A coarse-grained α-carbon protein model with anisotropic hydrogen-bonding

Abstract

We develop a sequence based α-carbon model to incorporate a mean field estimate of the orientation dependence of the polypeptide chain that gives rise to specific hydrogen bond pairing to stabilize α-helices and β-sheets. We illustrate the success of the new protein model in capturing thermodynamic measures and folding mechanism of proteins L and G. Compared to our previous coarse-grained model, the new model shows greater folding cooperativity and improvements in designability of protein sequences, as well as predicting correct trends for kinetic rates and mechanism for proteins L and G. We believe the model is broadly applicable to other protein folding and protein–protein co-assembly processes, and does not require experimental input beyond the topology description of the native state. Even without tertiary topology information, it can also serve as a mid-resolution protein model for more exhaustive conformational search strategies that can bridge back down to atomic descriptions of the polypeptide chain.

INTRODUCTION

Understanding the general energetic principles of protein self-assembly is a long-standing problem in biophysical chemistry. Recently, the framework of energy landscape theory has provided direction in the design of protein folding models that should exhibit correct folding thermodynamics by optimization of a funneled free energy surface.^1–3 The spatial resolution of the models do not have to be at full atomic detail since it is well known that models with topological features that correctly reproduce the spatial distribution of local and non-local contacts are sufficient for reproducing trends in thermodynamic and kinetic folding data.⁴

Inspired by early efforts of Thirumalai and co-workers,^5–8 we have developed a “minimalist” protein bead model that uses an α-carbon (C_α) trace to represent the protein backbone, in which structural details of the amino acids and aqueous solvent are integrated out and replaced with effective bead–bead interactions. These physics-based potentials are formulated so that there is still a connection between bead type and amino acid sequence in a reduced letter code, and hence stand distinct from Go-based potentials.⁹ We have successfully used the coarse-grained protein model to study the folding mechanism and kinetics of several proteins of the ubiquitin α/β topology,^10–13 to analyze folding simulation protocols,¹⁴ for competition between folding and aggregation in which we correlate differences in aggregation kinetic rates to differences in structural populations of unfolded ensembles,¹⁵ and most recently in aggregation processes relevant for the Aβ peptide indicted in Alzheimer’s disease.¹⁶

When the experimental folding and aggregation data to be understood is of higher spatial or timescale resolution, then isotropic interactions used in protein bead models may break down. One example is the study of early molecular origins of amyloid fiber formation for the Aβ peptide, in which the mature amyloid aggregate has a precise morphology of unbranched fibers composed of parallel intermolecular β-sheets.¹⁷ To understand these more complex protein assembly or co-assembly problems, it is important to both retain the efficiency of a single bead C_α model while incorporating some of the orientation-dependent properties of amino acids in protein structures. Several models formulated in this spirit include the extension of bead Go-potentials with orientation-dependent statistical potentials,¹⁸ or amino acid specific residue–residue distances.¹⁹

More closely related to this work are formulation of backbone hydrogen bond potentials in the context of off-lattice bead models.^3,20–22 Onuchic and Cheung incorporated an implicit hydrogen bond in terms of a pseudo-dihedral angle between four C_α centers straddling two separate beta-strands potential within their Go model that uses two centers per residue.³ However, their formulation incorrectly assumes that the strands’ C_α centers and hydrogen bonds lie in the same plane, when in fact hydrogen bonds are roughly perpendicular to the planes described by the C_α centers. Brooks and co-workers (private communication) use a three bead per residue model in which the C_α centers are straddled by additional centers embedded with a point dipole to represent the carbonyl and amide peptide linker. The work by Klimov and Thirumalai^21,22 approximates virtual positions of CO and NH moieties based on C_α positions, which are then used to determine whether the strands are well oriented to form hydrogen bonds. However, their implementation only takes into account hydrogen bond directionality and not hydrogen bond distance, and as a result the folding transition does not exhibit great cooperativity, with folding transitions occurring over a broad temperature range. Furthermore, their model is only effective for α-helical and anti-parallel β-sheet structures, but could not adequately describe parallel β-sheets. The protein model of Smith and Hall²⁰ uses a four center amino acid in which hydrogen-bonds are described as pseudo-bonds between residues to restrict both distance and orientation to realize α-helical and β-sheet structure. In all of these coarse-grained models, the additional centers per residue for a N residue chain scales up the computational cost by ~(cN)², where c is the number of centers per residue.

In this article we propose a reformulation of a one-site α-carbon model to introduce a potential of mean force hydrogen bonding term that encourages the cooperative formation of protein-like secondary structures. The orientation-dependent hydrogen bonding term is based on a similar functional form developed by Marcus and Ben-Naim²³ and later adopted by Silverstein et al.²⁴ to characterize hydrogen-bonding in a model of bulk water. Our protein model now incorporates a mean field estimate of the orientation dependence of the polypeptide chain that give rise to specific hydrogen bond pairing to stabilize α-helices and β-sheets. The model is first parameterized for protein G (PDB code: 2GB1),²⁵ and then validated using folding studies of protein L (PDB code: 2PTL).²⁶ As we show in the Results, the model shows improvements in designability and greater folding cooperativity, and kinetic rates and mechanistic outcomes consistent with experiment.

