Protein Folding Is Mechanistically Robust

Abstract

Markov state models (MSMs) have proven to be useful tools in simulating large and slowly-relaxing biological systems like proteins. MSMs model proteins through dynamics on a discrete-state energy landscape, allowing molecules to effectively sample large regions of phase space. In this work, we use aspects of MSMs to ask: is protein folding mechanistically robust? We first provide a definition of mechanism in the context of Markovian models, and we later use perturbation theory and the concept of parametric sloppiness to show that parts of the MSM eigenspectrum are resistant to perturbation. We introduce a new, to our knowledge, Bayesian metric by which eigenspectrum robustness can be evaluated, and we discuss the implications of mechanistic robustness and possible new applications of MSMs to understanding biophysical phenomena.

Introduction

Simulations have reached a level at which one can target molecular phenomena on timescales from microseconds to milliseconds (1,2) with atomistically detailed models. However, how accurate are the predictions that come from these models? As any model used is an approximation to reality, a key question must always be addressed: how robust are model properties to errors in the model, and how can one predict what properties might be robust in a given system?

To address these questions in the context of biomolecular simulation, we propose to harness efforts directed at deriving meaningful many-state models of protein folding dynamics. Advances in computational models have made folding simulations possible at vastly longer timescales than were previously reasonable (3–6). Discrete-time, discrete-space Markov state models (MSMs), which propagate a probability distribution using left multiplication on a transition matrix, have shown particular promise for simulating protein folding processes.

After grouping structures into metastable microstates, MSMs capture the rare transitions between a protein's local free energy wells. MSMs offer a statistical approach to simulation: instead of relying on single trajectories, MSMs simulate ensemble dynamics with a state population probability vector (7,8). Recent millisecond-timescale simulations of the 39-residue protein NTL9 (1) and an 80-residue fragment of the λ-repressor protein (2) demonstrate the ability of MSMs to model the large, slowly relaxing systems that are present targets in protein folding research.

Although protein folding mechanisms are intellectually interesting in their own right, the ability to understand folding mechanisms also has implications for studying processes like catalysis, inhibition, and allostery. A number of recent examples in the literature show that folding mechanism (through transitions to intermediate states) can play an important role in mediating biological processes (9,10).

Proteins are good examples of biological systems that demand robustness to changes in environmental parameters. Robustness in folding mechanism modulates the kinetically dependent processes in catalysis, inhibition, and regulation, which proceed in ordered steps, and moderates harmful phenomena like misfolding and aggregation (11). Aspects of this robustness have been well studied and are substantiated by experiments. A protein can reach its native state under a range of physical and chemical conditions, and systematic point mutations often have little impact on a protein's ability to find its final, functional structure (12–14). However, it is unclear where this robustness ends: which properties are most robust to perturbations, and which are the most vulnerable?

Beyond general questions concerning the robustness of protein folding mechanisms, we also aim to gauge the accuracy of MSMs constructed from simulated folding trajectories. Given the general robustness seen in real protein ensembles, we would hope to observe similar properties in the data derived from molecular dynamics (MD) simulations. However, errors due to discretization (in space and time) and finite sampling are inevitable and difficult to quantify (11). As a telling example, one might need to minimize the uncertainties of 100,000,000 parameters to create an informative model for a 10,000 state system (15). If the robustness of protein folding is any indicator, though, optimization of model parameters may be less important than previously thought.

To investigate the possibility of this robust behavior, we use a Hessian-based theory called parametric sloppiness (16–18). In general, biological systems have demonstrated a tendency to show a particular sloppy behavior in the face of parametric perturbation. Here, sloppiness describes the global behavior of a biological system with respect to local changes in environment. A sloppy system is insensitive to (perhaps even drastic) perturbations in the majority of its defining parameters, varying only with changes on a few stiff coordinates (16,17). This concept of sloppiness is related to system robustness: systems with sloppy sensitivities are invariant to many permutations in environmental conditions.

