Rare Dissipative Transitions Punctuate the Initiation of Chemical Denaturation in Proteins

Abstract

Protein unfolding dynamics are bound by their degree of entropy production, a quantity that relates the amount of heat dissipated by a nonequilibrium process to a system’s forward and time-reversed trajectories. We here explore the statistics of heat dissipation that emerge in protein molecules subjected to a chemical denaturant. Coupling large molecular dynamics datasets and Markov state models with the theory of entropy production, we demonstrate that dissipative processes can be rigorously characterized over the course of the urea-induced unfolding of the protein chymotrypsin inhibitor 2. By enumerating full entropy production probability distributions as a function of time, we first illustrate that distinct passive and dissipative regimes are present in the denaturation dynamics. Within the dissipative dynamical region, we next find that chymotrypsin inhibitor 2 is strongly driven into unfolded states in which the protein’s hydrophobic core has been penetrated by urea molecules and disintegrated. Detailed analyses reveal that urea’s interruption of key hydrophobic contacts between core residues causes many of the protein’s native structural features to dissolve.

Introduction

Proteins can be decommissioned through an array of chemical and physical means that induce function-impairing conformational changes in a biopolymer substrate. As with any nonequilibrium process that occurs in finite time, protein denaturation leaves an irreversible imprint on the universe in the form of a dissipative heat signature. The amount of heat dissipated during denaturation is dictated by the amount of entropy produced by the process (1, 2, 3, 4), a quantity related to the asymmetry of a system’s forward and time-reversed trajectories. Biopolymer pulling experiments—featuring protein and nucleic acid unfolding mediated by apparatuses like atomic force microscopes (5) and optical tweezers (6)—have leveraged corollaries of the fluctuation theorem, such as the Jarzynski Equality, which provide seminal connections between equilibrium free energies and dissipative work distributions (7, 8, 9, 10).

Although the forces driving pulling-induced protein unfolding experiments are, by design, straightforward to understand, denaturation mechanisms associated with so-called chaotropic chemical agents (like urea and guanidinium chloride) are not trivial to codify (11, 12, 13, 14, 15, 16). Physical pulling experiments have provided keen insights into protein unfolding processes because parameters like pulling force, pulling rate, and pulling direction can be carefully controlled to regulate dissipation. Such simple parameters do not pertain to chemically induced unfolding processes, wherein one is tasked with extracting phenomenological observations from a swarm of encounters between a protein and disordered solvent molecules. As with any nonequilibrium process, however, we do know that productive denaturation events will be accompanied by heat dissipated into the surroundings (3, 17). One is thus compelled to pose an unusual question: by tracking heat dissipation over the course of unfolding, can we design an experiment for systematically studying the chemical denaturation of proteins?

Although laboratory-based control of urea denaturation (outside the trivial tuning of concentration) is perhaps beyond our reach, recent advances in nonequilibrium simulation techniques may make such experiments tractable in silico. Network-based models of protein dynamics (often called Markov state models, or MSMs) have emerged as powerful analytical tools for understanding folding processes (18) and subtler conformational changes (19) that occur within massive molecular dynamics (MD) datasets (20, 21, 22). The power of the trajectory-generative properties of MSMs, however, has only very recently been harnessed. Coupled to large MD datasets and simple nonequilibrium theories or more complex large deviation theoretic machinery (23, 24, 25), MSMs can serve as engines for conducting classical path integral simulations over microsecond and millisecond timescales, providing descriptions of previously inaccessible nonequilibrium processes. In the past several years, biased trajectory ensembles drawn from MSMs have illuminated dynamical glass transitions on protein folding landscapes (26, 27, 28) and revealed fundamental connections between dissipative trajectories and activation mechanisms in signaling proteins (29, 30). After building an MSM of chemical protein denaturation, one could similarly apply the theory of entropy production to draw a system between different realms of dissipation. Studying equilibration dynamics in the highly dissipative regime, one might expect to highlight the forces that drive unfolding events.

In this work, we construct an MSM for the urea-induced denaturation of the small globular protein, chymotrypsin inhibitor 2 (CI2) (11, 31). By enumerating full entropy production probability distributions over trajectories seeded from the folded state, we observe distinct passive and dissipative regimes in the unfolding dynamics. Applying biased trajectory theory based on these transient entropy currents, we quantitatively characterize the trajectory ensemble found within the dissipative region. We see that this dissipative phase is indeed punctuated by urea-induced unfolding processes: we observe that prominent dissipative transitions are strongly correlated with urea intrusion events into the protein’s hydrophobic core. A detailed description of these results follows a brief explication of our theory, models, and methods.

