Confinement-Dependent Friction in Peptide Bundles

Abstract

Friction within globular proteins or between adhering macromolecules crucially determines the kinetics of protein folding, the formation, and the relaxation of self-assembled molecular systems. One fundamental question is how these friction effects depend on the local environment and in particular on the presence of water. In this model study, we use fully atomistic MD simulations with explicit water to obtain friction forces as a single polyglycine peptide chain is pulled out of a bundle of k adhering parallel polyglycine peptide chains. The whole system is periodically replicated along the peptide axes, so a stationary state at prescribed mean sliding velocity V is achieved. The aggregation number is varied between k = 2 (two peptide chains adhering to each other with plenty of water present at the adhesion sites) and k = 7 (one peptide chain pulled out from a close-packed cylindrical array of six neighboring peptide chains with no water inside the bundle). The friction coefficient per hydrogen bond, extrapolated to the viscous limit of vanishing pulling velocity V → 0, exhibits an increase by five orders of magnitude when going from k = 2 to k = 7. This dramatic confinement-induced friction enhancement we argue to be due to a combination of water depletion and increased hydrogen-bond cooperativity.

Introduction

While friction is considered a nuisance in many daily-life engineering applications, it is an integral and essential part of all nanoscale and biological processes involving self-assembled structures that are held together by noncovalent interactions (1). Besides the free energy landscape, it is friction that determines the lifetime of, e.g., receptor-ligand bonds (2) or the folding time of a protein (3). Two different scenarios must be distinguished:

• 1.In an iso-free-energetic transformation—for example, a kinesin motor sliding on a microtubulin filament (4) or an adsorbed macromolecule diffusing on a flat surface (5–7)—the free energy landscape is basically flat and only shows residual small corrugations or ripples. In principle, the time-dependent mean-square displacement of a random walk can be directly used to estimate the effective diffusion or friction coefficient.
• 2.In a typical protein-folding scenario, the native and unfolded states, in general, have different free energies and are separated by a pronounced barrier. In this situation, friction manifests itself more indirectly as the prefactor of the rate equation. The rate itself is typically dominated by the exponential Arrhenius factor, which depends to leading order on the free-energetic barrier height (8). However, when the energy barriers for the corresponding transition are low or even absent, friction again dominates the kinetics and one reaches the so-called speed limit of protein folding (9–11).

Clearly, in both of these situations, friction between different macromolecules or between different parts of a single macromolecule arises from an interplay of the surrounding viscous solvent (water in most cases) and the macromolecular conformational dynamics, and the separation of solvent and internal effects is, in general, not straightforward. The internal friction contribution can be pictured as arising from the multiple breaking and binding of noncovalent bonds between two adhering molecules or molecular parts as they slide against each other. For peptide chains in water, hydrogen bonds (HBs) were identified as the main contributors to internal friction (12–15), which reflects that HBs are numerous and fundamental for the stability of protein structures due to their very suitable range of formation free energy and their pronounced directionality. Note that HBs have also been used to design synthetic systems with interesting self-healing, adaptive, and tribological properties (16–18). Experimentally, the balance of solvent and internal friction contributions can be tuned by modifying the solvent viscosity by the addition of suitable viscogens, but because it is difficult to ascertain that viscogens do not modify the free energy landscape of the probed reaction coordinates, the separation of internal and solvent friction contributions has remained controversial (11,19,20). In recent simulations the water viscosity was tuned by changing the water mass. This method allowed us to study folding times of short peptides for water viscosities varying over two orders of magnitude, while the free energy landscape stayed strictly invariant (as can be shown using classical statistical mechanics laws) (21).

These simulations, in particular, showed that extracting internal friction effects from protein kinetics is subtle, because the functional dependence of folding or reconfiguration times on the solvent viscosity is complicated even for the simplest kinetic polymer model that incorporates internal friction. For a realistic protein system, the intricate coupling of solvent friction, dissipation within the peptidic backbone, and friction between residues that are distant in sequence space has not allowed formulation of a simple theoretical model thus far. The actual situation is even more complicated because the number of native contacts increases as the final folded state is approached and thus internal friction is expected to increase toward the native state (22–25), as was also demonstrated in coarse-grained simulations of homopolymeric globules (26,27). In other words, a fast initial low-friction collapse is expected to be followed by a much slower high-friction search for the native configuration (28,29). Hence, the friction coefficient defined along a reaction coordinate is generally state-dependent. Although reliable techniques for extracting the friction profile from dynamic trajectories have been proposed (30,31), the task of separating internal from solvent friction effects along the folding trajectory in a nontrivial free-energy landscape is a formidable task that has only been tentatively tackled so far (21).

To understand the microscopic mechanism of internal friction between adhering peptide chains, a simple model system is needed that circumvents at least some of the intricacies mentioned above. In previous simulations, we have studied the friction of various peptides on flat polar surfaces by pulling them in a steady state at prescribed mean sliding velocities V (15). We found that the friction force F_f in the viscous (i.e., low velocity) regime obeys the friction law

$$F_{\text{f}}={\gamma }_{\text{HB}}{N}_{\text{HB}}V$$ (1)

and is proportional to the number of hydrogen bonds N_HB between the peptide and the surface. The proportionality constant, the friction coefficient per HB γ_HB, was shown to take the value γ_HB = 10⁻⁸ kg/s independent of the polarity of the surface, the peptide type, and the normal force applied in the simulations to push the peptide onto the surface. All these factors were shown to basically change the number of HBs and the friction force in an identical manner, giving the friction law a certain generality. But how the internal friction depends on the local environment around the HBs has not been explored in those simulations, and in particular, it was not clear whether the results obtained for a peptide sliding over a surface apply to the case of peptide chains undergoing internal reconfigurations in the protein interior. In this article, we address that question and look at the friction within polyglycine bundles at varying degrees of aggregation. In particular, we study how the confinement of a peptide chain by other adhering peptides affects the dynamics in general and the friction coefficient in particular.