MODELS AND METHODS

Energy function

The modified minimalist model potential energy function is given by

$$\begin{array}{cc}\hfill & E=\sum _{\text{angles}}\frac{1}{2}{k}_{\theta }{(\theta -{\theta }_0)}^2\hfill \\ \hfill & +\sum _{\text{dihedrals}}\left[\begin{array}{c}A[1+\cos (\varphi +{\varphi }_0\left)\right]+B[1-\cos (\varphi +{\varphi }_0\left)\right]+\hfill \\ C[1+\cos 3(\varphi +{\varphi }_0\left)\right]+D\left[1+\cos \left(\varphi +{\varphi }_0+\frac{\pi }{4}\right)\right]\hfill \end{array}\right]\hfill \\ \hfill & +\sum _{i,j\ge i+3}4{\epsilon }_{\mathrm{H}}{S}_1\left[{\left(\frac{\sigma }{r_{ij}}\right)}^{12}-S_2{\left(\frac{\sigma }{r_{ij}}\right)}^6\right]+\sum _{\text{Hbonds}}{U}_{\mathrm{HB}}\hfill \end{array}$$ (1)

where θ is the bond angle defined by three consecutive C_α beads, ϕ is the dihedral angle defined by four consecutive C_α beads, and r_ij is the distance between beads i and j. The hydrophobic strength ε_H sets the energy scale. The bond angle term is a stiff harmonic potential with force constant k_θ = 20 ε_H/rad². The optimal bond angle θ₀ for bead i is set to 95° if bead i – 1 has helical dihedral propensity, and 105° otherwise.

Our model has been extended to now include new dihedral types in the turn region. As C_α-only models lack chirality, we introduce −/+90° turns (designated Q and P, respectively) to distinguish the native topology from its mirror image decoys, and 0° dihedral (designated U) to impose some rigidity in hairpin turns, beyond the original model turn T parameters. In accordance with the flexible nature of turn regions, these new dihedral types have lower barriers than their helical and extended counterparts. Each dihedral angle in the protein chain is then designated to be one of the following types: helical (H), extended (E), or one of the turns (T, P, U, or Q). The parameters A, B, C, D, and ϕ₀ in Eq. (1) are chosen to produce the desired minima (Table I). While all dihedral types encourage formation of the assigned secondary structures, they also allow access to other competing local secondary minima through manageable (~1–2.8 ε_H)barriers.

Table I.

Parameters for Various Dihedral Types

Dihedral type	A (εH)	B (εH)	C (εH)	D (εH)	φ0 (rad)	Local minima (global minima in bold)
H (Helical)	0	1.2	1.2	1.2	+0.17	−65°, +50°, 165°
E (Extended)	0.45	0	0.6	0	−0.35	−160°, −45°, +85°
T (Turn)	0.2	0.2	0.2	0.2	0	−60°, −60°, +180°
P (+90°)	0.36	0	0.48	0	+1.57	−155°, −25°, +90°
Q (−90°)	0.36	0	0.48	0	−1.57	−90°, +25°, +155°
U (0°)	0.36	0	0.48	0	+3.14	−115°, +0°, 115°

We have also increased the number of bead flavors from three of our original model to four in our new model. The third term in Eq. (1) represents nonlocal interactions between these four bead flavors: strong attraction (B), weak attraction (V), weak repulsion (N), and strong repulsion (L). The amino acid sequence of a protein can be mapped to its four-flavor sequence using the mapping rule shown in Table II, and the bead types determine the type of non-bonded interaction between two beads (Fig. 1). The parameters in Eq. (1) for attractive interactions B-B, B-V, and V-V all have S₂ = −1, while S₁ = 1.4, 0.7, and 0.35, respectively. For repulsive interactions, S₁ = 1/3 and S₂ = −1 for L-L, L-V, and L-B interactions; and S₁ = 1 and S₂ = 0 for all N-X interactions. The sum of van de Waals radii σ is set at 1.16 to mimic the large exclusion volume due to side chains.

Table II.

Mapping 20-Letter (20) Amino Acid Code to Coarse-Grained Four-Letter Code (4)

20	4	20	4	20	4	20	4
Trp	B	Met	B	Gly	N	Asn	L
Cys	B	Val	B	Ser	N	His	L
Leu	B	Ala	V	Thr	N	Gln	L
Ile	B	Tyr	V	Glu	L	Lys	L
Phe	B	Pro	N	Asp	L	Arg	L

Figure 1.

Non-bonded interaction energy as a function of pair-wise distance between bead i and j. Interactions BB, BV, and VV have attractive minima at r_ij = 1.3 while NX and BL/VL/LL interactions are purely repulsive.