Sloppy behavior is particularly prominent in complex biological systems, where the accuracy of an ultimate result is crucial, but associated kinetic pathways can be flexible. Previous work in this area has shown sloppy behavior in processes ranging from the Drosophila circadian rhythm to rat growth-factor signaling (16–18). For proteins, changes in sequence and environment that are inconsequential can be related to sloppy deviations in parameter space. As events that induce phenomena such as protein misfolding are few and difficult to detect, these processes can be associated with changes in stiff parameters under this framework.

Although this theory of sloppiness is useful, it is not intuitive from a physical standpoint. Accordingly, we preface our study with a discussion of eigenspectrum perturbation theory and its relation to sloppiness and sensitivity. We then use this perturbation theory and sloppiness theory to investigate MSM observable robustness to transition probability perturbation. We also develop a quantitative Bayesian metric by which robustness can be evaluated, and we discuss implications such robustness holds for applications of MSMs to biophysical phenomena.

Methodology

Exploring mechanism in an MSM context

A great challenge in the field of protein folding lies in understanding holistic folding mechanisms. Although determination of properties like folding rates and native-state structures has become common practice in both experiment and simulation, studies of mechanism are less established. Phi-value analysis, which extracts kinetic information using site-specific mutagenesis, is often used by experimentalists to study folding mechanisms. Although phi-value analysis has provided great insight into many systems, it still suffers from the imprecise meaning of intermediate phi-values and the obvious limitation of trying to extract kinetic information from thermodynamic data (19–21). Deriving mechanistic information from pure MD simulations also presents challenges due to difficulties in analysis and the existence of heterogenous folding pathways. In the case of simulation, however, one might look for MSMs to provide a means for extracting mechanistic information from MD simulations.

If we want to explore robustness in mechanistic properties derived from simulation, we first need to consider how mechanism should be defined in an MSM context. On first thought, one might determine that a protein's folding trajectory as seen in MD simulations represents its folding mechanism. However, we argue that this view of mechanism is overly restrictive: individuals within an ensemble experience different state-to-state transition sequences in the folding process. Although the MSM transition matrix defines which trajectories are possible, it also does not provide a clear picture of which pathways the ensemble prefers over short and long periods of time.

The eigenspectrum of the MSM transition matrix, however, provides both kinetic and thermodynamic information about the ensemble. With units of probability density, the transition matrix eigenvectors represent the normal modes of time evolution in the system. The stationary distribution, the eigenvector with unit eigenvalue, describes the equilibrium populations in the ensemble. The other eigenvectors, with subunit eigenvalues, describe changes in the system's population distribution at timescales set by their respective eigenvalues.

An MSMs probability distribution vector at any given timestep n has the nice property of being propagated by transition matrix eigenvalues and eigenvectors. This relationship is described by a simple equation involving the initial distribution vector ${\mathit{\pi }}^{\left(0\right)}$:

$${\mathbf{\pi }}^{\left(n\right)}\propto \sum _{i=1}^M{\lambda }_i^n\langle {\pi }^{\left(0\right)},{\mathbf{g}}_i\rangle {\mathbf{e}}_i,$$ (1)

where ${\mathit{\pi }}^{\left(n\right)}$ represents the system's nth probability distribution vector, λ_i denotes an eigenvalue of the $M\times M$ transition matrix, and g and e are the corresponding right and left eigenvectors of the transition matrix, respectively (11). The parenthetical $\left(n\right)$ is used to denote a discrete time index, whereas an n without parentheses indicates an exponent (as in the case of ${\lambda }_i^n$). Here, and below, the angle brackets are used to designate a dot product between the enclosed vectors: $\langle u,v\rangle ={\sum }_i{u}_i{v}_i$.

The previous expression describes how an arbitrary population distribution converges to the equilibrium distribution over time. As the number of timesteps n becomes large, all subunit eigenvalues (through the term λⁿ) and their eigenvectors decay to zero, and eventually only the stationary distribution multiplied by the unit eigenvalue remains.