We focus on the relative motion of short extended peptide fragments using extensive all-atom molecular-dynamics (MD) simulations in explicit water and infer the sum of the interpeptide and water-peptide friction contributions. By pulling one peptide out of a bundle consisting of a total of k peptide chains with k ranging from k = 2 (just two peptide chains binding to each other with water freely contacting both peptide chains) to k = 7 (here the central peptide chain that is pulled out of the bundle is completely surrounded by six neighboring peptide chains and no water can access the interface at which the friction is created), we modulate the degree of confinement. Our system is periodic along the pulling direction, which is central to our approach: Because the free energy landscape does not change along the reaction coordinate in the long-time limit, the forces we extract from the simulations are solely due to dissipative friction effects. We always vary the pulling velocity to extract the asymptotic linear-response friction coefficient γ_HB that is valid at vanishing velocity, which works fine for the moderately confined systems with k up to k = 5. For the fully confined peptide chain for k = 7, the direct extrapolation of the simulation data is not straightforward.

To reach the vanishing-velocity limit also in this case, we analyze our data obtained at finite pulling velocities down to V = 5 × 10⁻⁴ m/s using an adapted Fokker-Planck stochastic theory that accounts for nonequilibrium dissipative effects and allows robust extrapolation into the linear-response regime. For the least confined system with k = 2 we also perform equilibrium simulations without externally enforced pulling velocity, and find good agreement between the diffusion coefficient and our extrapolated linear-response friction coefficient from our nonequilibrium simulations at finite-sliding velocity. Based on our earlier results for a peptide chain sliding over a flat surface, we assume friction forces between peptide chains to be proportional to the interpeptide HB number in the low-velocity viscous limit. However, the proportionality constant, the friction coefficient γ_HB, depends sensitively on the degree of confinement and varies over more than five orders of magnitude with changing k. This confinement effect is argued to be due to a combination of the depletion of water from the interface between the moving peptide chains as well as increased HB cooperativity as the aggregation number of the peptide bundles increases.

Methods

For the initialization of our MD simulations, single polyglycine chains with lengths N = 10 or N = 20 amino acids (aa) are solvated in water for 1 ns. Next, water is removed, and k polyglycine chains are brought into contact in vacuum in their extended configuration. Then these structures are solvated by ∼1000 single-point charge (32) water molecules and equilibrated for at least 1 ns before pulling. For parallel peptide structures (which we will call bundles here), we also perform simulations for k = 2 perpendicular chains. In the pulling simulations, only one of the peptides is pulled out from the bundle, as schematically shown in Fig. 1. The other k − 1 chains are kept fixed by harmonic restraint potentials with spring constants of k_R ∼ 10⁵ pN/nm acting on one of their α-carbon atoms. We checked for the influence of the number of restrained atoms and found none as long as this number does not exceed 4 (see Fig. S1 in the Supporting Material).

Figure 1.

Illustration of peptide bundles made of periodically replicated N = 10 polyglycine chains. (A) Schematic picture of k = 2 parallel peptide chains. To obtain steady-state sliding, a harmonic spring that moves with constant velocity V is connected to one amino acid of one chain, while the other chain is held fixed at one atom position. (B) Parallel k = 3, 5, and 7 bundles. (Blue) Pulled peptide chain. For clarity, only a few water molecules are shown for the k = 7 case. (C) Perpendicular $k=2$ system. (Arrow) Pulled peptide chain. (D) The perpendicular case is shown together with its periodic embedding. (Dashed square) Simulated system. All snapshots are generated using VMD (46).

All peptide chains are fully periodic, i.e., a peptide backbone bond between the ith and Nth amino acids traverses the periodic box. The pulling is performed by attaching a one-dimensional harmonic spring with force constant K to the center of mass of a single aa of the pulled chain, and moving the other end of the spring at constant velocities ranging between V = 5 × 10⁻⁴ m/s and V = 50 m/s as shown in Fig. 1 A. In a small subset of simulations for k = 2, we pull the two peptides in opposite directions at velocities +V/2 and –V/2, yielding results that are consistent with our asymmetric pulling protocol (see Fig. S1). No force is exerted on the pulled peptide perpendicular to the pulling direction, so it is free to separate from the other chains during a pulling simulation.

From the average spring extension Δx the average total friction force F_f = KΔx is deduced. The spring constant is chosen such that the spring extension Δx is smaller than half of the box length in the pulling direction and ranges between K = 5 pN/nm and K = 500 pN/nm. The lowest pulling velocity achievable is mainly determined by the maximal duration of our simulations, which is 10 μs and gives for V = 5 × 10⁻⁴ m/s a total peptide displacement of 5 nm, which is barely larger than the box size. For MD simulations, the GROMACS MD software package (33) with the Gromos96 (34) force field for all bonding and nonbonding interactions is used. All simulations are run using periodic boundary conditions with constant particle number, constant mean pressure P = 1 bar enforced by isotropic box rescaling, and constant temperature T of 300 K using Berendsen’s method (35) with a coupling time constant of 1 ps. For long-range Coulombic interactions, the particle-mesh Ewald (36) method is employed. For the cutoff distance of nonbonded Coulomb and Lennard-Jones interactions, a value of 0.9 nm is used. All covalent bonds involving hydrogen atoms are constrained using the LINCS algorithm. Every 20 steps, the neighbor lists for nonbonded interactions are updated. The initial box dimension in the direction parallel to the peptide-longitudinal axes is set to N × 0.35 nm. The average simulation box sizes do not deviate much, and for the various setups, are given in Table 1. Box length fluctuations are <$1\%$. For the data analysis, typically the first $20\%$ of each trajectory is disregarded; all relevant quantities equilibrate on much shorter timescales, as demonstrated by explicitly calculated autocorrelation functions (see Fig. S3 and Fig. S6). Error bars are calculated via block averaging and shown only when they are larger than the symbol size. For exerting normal forces on the perpendicular peptides, a constant force on each atom including hydrogen atoms is applied that pushes the peptide strands against each other.