Materials and Methods

MD and Markovian network model of the denatured landscape

To conduct molecular dynamics simulations of urea-induced denaturation, a properly folded CI2 molecule was solvated in a water box containing 8 M urea and equilibrated at 333 K. This elevated temperature was applied to provide a moderate acceleration of the denaturation process. Six independent production runs were used to capture subsequent unfolding events, yielding ∼6 μs of simulation data in aggregate (see the Supporting Material for specific simulation parameters (32, 33, 34, 35, 36)). Although more extensive datasets are needed to capture the equilibrium dynamics of larger proteins like GPCRs, we expect the driven denaturation of the smaller CI2 to be well resolved within the trajectories sampled here.

These six trajectories were synthesized into a single MSM of denaturation dynamics (19, 37, 38, 39). In short, full system configurations—including the protein, water, urea, and ions—were assigned to N = 1040 discrete states based on protein structural characteristics alone (protein backbone RMSD (root mean square deviation); see the Supporting Material and Fig. S1 for details). Transitions among these states were counted within the trajectory dataset to yield an N × N transition probability matrix, $\mathbb{T}$, which serves as a propagator for state probabilities in discrete time: individual trajectories are advanced from a present state i to a subsequent state j with probability ${\mathbb{T}}_{ij}$. The MSM provides a network description of CI2’s unfolding dynamics from which statistically valid trajectories can be sampled at will. A Markov transition time (lag time) of 4 ns was estimated to be appropriate for our model; a total of 120,000 simulation frames (1 per 50 ps) was used in model construction. As controls of convergence and robustness, MSMs were also constructed using either half of the available trajectory data or one of two alternative distance metrics (core atom pair collective variables and protein backbone dihedral angles; see the Supporting Material for more details). Dynamical free energies (calculated as the negative logarithms of instantaneous probability distributions, which converge to the equilibrium free energy in the long-time limit of the model) can be computed by propagating state probability distribution vectors over the full matrix $\mathbb{T}$. Equilibrium free energy differences among Markov states can be defined by setting an arbitrary zero and evaluating the negative logarithms the stationary state probability ratios.

Dissipative trajectory analysis

From a functional perspective, heat dissipation represents the excess energy that is released over the course of some productive motion in a mechanical process. If some quantity of energy, E_in, is input into a biomolecular system that performs some amount of meaningful work, W, the amount of heat dissipated as a result of the process, Q_diss, is given by Q_diss = E_in – W. Whereas heat dissipation is physically realized as an exchange of energy between a system and its surrounding bath, dissipation is formally related to the entropy produced by the system, Ω, via a simple multiplicative factor of kT: Q_diss = ktΩ. Because entropy production is also a function of forward and reverse trajectory probabilities, it offers a simple means for characterizing dissipative processes in systems in which trajectory probabilities are readily computed (e.g., MSMs).

Our central hypothesis is that by studying the features of heat dissipated during the protein denaturation process, we will gain insight into the underlying productive work that is performed in the process (i.e., protein unfolding). If any particular molecular interaction or motion is accompanied by significant heat dissipation, one might suspect that important functional dynamics can be observed in nearby phase space. A potential challenge associated with this paradigm concerns the fact that minute amounts of heat are constantly being exchanged between a system and its bath, in both directions, which likely provide little information about functional dynamics. Methods for circumventing this frictional dissipation are discussed below; however, we first focus on the basic aspects of our dissipative framework.

The amount of entropy produced, Ω, by a microscopic trajectory x of some length t_obs can be computed from its forward and time-reversed probabilities (3):

$$\text{Ω}\left(x\left(t_{\text{obs}}\right)\right)=\text{ln}\left[\frac{p_{\text{forward}}\left(x\left(t_{\text{obs}}\right)\right)}{p_{\text{reverse}}\left(\tilde{x}\left(t_{\text{obs}}\right)\right)}\right].$$ (1)

Here, a tilde connotes time reversal. At equilibrium (a constraint that is approximately satisfied by our raw dynamics’ stationary distribution and ultimately enforced by detailed balance), the mean entropy production rate associated with urea denaturation is exactly zero. As long as no additional perturbations are applied after the initial addition of denaturant, protein/urea systems simply reach a new equilibrium state in the long-time limit. The entropy produced over the course of the transient unfolding process, however, is nonzero, and this quantity can be computed using a variety of techniques.

