Dynamics of Disordered Proteins under Confinement: Memory Effects

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Dynamics of Disordered Proteins Under Confinement: Memory Effects and Internal Friction Atanu Das, and Dmitrii E. Makarov J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.8b06112 • Publication Date (Web): 09 Aug 2018 Downloaded from http://pubs.acs.org on August 13, 2018

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The Journal of Physical Chemistry

Dynamics of Disordered Proteins Under Confinement: Memory Effects and Internal Friction Atanu Das† and Dmitrii E. Makarov*†‡ Department of Chemistry, University of Texas at Austin, Austin, Texas 78712, USA

†

‡

Institute for Computational Engineering and Sciences, University of Texas at Austin, Austin, Texas 78712, USA

ABSTRACT: Many proteins are disordered under physiological conditions. How efficiently they can search for their cellular targets and how fast they can fold upon target binding is determined by their intrinsic dynamics, which have thus attracted much recent attention. Experiments and molecular simulations show that the inherent reconfiguration timescale for unfolded proteins has a solvent friction component and an internal friction component, and the microscopic origin of the latter, along with its proper mathematical description, has been a topic of considerable debate. Internal friction varies across different proteins of comparable length and increases with decreasing denaturant concentration, showing that it depends on how compact the protein is. Here we report on a systematic atomistic simulation study of how confinement, which induces a more

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compact unfolded state, affects dynamics and friction in disordered peptides. We find that the average reconfiguration timescales increase exponentially as the peptide’s spatial dimensions are reduced; at the same time, confinement broadens the spectrum of relaxation timescales exhibited by the peptide. There are two important implications of this broadening: First, it limits applicability of the common Rouse and Zimm models with internal friction, as those models attempt to capture internal friction effects by introducing a single internal friction timescale. Second, the long-tailed distribution of relaxation times leads to anomalous diffusion effects in the dynamics of intramolecular distances. Analysis and interpretation of experimental signals from various measurements that probe intramolecular protein dynamics (such as single-molecule fluorescence correlation spectroscopy and single-molecule force spectroscopy) rely on the assumption of diffusive dynamics along the distances being probed; hence, our results suggest the need for more general models allowing for anomalous diffusion effects.

1. INTRODUCTION It has been recognized for decades that that elementary steps in protein folding are largely determined by the reconfiguration dynamics in the unfolded state . Thus motivated, many 1-3

experiments focused on the dynamics of unfolded proteins . But the significance of the dynamics 4-16

in the unfolded state goes far beyond the subject of protein folding. Many proteins are either disordered or contain long disordered segments under physiological conditions. To understand how quickly they can search for their biological targets, it is essential to study their dynamics . 17-18

Recent discovery that protein interactions may occur via highly dynamic, unstructured complexes

19

will likely further increase interest in the thermodynamics and dynamics of the unfolded state.


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One of the somewhat unexpected findings emerging from experimental studies , as well as 20-25

from theoretical studies and simulations , is that the observable dynamics and folding of proteins, 26-34

large macromolecules with heterogeneous chemical sequences, can be rationalized using the rather simple model of moving along a one-dimensional “reaction coordinate” R, which is usually synonymous with or related to the experimental signal . In application to folding or other large 35

conformational rearrangements, the existence of such a coordinate, which is descriptive of the conformational change of interest, is, of course, the fundamental premise of transition state theory, but transition state theory is not expected to apply to molecules as large, as complex, and as floppy as proteins. The limitations of transition-state theory in application to polyatomic molecules were recognized long ago by Kramers , who was the first to propose that the perturbation exerted by 36

many molecular degrees of freedom on the reaction coordinate can be modeled as friction similar to the hydrodynamic friction appearing in the theory of Brownian motion. Kramers himself, however, was well aware of the potential inadequacy of this model: while the origin of the hydrodynamic friction acting on micrometer-sized Brownian particles is well understood, similar justification for a friction force acting along a molecular reaction coordinate and being proportional to velocity does not exist. At best, certain, special coordinates, to which a one-dimensional diffusion picture is applicable, are anticipated theoretically . Those coordinates are, however, 37-40

complex functions of all the molecular coordinates. As such, they are a challenge to predict even computationally, and they are unlikely to coincide with, or be close to common experimental reaction coordinates. Despite these caveats, the Kramers model of folding kinetics was embraced by the protein folding community and led to many important insights. In terms of this model, the dynamics along an experimental observable R is completely described by a potential of mean force U(R) and by a