Table 1.

Average simulation-box sizes in our simulations taken from the simulations with the slowest pulling velocity

Peptide	N	System	(Lx × Ly × Lz) [nm3]
2 Glycine	10	Parallel	3.46 × 2.99 × 2.99
2 Glycine	20	Parallel	6.95 × 3.19 × 3.19
2 Glycine	10	Perpendicular	3.49 × 3.51 × 3.50
3 Glycine	10	Parallel	3.41 × 2.98 × 2.98
4 Glycine	10	Parallel	3.37 × 3.18 × 3.18
5 Glycine	10	Parallel	3.39 × 3.17 × 3.18
7 Glycine	10	Parallel	3.38 × 3.48 × 3.48

Results and Discussion

Parallel pulling of peptide bundles

Two-peptide system, k = 2

We first discuss friction for k = 2 parallel polyglycine chains as illustrated schematically in Fig. 1 A. During the course of our pulling simulation, the peptides slide against each other but stay adsorbed to each other. This can be attributed to the chain orientation and stretching that enhances the tendency of polyglycine to form compact structures (37,38). In Fig. 2, we show the average friction force per monomer, F_f/N; the number of peptide-peptide HBs per monomer between the two peptides, N_HB/N; and the number of peptide-water HBs between the pulled chain and the surrounding water molecules, $N_{\text{HB}}^{\text{PW}}/N$, as a function of the pulling velocity, V, for N = 10 (squares) and N = 20 (diamonds) polyglycine chains. The almost perfect agreement between the two data sets demonstrates that finite-size effects are already absent for the N = 10 system. Therefore, in the remainder, we only use the smaller peptide length N = 10. For low velocities, the friction F_f/N is clearly viscous; that means it is linearly proportional to V. Only for V > V₀ = 10 m/s are deviations from linearity seen in Fig. 2 A. The interpeptide HB number N_HB/N in Fig. 2 B, which is defined via the combined Luzar-Chandler angle-distance criterion (39), saturates for low velocities at a value of roughly N_HB/N ≈ 0.23, which shows that the two chains are far from forming the maximal number of possible HBs.

Figure 2.

Results for k = 2 parallel polyglycine chains consisting of N = 10 and N = 20 amino acids as a function of the relative sliding velocity V. (A) Friction force per monomer F_f/N. (Straight line) Slope of unity indicated for viscous limit, giving a monomer friction coefficient γ ≡ F_f/(NV) = 2 ± 0.5 × 10⁻¹² kg/s. (B) Average peptide-peptide HB number between the two polyglycine chains, N_HB/N. (C) Peptide-water HB number that the pulled chain makes with the surrounding water molecules $N_{\text{HB}}^{\text{PW}}/N$. Data at V = 0 is obtained for two freely diffusing chains.

The data show a marked decrease of N_HB/N only for velocities higher than ∼V_HB ≃ 10 m/s, as shown in Fig. 2 B. This can be easily understood based on a simple argument: Assuming an interpeptide HB lifetime of ∼τ_HB ≃ 20 ps (see Fig. S2) and a HB range of ∼a_HB = 0.2 nm, the critical velocity at which the pulling starts to conflict with the HB lifetime follows as V_HB = a_HB/τ_HB = 10 m/s. Note that the crossover in the friction force in Fig. 2 A roughly happens at the same velocity at V₀ = 10 m/s, but this is coincidental and for the more confined systems we generally observe V₀ ≪ V_HB. In Fig. 2, B and C, we also show equilibrium simulation results at V = 0 for both N = 10 and N = 20 chains, where all position and pulling restraints are removed so that both chains freely diffuse. The two parallel polyglycine chains stay adsorbed to each other for the entire simulation duration of 1 μs. The result N_HB/N = 0.23 at the very left side of Fig. 2 B is fully consistent with the pulling data at low velocities, which testifies to the convergence of our pulling simulations. The data for the peptide-water HBs in Fig. 2 C are much less influenced by the pulling, which shows that the hydration water can adjust more readily to the configurational changes enforced by the peptide sliding. Also note that the number of peptide-water HBs $N_{\text{HB}}^{\text{PW}}/N$ is much larger than the number of peptide-peptide HBs N_HB/N. The viscous straight line fit in Fig. 2 A gives a monomer friction coefficient γ ≡ F_f/(NV) = 2 ± 0.5 × 10⁻¹² kg/s, which is of the same order as the monomer friction coefficient in bulk (single-point charge) water, γ_b = 2 ± 1 × 10⁻¹² kg/s (15). We conclude that the presence of N_HB/N ≈ 0.23 interpeptide HBs per residue does not drastically increase the sliding friction within the errors when compared to the friction in bulk water. This is remarkable and will be rationalized later when we compare this with the more confined bundles.