Under the approximation of detailed balance invoked here, one can show that the entropy produced by a particular trajectory, Ω(x(t_obs = n)), only depends on its initial and final states (29):

$$\text{Ω}\left(x\left(t_{\text{obs}}\right)\right)=\text{ln}\left[\frac{{\pi }_{x_n}^{\text{eq}}}{{\pi }_{x_0}^{\text{eq}}}\right],$$ (2)

where π^eq is the equilibrium probability distribution for the model. Though the exclusive endpoint dependence of Eq. 2 implies that the entropy production of a given trajectory is time independent, the probabilities of the endpoints themselves change over time; entropy production distributions are therefore time dependent. For Markovian models with $\mathcal{O}$(1000) states, brute-force enumeration of trajectory endpoints is a relatively expedient task. To estimate the full probability distribution over Ω, P(Ω), one simply needs to bin entropy production values for individual trajectories according to the Chapman-Kolmogorov relation (29):

$$P\left(\text{Ω}\in \left({\text{Ω}}_i,{\text{Ω}}_j\right)\right)|x_0=\sum _{x_0,x_n}{\left[{\mathbb{T}}^{t_{\text{obs}}}\right]}_{x_0,x_n}\times \left(\theta \left(\text{Ω}-{\text{Ω}}_j\right)-\theta \left(\text{Ω}-{\text{Ω}}_i\right)\right),$$ (3)

where θ(Ω) is a Heaviside step function, $\mathbb{T}$ is the transition matrix, t_obs is the length of trajectories of interest, and x₀ and x_n are trajectory starting and ending points, respectively, which determine the matrix component being selected. This probability distribution is conditioned on the initial distribution over x₀, as the bar notation in Eq. 3 indicates. Because we are primarily interested in unfolding dynamics initiated from the native state, we enforce that initial condition by limiting the above sum to trajectories starting from that state.

To more quantitatively analyze P(Ω) distributions, one can define biased probability distributions centered around a chosen mean level of entropy production. To do so, one weights trajectories using a scalar field, λ, which takes the role of the inverse temperature, β, from equilibrium statistical mechanics. Here, one computes the statistics of the entropy production rate, ω = Ω/t_obs, using a cumulant generating function, F_ω, over a full λ-ensemble of trajectories, 〈exp(−λΩ(x))〉 (24, 40, 41):

$$F_{\omega }=-\frac{1}{t_{\text{obs}}}\text{ln}\langle \text{exp}\left(-\lambda \text{Ω}\right)\rangle .$$ (4)

The field λ acts as a control parameter on the entropy production rate of the system, drawing the dynamics through different levels of dissipation in various λ-ensembles of trajectories. Details concerning how one computes this cumulant generating function (via a so-called tilted transition matrix, ${\mathbb{T}}_{\lambda }$) and its dependent quantities are included in the Supporting Material. Most significantly, one can calculate the mean entropy production rate, 〈ω〉_λ, by taking a single derivative of F_ω:

$${\langle \omega \rangle }_{\lambda }=\frac{\partial F}{\partial \lambda }.$$ (5)

Because entropy production is not time extensive here, this mean entropy production rate goes to zero in the long time limit. Considering the mathematical relationship between F_ω and P(Ω) (see the Supporting Material for discussion), one expects the minima between peaks of P(Ω) to correspond to rising inflection points in 〈ω〉_λ. One can thus inspect finite-time λ-ensembles at λ-values above and below these inflection points to reliably sample from subdistributions of interest in P(Ω). In this work, we choose to focus our λ-ensemble analysis on two values of λ: λ = 1 (unbiased) and λ = 1.5 (dissipative). As shown in the Supporting Material, the properties of λ-ensembles are symmetric ∼λ = 0.5, meaning that studying unbiased dynamics at either λ = 0 or λ = 1 is equivalent.

Biased Markov state probabilities can also be computed within λ-ensembles of trajectories (see the Supporting Material for relevant equations) (26). Note that such calculations yield individual Markov state probabilities that are induced by the biased trajectory ensemble, not the probabilities of full trajectories themselves. Taking the negative logarithm of biased/unbiased probability ratios, one defines a dynamical free energy change for dissipative trajectories:

$$\text{Δ}{F}_{\text{diss}}=-kT\ln \left[\frac{P\left(x,\lambda =1.5,t,t_{\text{obs}}\right)}{P\left(x,\lambda =1,t,t_{\text{obs}}\right)}\right]=-kT\ln \left[R{P}_{\lambda =1.5}\right],$$ (6)

where RP is relative probability. RMSD projections and related radial distribution functions (presented below) can then be calculated according to a standard protocol.