Relating mechanism to an MSM eigenspectrum offers advantages over the alternatives that were previously discussed. The eigenvector decomposition method provides details about how entire probability distributions change, allowing for an idea of mechanism on an ensemble level. Large eigenvector entries represent states that are important to density transfer on the relaxation timescale of an associated eigenvalue. One can inspect the set of eigenvectors to find which individual states are mechanistically relevant at both fast and slow timescales. Information about trajectory (which folding pathways are most probable) and end result (how the state probability distribution converges to a stationary distribution) are intrinsic to the eigenspectrum.

Together, we extend, trajectory and end result define the essential parts of a folding mechanism. As such, we suggest that an MSM mechanism be defined in the context of eigenvector decompositions. To investigate mechanistic properties of MSMs, one should inspect the signs and magnitudes of transition matrix eigenvector elements for the eigenvalues that describe a given process. Furthermore, when one considers a folding mechanism, one is most interested in learning the important long timescale pathways between unfolded states and the native state. In MSMs, these slow processes are described by the eigenvectors with the largest eigenvalues. Therefore, the most salient information about folding mechanism can be extracted from eigenvectors that describe the system's long timescales.

To provide the reader with some intuition about how eigenvalues and eigenvectors can be related to mechanism, we provide a toy MSM example illustrated in Fig. 1 and Fig. 2. Fig. 1 shows a simple one-dimensional potential energy surface and its corresponding continuous probability distribution. To build an MSM on the toy surface, we discretize the potential in a natural manner wherein state boundaries are placed on the barriers between the wells in the surface. We also assume the potential is truly one-dimensional, i.e., transitions can only occur between neighboring wells.

Figure 1.

Top: Toy one-dimensional potential energy surface used to illustrate the role transition matrix eigenvectors play in describing mechanism. Vertical lines and shading indicate a natural partitioning of space wherein barriers divide states. Bottom: Probability distribution corresponding the toy potential energy surface with discretization.

Figure 2.

Eigenvalue spectrum and selected eigenvectors from the MSM built on the toy potential seen in Fig. 1.

Fig. 2 shows the transition matrix eigenvalue spectrum and selected eigenvectors for the nine-state MSM constructed on our toy potential. State assignments map directly onto the partition: the eigenvector components at left correspond to the states on the left side of the potential, etc. To make mechanistic assertions about dynamics on the surface, we simply need to inspect the magnitudes of eigenvector components. Each eigenvector component represents the relative flux into (if the component is positive) or out of (if the component is negative) the given state at the eigenvalue's timescale.

As seen in the top eigenvector ($\lambda =0.988$), the system's slowest mode describes the transfer of the population from the right side of the surface to the left side. In the system's second slowest mode ($\lambda =0.875$, middle eigenvector), probability density moves between the wells on the left side of the potential. In the third, high frequency eigenvector ($\lambda =0.385$, near the lag time rate), probability density is transferred from the shallow wells on the barrier to the system's most populated state.

When considering a folding mechanism, the system's two slowest modes provide the richest mechanistic insight. In an analogy to protein folding, the wells on the right side of the potential might represent the unfolded basin, whereas the state furthest left might represent a native-side intermediate. As such, the first eigenvector shows how population is transferred from the unfolded basin to the native basin at the folding timescale, and which states are important in that transfer; the second eigenvector shows how folding might proceed from a highly populated intermediate and the native state on a relatively slow timescale.

Although the fast eigenvector does provide information about how population descends the native-side barrier, one is presumably less interested in the dynamics between the highly transient states that eigenvector describes. We thus argue that when studying folding mechanism, the slow eigenvectors of an MSM are of fundamental interest.

Perturbation theory framework

With a working definition of folding mechanism now in hand, we can now explore whether or not these mechanisms are robust to perturbation. Before we introduce the Hessian-based method used for the bulk of this study, we will first lay out a simpler method for robustness evaluation that has roots in physics. This formalism is similar to the more statistically rigorous treatment used in later sections and can be used to draw parallels between physical and statistical methods.

We have already noted that perturbing the interactions of a protein is not new to the field of protein folding. Phi-value analysis (19–21) examines the rates of protein folding and unfolding when minor perturbations (e.g., point mutatations) are made to the protein experimentally. The fundamental assumption of this procedure is that minimal perturbations can probe the folding mechanism without altering it. However, it is natural to ask: what size of perturbation is small enough, and how can one build a framework for understanding these perturbations?