The last term in Eq. (1) represents a new distance and orientation-dependent potential that models backbone hydrogen bond explicitly, to describe a pair-wise mean force hydrogen bond interaction U_HB, which is inspired by the Mercedes Benz (MB) model of water first introduced by Marcus and Ben-Naim²³ and further developed by Silverstein and co-workers²⁴. In the original MB model, water molecules are represented as two-dimensional discs with three symmetrically arranged arms, separated by an angle of 120°. Water molecules interact through a standard Lennard-Jones term and an explicit hydrogen-bonding (HB) interaction that is favorable when the arm of one molecule aligns with the arm of another. We have adapted the functional form of the hydrogen bonding interaction to our three-dimensional minimalist protein model. The hydrogen bond potential between two beads i and j is given by:

$$\begin{array}{cc}\hfill U_{\mathrm{HB}}={\epsilon }_{\mathrm{HB}}F& (r_{ij}-r_{\mathrm{HB}})\hfill \\ \hfill & \times G\left(\mid {\mathbf{t}}_{\mathrm{HB},i}\cdot {\widehat{\mathbf{r}}}_{ij}\mid -1\right)H\left(\mid {\mathbf{t}}_{\mathrm{HB},j}\cdot {\widehat{\mathbf{r}}}_{ij}\mid -1\right)\hfill \end{array}$$ (2)

where

$$\begin{array}{cc}\hfill & F(r_{ij}-r_{\mathrm{HB}})=\exp [-{(r_{ij}-r_{\mathrm{HB}})}^2∕{{\sigma }_{\text{HBdist}}}^2],\hfill \\ \hfill & G\left(\mid {\mathbf{t}}_{\mathrm{HB},i}\cdot {\widehat{\mathbf{r}}}_{ij}\mid -1\right)=\exp \left[\left(\mid {\mathbf{t}}_{\mathrm{HB},i}\cdot {\widehat{\mathbf{r}}}_{ij}\mid -1\right)∕{{\sigma }_{\mathrm{HB}}}^2\right],\hfill \\ \hfill & H\left(\mid {\mathbf{t}}_{\mathrm{HB},j}\cdot {\widehat{\mathbf{r}}}_{ij}\mid -1\right)=\exp \left[\left(\mid {\mathbf{t}}_{\mathrm{HB},j}\cdot {\widehat{\mathbf{r}}}_{ij}\mid -1\right)∕{{\sigma }_{\mathrm{HB}}}^2\right]\hfill \end{array}$$ (3)

where r_ij is the distance and ${\widehat{\mathbf{r}}}_{ij}$ the unit vector between beads i and j, respectively. The distance dependent term F is a Gaussian function centered at the ideal hydrogen bond distance r_HB. For the direction dependent terms G and H, we use an exponential instead of a Gaussian function to ensure a smoother potential energy surface. The vectors t_HB,i and t_HB,j are unit vectors normal to the planes described by bead centers (i − 1, i, i + 1) and (j − 1, j, j + 1), respectively. The ideal hydrogen bond distance r_HB is set to 1.35 length units for α-helices and 1.25 length units for β-sheets in accordance with a survey of secondary structures in the PDB database. All other hydrogen bond parameters are identical for α-helices and β-sheets, with the width of functions F, G, and H set by σ_HBdist = σ_HB = 0.5.

The hydrogen bond potential is evaluated for all i-j bead-pairs capable of forming hydrogen bonds. Depending on its dihedral propensity, each bead is assigned a hydrogen bond forming capability from three possible types: helical (designated A), sheet (designated B), or none (designated C). For a bead assigned B, the hydrogen bond potential is evaluated between itself and all B-beads situated within a cutoff distance of 3.0 length units. For a bead assigned A, helical hydrogen bond potential is evaluated if its +3 neighbor is similarly assigned A. We find that the helical hydrogen bond is better modeled in a C_α-only model as an interaction between (i, i + 3) bead pairs, rather than (i, i + 4). From a survey of helices in the PDB, the distribution of r_i,i+3 has both a smaller mean and variance than r_i,i+4. Hence a potential using (i, i + 3) bead pairs is more stringent in discriminating between helical and non-helical geometry. The strength of the hydrogen bond is modulated by ε_HB, which is set to 0.7ε_H if the bead pair is B-B, B-V, or V-V. For L-X and N-X pairs, a higher ε_HB of 0.98ε_H is required to compensate for the non-bonded repulsion. This provides anisotropy in our C_α-only model: L and N residues can maintain closer contact with their hydrogen bonding partners, while remaining repulsive to beads in all other directions.

Protein model

The structural, thermodynamic, and kinetic properties of protein L and G have been well characterized experimentally.^27–35 Both proteins consist of an N-terminus hairpin, made up by β-strands 1 and 2, followed by a helix, and lastly a C-terminus hairpin made up by β-strands 3 and 4. Despite their similar topologies, L and G share only 15% sequence identity, and fold via different mechanisms. Experimental studies have shown that while the transition state of protein L consists of partially formed β-hairpin 1,^35,36 that of protein G comprises of partially formed β-hairpin 2.^30,37 Our existing sequence-based model has been shown capable of predicting the mechanistic differences in L and G folding,¹³ something not possible with Go potentials.

Here we show that our new model preserves this sequence-based feature, and can thus replicate the different folding mechanisms of L and G. In developing the model we optimized the potential energy parameters for protein G in order to reliably reach a global minimum corresponding to the native state topology using simulated annealing, as well as to yield reasonable thermodynamics, such as sharp cooperative melting curves and heat capacities. We then fixed those parameters to validate the model by characterizing the kinetic mechanism of protein G, as well as the thermodynamics and kinetic mechanism of protein L.