Analysis of λ-ensembles also helps one differentiate statistically meaningful dissipation events from noise. Any trajectory will feature many instances in which forward and reverse transition probabilities are imbalanced, and these microscopic transitions give rise to heat exchange with the simulated bath. However, forward-leaning transitions (for which T_ij/T_ji > 1) are often erased from the trajectory ensemble by compensating jumps in the reverse-leaning direction. Many of these forward-leaning frictional dissipative events thus contribute little to the ensemble-averaged entropy production, obscuring transitions consistent with this average behavior with noise. In probing ensemble-wide dissipation, we can thus define “statistical transitions” in terms of the dissipative free energies within λ ensembles: a microscopic transition, T_ij, is statistically meaningful to the dissipative average when it is forward-leaning and state j experiences a dissipative free energy decrease within the λ ensemble of interest. The dissipative free energy change observed in state i does not impact this designation. We apply these definitions in our analysis below.

Results and Discussion

Fig. 1 provides a cartoon representation of CI2’s crystallographic native state, which features mixed α/β secondary structures and a tightly packed hydrophobic core. Urea-induced disintegration of this hydrophobic core—comprised of Ile²⁰, Leu⁴⁹, and Ile⁵⁷, as described in the literature (11)—features prominently in later analysis. The snapshot of the early simulation box (Fig. 1) shows that 8 M urea consumes a large volume fraction of simulated configurations.

Figure 1.

Illustration of the simulated CI2 denaturation system. Heat dissipation is tracked over the course of CI2 simulations conducted in the presence of 8 M urea (rendered in a pink stick representation at left), highlighting interactions involving the protein’s hydrophobic core (shown at top right in a space-filling representation) and intruding urea molecules (depicted in a magenta surface representation, at bottom right). To see this figure in color, go online.

The MSM constructed from our six production trajectories describes the nature of the urea-induced unfolded ensemble seen in our simulations. The slowest relaxation half-life in the model, τ_relax ≈ 1.0 μs, suggests that most unfolding dynamics at this elevated temperature occur on submicrosecond timescales. Fig. 2 sketches the denaturation dynamics, in terms of dynamical free energies, which occur within the MSM after the system has been initiated in its most nativelike Markov state (Fig. 2 a). In the early stages of the simulation (illustrated in Fig. 2 b), probability density begins a diversion into two high-flux pathways: one in which the protein core remains largely intact but the protein’s backbone deviates from its native structure; and a second in which both the core and backbone are partially unfolded. A number of low probability states also arise in which the protein core, backbone, and secondary structures have largely abandoned their native configurations. The two central unfolding pathways persist as the dynamics pull density from nativelike states. Populations of the most denatured states wax in the tens of nanoseconds (Fig. 2 c) and peak after 100–200 ns (Fig. 2 d). After several hundreds of nanoseconds, the protein core refolds to a small degree as a new global free energy minimum emerges in the backbone-unfolded region (Fig. 2, c and d). Near-equilibrium dynamics dominate the system after ∼1 μs of model propagation. The denatured landscape (corresponding, in the long-time limit, to an equilibrium free energy surface) shown in Fig. 2 f features a significantly depleted native state and density distributed throughout a range of core- and backbone-unfolded configurations. Fig. S2 demonstrates that urea density progressively intrudes upon the protein center-of-mass over the course of equilibration, an observation consistent with previous studies of other proteins (42, 43, 44). These same unbiased dynamics were largely recovered in separate MSMs built from both half the trajectory data and with both alternative clustering metrics, suggesting proper convergence (Figs. S3–S5).

Figure 2.

Time evolution of CI2’s urea-induced denaturation landscape, presented in terms of a dynamical free energy over nonequilibrium state densities that evolve over the course of the unfolding process. Density (a) escapes the near-native basin along (b) core-unfolded and backbone-unfolded pathways, with states along the landscape diagonal (c and d) waxing and (e and f) partially refolding over hundreds of nanoseconds of dynamics. The near-equilibrium denatured surface (corresponding, in the long-time limit, to an equilibrium free energy surface) features density scattered throughout the landscape with a global minimum in which the core is partially folded. To see this figure in color, go online.

