Universality in the Timescales of Internal Loop Formation in Unfolded Proteins and Single-Stranded Oligonucleotides

Abstract

Understanding the rate at which various parts of a molecular chain come together to facilitate the folding of a biopolymer (e.g., a protein or RNA) into its functional form remains an elusive goal. Here we use experiments, simulations, and theory to study the kinetics of internal loop closure in disordered biopolymers such as single-stranded oligonucleotides and unfolded proteins. We present theoretical arguments and computer simulation data to show that the relationship between the timescale of internal loop formation and the positions of the monomers enclosing the loop can be recast in a form of a universal master dependence. We also perform experimental measurements of the loop closure times of single-stranded oligonucleotides and show that both these and previously reported internal loop closure kinetics of unfolded proteins are well described by this theoretically predicted dependence. Finally, we propose that experimental deviations from the master dependence can then be used as a sensitive probe of dynamical and structural order in unfolded proteins and other biopolymers.

Introduction

Conformations adopted by disordered polymers and the timescales of interconversion among them are thought to play key roles in biomolecular folding (1–5) and function (6). For example, the discovery that the folding rates of single-domain proteins are reasonably strongly correlated with various measures of their native state topology (7,8) has been interpreted in terms of the efficiency with which the largely unfolded polypeptide chain undergoes loop-closure and other polymer rearrangements as it stochastically searches for the correct, nativelike overall topology (4,9).

Likewise, the nature of the disordered state of biopolymers, including both polypeptides (reviewed in Oh et al. (10)) and single-stranded nucleic acids (reviewed in Lubin and Plaxco (11)) has further importance in the context of recent biosensor designs, which rely on target-induced changes in the dynamics and/or conformational ensembles of their component biopolymer probes (12,13).

Despite the importance of biomolecular dynamics, current understanding of the structural conformations and intramolecular dynamics in unfolded proteins and single-stranded oligonucleotides remains incomplete. For example, the amount of residual order in chemically unfolded proteins remains a contentious issue (14), with some experimental (15) and computational (16,17) evidence pointing to a significant degree of structural organization, whereas other measurements (18–22) exhibit Flory's random-coil scaling of the spatial dimensions with the polypeptide chain length N (number of monomers).

In an effort to better understand the dynamics of unstructured biopolymers, a number of groups have studied the dynamics of end-to-end collisions using both theoretical (2,23–37) and experimental (1,3,27,38–42) approaches. In contrast, studies of the rate with which two internal positions in a chain collide, or a terminus collides with an internal position, remain limited (43–51). Here, we explore this specific dynamic property of an unfolded biopolymer via simulations, experiments, and theory.

Experimentally, the frequency of such intrachain collisions can be measured by monitoring the quenching kinetics of a suitably chosen probe, which is attached to one monomer, by a near contact-limited quenching group that is linked to another monomer (1). In the diffusion-limited regime, where the intrinsic rate of quenching is so high that it occurs virtually instantaneously upon a collision between the monomers, the quenching kinetics is governed by the timescales of polymer motion and provides a measure of the collision frequency. This frequency depends on the location of the chosen pair of monomers within the chain. For a chain of N+1 monomers labeled 0,1, …, N, the central quantity considered here is the mean time τ_ij(N) to form a loop of length |i−j| as a result of a collision between the monomers i and j within the chain (Fig. 1). End-to-end (EE) collisions then correspond to i = 0, j = N, internal-to-end segment (IE) collisions to i = 0, j < N or j = N, i > 0, and internal-to-internal (II) collisions to 0 < i, j < N.

Figure 1.

Collision between monomers i and j within a polymer chain leads to the formation of a loop of length |i-j|. Here we describe how the timescale for this process depends on the monomer positions and the total chain length N. Of particular interest is the effect of tails, i.e., chain segments exterior to the loop. Appending a tail of length l to one of the chain ends (dashed line) increases the loop closure time. The tail effect is less obvious in a scenario where one of the monomers is moved into another position (dashed-line circle) such that the length of a tail is increased while the total chain length remains constant.