Below, we present a simple framework for understanding perturbations, building upon previous work in examining perturbations to a Hamiltonian in quantum mechanics. A popular method for approximating solutions to the Schrodinger equation involves splitting the system Hamiltonian into zeroth- and higher-order parts with expansion parameter ξ:

$$\mathcal{H}={\mathcal{H}}^0+\xi {\mathcal{H}}^{\prime }+{\xi }^2{\mathcal{H}}^{″}\dots .$$ (2)

If the eigenvalue problem for the zeroth-order Hamiltonian can be solved exactly, corrections to the eigenvalues and eigenvectors based on the perturbed Hamiltonian can be calculated with the well-known eigenspectrum perturbation theory (22).

As in the quantum mechanical problem, an MSM transition matrix could also be augmented by a perturbation operator. Suppose we would like to calculate the impact of a random perturbation on the eigenspectrum of the transition matrix. We could define a perturbed transition matrix 𝕋 (to first order) such that

$$\mathbb{T}\approx {\mathbb{T}}^0+\xi {\mathbb{T}}^{\prime },$$ (3)

where ${\mathbb{T}}^0$ is the original transition matrix and ${\mathbb{T}}^{\prime }$ is a matrix of random noise under the constraint that the sum ${\mathbb{T}}^0+\xi {\mathbb{T}}^{\prime }$ is row normalized. The first-order correction due to noise, ${\lambda }_m^{\prime }$, for each eigenvalue ${\lambda }_m^0$ of the transition matrix is given by the dot product

$${\lambda }_m^{\prime }=\langle e_m^0,e_m^0{\mathbb{T}}^{\prime }\rangle ,$$ (4)

where ${\mathit{e}}_m^0$ is the mth left eigenvector of the zeroth-order transition matrix (22). Corrected left eigenvectors are given by the formula

$$e_m=e_m^0+\sum _{j\ne m}\frac{\langle e_j^0,e_m^0{\mathbb{T}}^{\prime }\rangle }{{\lambda }_m^0-{\lambda }_j^0}{e}_j^0.$$ (5)

Using these corrections due to perturbation, one could gauge the impact of a random noise (or a more systematic) change in a transition matrix on its eigenspectrum. We later apply this perturbation theory to analyze the robustness of eigenvalues for a villin transition matrix. Fig. 3 shows the eigenvalue spectrum for this system. As with the toy model, the villin model has a few slow eigenvalues (above 0.5) that should be important in analyzing folding mechanism.

Figure 3.

Transition matrix eigenvalue spectrum for an MSM of the villin headpiece domain in explicit solvent. This eigenvalue spectrum is analyzed in the Perturbation theory section of this work.

In performing perturbation theory, we carry out a procedure that is conceptually not unlike that of phi-value analysis. We perform a small perturbation on the system (with added noise to the transition matrix analogous to a point mutation), and we analyze the impact that perturbation has on the mechanism (with differences in the eigenvalue spectrum analogous to free energy differences). Noise-like perturbations, of course, are unrelated to point mutations, but the two methods share many of the same ideas for investigating mechanistic properties. As in phi-value analysis, which elements of an MSM can we change without fundamentally altering the dynamics?

Need for a new framework

The method more extensively used in this study is similar to a classical perturbation theory. We perturb a transition matrix with noise, calculate the corrected eigenspectrum, and compare that eigenspectrum to the original. We decide to use an alternative method for two reasons. First, we would like to gauge the rate of change (called the sensitivity) in an eigenvalue or eigenvector with respect to the magnitude of perturbation. Furthermore, we would like to know this eigenspectrum sensitivity for each individual parameter in the model. These desires are not easily fulfilled with analytical perturbation theory. This work's method, drawn from the literature and tested on biological models, is designed to estimate such a rate of change (16–18).