The entropy production probability distribution for these unfolding dynamics is presented in Fig. 3, as a function of time. Initially constrained to zero, the entropy production distribution quickly bifurcates into two peaked subensembles: one strongly localized in the near-equilibrium region, and a second in which significant heat dissipation is observed. This dissipative peak spreads out as higher values of entropy production—beyond six nats—become more probable, but remains well defined at the point of convergence associated with equilibration (after ≈1 μs). These distributions suggest that heat dissipation during CI2 denaturation has clear binary characteristics: trajectories are likely to either produce several nats of entropy (corresponding to several kT of dissipated energy), or dissipate almost no heat at all.

Figure 3.

Time evolution of the entropy production probability distribution, P(Ω), during unfolding from the native state. (Top) Given here are snapshots of P(Ω) illustrating the spread of probability density to higher values of entropy production over time. (Bottom) Given here is a schematic of P(Ω) over a series of interesting time points (t = [4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 48, 64, 80, 100, 160, 200, 240, 360, 480, 720, 1000] ns). A narrow, near-equilibrium peak and broad, dissipative distribution emerge over the course of microsecond-long trajectories. To see this figure in color, go online.

Fig. 4 illustrates sample trajectories drawn from each of these passive and dissipative dynamical regimes. Backbone RMSD traces describe the general extent of unfolding over time; snapshots shown at regular intervals include a surface representation of urea molecules within 5 Å of the protein, rendered in magenta. In these data, we see our first hint at a relationship between unfolding and entropy production. Although some transient unfolding occurs in the middle of the low entropy production trajectory, those dynamics mostly feature a persistent native state ensemble. A near-native configuration is adopted at the final 1 μs observation time. By contrast, CI2 unfolds progressively throughout the dissipative trajectory, reaching a state in which the protein core has largely dissociated by observations’ end. Interestingly, both transient and persistent unfolding transitions do seem to be accompanied by greater urea intrusion (see the middle of the passive trajectory and the entire dissipative trajectory); this relationship between urea penetration and dissipation is explored further below.

Figure 4.

Example trajectories drawn from passive and dissipative regions of P(Ω). (Top) Given here is a trajectory representative of the narrow peak at near-zero entropy production, illustrated in the form of CI2 backbone RMSDs and configurational snapshots. Urea molecules within 5 Å of the protein are rendered in a magenta surface representation. Numbers indicate the average numbers of these urea molecules within the Markov state occupied at six time points: t = [12, 200, 400, 600, 800, 1000] ns. (Bottom) Given here is the trajectory pulled from the center of the dissipative P(Ω) distribution (Ω ≈ 3.5). To see this figure in color, go online.

An analysis of λ ensembles provides a more quantitative characterization of dissipative dynamics. Fig. S6 shows the mean entropy production rate, 〈ω〉, and its susceptibility, χ_ω, as a function of the λ control parameter over the first 100 ns of denaturation (t_obs = 100 ns). As expected from the morphology of P(Ω), we observe two distinct phases in the λ-ensemble dynamics: one dominated by the near-equilibrium peak of the P(Ω) distribution and another describing dynamics in its dissipative region. As noted above, the entropy production rate for this particular MSM goes to zero in the long-time limit. Nonetheless, the sharp crossover between the passive and dissipative regimes (coexistent with a large response function peak) represents a transient analog of entropy production phase transitions seen in simple nonequilibrium networks (41, 45). The passive phase (in which entropy production is suppressed) is punctuated by nonreactive trajectories that begin and end in the same (or a similar) state. Because these transitions are fundamentally uninteresting from the perspective of protein unfolding, we focus our analysis on the dissipative phase.

To more rigorously test our hypothesis that unfolding events are connected to highly dissipative trajectories, we can analyze time dependent state probabilities within a dissipative λ ensemble (specifically, at λ = 1.5). The free energy change plots shown in Fig. 5 demonstrate that unfolding events are indeed tightly coupled to the dissipative trajectory ensemble within the MSM. Starting from the native-localized initial state used above, entropy-producing dynamics quickly drive the system into the extremities of the unfolding order parameters (Fig. 5, a–c). The degree of population density enhancement decreases somewhat after hundreds of nanoseconds (Fig. 5, d and e), but considerably augmented probabilities (equivalent to 2–3 kT changes in the dissipative free energy) remain throughout the unfolded state after a full microsecond of relaxation. At their peak, dissipative free energy changes within core- and backbone-unfolded regions of configuration space reach −7 to −8 kT, corresponding to 1000-fold probability increases. The trajectories that dominate the dissipative ensemble thus clearly draw structures into the denatured region of the landscape, favoring many states in which CI2’s hydrophobic core has disintegrated. Entropy production of similar magnitudes was also localized in core- and backbone-unfolded states in our alternative models, albeit in slightly different distributions (Figs. S7 and S8); these observations illustrate the robustness of this dissipative analysis (see the Supporting Material for further discussion). The dissipative population enhancements observed in the unfolded state, of course, must be compensated by probability deficits in other states. In general, these deficits are subtle in magnitude (≤10%) and localized in the folded and near-folded states. The logarithmic scale of the dissipative free energy, which is important for differentiating the dramatic population changes seen in the unfolded state, diminishes the small relative deficits seen in near-native states in the dissipative ensemble. Comparison of Figs. 2 and 5 illustrates that no probability enhancements occur within ≈3 Å of either native order parameter; it is mostly from this region that unfolded probability density is borrowed.