The kinetic effect of the tails—the set of monomers external to the loop—has been previously studied experimentally (43,44), theoretically (46,52), and via simulations (29,48,49,52). In particular, when the loop length |i−j| = n was fixed and the tail length l was increased (Fig. 1), the IE collision time τ_0n(n + l) was found to increase monotonically to saturation, achieving the limit of a finite loop within a semiinfinite chain. Qualitatively, this result is easy to understand: more monomers have to move in order to close a loop whenever a tail is present (53). In addition, tails introduce steric clashes that reduce the probability and, consequently, slow the rate of loop closure (43,49,54). Although the experimental studies of IE loops used tails of finite length, the theoretical limit of infinite tails has also been studied, leading to scaling relationships between the loop formation time and its length (46). Finally, effects of chain flexibility on the interior loop formation within semiflexible chains have also been studied theoretically (52).

Although the above studies have provided initial forays into the dynamics of internal loop closure, current understanding of internal loop dynamics in biopolymers remains far from complete.

First, unfolded proteins and DNA typically used in experimental studies are far from an asymptotic infinite-chain limit that is commonly assumed by polymer theories. Finite-size effects, i.e., deviations from the behavior expected as N → ∞, as well as sequence-specific effects, thus may significantly affect the experimentally observed loop formation dynamics.

Second, even if the effects of the sequence and of finite chain length are neglected a number of theoretical issues remain to be resolved. To illustrate those, imagine two different experiments where the length of the tail adjacent to the monomer j is increased by l (Fig. 1). In the first one, the loop length remains constant while a chain segment of length l is appended to the polymer. As a result, τ_ij(N + l) increases with the tail length l, as established by the above-mentioned studies. In the second experiment, however, the monomer j is moved into a new position j′ = j−l closer to i while the total chain length remains constant.

Will the loop time τ_i,j−l(N) increase or decrease with increasing l?

The answer to this question involves a trade-off that has seen little previous exploration: as the tail grows longer, which will slow dynamics, the loop grows concomitantly shorter, which will speed it up and prior theory makes no prediction as to which of these two effects wins. In what follows we report on experiments—both in vitro and in silico—that explore the second of the above two scenarios, wherein the locations of the colliding entities are varied within a polymer chain of a given length. We will show that the resulting dependence of the loop formation time on the tail length may be nonmonotonic such that the loop formation time increases at first and then decreases with l.

More importantly, we will demonstrate the existence of a universal relationship between the loop formation timescale, the fundamental time- and length-scales of the entire polymer chain, and the relative positions of the colliding entities within the chain. Moreover, we will show that this observation is universal not only in a conventional, polymer-theoretical sense, but also applies, with a surprisingly good accuracy, to finite, experimentally addressable systems. That is, this relationship not only holds theoretically, in the asymptotic, infinitely long chain limit for polymers in a single universality class (e.g., Gaussian chains, excluded volume random flight chains, etc.), but is also seen experimentally across a diverse set of real biopolymers. Finally, we will argue that significant deviations from our theoretical predictions can be used as a sensitive probe to detect signatures of residual order in biopolymers (e.g., unfolded proteins).

Materials and Methods

Experimental measurements of loop formation times in homogeneous single-stranded DNA (polythymine)

All DNA constructs were purchased from Biosearch Technologies (Novato, CA) as purified, modified oligonucleotides and used as received. We employed ∼5 μM DNA in 100 mM NaCl/20 mM sodium phosphate pH 7 for all experiments. The temperature of the sample was controlled at 30 ± 1 C°.

Luminescence lifetime measurements were performed using a picosecond luminescence measurement system (C4780 system; Hamamatsu, Hamamatsu City, Japan) equipped with a nitrogen laser (LN203S2; Laser Photonics, Lake Mary, FL). We excited the ruthenium lumophore at 450 nm and collected integrated emission between 625 nm and 675 nm, which leads to improved signal/noise ratios over single-wavelength measurements. We confirmed the validity of collecting the emission from this broad wavelength by comparing the decay rate obtained from the emission at 650 nm, which produces rates within error of those observed using the integrated intensity (data not shown). The obtained data were fit to single exponential decays using Igor Pro (WaveMatrix, Norwood, MA). This contrasts with previous work, in which we reported that the lifetime decay of our ruthenium fluorophore is biexponential (41). Due to instrument limitations, however, the data reported here spanned 100–1000 ns and thus the more rapid, 18-ns decay process we reported earlier is not observed. Previously we have shown that this more rapid process occurs even in the absence of an attached quencher, suggesting that it is unrelated to collisional quenching. All reported error bars reflect estimated 95% confidence intervals as determined from triplicate measurements.