Second, sophisticated theory for error propagation in MSMs has been developed using a sensitivity-based analysis (23,24). These methods use Bayesian schemes to estimate uncertainty based on the available data. The nature of the sensitivities used to estimate such errors, however, has never been well characterized, and it would be useful to gain intuition about the relative magnitudes of these eigenspectrum sensitivities in recently constructed MSMs. Sloppiness-based techniques, as discussed below, provide an avenue to do so.

Sloppiness framework

In developing this more rigorous method for sensitivity analysis, we draw inspiration from theories first used in statistics and computer science. Hessian-based sensitivity studies are common in the statistical literature (25,26). These methods share the characteristic of calculating the Hessian (second-derivative) matrix of a particular function that gauges a model's dependence on a set of parameters. The eigenvalues of this Hessian matrix can then be used to estimate an observable's sensitivity to perturbation. Here, we adopt the notation and language of sloppiness used in recent Hessian-based sensitivity studies on systems biology models (16–18).

To investigate sloppiness in MSM transition probabilities, we start with the so-called model parameter cost function on the transition probability matrix. Given the perturbation of a certain system parameter, the cost function returns the induced sum-squared deviation in a dependent observable. In our case, we define a parameter to be a transition matrix element and an observable to be an eigenvalue or eigenvector. Adapted from the literature (16–18), the cost function $C\left(\mathbb{T}\right)$ is defined as

$$C\left(\mathbb{T}\right)=\frac{1}{N^2}\sum _i\sum _j\left(\sum _{k=1}^N{\left(e_k\left(p_{ij}^{\ast }\right)-e_k\left(p_{ij}\right)\right)}^2\right),$$ (6)

where 𝕋 is the $N\times N$ transition matrix, e is a left eigenvector of that matrix, p_ij is an individual transition probability defined by the model, $p_{ij}^{\ast }$ is a continuous variable representing a perturbation of p_ij, and $e_k\left(p_{ij}^{\ast }\right)$ represents an eigenvector entry as a function of $p_{ij}^{\ast }$ (16–18). To quantify the sensitivity of a model to changes in individual parameters, we use the concept of the sensitivity eigenvalue, λ_sens, of the cost function Hessian. For simplicity, the Hessian of $C\left(\mathbb{T}\right)$ is constructed as a diagonal matrix. Accordingly, each sensitivity eigenvalue is merely a Hessian matrix element evaluated at its corresponding parameter p_ij:

$${\lambda }_{sens}\left(p_{ij=t}\right)={\mathcal{H}}_{tt}\left(p_t\right)=\frac{{\partial }^{2}C\left(\mathbb{T}\right)}{{\partial p}_t^2}\left(p_t\right).$$ (7)

The sensitivity spectrum of the transition matrix, generated by plotting the sensitivities of all transition probabilities, gives a qualitative estimate of sloppy behavior, wherein a sensitivity spectrum that spans many orders of magnitude is said to indicate sloppiness (16). We should note that the functions $e_k\left(p_{ij}^{\ast }\right)$ lack an easily derived analytical form. In this study, such relations were determined by calculating eigenvectors at increments of $p_{ij}^{\ast }$ and fitting the numerical relationships $e_k\left(p_{ij}^{\ast }\right)$ to quartic polynomials.

Although the range of a sensitivity spectrum provides an intuitive estimate of sloppiness, a more quantitative metric would allow for better comparison of robustness within a given set of observables. A useful metric can be developed from individual terms in the cost function. We use a Bayesian approach to define, for a small perturbation, the expected deviation for an observable e:

$$\langle \delta \left(e\right)\rangle =N^2{\left(\Delta p\right)}^2\sum _i\sum _j\left(\frac{U_i}{{\sum }_m\left(U_m\right)}\right){\lambda }_{sens}\left(p_{ij}\right),$$ (8)

where $\Delta p$ represents the magnitude of a small transition probability perturbation and each U_i represents the relative uncertainty in a row of the transition matrix. A derivation for this equation is included in the Supporting Material (23,24).

The expected deviation quantifies robustness via a direct cost function variation estimate: for an observable e, $\langle \delta \left(e\right)\rangle$ represents a weighted average deviation due to perturbation over all components of the cost function $C\left(\mathbb{T}\right)$.