Peptide bundles k > 2

We now pull a single polyglycine chain out of parallel bundles consisting of k chains, as illustrated by the simulation snapshots in Fig. 1 B. Note that the pulled chain is always the chain that has maximal contact with the other chains, e.g., for the k = 7 bundle, the pulled chain (shown in blue in Fig. 1 B) is completely surrounded by neighboring peptide chains and thus isolated from water. As shown in Fig. 3 A, increasing the bundle aggregation number k drastically increases the friction force F_f/N: The difference in F_f between k = 2 (squares) and k = 7 (open-circles) for velocities V ∼ 0.1 m/s is almost three orders of magnitude. Note that for the k = 3 and k = 5 bundles (down-triangles and solid spheres in Fig. 3 A), for respective velocities <V ∼ 1 m/s and V ∼ 10⁻² m/s, the onset of viscous regimes with F_f ∼ V are observed, as indicated by straight-line fits of slope unity, similarly to the k = 2 case. For the fully confined bundle with k = 7, the viscous regime is not reached for the velocity range accessible in the simulations. This clearly demonstrates a fundamental problem of reaching equilibrium in atomistic MD simulations of peptide systems: the higher the confinement and therefore the friction, the lower the velocity below which nonlinear effects are absent! For high velocities, the friction forces of all bundles tend to converge. The increase of the friction force at low V with rising aggregation number k can only in part be traced back to an increase of the total number of HBs between the pulled and nonpulled peptides, N_HB/N. This is demonstrated in Fig. 3 B, where we plot the rescaled interpeptide HB number between the pulled chain and its neighbors, N_HB/[(k − 1)N], which is roughly constant for all different bundles considered. This means that the total number of HBs the pulled central chain makes with its k − 1 neighbors is roughly proportional to k − 1 itself, from which two conclusions can be drawn:

• 1.The hydrogen-bonding capacity of the central chain is far from being saturated for the constructs used by us.
• 2.More importantly, the friction force at low pulling velocities in Fig. 3 A is not solely proportional to the number of HBs between the pulled chain and its neighbors, simply because the three-order of magnitude difference in F_f between the k = 2 and k = 7 bundles at V ∼ 0.1 m/s cannot be explained by the sixfold increase in N_HB/N.

Figure 3.

(A) Friction force per monomer F_f/N. (B) Interpeptide HB number per monomer and per number of neighboring chains N_HB/[N(k − 1)]. (C) Peptide-water HB number between the pulled chain and surrounding water molecules for parallel N = 10 polyglycine bundles of different aggregation number k as a function of pulling velocity V. The central chain is pulled from the bundle, while the neighboring k − 1 chains are kept fixed.

The second conclusion is more clearly presented in Fig. 4 A, where we display the friction force per HB, F_f/N_HB, as a function of the aggregation number k, for three pulling velocities V = 0.1, 1, and 10 m/s. For the highest velocity, V = 10 m/s (squares), for which N_HB decreases from its equilibrium value due to nonequilibrium effects as shown in Fig. 3 B, k has only a weak effect on F_f/N_HB. However, as the velocity is decreased down to the more relevant value V ∼ 0.1 m/s (circles), a 100-fold difference emerges between the friction forces per HB for the k = 2 and k = 7 bundles. This confinement effect, which constitutes the main finding of our article, can in part be rationalized by an increase of the HB collectivity (15), meaning that HBs tend to rupture in larger groups as the confinement goes up (see Fig. S11 for an explicit demonstration and Fig. S12 for distributions). On the other hand, water depletion around the pulled peptide in the more confined bundles, which is clearly demonstrated in Figs. 3 C and 4 B where we plot the number of peptide-water HBs of the pulled chain, will tend to increase the free energy of a single HB (40), which has a similar effect and will therefore also tend to enhance friction effects. We will come back to this discussion later on.

Figure 4.

(A) Ratio of friction force and peptide-peptide HB number between the pulled chain and the fixed chains, F_f/N_HB. (B) Number of HBs between the pulled peptide and water, $N_{\text{HB}}^{\text{PW}}$ for three different pulling velocities as a function of the bundle aggregation number k.

At this point we have to mention one technical complication in simulations of highly confined peptides in k = 5 or k = 7 bundles at large pulling velocities: For pulling velocities above $V>5$ m/s, we observe that bundles deform after a few nanoseconds of simulations and the pulled peptide slips toward the surface of the bundle. In Fig. 5 we show trajectories for the friction force per monomer F_f/N and the HB number per monomer of the pulled chain N_HB/N for four different bundle sizes at high pulling velocity V = 10 m/s. The snapshots for the k = 5 bundle show that the bundle structure changes during the course of the simulation such that the pulled peptide (shown in blue), which initially was bound to all four peptides in the bundle, only adheres to three neighboring peptides (after 2 ns) and after 4 ns only to two neighboring peptide chains. These structural changes (far-from-equilibrium effects that we do not explore in this study) lead to a decrease in both F_f and N_HB; in fact, the red curves for k = 5 rather resemble the curves for smaller k after the restructuring events have taken place, in full agreement with the snapshots. We note that these restructuring events only occur for large V. They do not influence our results for the friction forces in the low-velocity regime, which is the main focus of this article. Nevertheless, to also produce reliable data in the high-velocity range, we average the k = 5 and k = 7 bundle data only for the initial time windows during which the peptide neighboring shell is intact and the bundles are not deformed.

Figure 5.

Trajectories of (A) the friction force F_f and (B) the peptide-peptide HB number N_HB at high velocity V = 10 m/s for parallel k = 2, 3, 4, 5 bundles made of N = 10 polyglycine chains, one of which is pulled relative to the other fixed k − 1 chains. Snapshots show deformed structures of the k = 5 bundle taken at times marked (arrows). The trajectories of N_HB are running averages over 50-ps intervals; the trajectories for F_f are data points recorded every 5 ps.