The reported end-to-end and internal-to-end collision rates were determined as reported previously (39,41) save that DABSYL (4-Dimethylaminoazobenzene-4′-sulfonyl) was employed as the electron-accepting quencher. Briefly, assuming that, when DABSYL and the ruthenium lumophore are in contact, quenching is significantly more rapid than the dynamics of the oligonucleotide, we estimate the collisional-quenching rate of the excited ruthenium complex by DABSYL via the subtraction of the other quenching processes (for example, contact with DNA bases, or inherent decay through radiative and nonradiative process) from the decay of the excited ruthenium complex. We confirmed that intermolecular quenching is not a significant contributor to the quenching rate associated with DABSYL from that the quenching rate is independent of the concentration of the ruthenium complex (data not shown).

Previously we have employed bipyridines as the quencher for such studies (39,41), which is known to quench via photoinduced electron transfer (55,56). In support of this we note that, although the bimolecular quenching rate is not experimentally accessible (due to the strong absorption of DABSYL), its redox potentials (−0.2 V and −0.4 V versus standard hydrogen electrode) (57) are higher than that of methyl bipyridine (−0.45 V) (58), indicating that electron transfer is more likely, and the absorption spectrum of DABSYL does not overlap with that of ruthenium lumophore, indicating that resonance energy transfer will not be significant. Further supporting the argument that DABSYL quenching occurs via photoelectron transfer, we find that the estimated end-to-end collision rates determined using DABSYL as the electron acceptor are somewhat more rapid than those observed when methyl bipyridine is the acceptor, and the latter is known to exhibit diffusion-limited quenching with a closely similar ruthenium lumophore (55,56).

We also find that the chain-length dependence of the end-to end collision rate with DABSYL as the quencher, which has an exponent of 3.34 ± 0.10, is experimentally indistinguishable from the 3.49 ± 0.13 found previously (41) using methyl bipyridine as the quencher . Finally, we find that the end-to-end collision rate is linearly dependent on solution viscosity, which confirms that quenching of ruthenium compound by DABSYL is diffusion-limited, as in our previous study (41). Further technical details including the raw experimental data and the structure of the constructs used are provided as the Supporting Material.

Simulation studies of loop formation in polymers

Our model for a polymer chain consists of N+1 beads of mass m connected by N springs. Consecutive (bonded) beads separated by a distance r are connected by a harmonic interaction potential,

$$V_{bonded}\left(r\right)\mathrm{=}\frac{1}{2}{k}_{bond}{\left(r\mathrm{-}\sigma \right)}^{\mathrm{2}},$$

where σ is the equilibrium bond length and

$$k_{bond}=100.0\varepsilon /{\sigma }^2.$$

Excluded volume interactions are represented by a repulsive Lennard-Jones potential between nonconsecutive (nonbonded) beads separated by a distance r:

$$V_{nonbonded}\left(r\right)\mathrm{=}4\varepsilon \left[{\left(\sigma /r\right)}^{\mathrm{12}}\mathrm{-}{\left(\sigma /r\right)}^{\mathrm{6}}\right]\mathit{\theta }\left({\mathrm{2}}^{\mathrm{1\; /\; 6}}\sigma \mathrm{-}r\right).$$

Here ɛ is a characteristic energy scale and θ is the Heaviside step function that truncates the attractive portion of the Lennard-Jones potential. The dynamics of each bead is governed by the Langevin equation

$$m{d}^2{\mathbf{r}}_{\mathit{i}}\mathrm{/}d{t}^2\mathrm{=-}\partial V/\partial {\mathbf{r}}_{\mathit{i}}\mathrm{-\xi }d{\mathbf{r}}_{\mathit{i}}/dt\mathrm{+}{\mathbf{F}}_i\left(t\right),$$

where r_i(t) is the position of the bead, V is the total interaction potential, ξ is a friction coefficient, and F_i(t)is a random force satisfying the fluctuation-dissipation theorem. A friction coefficient of

$$\xi =2.0{\left({\sigma }^2/m\varepsilon \right)}^{-1/2}$$

was chosen such that the dynamics were in the overdamped regime (59,60). For the case where hydrodynamic interactions were included, we used the Ermak-McCammon Brownian dynamics algorithm (61) with the Rotne-Prager-Yamakawa diffusion tensor (62). The hydrodynamic radius for each monomer bead was set at 0.35σ. All simulations were performed at a temperature of

$$T=1.0\varepsilon /k_B.$$

The results reported here are given in dimensionless units, where

$${\tau }_0={\left(m{\sigma }^2/\varepsilon \right)}^{1/2}$$

sets the unit of time.