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friction coefficient γ (where −γ dR /dt is the friction force), or, equivalently, by a diffusion coefficient, which is related to the friction coefficient via the Einstein-Smoluchowski relationship D = kBT / γ . The dynamics of the protein in the unfolded state is then completely characterized

by introducing a characteristic timescale (known as the reconfiguration timescale τ r ) and a characteristic lengthscale l (which, for example, can be taken to be the polypeptide’s radius of gyration or the root mean square of its end-to-end distance); in terms of these two scales, the diffusion coefficient can be expressed as l2 D~ . τr

The rate of protein folding k f can be further rationalized by employing the Kramers formula, which gives an Arrhenius-type law ΔU

1 −k T kf ~ e B , τr

where is ΔU the free energy barrier encountered along the potential of mean force. Further refinements of this result are possible to account, for example, for the spatial variation of the diffusion coefficient D = D(R)

28, 41-45

. Given its central role in this diffusive model of folding

dynamics, understanding the physical nature of the diffusion coefficient (or quantities related to it) became the subject of many experimental studies and remains a topic of active research at present

25, 46-49

.

In particular, does the effective friction along the reaction coordinate originate from intrachain interactions within the polypeptide (internal friction) or from interactions with the surrounding water (solvent friction) ? To answer this question, one can measure the viscosity dependence of 50-53

the friction coefficient and identify the viscosity independent component as internal friction and


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viscosity dependent component as solvent friction . The relative contribution of internal friction 52

has indeed been found to be significant; it is also strongly dependent on the system and on the timescale of interest . 54-60

The physical origin of internal friction has been a subject of some debate, and many different (but not mutually exclusive) mechanisms have been proposed, including roughness of the energy landscape , hydrogen bonding , salt bridges , solvent memory effects , and crankshaft-like 61-62

63-64

65

66-67

dihedral rotations . 68-69

In polymer physics, the subject of internal friction (or internal viscosity) dates back to the work of Kuhn (see ref.

70-74

for a survey of this topic); many recent efforts to understand internal friction

have thus turned to polymer physics ideas

58-59, 63, 68-69, 75-76

. The relevance of simple polymer theories,

usually based on homopolymer models, as a description of polypeptide chains with highly heterogeneous sequences is far from obvious . Yet large body of evidence indicates that such 77

models provide a good first order approximation for the chain statistics

78-82

and dynamics

58, 76, 83-85

in the

unfolded state. In particular, the experimentally observed dynamics of proteins that are either disordered intrinsically or unfolded using denaturants can often be described using remarkably simple models from polymer physics based that are extensions of the classic Rouse and Zimm models

58, 73, 76, 84

. In the

Rouse or Zimm model with internal friction (RIF and ZIF), the effect of internal friction is captured by a single, and experimentally measurable , internal friction timescale τ i , which is equal to the 76

polypeptide’s relaxation timescale extrapolated to zero solvent viscosity. RIF and ZIF have been the focus of many recent theoretical studies

84, 86-90

experimental observations

58, 76, 85, 91

and have been used to rationalize a number of

.


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Further modification of the classic Rouse and Zimm models is required to account for the denaturant-induced changes in the polymer statistics and dimensions. For example, a protein that is unfolded under native conditions is expected to have a smaller radius of gyration than the same protein at a high concentration of denaturant. Moreover, while highly denatured proteins are expected to behave like excluded-volume like random coils , they are often close to the theta point 79

under native conditions . Even under the native conditions, there is large variation in the 92-98

compactness of the unfolded state, with highly charged intrinsically disordered proteins often being more expanded than unfolded proteins that have a well defined folded state. A simple way to account for these effects is to introduce an attractive potential; for example, in our earlier work

76,

84

we envisioned placing the protein in a trap that exerts an attractive central parabolic potential on

each monomer, whose strength is adjusted to match the experimentally observed dimensions of the chain at different denaturant concentrations. Although, of course, such an ad hoc model does not capture the physical mechanisms of a denaturant’s action , it is consistent with more realistic 99-100

models of the coil-globule transition and it captures the experimentally observed effect of 76

denaturants on polypeptide dynamics while preserving linearity, and thus simplicity, of the Rouse 84

and Zimm models. The model that mimics the effects of a denaturant via a confining potential captures the speedup of intrachain dynamics with decreased denaturant concentration

12-13, 84

as long as internal friction

effects are negligible; when they are not, they result in the opposite trend, where increased compaction of the chain leads to an increase in the internal friction timescale τ i thus slowing intrachain dynamics . 76

Similar correlation between the internal friction timescale and the

compactness of the unfolded chain is also observed across different proteins , with highly charged 76

and thus expanded intrinsically disordered proteins showing little internal friction even under