Perpendicular pulling of two peptides

We next discuss the scenario where we pull two polyglycine chains perpendicular to each other, as schematically shown in Fig. 1, C and D. Similar to the parallel pulling setup, one of the chains is pulled by applying a force on a single aa while the other chain is fixed via a restraint acting on one atom. In Fig. 6, parallel and perpendicular pulling simulations are compared as a function of pulling velocity V. Before we discuss the data, some specificities of the perpendicular pulling simulations must be discussed: Because the contact between the two peptide chains is reduced in the perpendicular geometry, the adhesive energy is smaller and therefore the peptide chains detach from each other from time to time. This results in time-spans where the interpeptide HB number is zero and the friction force is purely due to solvent friction (see Fig. S4 for time traces of the HB number) because the two peptides have no contact. Therefore, in all averages that are presented for the perpendicular case, we have eliminated the time-spans within which N_HB(t) = 0. As an alternative way of eliminating detached configurations, we applied normal forces, F_N = 0.12 pN and F_N = 1.2 pN per aa, in opposite directions on both chains (normal to the plane spanned by the two chain tangents), in which case we have not eliminated the time-spans with N_HB = 0 from our further data analysis.

Figure 6.

Comparison of parallel and perpendicular pulling geometries. In the perpendicular case we also show data for two different values of the normal force F_N, defined as the normal force applied on each amino acid. (A) Friction force per monomer F_f/N between two N = 10 polyglycine chains, one of which is pulled perpendicular or parallel to the other fixed chain. The lines have a slope of unity and correspond to friction coefficients of γ ≡ F/(NV) = 2 ± 0.5 × 10⁻¹² kg/s for the parallel case and γ ≡ F/(NV) = 1.2 ± 0.3 × 10⁻¹² kg/s for the perpendicular case. (B) Peptide-Peptide HB number N_HB for the perpendicular case. For the V = 0 data both chains can freely diffuse.

As seen in Fig. 6 A, the friction forces per monomer F_f/N are quite similar for the parallel and perpendicular cases. Also, a finite normal force does not yield a drastic effect in the friction force for F_N = 0.12 pN (stars) or F_N = 1.2 pN (solid squares) in the perpendicular case. Linear fits in the low-velocity regime lead to the friction coefficients per monomer of γ = F_f/(VN) ≈ 2.0 ± 0.5 × 10⁻¹² kg/s for the parallel and γ = F_f/(VN) ≈ 1.2 ± 0.3 × 10⁻¹² kg/s for the perpendicular geometry, which are of the same order as the friction coefficient in bulk water of γ_b ≈ 2.0 ± 1 × 10⁻¹² kg/s, as already discussed above. Clearly, the total excess friction due to interpeptide interactions should scale extensively (i.e., proportional to N) in the parallel case but be independent of N in the perpendicular case, because the contact zone between the two perpendicular peptide strands does not depend on the chain length. We will soon see that a more appropriate analysis uses the friction force per HB, which fully accounts for the difference of the scaling with N in the two geometries.

The data for the HB numbers in Fig. 6 B show that in the perpendicular case N_HB ≈ 0.5 for F_N = 0 and F_N = 0.12 pN, which increases up to N_HB ≈ 0.8 for F_N = 1.2 pN. These values are significantly larger than the value N_HB/N ≈ 0.23 for the parallel case as shown in Fig. 2 B. A simple interpretation is that, in the perpendicular geometry, the contact between two peptides is established via a few amino acids with the same HB number per aa as in the parallel case. This is confirmed by an analysis of the amino acids that contribute to the HBs between the perpendicular chains (see Fig. S5) and further validated by the fact that the HB free energies as deduced from the HB lifetimes are very similar for the parallel and perpendicular geometries (see Fig. S2). The data points at V = 0 in Fig. 6 B are again obtained for freely diffusing chains in the absence of an externally imposed sliding. These data confirm that, as far as HB numbers are concerned, our data at finite pulling velocities are well converged and correspond to the linear-response regime.

Extracting friction coefficients in the viscous regime

As shown in Fig. 3 A, for the k = 2, 3, and 5 bundles, the viscous regime where F_f ∼ V holds is reached in simulations at low velocities, and for these low-confinement bundles the viscous friction coefficient can be directly read off from the data. For the more highly confined bundle with k = 7, the viscous regime is clearly not reached even for the lowest pulling velocity achieved in our simulations. In the experimental situations we aim to address in this work, macromolecules diffuse in equilibrium and slide against each other, such as in protein folding. In other scenarios, external forces are applied and the resulting velocities are in the μm/s range, such as in force-spectroscopic experiments or when biological single-molecule motors are active. In essence, we are experimentally always in the viscous linear response regime where friction forces are proportional to velocities. To also extract the viscous friction coefficients for the fully confined k = 7 bundle, we have to extrapolate the data. In analogy to our previous analysis (15) where we showed that friction forces for peptides adsorbed on surfaces are proportional to the number of HBs (and which led to the formulation of the viscous friction law valid for hydrogen-bonded matter presented in Eq. 1), we define the friction coefficient per HB as

$${\gamma }_{\text{HB}}\equiv \frac{F_{\text{f}}}{\left(V{N}_{\text{HB}}^{\text{eq}}\right)},$$ (2)

which is shown in Fig. 7 as a function of the rescaled friction force F_f/N for all parallel bundles. That HBs cause friction also in this case is shown by the pronounced and long-lived cross-correlations between the friction force and the HB number (see Fig. S6). Note that in the definition of γ_HB we use the equilibrium HB number $N_{\text{HB}}^{\text{eq}}$ because for V < 5 m/s, $N_{\text{HB}}\approx N_{\text{HB}}^{\text{eq}}$, as can be seen in Figs. 2 and 3. For the low-confinement k = 2, 3, and 5 bundles, the friction coefficient γ_HB in Fig. 7 shows a pronounced plateau for F_f/N < 10 pN, which confirms that, for these bundles, we reach the relevant linear-response regime in our simulations. The data point for the k = 2 bundle at vanishing friction force F_f/N = 0 to the far left in Fig. 7 is obtained for two freely diffusing polyglycine chains without positional restraints. The data point is obtained from the diffusion equation