To compute the loop formation time between a selected pair of monomers, i and j, we have assumed that a loop is formed instantaneously whenever the monomers are within a distance R_c of one another. In practice, we introduce a distance-dependent rate (60) given by

$$k_{ij}\left(r_{ij}\right)\mathrm{=}{k}_0\mathit{\theta }\left(R_c\mathrm{-}{r}_{ij}\right),$$

where r_ij is the distance between monomers i and j and k₀ is the intrinsic rate, which is chosen to be large enough that the diffusion-controlled limit is reached and the results of the simulations are independent of k₀.

The mean time τ_ij for diffusion-controlled collisions between a pair of monomers i and j is computed from the equation

$${\mathit{\tau }}_{\mathit{ij}}\mathrm{=}{\int }_{\mathrm{0}}^{\mathrm{\infty }}{\mathit{S}}_{\mathit{ij}}\left(t\right)dt,$$

where

$$S_{ij}\left(t\right)\mathrm{=}\langle \exp \left(\mathrm{-}{\int }_0^t{k}_{ij}\left[r_{ij}\left(t^{\prime }\right)\right]d{t}^{\prime }\right)\mathrm{\times }\mathit{\theta }\left[r_{ij}\left(0\right)\mathrm{-}{R}_c\right]\rangle \mathrm{/}\langle \mathit{\theta }\left[r_{ij}\mathrm{-}{R}_c\right]\rangle$$

is the probability that no collisions between i and j took place from time 0 to t provided that the initial distance between the two monomers,

$$r_{ij}\left(t=0\right),$$

exceeds R_c. Here the angular brackets denote averaging over the canonical ensemble of initial polymer configurations. See Cheng and Makarov (59) for further computational details.

Results and Discussion

End-to-end versus internal-to-end loops: experiment

In a previous study (41), we have measured the diffusion-controlled end-to-end loop formation kinetics in unstructured, single-stranded DNAs as a function of chain length. Here we have extended these studies by measuring the internal-to-end loop formation kinetics of single-stranded DNA consisting of 27 monomers (Fig. 2). In doing so we find that, in agreement with both previous experiments (43,44) and simulations (29,48,49), the IE loop formation time τ_ij(N) = τ_0j(N) is always longer than the end-to-end collision time τ_0j(j) for a loop of the same length j (Fig. 2). At the same time, for chains of a given total length N, the loop formation time decreases as the length of the tail, N−j, increases, a trend opposite that observed in the constant-loop-length measurements. We also find that, except for the shortest loops, the experimental loop length dependence of the loop formation time can be fitted by a power law for both the EE and the IE cases,

$${\mathit{\tau }}_{ij}\mathrm{\propto }{\left|i\mathrm{-}j\right|}^{\mathit{\delta }},$$ (1)

with exponents of

$$\delta =3.34\pm 0.1\mathrm{and}\delta =2.83\pm 0.5,$$

respectively. We emphasize that Eq. 1 should be regarded here as a fit of experimental data rather than a true scaling law. Indeed, the experimental value of the exponent δ in the EE case is considerably larger than the value predicted by polymer theory in the asymptotic limit of very long chains (46). As discussed in an earlier article (41), this disagreement between polymer theory and the experimentally observed loop length dependence is likely to arise due to finite-size effects. Specifically, the internal dynamics of our single-stranded DNA constructs are affected by both electrostatic interactions within these relatively short chains and differences between the DNA and the linkers that connect it to the lumophore and quencher (41).

Figure 2.

Loop length dependence of the loop formation times in single-stranded DNA. Except for the shortest loops, the loop length dependence of both internal-to-end (IE) and end-to-end (EE) collision times can be fitted by a power law of the form τ_ij ∝|i−j|^δ, where δ = 3.34 ± 0.1 for EE loops and δ = 2.83 ± 0.5 for IE loops (straight lines). The deviations observed for the shortest loops presumably arise because of the linker effects, as previously reported (41). Given the same loop length, the loop formation time for an IE loop is always longer than that for an EE loop.