The amino acid sequences of proteins L and G were mapped to reduced minimalist code as per Table II. The dihedral angle propensities were assigned according to their respective PDB structures, with the hairpin turns described using P, U, and Q to encourage the correct chirality. Since we wish to focus on whether differences in the folding behaviors are due to sequence, we assign identical dihedral propensities to hairpins in both L and G. However, the first hairpin turn in protein L (Phe, Ala, Asn, Gly, Ser) is one residue longer than that of protein G (Gly, Lys, Thr, Leu). To address this we use a modified sequence for protein L in which the 11th residue (Asn) is omitted. Dihedral propensities in the hairpins in both proteins can now be similarly assigned for fair comparison, although the model can be reformulated with this extra bead if desired. The hydrogen bond forming capability (A, B, or C) follows the dihedral specification above. The mapped sequence, dihedral propensity and hydrogen bond assignments are listed in Table III.

Table III.

Sequence, Dihedral, and Hydrogen Bond Assignments for Proteins L and G

Protein L
1° 2PTL	VTIKANLIFANGSTQTAEFKGTFEKATSEAYAYADTLKKDNGEYTVDVADKGYTLNIKFAG
1° 2PTL (without Asn-11)	VTIKANLIFAGSTQTAEFKGTFEKATSEAYAYADTLKKDNGEYTVDVADKGYTLNIKFAG
1° model L (mapped):	BNBLVLBBBVNNNLNVLBLNNBLLVNNLVVVVVLNBLLLLNLVNBLBVLLNVNBLBLBVN
1° model L (optimized):	NNBLVNBNVNNNNLNVLVLNNBLLVNNLVVVVBNNVLLLLNLVNVLVVLLNVNBLBLBNN
2° model L:	EEEEEEEQUPEEEEEEETPTHHHHHHHHHHHHHHHTPUEEEEEEEEEPUQEEEEEEE
Hbond model L:	BBBBBBBBCCBBBBBBBBCCAAAAAAAAAAAAAAACCCCCBBBBBBBCCCBBBBBBBB
Protein G
1° 2GB1	MTYKLILNGKTLKGETTTEAVDAATAEKVFKQYANDNGVDGEWTYDDATKTFTVTE
1° model G (mapped):	BNVLBBBLNLNBLNLNNNLVBLVVNVLLBBLLVVLLLNBLNLBNVLLVNLNBNBNL
1° model G (optimized):	VNVLBNBLNLNVLNLNNNLVBLVNNNLLVBLLVVLLLNVLNLVNVLNVNNNBNBNN
2° model G:	EEEEEEEQUPEEEEEEETTTQHHHHHHHHHHHHHHTTTTTEEEEEPUQEEEEE
Hbond model G:	BBBBBBBBCCBBBBBBBBCCCAAAAAAAAAAAAAAACCCCBBBBBBCCBBBBBB

Mutations made are indicated in bold.

The initial mapping of the primary sequence from the 20-amino acid code to the 4-letter minimalist code contains some ambiguity. For instance, lysine has both a long hydrocarbon chain and a charged amine group, and could be treated as either hydrophilic or hydrophobic. The initial energy landscape contains many competing local minima due in part to such ambiguity. Sequence design based on the minimal frustration principle is done to smooth the potential energy surface and improve foldability. Our sequence design strategy is based on the theoretical criterion^1,2 that a foldable heteropolymer sequence has a significant energy gap ΔE between its native-state energy E_native and average misfold energy ⟨E_misfold⟩. Using our initial mapping sequence, we generate a library of misfolded (non-native) structures from simulated annealing. To obtain a better folding sequence, we generated sequences with various single mutations, threaded them to structures in the misfold library, and select the mutant sequence that maximizes the energy gap ΔE. To minimize drift from the original sequence, we allow only single mutations of types B↔V, V↔N, or N↔L, or dihedral mutations. The mutation process is repeated until we obtain a foldable sequence that finds the native state reliably 50% of the time using simulated annealing.

Simulation protocol

All simulations are performed in reduced units with mass m, energy ε_H, length σ₀, and k_B set to unity. The bond length between adjacent C_α beads serves as the unit of length σ₀, and is held rigid by using the RATTLE algorithm.³⁸ Reduced temperature and time are given by ^T* = ε_H/k_B and $\tau ={({m\sigma }_0^2∕{∊}_{\mathrm{H}})}^{1∕2}$, respectively. We use constant-temperature Langevin dynamics with a friction coefficient of 0.05τ⁻¹, and a timestep of 0.005τ to perform simulations for characterizing the thermodynamics and kinetics of folding.

For each simulated annealing run we launch 50 trajectories at a high temperature (T* = 1.6) and evolve them for 1250τ to generate uncorrelated, unfolded conformations, then gradually cool these trajectories to T* = 0.1 for 7500τ. The trajectories are then annealed at T* =0.45 for 50τ, and cooled for 5000τ to T* = 0.1, and the anneal-cool cycle repeated once more before the resulting structure is quenched from T* = 0.1 to T* = 0.