$$\langle \Delta r{\left(t\right)}^2\rangle =\frac{2k_{\text{B}}Tt}{\left(N_{\text{HB}}^{\text{eq}}{\gamma }_{HB}\right)}$$

by measuring the time-dependent mean-squared relative displacement between two chains 〈Δr(t)²〉 in the absence of an external driving force (see Fig. S7). Although the rather small friction effect for k = 2 in connection with the pronounced numerical errors does not allow for a critical comparison between the data, we note that the free-diffusion data is consistent with the finite velocity data for k = 2 within error bars. We stress that the determination of the friction coefficient from equilibrium simulations is much more time-consuming than the nonequilibrium simulations at finite external pulling force. Even for the least confined case, k = 2, a 2-μs-long simulation was barely sufficient to reach a clear diffusive regime (see Fig. S7 for the mean-squared relative chain displacement). Hence, results for higher confinement k > 2 levels cannot be obtained via equilibrium simulations. This explains why we went through the more elaborate procedure of performing simulations far from equilibrium, which are subsequently extrapolated into the linear-response regime.

Figure 7.

Comparison of the simulated friction coefficient per HB, ${\gamma }_{\text{HB}}=F_{\text{f}}/\left(V{N}_{\text{HB}}^{\text{eq}}\right)$, for k = 2, 3, 5, 7 (data points, bottom to top), with the scaling form Eq. 3 as a function of F_f/N. The bundles are formed by k parallel N = 10 polyglycine chains. (Inset images) Corresponding representative snapshots for each bundle. (Red) Pulled chain. Note that there are two similar fits for each data set except k = 2, with the first assuming vanishing cooperativity m = 1 (dashed lines) and the second assuming constant HB energy U_HB/k_BT = 4.5 (solid lines), respectively. The data point at vanishing friction force F_f/N = 0 to the far left is obtained for k = 2 freely diffusing chains.

In contrast to the results for k = 2, 3, and 5, for the k = 7 bundle, even for the slowest velocity of V = 0.002 m/s, γ_HB does not reach a plateau value in Fig. 7. To observe the linear-response regime directly in simulations for these bundles, velocities far lower than V = 0.002 m/s would be required (15), which is too slow to obtain converged data (as argued in the Methods). This is a generic feature of nonequilibrium MD simulations at finite applied force (41–44) and not a short-coming of our specific simulation setup.

To extract the friction coefficient in the limit F_f → 0, we use the mapping on a model that was introduced in the context of protein folding in a one-dimensional reaction coordinate (45) and used previously by us to model peptide-surface friction (15). In this model, the diffusive motion of a particle in a corrugated potential under the action of a finite applied force is described by the Fokker-Planck (FP) equation. The corrugated potential is assumed as sinusoidal,

$$U\left(x\right)=m{U}_{\text{HB}}\frac{\left(\text{cos}\left[2\pi x/a\right]-1\right)}{2},$$

with a lattice constant a and a strength of mU_HB. Here U_HB is the free energy to break a single HB, and the factor m describes the cooperativity of the friction process and is a measure of how many HBs break collectively. As shown in the Supporting Material, the friction coefficient per HB can be written in a scaling form as (15)

$${\gamma }_{\text{HB}}=\frac{N{\gamma }_0}{N_{\text{HB}}^{\text{eq}}}+\frac{{\gamma }_0}{m}\Psi \left(\frac{ma{F}_{\text{f}}}{k_{\text{B}}T{N}_{\text{HB}}^{\text{eq}}},\frac{m{U}_{\text{HB}}}{k_{\text{B}}T}\right).$$ (3)

The first term on the right side describes the friction in the high-velocity limit where the effects of the corrugated potential vanish. From fits to our data we estimate γ₀ = 10⁻¹² kg/s. The second term describes the friction due to the corrugated potential and is proportional to the scaling function Ψ. It takes nonequilibrium effects into account and describes the friction coefficient in units of γ₀ of one cooperative unit consisting of m HBs, subject to the driving force $m{F}_{\text{f}}/N_{\text{HB}}^{\text{eq}}$ and diffusing in the sinusoidal potential U(x). The function Ψ follows from the closed-form solution of the FP equation (45,46) (see the Supporting Material for the derivation and alternative ways of data fitting in Fig. S10 and Fig. S13). Note that we assume the total friction force F_f to be equally shared by all N_HB HBs, which was shown to be an accurate approximation (15). Our model has three parameters: the lattice constant a; the energy per HB U_HB; and the cooperativity factor m. But basically only the two parameter combinations ma and mU_HB can be reliably extracted from the simulation data, meaning that the third parameter has to be inferred from an additional hypothesis, as we show next.

In Fig. 7, the scaling form Eq. 3 is used to fit the data for all bundles by using two different fitting procedures for m, a, and U_HB. Note that in the high force regime, the curves saturate at different values due to the different values of $N/N_{\text{HB}}^{\text{eq}}$ in the first term in Eq. 3. In the first scheme, we set $m=1$, that is, we neglect cooperativity and assume that each HB moves independently from all other HBs in the corrugated potential U(x) of strength U_HB. The resulting fits are shown as dashed lines and describe the data very well, and in particular cover the steep increase of γ_HB as the friction force decreases. For k = 2, 3, and 5, the fit is unambiguous: a controls the lateral position of the scaling function in Fig. 7, which can be easily appreciated from the fact that a and F_f only appear in the bilinear form aF_f in Eq. 3; the potential height U_HB mainly sets the value of γ_HB in the viscous limit for F_f → 0, which can be realized from the fact that Ψ(x,y) vanishes as x → ∞ (as has been shown before in Erbaş et al. (15)). For k = 7, the fit of a is robust, but for U_HB only a lower limit can be estimated. The fit values corresponding to the dashed lines are a = 0.36, 0.97, 2.35, 3.13 nm and U_HB/k_BT = 4.5, 7.2, 12.1, 18.5 for k = 2, 3, 5, 7, respectively.