End-to-end versus internal-to-end loops: simulations

In addition to the above experiments we have also performed Langevin dynamics simulations of loop formation within bead-and-spring polymer models, using the usual assumption (see, e.g., Toan et al. (33) and references therein) that a loop forms between monomers i and j whenever the distance between the two becomes shorter than a certain capture radius R_c. Note that more general, distant-dependent reaction rates may lead to deviations from diffusion-controlled kinetics (60), which is neglected in this study. Our simulations of EE and IE loops show trends similar to those observed experimentally (Fig. 3). When the tail is longer than ∼35% of the loop length, the loop formation times become independent of the length of the tail. Such saturation in the tail length dependence has been anticipated by theory (46), observed in previous simulations (47,48) and demonstrated in a recent experimental study (45).

Figure 3.

Loop length dependence of the loop formation times from simulations of bead-and-spring polymer models. The loop length dependence of both IE and EE collision times can be fitted by a power law of the form τ_ij ∝|i−j|^δ, where δ = 2.38 for EE loops and δ = 2.55 for IE loops (straight lines). For sufficiently long tails, the IE loop formation time is independent of the tail length (and thus the total chain length). Here, the simulations were performed for Rouse chains with excluded volume interactions. The collision between monomers was assumed to take place whenever the two monomers were within R_c = 2.5 equilibrium bond lengths from one another.

In this long-tail limit, the loop length dependence of the IE loop formation time follows a power law of the form of Eq. 1. Curiously, unlike the case of EE loops in single-stranded DNA, where the experimental value of δ is significantly higher than that for simple bead-and-spring models (41), the experimental scaling exponent for IE loops observed in Fig. 2 agrees well with the bead-and-spring value of δ ≈ 2.55 (Fig. 3).

This agreement, however, has to be taken with a grain of salt given that the properties of relatively short, single-stranded DNAs are known to deviate from those of idealized random-coil models due, for example, to the importance of electrostatic effects over short length-scales (41). We also note that the scaling exponents observed in Fig. 3 for both EE and IE loops are in reasonable agreement with the renormalization group predictions of Friedman and O'Shaughnessy (46) as well as with earlier simulations (64). Indeed, in both cases the value of the scaling exponent predicted by renormalization group theory is (46,64)

$$\delta =2v+1\sim 2.2,$$

where ν is Flory's scaling exponent. Considering potentially significant finite size effects that are often observed in simulations of such systems (59,60), this value appears reasonably close to the δ = 2.38 and δ = 2.55 estimated for EE and IE loops, respectively.

The simulated IE loop formation times are longer than the EE times for the loops of the same length, which is in accord with our experimental data. Assuming IE loops with infinite tails, the ratio

$${\tau }_{0j}\left(\mathrm{\infty }\right)/{\tau }_{0j}\left(j\right)$$

of the IE and EE loop formation times is not constant but weakly dependent of the loop length j, in contrast to a constant ratio of 4 predicted by an earlier study of position-dependent reconfiguration times in Rouse chains (65). Although our simulation results are qualitatively consistent with the experimental data of Fig. 2, they also predict a maximum in the dependence of the time τ_0j(N) on j (Fig. 3), which is not observed experimentally (Fig. 2). This discrepancy is further discussed below.

Dimensional analysis of intrachain loop formation kinetics

Although the qualitative agreement we observe between experiments and simulations is reassuring, development of a more quantitative model for intrachain loop formation is hindered by the fact that experimental loop formation rates depend on the photophysics of the probes used to measure them as well as on the structural properties of the linkers connecting the probes to the molecules of interest (41). It is difficult to realistically capture such features within coarse-grained bead-and-spring polymer models.

Instead, in our simulations we have assumed that all these effects can be lumped into an effective capture radius R_c that defines a collision. Although this assumption appears plausible, an estimate for the appropriate numerical value of R_c is not readily available for the specific optical reporting groups and single-stranded DNA polymers we have employed experimentally. Fortunately, dimensional arguments and numerical examples presented below establish that simulation results can be rescaled to assume a form that is independent of the chain length and is only weakly dependent on the capture radius, thus allowing a direct, quantitative comparison between the loop formation times in idealized, long homopolymer chains and real biopolymers.

Within the model adopted here, the loop formation time τ_ij(N) depends on the capture radius R_c, the solvent viscosity η, the polymer length N, and the positions of the probes i and j within the polymer chain. Generally, the characteristic timescales of polymer dynamics become longer with increasing chain length. In the Rouse model of chain dynamics, for example, these timescales are proportional to N² (see, e.g., (53,66)).