The free energy landscape is characterized with the multidimensional histogram technique.^39,40 We collect multiple nine-dimensional histograms over energy E, radius of gyration R_g, number of native contacts formed Q, number of native contacts formed between strand 1 and strand 2 (Q_β1), number of native contacts formed between strand 3 and 4 (Q_β2), and native-state similarity parameters χ, χ_α, χ_β1, and χ_β2, where χ is given by

$$\chi =\frac{1}{M}\sum _{i,j\ge i+4}^{N}h\left(\epsilon -\mid r_{ij}-r_{ij}^{\text{native}}\mid \right)$$ (4)

The double sum is over beads on the chain, and r_ij and $r_{ij}^{\text{native}}$ are the distances between beads i and j in the state of interest and the native state, respectively; h is the Heaviside step function, with ε = 0.2 to account for thermal fluctuations away from the native-state structure. M is a normalizing constant to ensure that χ = 1 when the chain is identical to the native state and χ ≈ 0 in the random coil state. The remaining χ parameters are specific to their respective elements of secondary structure. That is, χ_α involves summation over beads in the helix, and χ_β1 and χ_β2 involve summation over beads in the first and second β-sheet regions, respectively.

From the histogram method, we get the density of states as a function of a set [O] of nine order parameters, Ω([O]) = Ω(E, R_g, Q, Q_β1, Q_β2, χ, χ_α, χ_β1, χ_β2), which can be used to calculate thermodynamic quantities. In constructing the free energy surfaces, we collect histograms at 14 different temperatures: 1.30, 1.00, 0.80, 0.60, 0.50, 0.40, 0.38, 0.36, 0.34, 0.32, 0.30, 0.25, 0.20, and 0.15. We run five to eight independent trajectories at each temperature and collect 4,000 data points per trajectory. The potential of mean force W along reaction coordinate Q is given by

$$W\left(Q^{\prime }\right)=-kT\ln \left[\int d\left[O\right]\delta (Q-Q^{\prime })\Omega \left(\right[O\left]\right)e^{-E∕k_{B}T}\right]$$ (5)

The folding kinetics is studied using mean first passage time (MFPT) based on a native state cut-off. With the MFPT method, we decorrelate 2000 independent trajectories at T* = 1.6 for 1250τ, jump to the temperature of interest, and continue evolving the trajectories. We recorded the time τ_i that each trajectory took to enter the native basin of attraction, defined as Q > 0.8. The fraction of trajectories folded at time t is then calculated by P_nat(t) = (no. of trajectories with τ_i < t)/N. Analysis of the P_Nat(t) kinetic data are detailed in Results and Discussions.

Studies of transition state (TS) ensembles are performed using the P_fold analysis method.⁴¹ We first identify putative transition states from various projections of order parameters onto the free energy surface. Because we are vetting the new model against a known mechanism, we focused our free energy projections for protein L and G along the order parameters Q and/or χ_β1 and Q and/or χ_β2, respectively, in order to collect putative TS structures. P_fold analysis is then performed: for each putative TS structure, we launch 100 trajectories at the folding temperature, evolve them for 1000τ, and evaluate the probability (P_fold) that these trajectories fall into the folded basin (defined as Q > 0.8). Structures with 0.4 ≤ P_fold ≤ 0.6 are considered to be part of the TS ensemble.

RESULTS AND DISCUSSIONS

Sequence design and native structures

We obtained an optimized sequence for protein L after 12 sequence mutations and three dihedral mutations, while the optimized sequence for protein G consists of 11 sequence mutations and one dihedral mutation. Table III compares the optimized sequences to their original mapping. We find that the original mapping is robust since 50% of the sequence mutations involved ambiguous definitions of valine (B or V) or alanine (V or N), and thus could be explained by these amino acids being “borderline” on the hydrophobic scale. We find a trend that valines and alanines in the core tend to be retained as B and V (more strongly hydrophobic), while those on the periphery are mutated to V and N (less hydrophobic).

We performed simulated annealing using these optimized sequences to obtain the lowest energy structures (Fig. 2). First we compare the structural similarity of the native state of our protein L and G models with the experimental structures using the Combinatorial Extension (CE) method.⁴² The CE algorithm excludes loop α-carbon positions to align the model and solution structures despite the different lengths of the loop regions. Using the CE method the new model gave RMSDs of 2.6 Å for Protein L and 3.0 Å for protein G, compared to the old model RMSDs of 4.4 Å for Protein L and 5.3 Å for protein G.¹³ We also calculated the root mean square distance (RMSD) of C_α atoms between these simulated native structures and their NMR counterparts using the rms.pl script from the MMTSB toolbox.⁴³ To ensure a stringent comparison, this time we do not allow gaps or deletions in our alignments, although we modified the 2PTL coordinate file to omit Asn-11 to allow a bead-to-bead comparison with our 60-bead model of protein L. The calculated RMSDs of our simulated native structures are 4.4 Å for Protein L and 3.0 Å for protein G using the alignments with no gaps.

Figure 2.

Simulated Annealing Results for Protein L and G. (a) PDB structure of Protein L (2PTL) with N-terminus loop region (residue 1–17) omitted. (b) Lowest energy structure from simulated annealing of 60-residue optimized sequence of Protein L. RMSD between 2PTL and our model protein L is 4.4 Å (c) PDB structure of Protein G (2GB1). (d) Lowest energy structure from simulated annealing of 56-residue optimized sequence of Protein G. RMSD between 2GB1 and our model protein G is 3.0 Å.