Figure 5.

Dynamical free energy changes within the dissipative ensemble of trajectories (λ = 1.5) contributing to urea-induced denaturation dynamics, as a function of time. Probabilities of unfolded states along the landscape diagonal (a–d) are greatly enhanced in dissipative trajectories; dynamical free energy changes of −2 to −3 kT persist within many unfolded states over a full microsecond of dissipative dynamics (e and f). To see this figure in color, go online.

Given this statistically validated connection between heat dissipation and denaturation, one is compelled to study the roles solvent molecules adopt within the dissipative ensemble. Fig. S2 demonstrates that significant urea intrusion upon the CI2 core is observed in the unbiased dynamics. As Fig. 6 indicates, entropy-producing trajectories favor even higher urea densities in the vicinity of the protein core. In one striking example (t_obs = 72 ns), the dissipative ensemble-averaged urea oxygen radial distribution function (Fig. 6 a) features a threefold increase in maximum density alongside other local concentration enhancements within the core’s center-of-mass region. Fig. 6 b contains a time profile of ratios of dissipative and unbiased ensemble cumulative distribution functions, integrated from 0 to 5 Å from the protein core. At early times—when nativelike structures remain largely intact—successive urea intrusion events define the dissipative landscape, yielding nearly 40% cumulative urea density increases within the hydrophobic core (as compared to the already urea-enriched unbiased dynamics). As denatured structures become more prominent and fluctuate in structure (after ∼250 ns), the cumulative urea density in dissipative trajectories dips slightly below that seen in unbiased trajectories; after ∼600 ns, however, the dissipative urea density stabilizes at a value modestly above that observed in the unbiased ensemble.

Figure 6.

Urea-core intrusion in dissipative denaturation dynamics. (Top) Shown here is a comparison of ensemble-averaged urea oxygen radial distribution functions from dissipative (λ = 1.5) and unbiased (λ = 1) trajectories early in the relaxation dynamics (t_obs = 72 ns). (Bottom) Given here are ratios of urea and water oxygen cumulative distribution functions, integrated from 0 to 5 Å and computed from the same dissipative and unbiased trajectory ensembles, as a function of time. The horizontal dashed line corresponds to a unit ratio. To see this figure in color, go online.

By contrast, water exhibits a deficit of density throughout nearly all of the dissipative dynamics, suggesting that the protein core remains dryer in dissipative processes than it does in unbiased trajectories. One peak indicating a slight (≈3%) surplus of water density does appear within the first portion of the time profile; however, this water density enhancement directly follows a series strong urea intrusion events, suggesting that the insertion of water there could simply be facilitated by preceding urea-core insertion. Once dynamics reach a microsecond, the CI2 core is 20% dryer in the dissipative ensemble than in unbiased trajectories. These results are perhaps consistent with previous observations of “dry globule” denatured states created by urea permeation events that are exclusive of water molecules (42). Regardless, we have found that heat dissipation is strongly correlated with dominant unfolding processes that are, in particular, characterized by enhanced urea penetration into CI2’s hydrophobic core.

To provide clearer examples of dissipative urea intrusion events, we turn to an analysis of individual unbiased trajectories. A representative MSM trajectory is featured in Fig. 7. Aforementioned frictional dissipation events, in which microscopic transition probabilities are simply forward-leaning, occur approximately once every three time steps (where a time step, here, is defined by the Markovian lag time). However, little noticeable unfolding occurs over the first 90% of the trajectory, suggesting that such frictional transitions are largely noise from the perspective of understanding protein denaturation. Subjected to the stricter filter of the statistical dissipation event criterion (defined above), most such frictional events indeed disappear. Strikingly, the dissipative transitions that remain correspond to dramatic unfolding events driven by the intrusion of urea density into CI2’s hydrophobic core. One exemplary process, pulled from the statistical dissipation events near the end of the trajectory, is shown at right in Fig. 7.