6

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native conditions . This correlation suggests that nonlocal sequence-distant interactions (which 58

increase upon chain compaction) are responsible for internal friction. Thus assertion, however, appears to be at odds with the finding that local properties of the polypeptide’s energy landscape such as the barrier to dihedral rotations control the magnitude of internal friction . Specifically, 68-69

decreasing (increasing) the magnitude of the dihedral barrier Udih in simulations of an unfolded protein has led to a smaller (greater) value of τ i ; moreover, the dependence of τ i on Udih is close U dih

to exponential, τ i ∝e

kBT

, consistent with the picture that dihedral rotations control the protein’s

reconfiguration dynamics in the limit where internal friction effects dominate. But since changing U does not significantly change the dimensions of the unfolded chain, the role of the chain dih

compactness is not explained within this picture. Thus complete understanding of the molecular mechanisms leading to internal friction in unfolded proteins is still lacking. Another open question about the dynamics of unfolded proteins is concerned with validity of the model of one-dimensional diffusion along an experimental reaction coordinate. Several theoretical and experimental 101

21-22

studies attacked this question directly, concluding that this model

is adequate, at least for a subset of experimental systems studies by single-molecule force spectroscopy. At the same time, there is overwhelming evidence from molecular simulations that the dynamics of inter-monomer distances and other commonly used reaction coordinates cannot always be modeled as a memoryless diffusion process

32, 40, 68, 102-109

report anomalous diffusion

110-111

.

, and several experimental studies also

In particular, subdiffusive dynamics, where mean square

displacements grow more slowly than linear in time, is often observed in simulation studies. Such subdiffusive behavior is not a short-time effect that becomes negligible when longer-time phenomena are studied: subdiffusive character of the dynamics, for example, drastically affects


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the folding dynamics of a fast folding protein at timescales of several hundreds of nanoseconds , 112

106

which are longer than its average transition path time. Subdiffusive monomer dynamics is to be expected for the Rouse and Zimm models at intermediate timescales

87, 113-114

; however, analysis of

RIF/ZIF dynamics shows that diffusive behavior should be recovered in the limit of high internal friction . But direct simulations of protein dynamics in this limit show that this is not so, calling 84

68

validity of RIF and ZIF models into question. Motivated by the above questions, we seek here to understand how reconfiguration dynamics of disordered polypeptides depend on both their dihedral energy landscape and on their compactness, how the dynamics in a polypeptide’s conformational space is manifested when further projected onto a one-dimensional reaction coordinate, and why (and if) this one-dimensional dynamics is subdiffusive. We chose the Gly-Ser repeat peptide as a model system, because its properties are close to those of a random coil

13, 84

and because it is implicated in cellular processes . We induce a 115

more compact state of the chain by simply placing within a cubic trap made from Argon atoms, as shown in Figure 1. We note that, in addition to elucidating how protein compactness affects internal friction, our simulations also offer insight as to how cellular crowding or protein encapsulation within, e.g., nanoparticles might affect protein dynamics. A key finding of our study is that, although compaction leads to higher internal friction (as found in earlier studies), this effect cannot be simply described in terms of a single internal friction timescale τ i assumed by simple polymer models. Rather, a distribution of timescales, which is shifted toward longer times while, at the same time, becomes more heavy-tailed, must be considered in order to account for anomalous diffusion effects observed in the dynamics of intermonomer distances.


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2. METHODS 2.1. Simulations of Unconfined Peptides. We performed simulations of an 11-residue peptide fragment with the Gly-Ser repeat, which is often viewed as a model polypeptide with properties close to those of a random coil . The random-coil behavior of this peptide is confirmed by the 13, 84

distribution of its end-to-end distance, which is close to that expected for a random coil (Fig. S1). Molecular dynamics simulations were performed using the GROMACS software package, version 4.5.5 . Parameters were taken from the Amber03 parameter set . TIP4P-D explicit water model 116

117

was used to represent the solvent, because this water model is known to accurately reproduce experimentally observed conformational ensembles for disordered proteins . The peptide was 118

solvated in a cubic 5.2 ´ 5.2 ´ 5.2 nm box. Counterions were added using the genion module of 3