Thermodynamics

Figure 3 plots the thermodynamic averages of percentage folded P_Nat [Fig. 3(a)], heat capacity C_v [Fig. 3(b)], and radius of gyration R_g [Fig. 3(c)] against temperature for Protein L and G. Compared to results from our old model with fewer flavors and without the hydrogen bond potential,¹³ the new model demonstrates improved folding cooperativity. The folding temperature T_f, defined as the temperature at which P_Nat = 0.5, is 0.36 for protein L and 0.325 for protein G. The thermal stability plots show sharp transitions about T_f, a sign of greater folding cooperativity. The heat capacity and radius of gyration plots likewise show distinct transitions. The collapse temperatures are T_θ = 0.36 for protein L and T_θ = 0.335 for protein G, indicating that folding (T_f) is concomitant with collapse (T_θ).

Figure 3.

Thermodynamics averages for proteins L and G as functions of temperature. (a) Percentage folded P_Nat, (b) heat capacity C_v, and (c) radius of gyration R_g.

The thermal stability P_Nat plot suggests that Protein L is more stable than protein G at any given temperature. This disagrees with experimental findings that protein G is marginally more stable than protein L under various denaturant conditions.^30,35 It has been suggested that protein L’s instability arises in part from torsional strain in the second hairpin.³⁶ Since we have adopted identical dihedral propensities for hairpins in our model L and G to focus on sequence effects, our models do not take into account this torsional destabilization and could explain why our model protein L appears more stable than protein G. The heat capacity peak for protein L has a larger magnitude than that of protein G, which could be explained by protein L forming more hydrophobic contacts and hydrogen bonds in its native state than protein G.

To examine the free energy landscape, we project the potential mean force W along various order parameters. Figure 4(a,b) show the projections along Q for protein L and G at different temperatures. At their respective folding temperatures, proteins L and G each have two minima (denatured and native), suggesting a two-state folding mechanism. Figure 4(c,d) show the two-dimensional (2D) projections along χ_β1 and χ_β2 for L and G at their folding temperatures. For Protein L, the minimum-energy path proceeds through a transition state in which β-hairpin 1 is partially formed while β-hairpin 2 is structureless, before reaching the native state. Protein G, on the other hand, has a minimum energy path that involves formation of a native-like β–hairpin 2, before crossing the transition state to reach the native state. The 2D projections are in agreement with experimental evidence that the denatured state ensemble (DSE) and transition state ensemble (TSE) of protein L consist of partially formed β-hairpin 1,^35,36 while those of protein G involve a partially buried β-hairpin 2.^30,37 However, P-fold analysis is needed to determine whether transition state ensembles obtained from the free energy projections are meaningful with respect to folding mechanism.

Figure 4.

Free energy surface projections onto different reaction coordinates. (a) Projection of protein L’s free energy along reaction coordinate Q over temperature range of 0.32 < T < 0.39. (b) Projection of protein G’s free energy along reaction coordinate Q over temperature range of 0.29 < T < 0.36. (c) Projection of protein L’s free energy surface onto χ_β1 and χ_β2 at T_f = 0.36. (d) Projection of protein G’s free energy surface onto χ_β1 and χ_β2 at T_f = 0.325. Contours for (c) and (d) are spaced 0.5 kT apart.

Transition states analysis

The 2D free energy projections along χ_β1 and χ_β2 [Fig. 4(c,d)] suggest different minimum free energy paths for the folding of L and G. From these projections, highest energy state for protein L appears to have a partially formed β-hairpin 1, while that of protein G has a partially formed β-hairpin 2, although the relevant transition state ensemble (TSE) may be of higher dimension than suggested by simpler reaction coordinates χ_β1 or χ_β2. In fact these simpler reaction coordinates proved not to be saddle points on the multi-dimensional energy landscape according to P_fold, and therefore we needed to collect putative transition states for more complicated reaction coordinates. We found that the collective Q coordinate combined with χ_β1 and χ_β2 for proteins L and G respectively were sufficient to determine the TSE. According to the Q-χ_β1 projection for protein L, the putative TSE structures are collected for structures with 0.4 < Q < 0.6 and 0.5 < χ_β1 < 0.7 [Fig. 5(a)]. According to the Q-χ_β2 projection for protein G, putative TSE structures are collected for structures with 0.6 < Q < 0.8 and 0.35 < χ_β2 < 0.8 [Fig. 5(b)]. P_fold analysis was performed (see Methods) and we identified the true TSEs for proteins L and G [Fig. 5(c,d) respectively]. Comparing the transition state contacts (red contours) for protein L and G, it is evident that the TSE of protein L consists of more native-like contacts in β-hairpin 1, while the TSE of protein G has more native-like contacts in β-hairpin 2. This is consistent with experimental studies using ϕ-value analysis.^30,36 The contact maps also show some contacts between strand 1 and 4, which are consistent with experiments. However both TSE contours indicate well-formed helices for L and G, while mutagenesis studies have suggested helices are relatively disrupted in TSEs.