Before moving on to our results, we should note that under both perturbation schemes the perturbed transition matrix will violate the detailed balance condition $p_{ij}{\pi }_i=p_{ji}{\pi }_j$. Such matrices thus describe only near-equilibrium steady states of the perturbed system. As a physical analogy, our perturbation schemes do not represent reversible changes in activation barrier heights, but rather correspond to nonequilibrium experiments in which energy is added to break detailed balance. These nonequilibrium results are then compared among our observables of interest.

Results and Discussion

Perturbation theory framework

As an instructive example, we first use classical perturbation theory to gauge robustness in eigenvalues of the villin MSM transition matrix. The stationary eigenvalue will not change with any transition matrix perturbation, because the unit eigenvalue is a property of all regular stochastic matrices. All other eigenvalues, however, do depend on the particular values of transition matrix elements. To measure how much these kinetic eigenvalues change upon perturbation, we perturb the transition matrix and calculate the first-order eigenvalue correction using Eq. 4. Note that, in this case, calculated eigenvalue corrections are exact, as the perturbed transition matrix is itself exactly first order in nature.

In this case, the villin count matrix was perturbed by additive Gaussian noise ($\mu =5$, $\sigma =3$) to 1% of matrix elements, constrained to positivity, and then renormalized to yield a perturbed transition matrix. To find the matrix ${\mathbb{T}}^{\prime }$ in Eq. 4, we subtract the perturbed matrix from the original matrix. Mean eigenvalue corrections were computed over 1000 random perturbations of the kind just described. Fig. S1 illustrates the relationship between mean eigenvalue correction and eigenvalue relaxation timescale. The eigenvalues at long timescales (i.e., eigenvalues with large magnitudes) require quite small corrections due to the random perturbation, whereas the eigenvalues at shorter timescales (corresponding to high frequency modes in the system) change to a greater extent when the transition matrix is perturbed.

We should note that while large eigenvalues change less when perturbed than their small counterparts, the relaxation timescales derived from these eigenvalues exhibit the opposite trend. The reason for this discrepancy arises from the nonlinear way in which physical timescales are calculated: a relaxation timescale is proportional to one over the logarithm of its corresponding eigenvalue (see Fig. S1 caption). Because the system's largest eigenvalues are near a singularity in the timescale function, even modest changes in those eigenvalues translate to large changes in timescale. Thus, whereas the largest eigenvalues ($\approx 0.99$) change only by a few parts per thousand upon perturbation, their timescales still change by ∼25% ($\approx 150\text{ns}$). Fig. S2 shows the mapping between fractional timescale correction and relaxation timescale for all eigenvalues being considered.

Because slow timescales are often those most important for model interpretation, this intrinsic deficit in slow timescale robustness should be considered in future MSM analyses. However, we maintain that the relatively small deviations in large eigenvalues still allow for meaningful analysis. A 25% change in timescale, while significant, does not drastically alter the physical interpretation of a relaxation process. If the largest eigenvalues changed to the extent that many smaller eigenvalues change, the longest timescales could deviate by an order of magnitude or more, and any conclusions based on those data would be suspect. The fact that the slow timescales do not change so dramatically is comforting from the standpoint of ongoing Markov state modeling.

Sloppiness framework

Having demonstrated the use of perturbation theory in analysis of transition matrix eigenvalues, we now apply the more sophisticated sloppiness theory in analyzing the eigenvectors of MSMs. Transition matrix eigenvector sensitivities were analyzed for MSMs of Fs-peptide (in explicit solvent, lumped to 19 macrostates) and the villin headpiece domain (in both explicit and implicit solvent, lumped to 500 macrostates) (24,27,28).

For a preliminary illustration of sensitivity eigenvalues, Fig. 4 shows stationary distribution sensitivities for all transition probability parameters of the Fs-peptide MSM. Clearly, the magnitudes of sensitivities vary greatly from state to state, suggesting that the stationary eigenvector is much more sensitive to some states than it is to others. The largest sensitivities often, though not always, correspond to parameters leading to the model's most populated states (States 13 and 14) and those along the matrix diagonal. In this case, no sensitivities are particularly large (at most ∼$1\times {10}^{-3}$), indicating the distribution will not change drastically upon perturbation.