Figure 7.

Dissipative trajectory dynamics featuring urea intrusion into the CI2 protein core. (Left) Demarcation of frictional and statistical dissipation events within a representative (and relatively inactive) MSM trajectory. (Right) Illustration of the statistical dissipation process beginning at the time t_start indicated at left. Here, passive transitions are designated as any hops that are not classified as statistical dissipative transitions, as defined in the Materials and Methods. Urea (rendered in a magenta surface representation) penetrates into the CI2 core around t_start, leading to a series of dissipative events in which the protein core components are driven apart, and component secondary structural elements are separated. The dissipative cascade ends as CI2’s hydrophobic core reassociates. To see this figure in color, go online.

Before the first dissipative transition, one notes a considerable accumulation of urea molecules near contact interfaces among core residues. This urea accumulation, in turn, leads to an unraveling of the hydrophobic core and accompanying heat dissipation into the bath. Heat continues to be released as the three core residues are pulled in opposite directions by the unstructured protein chain. After ∼20 ns, the residues highlighted in Fig. 1 regain contact, and a passive conformational transition finally occurs as the core reforms. Similar events—in which urea incursion leads to the dissipative separation of the core and subsequent reassociation—are illustrated in the Supporting Material (Figs. S9 and S10). This hydrophobic residue-centric mechanism is well aligned with a recent modular study of urea-peptide interactions: Guinn et al. (46) reported that urea interacts favorably with both aromatic and aliphatic side chain carbons. Though these experiments indicate that urea accumulation is more pronounced around aromatic carbons, their data suggest that the high density of aliphatic groups in the CI2 core should facilitate direct urea interactions. Our dissipative trajectory analysis certainly supports this notion.

We hypothesized that by tracking entropy production (heat dissipation) of the course of protein denaturation, we could gain insight into the underlying forces that drive urea-induced protein unfolding. In the case of CI2, we see that heat dissipation indeed provides some insight into the denaturation process. Urea molecules penetrate the hydrophobic core and outcompete key residues in forming van der Waals contacts with their native binding partners; heat is evolved into the bath as the core disintegrates. Dramatic dissipative events become less likely as the system equilibrates toward a broad denatured ensemble in which native structural features, including the hydrophobic core, are sporadically maintained. Persistent dissolution and reformation of the core, as seen in Fig. 7, have the auxiliary effect of diminishing secondary structural elements throughout the entire protein, which garner stability from tertiary contacts within the protein’s native state. In particular, Figs. S11 and S12 demonstrate that the disassembly of CI2’s native β-sheets (which incorporate the core’s Leu⁴⁹) occurs without much concerted heat dissipation. Such unfolding, however, does immediately succeed core-localized dissipation: cross-referencing Fig. S11 and Fig. 5, one sees that dissipative probability enhancements in the core-unfolded region occur in the frames before significant (unbiased) accumulation in β-unfolded states. Dissolution of these β-rich regions thus seems to proceed via a near-equilibrium (i.e., only mildly dissipative) mechanism, a process initiated by highly dissipative events in the protein core that afford key residues (i.e., Leu⁴⁹) more conformational flexibility. Interestingly, nativelike configurations of Leu⁴⁹ have been identified as important features of the unfolding transition state for CI2 in experiments, as Leu⁴⁹ exhibits a large fractional Φ-value (47). The conformational persistence seen within the β-sheet region in our simulations is thus likely consistent with these past experimental observations of CI2 denaturation.

It is important to note that experimental rate constants for the denaturation of CI2 are much smaller than can be accessed in our simulations, on the order of inverse seconds (31, 48). CI2 unfolding in vitro is complicated by the presence of the five prolines in its sequence, which are fixed in trans configurations in its native state but exist in an equilibrium mixture of cis/trans isomers in its denatured state (31). No treatment of these slow equilibration steps is included in this work: prolines are simply kept in their nativelike trans configurations. Delayed proline isomerization within the unfolded state could certainly result in further loss of native structure. Even considering the absence of proline isomerization, however, microsecond trajectories are probably insufficient to capture the full CI2 denatured ensemble. Jackson and Fersht (31) reported a weakly structured hydrophobic core in CI2’s denatured state. The simulations presented here likely represent a structural ensemble closer to the transition state than is the fully equilibrated unfolded state, as our data show a partially assembled hydrophobic core (in good agreement with the Leu⁴⁹ Φ-value) (31). Despite these remaining questions about the final denatured state of CI2, our results nonetheless highlight a dramatic role for entropy production in the initial stages of CI2’s denaturation process.