Figure 5.

P_fold analysis of proteins L and G. Putative transition state ensembles are identified from free energy projections along (a) Q-χ_β1 and (b) Q-χ_β2 for proteins L and G, respectively. Contact maps of transition state ensembles from P_fold for (c) Protein L and (d) Protein G. Black contours denote native contacts. Red contours denote contacts made by 90% of structures in the transition state ensembles.

To explore how our model TSE correlates with mutagenesis experiments at a residue level, we perform single mutations on the optimized sequence of protein L and monitor how its transition state is perturbed by each mutation. From the mutations done by Kim et al.,³⁶ we performed 16 single-site mutations which can easily be represented by our model (Table IV). Instead of a full free energy calculation for each mutant, we instead evaluated the importance of the mutation for perturbing the N_TSE conformational members of the TSE of the WT sequence. For each conformation of the TSE we performed the relevant mutation and ran a P_fold calculation in order to compute 1 – N_TSE(MUT,i)/N_TSE, where N_TSE(MUT.i) refers to the number of conformations collected with 0.4 ≤ P_fold ≤ 0.6.

Table IV.

Mutations Performed on Protein L

Experimental mutation36	Model mutation	Experimental φ-values36	1 − NTSE(MUT,i)/NTSE	R
K7A	L4V	0.70	0.61	0.80
A8G	V5N, E5Ta	0.43	0.39	0.40
G15A	N11V	0.86	0.24	0.20
T17A	N13V	0.42	0.36	0.40
T19A	N15V	0.17	0.27	0.20
E21A	L17V	1.08	0.61	0.80
K23A	L19V	0.57	0.39	0.40
G24A	N20V	0.20	0.33	0.20
T30A	N26V	0.14	0.88	0.80
N44A	L40V	0.08	0.27	0.20
G45A	N41V	−0.10	0.39	0.40
T48A	N44V	0.44	0.30	0.20
G55A	N51V	0.18	0.33	0.20
T57A	N53V	0.07	0.42	0.40
N59A	L55V	0.12	0.39	0.40
K61A	L57V	0.18	0.33	0.20

Note that residue indices of model L differ from experiment.

^aMutation in dihedral sequence.

To compare against ϕ-values, we define a parameter R_i

$$R_i=\{\begin{array}{cc}0.2;\hfill & 0\le 1-N_{\mathrm{TSE}(\mathrm{MUT},i)}∕N_{\mathrm{TSE}}\le 0.33\hfill \\ 0.4;\hfill & 0.33<1-N_{\mathrm{TSE}(\mathrm{MUT},i)}∕N_{\mathrm{TSE}}\le 0.5\hfill \\ 0.8;\hfill & 0.5<1-N_{\mathrm{TSE}(\mathrm{MUT},i)}∕N_{\mathrm{TSE}}\le 1.0\hfill \end{array}\phantom{\}}$$ (6)

to simplify outcomes into low, medium, and high perturbations to the WT TSE. Figure 6 shows the correlation between the experimental ϕ-values and R_i. While there are three outliers (N11V, N26V, and N41V) that deserve to be investigated further, the general trend is consistent with the experimental findings that residues in β-hairpin 1 are more important in the transition state then those in β-hairpin 2. This encouraging result suggests that we can pursue more rigorous free energy calculations of ϕ-values to build on the approximate approach used here.

Figure 6.

Correlation between experimental ϕ-values and perturbation to transistion state R for protein L. There is general agreement between experiment and R, with some outliers (N11V, N26V, N41V). Both experiment and our simulation indicate residues in β-hairpin 1 are more important for the transition state than those of β-hairpin 2.

Kinetics

To rule out the possibility of glassiness, we evaluate the glass transition temperature, T_g, for our model. Wolynes and co-workers⁴⁴ have shown that a foldable, minimally frustrated heteropolymer has a folding temperature well above its glass transition, so that a ratio of T_f/T_g should be greater than one. A working definition of the kinetic glass temperature T_g is the temperature at which average folding time ⟨τ_f⟩ is midway between τ_min, the fastest (minimum) folding time achievable, and τ_max the simulation cutoff time chosen to greatly exceed the observable folding times⁴⁵ (set to 100,000τ in this work). In Figure 7 we show that this occurs at T_g = 0.14, so that T_f/T_g ~ 2.3 for our model of Protein G, indicating that the energy landscapes is sufficiently smooth down to fairly low temperatures.

Figure 7.

Determining the kinetic glass temperature T_g of protein G. The temperature at which average folding time ⟨τ_f⟩ is midway between τ_min, the fastest (minimum) folding time achievable, and τ_max the simulation cutoff time chosen to greatly exceed the observable folding times⁴⁵ (set to 100,000τ in this work). We determine that T_g = 0.14, so that T_f/T_g ~ 2.2 for our model of Protein G, indicating that the energy landscapes is sufficiently smooth down to fairly low temperatures.