GROMACS, which randomly replaces water molecules with counterions in favorable locations determined by computing the electrostatic potential at the insertion site. Each system was energy minimized by, first, the steepest descent algorithm and then by a conjugate gradient algorithm, to arrive at a conformation with no steric clashes. Each of these minimized conformations was equilibrated in two steps, with position restraints applied to all the heavy atoms throughout. The first step involved simulating each system for 500 ps under a constant volume (NVT) ensemble. All the atoms were coupled to a bath, with the temperature kept at 300 K using the Berendsen weak coupling method. Following NVT equilibration, 500 ps of constant pressure (NPT) equilibration was accomplished to maintain pressure isotropically at 0.138 atm. All the simulations were performed in a cubic cell employing periodic boundary conditions with the standard minimum image convention in all three directions. Long-range electrostatics was treated with the particle mesh Ewald method . The cutoff used for Lennard-Jones interactions was 9 Å. Particle 119

mesh Ewald method with a real space cut-off at 9 Å was used to account for the electrostatic


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interactions. All bond lengths and all angles involving hydrogen atoms were constrained using the LINCS algorithm . An integration time step of 2 fs was used for all the simulations. Production 120

MD runs were performed at T=300 K and P=0.138 atm in the absence of any restraints; these conditions are close to those employed in previous experimental and theoretical 76

68-69

studies. The

modified Berendsen thermostat was used to maintain temperature, and the Parrinello-Rahman 121

barostat was used to isotropically regulate pressure during the production runs. The total length 122

of a production run varies from ~2 µs (for the unconfined peptides) to ~10 µs (for the confined peptides). A set of three independent trajectories was run for all the systems to estimate the error bars on the relaxation times and the exponent α .

2.2. Simulations of Confined Peptides. To study how compaction affects peptide dynamics, we placed peptides inside a cubic argon box with a layer of argon atoms on all the three sides of the box. We chose argon to minimize any specific interactions between the peptide and the box. Argon parameters were taken from OPLS-AA parameter set.

123-124

Four different cubic argon boxes

were used, with edge lengths of 4.2 nm, 3.6 nm, 3.0 nm, and 2.4 nm. Each of these systems was then placed inside a cubic cell whose edge length was 5.2 nm, as in the case of unconfined peptides. The molecules were solvated with TIP4P-D water molecules, and counterions were added following the same procedure as in the case of unconfined peptides. Minimization, equilibration, and production runs were performed as described above for the case of unconfined peptides, the only difference being that position restraints were applied to all the argon atoms throughout the production run in order to maintain the size and the shape of the argon box.


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2.3. Simulations with Modified Dihedral Barrier Heights. The potential energy function describing the backbone dihedral angles has the form:

U U(θ ) = n [1+ cos(nθ − γ )] 2

(1)

where θ is the dihedral angle (either φ or ψ ) and Un is the corresponding force constant. The phase angle γ takes values of either 0° or 180°, and n is an integer that determines the periodicity of the potential . 117

To explore the connection between internal friction and the dihedral energy landscape, we repeated our simulations for the 11-residue Gly-Ser repeat peptide using two stronger and two softer dihedral potentials by rescaling all dihedral force constants by factors of 1/2, 3/4, 3/2, and 2. Since the dihedral barrier heights are somewhat different (but comparable) for φ and ψ (see Fig. S2), the value for the dihedral barrier Udih reported above is that for the angle φ .

2.4. Calculation of the Relaxation Times. The relaxation time of the quantity of interest was computed from its autocorrelation function (ACF) C(τ ) , according to ∞

τ r = ∫ CN (t)dt , (2) 0

where C(τ )− C(∞) C N (τ ) = C(0)− C(∞)

(3)

is the normalized autocorrelation function. For a dihedral angle θ(t), the corresponding autocorrelation function was defined as : 125

C (τ ) = cos ⎡⎣θ ( t ) − θ ( t + τ ) ⎤⎦

(4)


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Normalized dihedral ACF was further averaged over all the dihedrals in the peptide before computing a single dihedral relaxation time from Eq. 2. For the end-to-end vector R, the ACF was defined as C (τ ) = R (t ) R (t + τ ) , and for the endto-end distance R we have C (τ ) = R ( t ) R ( t + τ ) . Because of the noise in the raw ACFs, the integral of Eq. 2 was evaluated using analytic fits of C N (t) . Single exponential, bi-exponential and stretched exponential fits were, respectively, used

for the end-to-end vector, dihedral angle and end-to-end distance ACFs. Those fits along with the raw autocorrelation functions are shown in Fig. S3-S51.

Figure 1. A snapshot of an 11-residue Gly-Ser repeat confined within an Argon box.