As the dynamics within the dissipative trajectory ensemble illustrate, entropy production serves as a reliable harbinger of urea intrusion events. In effect, we have used nothing but the basic principles of entropy production (those concerning the irreversibility of trajectories in a nonequilibrium system) to pinpoint microscopic interactions between urea and protein residues central to CI2’s stability. This observation strengthens previous assertions regarding the connection between entropy production and productive work in biophysical systems (17). Perhaps unsurprisingly, highly dissipative events localized to particular regions of a protein’s phase space tend to correspond to dramatic conformational changes in nearby residues. Dissipative events associated with high-frequency thermal fluctuations are generally annihilated by opposing processes within an ensemble average of trajectories. It thus stands to reason that the dissipative processes that prevail within a mechanical system will often occur in concert with those engaged in productive, functionally relevant motion (which, for urea-protein systems, concerns the destabilization and unfolding of protein structures). The power of the in silico experiments presented here resides in the foundational relationship between equilibrium and dissipative work: by tuning how likely rare dissipative events are to occur, one also modulates the frequency with which functional dynamics, and the fundamental interactions that drive them, proceed. In the case of the urea-induced unfolding of CI2, destructive interactions seem to evolve a distinctive heat signature that points to their functional relevance in generating the denatured protein ensemble.

Conclusions

We have thus designed and executed in silico experiments that allow the degree of heat dissipation to be characterized over the course of chemically induced protein denaturation. We find that, within dynamics in the high entropy-production regime, productive unfolding processes are correlated with heat dissipation events derived from urea’s interactions with core hydrophobic protein residues.

Because the entropy produced by transitions between two specific states is simply determined by their free energy difference, the dissipation associated with any single trajectory is somewhat trivial. The utility of dissipative analysis lies in performing a path integral over all possible trajectories: by summing over dissipative weights, intriguing behavior emerges that would not be evident from equilibrium free energy landscapes. The time evolution of dissipative trajectories tells us the pathways by which particular high free energy states are reached, and highlights the mechanism by which urea brings about the denatured ensemble. The idea of differentiating near-equilibrium and far-from-equilibrium dynamics is also not easy to establish based on equilibrium free energy surfaces: for example, although β-unfolded states and core-unfolded states both have reasonably high free energies, urea’s relationship with these two states is drastically different. Dissipative trajectory analysis played a critical role in delivering these insights.

As noted above, dissipative trajectory experiments operate with complete independence from other physical characteristics a system may possess. In this sense, the generality of entropy production affords one a unique platform for probing physical interaction mechanisms with minimal expectation bias. Net forces that have entropic origins, for example, are notoriously difficult to classify based on the analysis of microscopic forces/energies and individual system configurations. To a large degree, however, the average entropy production points in the direction of equilibration (or, in the case of driven nonequilibrium systems, the direction of the energetic bias). The statistics of dissipation and their relation to a system’s phase points should thus provide a broad perspective on physical driving forces on an ensemble level, capturing the effects of both destructive local processes and collective phenomena. In the case of urea-based denaturation, complex interactions occurring among water, urea, and protein molecules have muddled interpretations of the driving forces behind unfolding. Not only do both water and urea molecules directly interact with protein substrates, but the chaotropic properties of urea also fundamentally alter the diffusive and hydrogen bonding properties of water. The effects of urea-disordered water on protein denaturation have been emphasized in the past (11). Our results do not contradict all roles for disordered water in denaturation; indeed, the relative drying of the hydrophobic core that occurs in the dissipative ensemble implies a disruption of local water-protein hydrogen bonds, and perhaps a correspondent restoration of the hydrogen bond network in bulk solvent. This hypothesis suggests an indirect favorable effect of urea intrusion upon the hydrophobic core. Nonetheless, heat dissipation also suggests an intriguing connection between direct urea/protein hydrophobic core interactions and denaturation. The possibilities for conducting similar dissipative trajectory experiments abound within the sphere of intermolecular physics, wherein the arrow of entropy production could shed light on the complicated mechanisms governing a range of denaturation, solvation, aggregation, and self-assembly processes.

Author Contributions

J.K.W. and R.Z. designed research. J.K.W., S.-g.K., and R.Z. performed research. J.K.W. and S.-g.K. contributed analytical tools. J.K.W. analyzed data. J.K.W. and R.Z. wrote the article.

Editor: Nathan Baker.