The protein L and G models were next analyzed for the kinetic rates and mechanism of folding at their folding temperatures T_f = 0.36 and T_f = 0.325, respectively. During folding simulations, there is a finite equilibration time during which trajectories equilibrate from the initial free energy surfaces at T = 1.6 to those at their target temperatures. The conventional treatment is to include a fitting parameter for dead time τ_D when fitting P_Nat(t)

$$P_{\mathrm{Nat}}\left(t\right)=1-\sum _i{A}_i\exp [-(t-{\tau }_{\mathrm{D}})∕{\tau }_i]$$ (7a)

where A_i is the population for average timescale process τ_i. The parameters used to fit the kinetic data for proteins L and G using Eq. (7a) are listed in Table V.

Table V.

Kinetic Fit Parameters

		Sequential fit46 (Gaussian relaxation followed by single exponential)*
Conditions		μ	σ	τ 0	χ 2
1.	L, T* = 0.36	712	305	11,895	0.0408
2.	G, T* = 0.325	741	450	3,963	0.0935
		Single exponential fit with dead time†
Conditions		τ D		τ 0	χ 2
3.	L, T* = 0.36	694		11,928	0.0506
4.	G, T* = 0.325	641		4,142	0.3436

^*Fitted using Eq. 7c.

^†Fitted using Eq. 7a.

We have shown in previous work⁴⁶ that, instead of using a constant deadtime, the initial equilibration to the new folding conditions could be better modelled as a relaxation process with Gaussian distributed probability. The overall kinetic data could hence be modeled as a sequential process with (a) initial Gaussian relaxation followed by (b) subsequent (multi)exponential kinetics

$$P_{\mathrm{Nat}}\left(t\right)={\int }_{u=0}^t{\int }_{s=0}^{t-u}\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\left(\frac{{(u-\mu )}^2}{2{\sigma }^2}\right)}\cdot \alpha e^{-\alpha s}dsdu$$ (7b)

Integration of Equation 7b leads to

$$\begin{array}{cc}\hfill P_{\mathrm{Nat}}\left(t\right)=\frac{1}{2}& \left[1+\mathrm{erf}\left(\frac{t-\mu }{\sigma \sqrt{2}}\right)\right]\hfill \\ \hfill & -\sum _i\frac{A_i}{2}\left[1+\mathrm{erf}\left(\frac{t-B_i}{\sigma \sqrt{2}}\right)\right]e^{-\frac{t}{{\tau }_i}{e}^{-\frac{D_i}{2{\sigma }^2}}}\hfill \end{array}$$ (7c)

where B_i = (μ + α_iσ²) and D_i = μ² − (μ + α_iσ²)², with mean μ and variance σ, and α_i is the kinetic folding rate for average timescale process τ_i. The fitting parameters using the sequential fit are also listed in Table V. Comparing the fit quality, it is evident that the sequential mechanism provides a better fit than the dead time treatment, and Figure 8 shows the quality of fit for P_Nat(t) for Protein L and G at their respective folding temperatures. Beyond the equilibration phase, the P_Nat(t) data of Protein L [Figure 8(a)] fits to a single exponential, in agreement with experimental data³⁵. The P_Nat(t) data of Protein G at T_f = 0.325 also fits a single exponential [Figure 8(b)], agreeing with single exponential kinetics reported for protein G at its denaturant midpoint.³² The folding time constants for L and G are 11,895τ and 3963τ, respectively. This is in qualitative agreement with experimental data^30,35 that protein G folds faster than L.

Figure 8.

Kinetics data with fits for L and G at their respective folding temperatures using mean first passage time (MFPT) data. (a) Percentage of trajectories folded (P_Nat) as a function of time for protein L at T_f = 0.36. (b) Percentage of trajectories folded (P_Nat) as a function of time for protein G at T_f = 0.325. Both set of data are fitted to both a sequential and dead time model (see text). Fit parameters are listed in Table V. The sequential process is seen to give a better fit to the kinetic data.

CONCLUSIONS

We have presented an improved coarse-grained model capable of modeling directional hydrogen bonding. The model retains a strong connection between sequence and folding mechanism for proteins L and G, and shows increased folding cooperativity. The model native states also exhibit a greater structural faithfulness to experimentally solved structures. The addition of a fourth bead flavor (V) also provides an improvement over the old model by providing a more graded spectrum of attractive interaction energies (Fig. 1). Overall the improvements to the original model, without introducing greater computational cost, translate to a smoother energy landscape and improved T_f/T_g ratios. The thermodynamic data presented demonstrate that our model assembles more cooperatively and preserves the sequence information that result in different free energy pathways for proteins L and G. This finding is further reinforced by kinetic P_fold analysis of their respective TSEs, which show good agreement with experimental mechanisms of protein L and G folding, and decent correspondence with ϕ-value mutation study. The kinetics performed at their melting point (T = T_f) showed that both L and G fold via two-state mechanisms, consistent with experimental consensus under these midpoint denaturant conditions.^32,35

We believe the model shows promise in application to other protein folding studies. One interesting outcome of the new model is our observation of kinetic complexity and burst phase kinetics under more strongly folding conditions for protein G that we hope to report in a future paper. The computational efficiency of the model has also permitted us to develop molecular models of the Alzheimer’s Aβ_1–40 fibril in order to determine the critical nucleus, stability with chain size, and fibril elongation,¹⁶ opening opportunities for other protein-protein co-assembly processes.