3. RESULTS 3.1. Both the End-to-End Relaxation Time τ ee and the Dihedral Relaxation Time τ dih of a Confined Polypeptide Depend Strongly on Its Radius of Gyration (Fig. 2). As seen in Figure


12

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2, both the end-to-end distance relaxation time and the dihedral relaxation time of a confined peptide increase by two orders of magnitude when its radius of gyration R is decreased by ~25%. g

The sharp increase in the end-to-end relaxation time upon compaction found in Figure 2 is similar to the R dependence of the internal friction time τ i found for several proteins and for varied g

denaturant concentrations, see Figure 5 in ref. . The R dependence of the dihedral relaxation time 76

g

can be fitted by the exponential function τ dih = ae

− R g /Reff

(see Fig. 2), with a characteristic length

Reff that is approximately equal to ~5.0 Å.

Figure 2. End-to-end relaxation time τ ee and dihedral relaxation time τ dih plotted as a function of the polypeptide’s radius of gyration (normalized by the radius of gyration of the same peptide in the absence of confinement). The dashed line represents exponential fit to τ dih . The inset shows the same relaxation times on a logarithmic scale.


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3.2. Both the End-to-End Relaxation Time τ ee and the Dihedral Relaxation Time τ dih of a Confined Polypeptide Increase with the Increasing Barrier to Dihedral Rotations (Fig. 3). As observed in earlier studies , the dihedral relaxation time increases exponentially with the height 68-69

of the dihedral barrier, indicating Arrhenius-type barrier crossing dynamics (Fig. 3). Indeed, the slope of the vs ln τ dih vs Udih dependence observed in Figure 3a is close to (kBT )−1 . For the largest box used in the simulations the effects of confinement are essentially negligible. In this case, the end-to-end distance relaxation time shows an upturn in its dihedral barrier dependence, where it is nearly independent of the dihedral barrier Udih when this barrier is low enough ( U dih ≤ 9 kJ/mol), but increases exponentially (and proportionally to τ dih ) for higher values of U dih (Fig. 3b). This behavior is indicative of a transition between the Rouse/Zimm type of dynamics for low dihedral barriers, where the relaxation time is controlled by the solvent friction, to Kuhn internal friction regime, where dihedral dynamics controls global relaxation of the polymer coil . Consistent with this assertion, the end-to-end distance relaxation time (indicative of 68

internal dynamics) is comparable to the end-to-end vector relaxation time (indicative of rotational dynamics) in the low-barrier case, as expected for the Rouse and Zimm models (see Table S1). In 68

contrast, for higher dihedral barriers the end-to-end distance relaxation time is much longer than the end-to-end vector relaxation time, indicating that the internal dynamics is much slower than the rotational dynamics – this is a signature of the internal-friction dominated regime . 68


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Figure 3. Relaxation times, τ dih and τ ee , vs. dihedral barrier height for different sizes L of the confining box. The vertical black dashed line shows the normal dihedral barrier height set by the force field used. The inset shows the same relaxation times on a logarithmic scale.

3.3. Dihedral and End-to-End Distance Relaxation Times Are Linearly Correlated. When the dihedral relaxation time is plotted against the end-to-end distance relaxation time (Fig. 4 – same data as in Fig. 3, with two additional data points for L = 3.0 nm and L = 2.4 nm, with normal dihedral barrier height) for different values of the dihedral barrier and different values of the box size, we see that the two times are approximately linearly correlated. When the dihedral barrier is increased, this slows down the dihedral hopping dynamics. Consistent with the picture where internal friction results from dihedral rearrangements , this also increases the global relaxation 68-69

time as quantified by the end-to-end distance relaxation time τ ee . Likewise, when the chain becomes more compact, the dihedral dynamics becomes slower and, accordingly, τ ee decreases. In fact, the effect of compaction on the dihedral relaxation time can be understood as a result of increased free energy barrier to dihedral rotations in a compacted chain (Fig. 5).


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Figure 4. Dihedral angle relaxation time vs. end-to-end distance relaxation time. Different colors correspond to different box sizes, as shown in the legend. The black dashed line shows a linear fit to the entire data set.

Figure 5. The effect of confinement on the potential of mean force measured as a function of the

f dihedral angle of GLY-5: The free energy barrier to dihedral rotation increases as a result of chain compaction. A detailed analysis of all the backbone f and y dihedral angles is given in the Supporting Information (Fig. S52).