Figure 4.

Sensitivity eigenvalue matrix for the stationary distribution of the Fs-peptide transition matrix. Indices on the right of the plot indicate states from which a transition originates, whereas indices on the left indicate where a transition terminates. Though all sensitivities are relatively small ($\le 1\times {10}^{-3}$), the largest sensitivities often correspond to parameters that describe transitions into highly populated states (i.e., States 13 and 14) and self-transition probabilities.

The biophysical meaning of the transition matrix perturbations carried out in this work requires some thought. Given that transition probabilities are held constant over the course of a simulation, these perturbations are unlike the thermal fluctuations that cause Brownian motion, because such fluctuations occur on ultrashort timescales. Rather, time-independent perturbations are more like probes present in a nonequilibrium experiment, or, with the enforcement of detailed balance, equilibrium phenomena like interactions with ligands or denaturant. Systematic perturbations, and an analysis of how eigenvectors are affected by these perturbations, might thus provide a means of simulating such interesting processes.

The main purpose of this study, however, is to investigate general mechanistic properties of MSMs. We first look to eigenvector sensitivity spectra to provide a qualitative picture of mechanistic robustness to perturbation: Fig. S3 shows the sensitivity spectra for three selected eigenvectors of the villin implicit solvent model. Because the sensitivities in each spectrum are spread quite evenly over many orders of magnitude, the spectra meet our qualitative criterion for sloppiness. It should be observed that sensitivities near the maximum sensitivity eigenvalue are related to stiff directions in parameter space, as changes in those parameters cause the greatest relative changes in model behavior. For the most part (as seen in all three spectra in Fig. S3), sensitivity values are sparse near the maximum sensitivity eigenvalue, suggesting that stiff parameters are few in number.

Notably, the sensitivity spectrum for the stationary distribution of the villin implicit solvent model (shown in Fig. S3) spans nearly six more orders of magnitude than do the spectra related to other eigenvectors. For the following analysis, suppose 10% of rows in villin's transition matrix are perturbed by noise. Evaluating the stationary distribution under our Bayesian robustness scheme ($\Delta p=0.01,N=50)$, we see that the stationary distribution is three orders of magnitude more robust to perturbation than the rate eigenvectors: $\langle \delta \left(\mathit{\pi }\right)\rangle =4.96\times {10}^{-4}$, $\langle \delta (\mathit{e},\lambda =.729)\rangle =0.10$, $\langle \delta (\mathit{e}$, $\lambda =.510)\rangle =0.12$. A similar gap between stationary and rate eigenvectors was seen in both of the other models analyzed. However, the slow rate eigenvectors are still robust under our metric: expected deviations on the order of 0.1 are quite small in eigenvectors over all 500 components. It should also be noted that a transition probability perturbation of 0.01 is not insignificant: transition matrix elements in these models often fall in the range of 0.001–0.05. As discussed below, the so-called slow eigenvectors of all three models were observed to be similarly robust. We observe in general that the thermodynamic observables of MSMs are much more robust to perturbation than their kinetic counterparts. These data help to justify previous observations that equilibrium properties converge more quickly than dynamical ones under rapid conformational sampling (27).

In Fig. 5 and Fig. 6, we use our Bayesian metric on a variety of spectra (again, with $\Delta p=0.01$) to compare eigenvector robustness as a function of rate within the three systems. Rate is defined as the inverse of an eigenvector's relaxation timescale at the model lag time τ_lag (${\tau }_{lag}=10\text{ns}$ for villin, 2 ns for Fs-peptide), with ${\tau }_{relax}=-{\tau }_{lag}/\text{ln}\lambda$. In each case, λ is the eigenvalue of the transition matrix corresponding to the eigenvector being analyzed (8).

Figure 5.

Expected deviation, $\langle \delta \left(\mathit{e}\right)\rangle (N=2,\Delta p=0.01)$, versus rate for eigenvectors of Fs-peptide. The expected deviation for the stationary distribution corresponds to the point at zero rate, and $N=2$ was chosen to represent 10% of states. For Fs-peptide, expected deviations in eigenvectors increase loosely with increasing rate.