As one can see, both the periodicity a as well as the HB strength U_HB increase almost linearly with the aggregation parameter k. Although an increase of the HB strength with increasing confinement is expected due to water depletion effects, based on the fact that the HB strength in vacuum is higher than in water, the strength of U_HB/k_BT = 18.5 observed for the fully confined situation k = 7 is clearly too high and therefore unrealistic. Even more to the point, the pronounced increase of the corrugation periodicity a with growing confinement is dubious, as it is not clear what the periodicity a = 3.13 nm for k = 7, which almost equals the entire peptide length, means. The assumption of the cooperative breaking of m HBs is a simple and intuitively appealing way of dealing with this situation, because if one assumes that m HBs act coherently, both the wavelength a and the HB strength U_HB are reduced because only the products ma and mU_HB appear as arguments in the scaling function Ψ in Eq. 3.

Therefore, according to our second fitting scheme, we fix the individual HB strength to the value U_HB/k_BT = 4.5, which is the strength that we obtained in the first fitting scheme for k = 2 (and which agrees with the perpendicular pulling scenario and also our previous work on peptide sliding on solid surfaces, as will be further explained below), and treat m and a as fitting parameters. The resulting fits are shown as solid lines in Fig. 7, and basically describe the simulation data with the same accuracy; the extracted fitting parameters are a = 0.36, 0.55, 0.78, 0.67 nm and m = 1, 1.74, 2.94, 4.38 for k = 2, 3, 5, 7, respectively. The potential periodicity a now increases much more modestly with growing confinement and varies between one and two aa contour lengths. We note that the extrapolated friction coefficients γ_HB turn out to be independent of the fitting scheme, as can be seen from the close agreement between the dashed and solid curves in Fig. 7, yet the suggested physical mechanism behind the friction-increase in confined bundles is vastly different. Clearly, the truth will lie somewhat in the middle, and the increased friction will be due to a combination of both water depletion effects (which raise the HB free energy) and cooperative effects, as we argue in detail in the Conclusions.

In Fig. 8 we plot the viscous friction coefficient γ_HB obtained via extrapolation of the FP fits, assuming a constant HB energy of U_HB/k_BT = 4.5 (second fitting scheme denoted by solid lines in Fig. 7) to the vanishing-force limit as a function of the bundle aggregation number k. The HB friction coefficient spans a range of five orders of magnitude, depending on the confinement in the bundle. For comparison, we also add the value γ_HB = 10⁻⁸ kg/s for a polyglycine chain pulled over a polar surface as obtained earlier in Erbaş et al. (15) as a horizontal line. In that work, the friction coefficient value was obtained by a fit using an energy barrier of mU_HB = 13.8 k_BT, which together with our estimate U_HB = 4.5 k_BT gives a cooperativity factor m = 3.1. Interestingly, this value for m is close to what we obtain for a k = 5 bundle, which, looking at the simulation snapshots in Fig. 8, can almost be considered as forming a planar substrate on which the pulled peptide is sliding, and therefore gives a hint why we obtain the same cooperativity factors in the two different simulation setups.

Figure 8.

Friction coefficient ${\gamma }_{\text{HB}}=F_{\text{f}}/\left(V{N}_{\text{HB}}^{\text{eq}}\right)$ in the viscous limit V → 0 for parallel bundles, obtained via extrapolation of the curves in Fig. 7, as function of the bundle aggregation parameter k. (Solid line) Guide to the eye. (Inset) Resulting cooperativity factor m as a function of k. (Horizontal line) γ_HB for a glycine chain being pulled over a hydrophilic hydroxylated surface (15).

The strong increase of γ_HB with confinement is independent of the effective corrugation wavelength a and therefore exclusively associated with the increase of the effective potential corrugation amplitude mU_HB. In fact, in the double asymptotic limit mU_HB/k_BT ≫ 1 and $ma{F}_{\text{f}}/k_{\text{B}}T{N}_{\text{HB}}^{\text{eq}}\to 0$, i.e., for strong corrugation and in the viscous low-force limit, the scaling function Ψ in Eq. 3 simplifies and we obtain

$${\gamma }_{\text{HB}}\simeq {\gamma }_0\frac{\Psi }{m}\simeq {\gamma }_0\frac{e^{m{U}_{\text{HB}}/k_{\text{B}}T}}{\pi m^2{U}_{\text{HB}}/k_{\text{B}}T}.$$ (4)

In the above expression we neglected the solvent friction contribution, in line with our assumption mU_HB/k_BT ≫ 1. Note that in this limit, γ_HB is independent of a and increases to leading order exponentially with mU_HB/k_BT, compared to which the factor m₂U_HB/k_BT in the denominator is negligible. In particular, the cooperativity factor m itself, although it appears in the denominator, has only minor significance compared to the parameter combination mU_HB. This explains why the different fitting schemes, indicated by solid and dashed lines in Fig. 7, lead to very similar curves.

An independent approach to determine a realistic value of U_HB and thus the cooperativity factor m is based on the perpendicular pulling scenario. Here we expect cooperativity to be absent, i.e., m = 1, because the interpeptide HB number N_HB rarely exceeds 1 (see Fig. S4 for distributions), meaning that either one HB or no HB is present between two perpendicular peptide strands. In Fig. 9, we compare the friction coefficient for the perpendicular pulling scenario for different normal forces F_N = 0 (spheres) and F_N = 1.2 pN (solid squares) with the parallel k = 2 (open squares) and k = 3 (triangles) bundles. For both perpendicular cases, data extrapolate to a value of roughly γ_HB ≃ 2 ± 1 × 10⁻¹¹ kg/s as V → 0, which is slightly higher than the value of γ_HB for the parallel k = 2 bundle and significantly lower than that for the k = 3 bundle. The barrier heights used in the fits for the perpendicular cases (black and red curves) are consistent with the fitted value of U_HB ≈ 4.5 k_BT for the parallel k = 2 case, showing that our fitting procedure for U_HB is robust.