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3.4. Confinement Increases Non-Markov, Subdiffusive Character of End-to-End Distance Dynamics. In the case of Rouse or Zimm dynamics, all the dynamical timescales of the polymer scale proportionally to the solvent friction. Given the nearly linear correlation between the dihedral relaxation time and the global end-to-end distance relaxation time, and given that the dihedral relaxation time is proportional to the internal friction timescale τ i , it is tempting to propose that 68

internal friction has a similar effect to that of solvent friction, leading to proportional scaling of all relevant timescales. This is, however, not the case; instead, the nature of dynamics changes upon compaction of the polypeptide. Indeed, consider the behavior of the mean square displacement measured along a polypeptide’s trajectory 2

2

ΔR (Δt)

dt $%R(t + Δt)− R(t)&' ∫ = ∫ dt

,

(5)

where R(t ) is its end-to-end distance as a function of time. If the dynamics of the end-to-end distance is diffusive, then, at short enough times (where the effects of the chain connectivity or, equivalently, of the associated potential of mean force are negligible), one has

ΔR2(Δt) = 2DΔt , (6) where D is a diffusion coefficient. For free diffusion, the above equation holds for any time Δt , but, of course, chain connectivity precludes free diffusion of one chain end relative to the other. For short enough times, however, chain connectivity effects become negligible and the diffusion coefficient D can be estimated.


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The assumption of diffusive dynamics is, however, incorrect for the end-to-end dynamics of the peptides studied here (Fig. 6). Instead, their dynamics is subdiffusive, with the mean square displacement growing with time as a power law,

ΔR2(Δt ) ∝ Δt α

(7)

Such subdiffusive behavior has been reported in earlier studies

40, 68, 102, 105-106, 126-127

. A remarkable feature

observed in Figure 6 is that the dynamics becomes more subdiffusive as confinement is increased, with the exponent α decreasing from ~0.6 for the unconfined peptide to ~0.1 for the most compact peptide. This finding shows that compaction does not merely lengthen the internal friction timescale, but fundamentally alters the nature of protein dynamics. To further emphasize this point, let us reiterate that the effect of compaction on both the dihedral ( τ dih ) and end-to-end distance ( τ ee ) relaxation times can be interpreted as a result of increased free energy barrier to dihedral rotations. In other words, both times can be changed, in a similar way, by either changing the enthalpy barrier associated with dihedral rotation or by changing the volume within which the chain is confined. In contrast, the exponent α is essentially unaffected by the dihedral enthalpy barrier (Fig. 7), while it is significantly affected by confinement. Thus a single internal friction timescale τ i (which is proportional to τ dih ) cannot provide a complete description of dynamics in the internal-friction dominated regime. As will be further argued in the next Section, the distribution of relaxation times, rather than merely their mean, must be considered to understand how peptide dynamics is altered by confinement. Anomalous diffusion observed in the time evolution of inter-monomer distances is a fingerprint of non-Markovian dynamics. The increasingly non-Markovian character of the trajectories R(t )


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observed with the increased confinement can also directly be seen by applying the “nonMarkovianity” test proposed in ref. . In this test, one defines an interval (a,b) within which the 128

motion is investigated and measures the probability P(a → b|R) that a specified point R, a < R < b , belongs to a path that enters this interval from the left through the boundary a and exits the interval to the right through the boundary b. For a Markov process, the maximum value of P(a → b|R) is achieved at some intermediate point within the interval and is equal exactly to 1/4.

For a non-Markov process, however, the maximum value is below 1/4, with a decreasing maximum value indicating increasingly pronounced memory effects. We applied this test to each of the peptide trajectories (Fig. 8). Because the range of values of the end-to-end distance R spanned by each peptide depends on confinement, we chose, in each case, a value of a that corresponds to the maximum of the probability distribution of R. The value of b was then chosen equal to a plus the variance of the distribution of R.

4. DISCUSSION AND CONCLUSIONS Let us restate two key findings that emerge from our results: (i) Both increased confinement and increased enthalpic dihedral barrier slow down the global dynamics of peptides, an effect that can be interpreted as increased internal friction. (ii) Yet increased confinement fundamentally changes the character of the dynamics of intramolecular distances, making them more subdiffusive. In contrast, simply changing the dihedral barrier does not. These observations lead us to conclude that peptide dynamics cannot be fully characterized in terms of the internal friction timescale τ i alone. To further emphasize this point, we recall that the Rouse and Zimm models with internal friction predict that an increased internal friction timescale results in less subdiffusive dynamics of intermolecular distances , a prediction opposite what we observe here. 84


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To understand what causes this discrepancy, we need to examine the microscopic origins of subdiffusive dynamics. Many distinct physical mechanisms may account for the behavior expressed by Eq. 7 ; a common feature of the systems displaying such anomalous diffusion is that 129

they lack a single characteristic timescale. In particular, conformational trapping has been proposed to explain anomalous diffusion effects in proteins

102-103, 126

. This explanation is appealing in

the present context because confinement is expected to increase the likelihood that the peptide becomes trapped in a local energy minimum, as escaping the minimum may require a large conformational rearrangement of the chain impeded by the confining volume

130-131

.

