segment longer and more stable. Forces beyond that threshold unravel the central helical segment. The probabilities of all possible ij helical windows relative to a completely random-coil conformation Kij (probability of forming a j residue long helical window initiated at the ith residue) are shown in Figure 8. When no tension is applied, the
average about 50% helical at pH = 2.0. As we start pulling on the chain, the average helical content initially increases. This phenomenon has been previously predicted by other theoretical models13,29 and also experimentally observed.30 The favorable energetic effect of helix formation explains this rather counterintuitive behavior of force induced helicity at low forces. As shown in Figure 4, for forces below 10 pN, a helical 7953
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Figure 8. Probability of all the possible helical windows Kij’s within (AEAAKA)10 with unprotected termini for several different tensions. Kij is the probability of forming a j residue long helical window initiated at the ith residue.
polypeptide chain spends most of its time in a conformation formed of two random-coil segments on both ends and a large central helical segment that fluctuates in both length and position. The length of the central helical window j fluctuates between 45 to 60 residues (the entire molecule) while the lengths of the two frayed ends fluctuate between 0 to 10 residues. As shown in Figure 8, as the helix starts to unravel under tension, the helical window probabilities tend to spread out more evenly among helices of different lengths that are formed at different locations along the chain. C. AFM Single Molecule Experiment. The results presented in the previous sections are within a constant force ensemble. Another common set up for single molecule pulling experiments is with an AFM or optical tweezers where one end of the molecule is trapped in a parabolic potential while the other end is tethered to a fixed surface (Figure 3). Figure 9 shows model results for AFM pulling of (AEAAKA)10 with unprotected termini at pH of 7.0, temperature of 295 K and ionic strength of 0.1 M, for cantilever stiffness values of 0.5 pN/ Å and 5.0 pN/Å. For comparison, the force−extension curve of the constant force ensemble model for the same molecule and at the same conditions is also included in the plot. As explained in section II.D, our model assumes that the extension fluctuations within the polypeptide of a given helical conformation are negligible relative to those caused by helix− coil transitions. As can be seen in this plot, the difference between the three force−extension curves is more pronounced
Figure 9. Model predictions of force−extension curves of (AEAAKA)10 for single-molecule pulling experiments with AFM and different cantilever stiffness values.
in the helix unraveling region. This is where the molecule’s extension shows large fluctuations. The three force−extension curves coincide for forces above 40 pN where the molecule is entirely in a random-coil conformation. D. Potential of Mean Force. As discussed in the model development section, we can calculated the potential of mean force (PMF) for a helix forming polypeptide based on the force−extension curve of the AFM model with a large cantilever spring constant k. Figure 10 shows the helical content of (AEAAKA)10 as a function of average extension for 7954
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Figure 10. Model predictions of average helical content of (AEAAKA)10 for single-molecule pulling experiments for AFMs with different cantilever stiffness. Figure 12. Plot of ϕ(ξ) − fξ for several values of force between 15 and 35 pN. For forces between approximately 25 and 30pN the plots display two distinct minima that indicate a bistability of extension for those conditions.
different values of the cantilever spring constant. As shown in this plot, the results for spring constants larger than 500 pN/A are almost identical. Therefore, a spring constant larger than 500 pN/A should result in a correct evaluation of the PMF. Here we use the large spring constant value of 5000 pN/A. The force−extension curve for k = 5000 pN/A is also presented in Figure 10. The integral of this curve from zero to a given extension is the PMF at that extension. Figure 11 shows the PMF and helical content of (AEAAKA)10 as a function of the molecule’s extension.
contraction cycle, the molecule stays fairly close to equilibrium and there is no significant force−extension hysteresis between the forward (unfolding) and reverse (folding) directions. In this way, almost all the work done on the molecule upon pulling is restored upon contraction.