To study spatio-temporal correlations within chains of different length N, it is sensible to factor out this chain length dependence so that intrachain loop formation times are measured relative to each chain's own characteristic reconfiguration timescale. One could, for example, normalize τ_ij(N) by the slowest relaxation time for the chain, such as the Rouse (Zimm) time in case of chains described by the Rouse (Zimm) model (53). This choice is, however, impractical because polymer relaxation times are not directly accessible by experimental measurements of loop formation kinetics. Instead, we choose a related timescale (24,36,46), the end-to-end loop formation time τ_0N(N), as a measure of the global reconfiguration timescale for the entire chain. We thus introduce a dimensionless loop formation time given by

$$T_N\left(i\mathrm{/}N,j\mathrm{/}N\right)\mathrm{=}{\mathit{\tau }}_{ij}\left(N\right)/{\mathit{\tau }}_{\mathrm{0}\mathit{N}}\left(N\right).$$ (2)

Unlike τ_ij(N), the quantity defined by Eq. 2 is independent of the solvent viscosity because both the numerator and the denominator of Eq. 2 are proportional to the viscosity (60) in the diffusion-controlled limit. Nondimensionality of T_N(i/N, j/N) further imposes a restriction on its capture radius dependence. Indeed, if we assume that the only two relevant length-scales of the system are the capture radius R_c and a typical length scale R of the polymer, which can be chosen equal to the root-mean-square (RMS) end-to-end distance of the chain,

$$R={\langle {\left({\mathbf{r}}_N-{\mathbf{r}}_0\right)}^2\rangle }^{1/2},$$

then T_N(i/N, j/N) must depend only on the dimensionless ratio, R_c/R, of these two length-scales. We finally conjecture that the dimensionless time T_N(i/N, j/N), written as a function of chain length and of the monomer positions rescaled by the total length, does not explicitly depend on N so that its subscript can be dropped, i.e., T_N(i/N, j/N) = T(i/N, j/N). This amounts to the assumption that the scaling of τ_ij(N) with loop length for all self-similar loops (i.e., loops sharing the same values of i/N, j/N, and R_c/R) is the same and coincides with that of τ_0N(N). Such self-similarity has been shown to hold for equilibrium intermonomer distance distributions in excluded-volume polymer chains (46,67) and for reconfiguration times in Rouse-type chains as determined via fluorescence energy transfer experiments (65).

To summarize the above-dimensional analysis, our conjecture is that the internal loop formation time normalized by the end-to-end loop formation time,

$${\mathit{\tau }}_{ij}\left(N\right)/{\mathit{\tau }}_{\mathit{end}\text{-}\mathit{to}\text{-}\mathit{end}}\left(N\right)\mathrm{=}T\left(\frac{i}{N},\frac{j}{N},\frac{{\mathit{R}}_c}{R}\right),$$ (3)

is independent of the polymer length and is a universal function of the length of each tail relative to the total polymer length and of the capture radius normalized by the RMS polymer end-to-end distance. Moreover, although both internal and end-to-end loop formation times may significantly depend on the capture radius, it seems probable that such dependence would at least partially cancel out in a ratio of the two timescales resulting in a weak, or even nonexistent dependence on capture radius.

To test the above conjecture we have computed normalized loop formation times for chains of different length N. When the value of the capture radius was chosen to scale proportionally to the RMS end-to-end distance so as to keep the ratio R_c/R constant, the resulting dependences on the monomer positions (again, normalized by polymer length) for all polymers belonging to the same universality class collapsed to a single master curve (Fig. 4 demonstrates this for IE loops).

Figure 4.

Universal dependence of the IE loop formation time on the monomer positions and the chain length. When the loop formation times are normalized by the end-to-end collision times and the ratio of the capture radius R_c to the end-to-end RMS distance, R, remains fixed (here, R_c/R = 0.2), the simulation data for chains of different lengths collapse onto a single curve. The precise shape of the curve, and, in particular, its power-law scaling with the loop length in the limit of short loops (inset), depends on the polymer's universality class (here we compare Rouse chain with and without excluded volume interactions). Remarkably, when normalized by the end-to-end loop formation time, the experimental data for loops in single-stranded DNA (same data as in Fig. 2) agrees quantitatively with the simulation results.