A quantitative view of how trapping leads to anomalous diffusion is provided by continuous time random walk (CTRW) type models , where the system undergoes discrete jumps in 129

conformational space. An essential feature that leads to non-diffusive behavior in these models is a broad distribution of the waiting times between successive jumps . In particular, slowly decaying 126

waiting time distributions with power-law tails are known to lead to subdiffusive dynamics

129, 132

.

Is the peptide dynamics studied here consistent with the CTRW picture? To answer this question, we have examined the waiting time distributions between distinct structural jumps exhibited by the peptides. The structural jumps were identified in two ways. First, we have defined them as distinct events where one of the dihedrals of the chain transitions into a different region of the Ramachandran plot (see Fig. S53). Second, we employed principal component analysis and considered the time evolution of the first principal component (PC1) . Distinct jumps in PC1 were 133

then identified using the change point algorithm of Watkins et al. . In each case, the distribution 134

of the waiting times between jumps was computed (Fig. 9a,b). As seen from Fig. 9, the distribution of the waiting times between subsequent jumps is generally nonexponential. Although insufficient statistics prevents us from establishing its precise functional


20

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form, we can quantify its nonexponentiality by evaluating the coefficient of variation (CV) defined as the ratio of the standard deviation of a distribution to its mean,

CV =

⎛ t2 − t 2⎞ ⎝ ⎠ t

1/2

(8)

For an exponential distribution, we have CV=1, values of CV significantly exceeding 1 indicate a long-tailed distribution . Starting from unconfined peptide and decreasing the box size to 4.2, 135

3.6, 3.0, and 2.4 nm, the values of CV calculated for the distributions shown in Fig. 9 are 2.5, 3.7, 4.2, 6.9, and 29.9 for the waiting time analysis based on PCl; they are equal to 3.79, 4.63, 6.78, 12.7, and 16.9 for the waiting time distributions of the dihedral jumps. Therefore, the distributions increasingly deviate from the exponential form as the peptide conformations become more compact, consistent with the observation that the dynamics becomes less Markovian. This finding lends support to the CTRW-type mechanism of subdiffusive dynamics, where confinement broadens the distribution of barriers to transition among different conformations thereby also broadening the waiting time distribution. Despite evidence in support of the conformational trapping mechanism (see also ref. ), our 102

understanding of the physical origins of subdiffusive dynamics is incomplete. In particular, the standard CTRW model with a power-law distribution of waiting time leads to a subdiffusive behavior for the mean square displacement averaged over an ensemble of trajectories R(t )

⎡⎣ R(Δt )− R(0)⎤⎦

2

129

∝ Δt α (9) ensemble

Because CTRW dynamics is not ergodic , this average differs from the time average performed 129

along a single trajectory, as prescribed by Eq. 5. In fact, the average of Eq. 5 for a CTRW in free space still shows linear growth with Δt , which contradicts our observations. Noisy CTRWs or 129

129


21


Page 22 of 46

CTRW models with non-power-law waiting time distributions may be better candidates for models describing our data and should be explored in the future. Chemically denaturated proteins are often well described as Flory random coils

, even when

79-80, 95

residual structure is non-negligible . Somewhat surprisingly, simple models treating proteins as 81-82

random homopolymers often provide a good description of intrinsically disordered proteins and of proteins unfolded under native conditions

93, 136-137

, an observation also supported by computer

simulations . A key breakthrough in understanding the dynamics of unfolded proteins, which 58, 69

typically occurs at a nanosecond timescale, came from single-molecule fluorescence correlation spectroscopy (FCS) experiments

20, 138-142

. Interpretation of such studies, however, faces a challenge:

the dynamics of even the simplest Flory random coil, as described by Rouse and Zimm models, exhibits a spectrum of timescales corresponding to various collective modes , while experimental 143

information extracted from FCS studies usually involves a single timescale corresponding to the relative motion of a pair of monomers. By placing probes at different positions along polypeptide chains

, by studying the viscosity and denaturant dependence of the dynamics , and by

76, 144

76

combining FCS measurements with other techniques such as quenching studies and with molecular simulations , it was possible to tease out certain details about spatio-temporal correlations within 58

unfolded proteins. Yet findings reported here suggest that a comprehensive picture is still lacking. Outlined below are the specific challenges that we believe still should be addressed in future work: (i) The most successful polymer physics model of protein dynamics as observed in FCS and contact quenching studies is the Rouse/Zimm model with internal friction (RIF/ZIF)

73, 76, 84

. Many

predictions of this model are consistent with experimental measurements . However findings 58, 76

presented here and in an earlier study indicate that certain predictions from ZIF/RIF disagree with 68

simulations. Specifically, RIF/ZIF predicts that inter-monomer dynamics must become purely