Figure 9.

Comparison of the friction coefficient per HB ${\gamma }_{\text{HB}}=F_{\text{f}}/\left(V{N}_{\text{HB}}^{\text{eq}}\right)$, with the scaling form Eq. 3 as a function of the friction force per monomer F_f/N for parallel and perpendicular pulling scenarios.

Conclusion

Using fully atomistic MD simulations in the presence of explicit water, we obtain the viscous friction coefficient γ_HB per interpeptide HB as a function of the bundle aggregation number k. The confinement-dependent HB friction coefficient ranges from γ_HB ≃ 10⁻¹¹ kg/s for two polyglycine chains sliding against each other to γ_HB ≃ 10⁻⁶ kg/s for a polyglycine being pulled out from a close-packed bundle with six neighboring chains. This dramatic increase of the friction coefficient means that internal friction sensitively depends on the local environment that a peptide chain experiences: A peptide chain at the periphery of a protein globule will be subject to much less friction than a peptide chain in the close-packed interior of a protein. Because the friction coefficient is proportional to the diffusion coefficient, our results show that confinement can slow down the diffusional dynamics by greater than five orders of magnitude as one goes from a fully solvent-exposed residue to a completely buried solvent-depleted residue. We hasten to add that the friction coefficient γ_HB by definition is a measure of the friction per HB; because the number of HBs is higher in more confined regions, the resulting friction will even be enhanced beyond the confinement effects predicted for γ_HB.

There are different ways of lending intuitive meaning to the rather abstract notion of a friction coefficient per HB (15). Stokes law predicts that the friction coefficient of a sphere with radius R moving in a viscous fluid with viscosity η is given by γ_St = 6πηR. The equivalent radii for a single HB predicted from the viscous friction coefficients γ_HB from Stokes law amount to R ≃ 10,100 nm for k = 2, 3 and R ≃ 1, 1000 μm for k = 5, 7, respectively. In the fully confined bundle for k = 7, the friction of a single HB corresponds thus to the equivalent viscous friction of a sphere with a radius of R = 100 μm moving in water, which is an enormous amplification of the effective size. Another revealing quantity is the reptation time, i.e., the time it takes for an assembly of N_HB HBs to diffuse over a length corresponding to the contour bN_HB. According to the one-dimensional diffusion equation, this time is given by $\tau \simeq N_{\text{HB}}^3{b}^2{\gamma }_{\text{HB}}/\left(k_{\text{B}}T\right)$. Taking b = 0.2 nm and γ_HB ≃ 10⁻⁸ kg/s as obtained for a peptide in a k = 5 bundle, we obtain $\tau \simeq N_{\text{HB}}^3\times 100$ ns, thus for N_HB = 5 we obtain a diffusional time of 10 μs. For the fully confined peptide in the k = 7 bundle we obtain for N_HB = 5 the enormous diffusional time of 1 ms. This shows that HB friction associated with the diffusive escape from misfolded motifs can account for very long times up to the millisecond scale, which also means that, for efficient and fast folding, such traps have to be avoided by sequence design.

In extracting the asymptotic viscous friction coefficient, we used two different fitting schemes: in one, we neglected cooperativity by setting m = 1 and extracted the HB energy U_HB, which in this scheme was shown to increase with growing confinement but to take on unrealistically high values for the fully confined bundle with k = 7. In the other scheme, we fixed the HB energy at U_HB = 4.5 k_BT and extracted the cooperativity factor m which was shown to increase almost linearly with the bundle aggregation number k. Both fitting schemes basically led to the same extracted friction coefficient per hydrogen bond, so our results for γ_HB are robust. Clearly, reality lies in-between these two limiting cases.

It is very plausible that the HB energy increases as the confinement goes up, because the solvent-accessible surface of the pulled peptide goes drastically down as the bundle size increases (see Fig. S9 for surface-accessible areas as a function of k). It is known that in the aqueous environment, single HB energies range between U_HB ≃ 2–3 k_BT, whereas in vacuum, they go up to U_HB ≃ 10 k_BT (40). In fact, in the Supporting Material, we estimate the effective HB strength $U_{\text{HB}}^{\text{eff}}$ in bundles of varying degree of confinement based on the HB lifetime using a simple Kramer’s approach in the absence of pulling. We find $U_{\text{HB}}^{\text{eff}}$ to increase from $U_{\text{HB}}^{\text{eff}}=5k_{\text{B}}T$ for k = 2 to $U_{\text{HB}}^{\text{eff}}=11.2k_{\text{B}}T$ for k = 7 (see Fig. S3)—indeed intermediate between our two fitting approaches presented above. But because the Kramer’s approach assumes the kinetic prefactor to be independent of k, the result for $U_{\text{HB}}^{\text{eff}}$ should be considered with care as well.

It is, on the other hand, also plausible that the cooperativity increases with growing confinement, because the distance between neighboring HBs that the pulled peptide forms, goes down (as follows from the fact that the HB number the pulled peptide forms grows approximately linearly with the number of neighbors). Enhanced cooperativity is also expected due to the increased mechanical stiffness in bundles of higher aggregation number k. In essence, we conclude that both water depletion effects and cooperativity effects contribute to the increase of friction as the confinement goes up, which is fully reflected by the fact that the main parameter in our extrapolation model is the product mU_HB of the cooperativity factor and the single HB energy. The unambiguous disentanglement of these two effects is left for future work.