The precise shape of the curve depends on the polymer's universality class. In particular, for short IE loops such that j/N << 1, the dependence of T(0, j/N, R_c/R) on j/N is a power law, with a scaling exponent that is different in the case of Rouse chains (no excluded volume) and excluded-volume random coils (Fig. 4, inset). This extreme case is, however, inaccessible by our experimental measurements, where 0.3 < j/N ≤ 1. In the experimentally relevant regime, the excluded volume interactions have a relatively weak effect on the overall shape of the curve of T(0, j/N, R_c/R) versus j/N. This finding is not significantly altered when hydrodynamic interactions within the polymer chain are taken into account: Although those interactions significantly alter the magnitudes of the un-normalized loop formation times, they only weakly affect the shape of the master curve (see the Supporting Material).

This weak effect of hydrodynamic interactions may seem somewhat surprising, especially considering the existing theoretical view that the diffusion-controlled regime no longer holds after hydrodynamic effects are introduced (46,64,68,69). This apparent contradiction, however, is at least partly semantic: The operational definition of the diffusion-controlled limit adopted here is that the timescale of the process is proportional to the solvent viscosity. This definition is both appealing intuitively and testable experimentally (27,41), but it is subtly different from that adopted in the literature (46,64,68,69). Specifically, those articles differentiate between the diffusion-controlled limit and what they refer to as the law-of-mass action (LMA) regime.

The latter regime, which is predicted to occur, e.g., for end-to-end collision kinetics when both excluded volume interactions and hydrodynamic effects are present, results in a loop formation rate that is proportional to the equilibrium probability of forming the loop. However, in our language, LMA regime is still diffusion-controlled as long as the proportionality factor exhibits inversely proportional solvent viscosity dependence, as in the Szabo-Schulten-Schulten theory (27,28,32).

A measurable signature of the LMA regime is a different value of the scaling exponent δ (compare to Eq. 1), although in practice this difference is relatively small and can be obscured by effects arising from the finite size of our polymers (64). Indeed, the value of δ estimated from our simulations of excluded-volume chains in the presence of hydrodynamic interactions (see the Supporting Material) is in agreement with the LMA predictions (46,64). We thus conclude that our results agree with previous theoretical views (46,64,68,69) and that the shape of the master curve observed in Fig. 4 is not significantly altered by the LMA regime, which still falls within our definition of diffusion-controlled kinetics.

Although the above dimensional analysis requires that the capture radius R_c be proportional to the polymer's end-to-end distance in order to observe a chain length independent master curve, experimentally R_c is not a free parameter. Moreover, its precise value is generally unknown, thus introducing an uncertainty as to which value to use when comparing the theoretical master curve with experimental data. Fortunately, as anticipated above, the dependence of T(i/N, j/N, R_c/R) on the capture radius is fairly weak.

Specifically, unlike power laws typically observed for the loop length dependence of the collision times, the dependence of T(0, j/N, R_c/R) on R_c/R is logarithmic (Fig. 5). This weak, logarithmic dependence holds even for values of the capture radius that are unrealistically (unphysically) large; e.g., for values so large that the concept of a loop is no longer well defined. Of note, excluded volume effects lead to a stronger capture radius dependence as compared to that for Gaussian chains whereas hydrodynamic interactions do not have any significant effect (Fig. 5). We likewise note that in contrast to the rather weak dependence of the rescaled time, T(0, j/N, R_c/R), the unnormalized loop formation time τ_ij exhibits a much stronger capture radius dependence (compare to Fig. S2 in the Supporting Material). This, again, highlights the advantage of rescaling the data according to Eq. 3 for a meaningful comparison between experiments and theory.

Figure 5.

Effect of the capture radius on the internal-to-end loop formation kinetics: T (i/N, j/N, R_c/R) has a relatively weak, logarithmic dependence on R_c/R. Here we illustrate this dependence for the loops formed between the middle of the chain and one of the chain ends, i.e., i = 0, j = N/2. See the Supporting Material for further analysis of the capture radius effects.

Our experimental data also agrees with the above scaling conjecture. Indeed, normalized experimental loop formation times in single-stranded DNA are in a remarkably good agreement with the scaling predictions (Fig. 4). The only significant discrepancy between simulations and experiments is found when j > 0.9N; i.e., when the probe is placed within ∼10% of end of the chain. That is, although the theoretically predicted dependence T(0,j/N) shows a maximum near j ∼ (0.9–0.95) × N that is similar to the rollover behavior that has been previously predicted for FRET-derived position-dependent chain reconfiguration times (65), the experimental data plateaus for j/N > 0.7. This discrepancy likely originates from the limited flexibility of even single-stranded DNA, whose Kuhn segment is expected to consist of n_K ∼ 5–8 monomers.