22

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diffusive in the limit where internal friction dominates . Here, in contrast, we find that the inter84

monomer dynamics becomes increasingly non-Markov and subdiffusive as the internal friction effects are increased as a result of confinement. This disagreement can be traced back to the fact that RIF/ZIF predict that the internal-friction-dominated regime can be characterized by a single characteristic timescale τ i . At odds with this assertion, we see a long-tailed distribution of timescales that becomes broader as internal friction increases. Is it possible to generalize RIF/ZIF to account for a distribution of internal friction timescales? Can alternative polymer models

59

describe this behavior better? (ii) Results reported here and in earlier studies

68, 106

suggest that the dynamics of common

experimental observables such as interatomic distances are almost universally subdiffusive. Yet analysis of experimental studies often relies on the model of diffusive dynamics

138-140

. To ensure

internal consistency between data analysis and experimental conclusions, more general models (either those describing the dynamics of one-dimensional reaction coordinates or models of polymer dynamics) are desirable. (iii) Related to the above two points, given that the assumption of one-dimensional diffusive dynamics along intramolecular distances and other common reaction coordinates breaks down 106

(at least in simulation studies), is there a more general theoretical model that can be used instead? Several generalizations, in fact, exist that account for subdiffusive dynamics. Those include the generalized Langevin equation with a memory kernel and a number of different generalizations of the Smoluchowski equation

106, 126, 132, 145-147

. Although all of them capture anomalous diffusion, their

specific predictions are different; for example some of them show nonergodic behavior while others do not. Which one (if any) is consistent with both microscopic dynamics seen in simulations and with experimental observations?


23


Page 24 of 46

Much of our current understanding of the unfolded state of proteins, of its thermodynamics, and of its dynamics is a result of closely coupled efforts from experimentalists and theoreticians. The authors, likewise, hope that the above questions will be answered in the near future through synergy of experimental, computational, and theoretical efforts.

Figure 6. Mean square displacement of the end-to-end distance as a function of time plotted on a log-log scale. The slope of the plot is the exponent α defined by Eq. 7. Different colors correspond to different box sizes. The peptide shows increasingly subdiffusive behavior (i.e., smaller values of α ) as the confinement increases.


24

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Figure 7. Exponent α plotted as a function of the dihedral barrier for free and confined peptides. Dashed lines show linear fits of the data.

Figure 8. “Markovianity test” : the probability P(a → b|R) that a point R belongs to a transition 128

path traveling from a point a to a point b>a (a chosen at the maximum of the distribution of R and b-a chosen to be equal to the variance of the distribution) peaks at a value that becomes smaller as confinement is increased. While in the absence of confinement this peak value is close to 0.25, a value expected for Markov dynamics, progressively low values observed as confinement is increased indicate increased memory effects.


25


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Figure 9. Waiting time distributions for jumps in (a) first principal component PC1 and (b) dihedral angles.

ASSOCIATED CONTENT Supporting Information. Random coil behavior of the Gly-Ser repeat peptide; force field defined torsional barrier height; end-to-end distance, end-to-end vector, and dihedral angle autocorrelation functions; effect of compaction on individual dihedral angles; waiting time estimation protocol.

AUTHOR INFORMATION Corresponding Author *E-mail: [email protected]. Notes


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The authors declare no competing financial interest. Author Contributions The manuscript was written through contributions of both authors. Both authors have given approval to the final version of the manuscript.

ACKNOWLEDGMENT We are indebted to Haw Yang for providing us with his change point detection code and for help with this code, and to Hagen Hofmann, Rohit Satija, and Benjamin Schuler for comments and discussions. This work was supported by the Robert A. Welch Foundation (Grant No. F-1514) and the National Science Foundation (Grant No. CHE 1566001).

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TOC. Effect of confinement on peptide conformation.


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Biographies Atanu Das received his B.S. and M. S. degrees from University of Calcutta in West Bengal, India. He earned his Ph.D. in 2010 from University of Calcutta in the group of Chaitali Mukhopadhyay. He is currently a postdoctoral fellow in the Department of Chemistry at the University of Texas at Austin. Dmitrii E. Makarov graduated from the Moscow Institute of Physics and Technology in 1990 and earned a PhD in theoretical physics from the Institute of Chemical Physics in 1992. He is currently a Professor of Chemistry at the University of Texas at Austin. His expertise is in theoretical and computational chemical physics and in biophysics.


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Dynamics of Disordered Proteins under Confinement: Memory Effects

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