IV. SUMMARY AND DISCUSSION Coupling the random-coil polypeptide elastic model of iPFRC with the helix−coil transition model of AGADIR, we have developed a statistical mechanical model of force−extension behavior of α-helical polypeptides. The model is developed for a polypeptide chain void of tertiary structures (i.e., relatively short polypeptide chains or polypeptide segments for which the tertiary structure has been disrupted first). The presence of tertiary structure would possibly alter helix−coil transition equilibrium and force−extension behavior of the polypeptide. The model explains the force−extension behavior associated with a quasi-equilibrium pulling velocity and includes only (i, i + 4) α-helices. As explained in section III.E, due to the special characteristics of α-helical structures, for experimentally and biologically relevant pulling velocities (below 10−2 Å/ns), the force−extension of a helical polypeptide is reversible and close to equilibrium. Fast and nonequilibrium pulling velocities within MD simulations might lead to formation of the thermodynamically unstable (i, i + 3) 310-helices.31 AGADIR and therefore our model only include α-helical structures and do not take into account the 310 helical structures. Also since AGADIR is only parametrized for an aqueous solution, our model at this point is not able to handle pulling experiments within organic solvents. The effect of organic solvents on the helix−coil transition32 and the comparison of AGADIR with other helix−coil transition models18,24 have been widely studied in literature. The residue level predictions of our model (see section III.C) reveal that forced unraveling of an α-helical polypeptide occurs via helix−coil transitions among numerous likely helical windows formed along the polypeptide chain. Therefore, although a two-state model can be applied to the tertiary structure unfolding of proteins, modeling the forced unraveling of an α-helical polypeptide with a two-state model of fully helical and fully random-coil states would result in invalid
Figure 11. PMF and average helical content of (AEAAKA)10 as a function of polypeptide extension.
The force−extension curve in Figure 10 and the PMF in Figure 11 demonstrate the characteristics of forced unfolding of the secondary structure of a helical polypeptide. As can be seen in Figure 10 the first inflection point of the force−extension curve corresponds to the point at which unraveling of the helical structure starts. Beyond this point the molecule becomes less stiff upon pulling ((∂2f/∂ξ2) < 0). The concave region of the PMF that corresponds to negative slope in the force− extension curve is indicative of mechanical bistability. As shown Figure 12, for forces between 25 and 30 pN ϕ(ξ) − fξ displays two distinct minima. Therefore, under a fixed tension in this region, the extension fluctuates about two different values (here between about 100 and 140 Å). In one state the molecule is mostly helical and in the other mostly in a random-coil conformation. The free energy barrier between the two minima determines the time scale of the transition between the two states. The small free energy barrier (less than 2kBT), as can been seen in Figure 12, indicates a reversible or elastic behavior of a helical polypeptide. Therefore, a helical polypeptide unravels, upon relatively fast pulling, without dissipating a large amount of work. In other words, within a pulling and 7955
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Office of Science, Office of Basic Energy Sciences under Award Number DE-SC0000989.
conclusions that are mainly artifacts of a large pulling velocity that is used in some MD simulation studies.9,10 The potential of mean force as a function of molecular extension predicted by our model reveals that, in the region of bistability, only a small free energy barrier develops between the extended (mainly random-coil) and contracted (mainly helical) conformations. This explains the highly elastic and reversible force−extension behavior of the helical polypeptides. Helical polypeptides can unravel upon relatively fast pulling velocities (up to 10−2 Å/ns) without dissipating a large amount of work. In this way, almost all the work done on the molecule upon pulling can be restored upon contraction. This characteristic is what makes them ideal spring components in some mechanically functioning proteins. This reversible extension behavior has been experimentally demonstrated for the myosin coiled-coil3 and the α-helical linker between the C-terminal immunoglobulin (Ig) domains My12 and My13 of myomesin.4 On the other hand when dissipative or shock absorbing behavior is required, nature chooses mainly nonequilibrium unfolding of tertiary structures, like that in the giant protein titin that is responsible for the passive elasticity of the muscle.1 As opposed to previously developed models13,14,29 that were mostly concerned with a qualitative explanation of the elastic behavior of α-helical homopolypeptides, our model is capable of making quantitative predictions of the force−extension behavior of a given polypeptide sequence for given solution conditions of pH, temperature and ionic strength. This makes the model a valuable tool for single-molecule experimental studies of forced protein unfolding. As shown in section III.A the force−extension curve of a helical polypeptide deviates from that of a nonhelical polypeptide for forces below 30pN (see Figure 4). The plateau developed in the force−extension curve, due to unraveling of helical structures, has been observed in a recent high resolution single molecule pulling experiment.4 Although a low resolution AFM experiment might not reveal the helix−coil transition plateau, it has been shown that the WLC model does not perfectly fit the force−extension data of a helix forming polypeptide33. Our model provides a more accurate force−extension curve with a minimal computational cost. The source codes of the model would be provided upon request and a web application will be available for fast online calculations at www.nanoHUB.org.
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ASSOCIATED CONTENT
S Supporting Information *
Detailed derivations of eqs 9, 11, and 22 and a comparison between models with the 1s and 2s approximation for two different polypeptides. This material is available free of charge via the Internet at http://pubs.acs.org/.
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REFERENCES
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We would like to thank Professor Luis Serrano and his coworkers for providing us with the source codes of AGADIR. This work was supported by the Nonequilibrium Energy Research Center (NERC) which is an Energy Frontier Research Center funded by the U.S. Department of Energy, 7956
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