The last n_K nucleotides in our experimental chains thus constitute a nearly rigid segment that moves in a concerted fashion. We therefore expect τ_0j to be approximately the same for all j such that j > N − n_K. This effect is not captured by the simulations used to generate the dependences seen in Fig. 4, which employed a highly flexible chain of Kuhn length comparable with the monomer size. Thus, to experimentally observe the maximum occurring in our simulations at j ∼ (0.9–0.95) × N, it would be necessary to use DNA chains long enough that their tails (of length 0.05 – 0.1N) are considerably longer than their Kuhn length. Limitations in the synthesis of the necessary DNA constructs, however, preclude experimental investigation of this hypothesis.

Intrachain dynamics in unfolded proteins: comparison with simulations

The above-described comparison between theory and experiment requires knowledge of both the end-to-internal and end-to-end collision rates of each construct. In addition to our oligonucleotide studies, such data are also available for a limited set of polypeptide constructs. Specifically, Reiner et al. (50) have used triplet-triplet energy transfer to measure the diffusion-controlled rates of both internal and internal-to-end loops within an unfolded 36-residue-long protein, the villin headpiece.

To our knowledge, their measurement of a loop between i = 7 and j = 23 is the only internal-to-internal collision rate reported in the literature to date. Unlike IE collisions, for which a plot of T versus j/N is sufficient for comparison between theory and experiment, a comparison with internal loops requires the entire surface of τ_ij(N)/τ_0N(N) as a function of i/N and j/N. In doing so (Fig. 6) we find that, despite differences in the method used to probe the loop formation and in the physics of the polymer itself (unfolded protein versus single-stranded DNA), the data of Reiner et al. (50) also agree well with our predictions, with the exception of the i = 23, j = 35 pair, for which the corresponding loop formation time is considerably longer than expected. As pointed out by Reiner et al. (50), however, this anomaly presumably arises because of the residual helical order between these two residues. Indeed, a helix intervening between a pair of residues would preclude their short-ranged contact so that the measured loop formation time would be controlled by the timescale of helix unfolding (65,70) and would be longer than for a fully random polymer.

Figure 6.

When rescaled according to Eq. 3, experimental measurements of internal loop formation times in an unfolded protein (50) agree with the simulation-derived dependence T (i/N, j/N, R_c/R) (here we have assumed R_c/R = 0.2). The outlier observed for i = 23, j = 35, is due to residual helical order between those monomers.

Conclusions

Life requires that its building blocks, biopolymers, populate highly specific, nonrandom conformational ensembles. Generic polymer chains, on the other hand, commonly exhibit universal scaling laws relating their various properties to their length. Those laws are independent of the details of intramolecular interactions, provided that the chains are long enough. Such polymers therefore provide a reference for measuring the structural organization within functional biomolecules. Unfortunately, comparison between universal scaling laws and the behavior of real biopolymers is obscured by finite-size effects: whereas scaling laws are only valid asymptotically in the infinitely long chain limit, biomolecules of interest are often not long enough to display such scaling laws or even to allow their reliable verification.

Here we have exposed a different type of general behavior in the dynamics of internal loop formation within disordered polymer chains and showed that the timescale of forming such loops, when properly renormalized, obeys a simple universal dependence on the position of the loop extremities. In contrast to the chain length dependence, this dependence is in practice rather forgiving of the experimental constraints such as limited chain length or lack of microscopic knowledge of the probes used to make the measurements. At the same time, it is highly sensitive to any residual order found within the polymer chain, much more so than equilibrium properties such as the average distance between the monomers that form the loop. Indeed, average dimensions of unfolded proteins can exhibit Flory's random-coil scaling with chain length even when significant partial order is present (71,72). On the other hand, intervention of a rigid residual structure (e.g., a helix) between a pair of residues can suppress their respective collisions, thus considerably slowing down the observed loop formation time.

The robustness of the dependence of the loop formation time on the location of the loop-forming monomers established here for random coils on one hand, and its sensitivity to residual structural order on the other hand, suggest that spatio-temporal correlations inferred from measurements of timescales of interior loop formation in biopolymers can be used as a sensitive probe of both their structure and dynamics.