Figure 6.

Expected deviation, $\langle \delta \left(\mathit{e}\right)\rangle (N=50,\Delta p=0.01)$, versus rate for eigenvectors of villin in explicit (open circles) and implicit (solid circles) solvent. Again, the expected deviations for the stationary distribution are represented at zero rate. In both cases, expected deviation appears to increase with increasing rate; the explicit model seems to be slightly more robust at rates near the lag time rate of 0.1 ns.

All three plots show a similar increase in expected deviation as eigenvector frequency increases up to (and in the case of Fs-peptide, beyond) the relaxation timescale rate $k=1/{\tau }_{relax}$ ns⁻¹. In general, eigenvectors at each system's slowest timescales are 1.5 to 2 times more robust than those near the lag time rate. Absolute robustness, as measured here, appears to be roughly independent of system size: although the villin MSM contains many more states than does the model for Fs-peptide, the magnitudes of deviations seen in Fig. 5 and Fig. 6 are comparable.

This trend in robustness versus rate is pleasing from a physical point of view. Changes in the transition matrix divert the ensemble's walk to pathways not described by the original model. These diversions, one would expect, might have a large impact on ensemble dynamics at a system's shortest timescales. We see that the transfer of density between states at quite long timescales, however, is less dependent on these high frequency trajectory diversions. Indeed, in processes like protein self-assembly that occur with continual environmental fluctuation, adaptivity over long timescales is needed to ensure reproducible results. Our data show that many protein folding MSM eigenvectors exhibit a similar resistance to parametric perturbation.

To provide a specific example of mechanistic changes at short and long timescales, Fig. 7 contains difference maps for various left eigenvectors of a perturbed Fs-peptide model. The Fs-peptide transition matrix was perturbed in a similar fashion to that used for villin: counts were added to ∼5% of count matrix elements using Gaussian noise ($\mu =5$, $\sigma =3$). The matrix was then constrained to positivity and renormalized to yield a perturbed model. The difference maps in Fig. 7 simply represent the difference between left eigenvectors of the perturbed model and those of the original model with the indicated eigenvalues λ (after perturbation).

Figure 7.

Left eigenvector difference maps for a perturbed model of the Fs-peptide. Quantities on the vertical axes are unitless and describe the difference in flux into a state in the perturbed model relative to the original model. As the maps indicate, the fluxes at long timescales are less affected by perturbation. Dynamics into and out of states with low populations (e.g., State 9 and State 17) fluctuate largely at short timescales but are stable in the slowest eigenvectors.

A first observation drawn from Fig. 7 lies in the relative magnitudes of eigenvector deviations: in agreement with Fig. 5, the sum-squared deviations for faster eigenvectors are much larger than those seen for slower eigenvectors. With respect to gaining specific mechanistic insight, changes in the slowest eigenvector (λ = 0.785, ${\tau }_{relax}\approx 8$ ns) are relatively uniform, with the exception of a small increase of flux into State 18 and a small increase in flux out of State 4, both intermediate states connected directly to the folded helix.

In the two eigenvectors at fast timescales, however, significant changes in flux occur for a large percentage of the states in the model. In the case of the intermediate eigenvector ($\lambda =0.361$, ${\tau }_{relax}\approx 2$ ns), States 5 and 6 (both intermediates connected to the native state) lose the most population flux, whereas many other intermediate (directly connected to the native state) and unfolded (not directly connected with the native state) states gain flux. For the fastest eigenvector ($\lambda =0.032$, ${\tau }_{relax}\approx 0.6$ ns), large losses in flux occur for States 9 and 17, which are sparsely populated unfolded states not connected to the native state.

These observations together support our conclusions about mechanistic robustness versus rate. The mechanistic characteristics described by the slow eigenvectors change very little, while fluxes in the faster eigenvectors change drastically. Particularly, fluxes in and out of lowly populated states (like State 17), which one would expect to have little impact on the overall folding mechanism, can change considerably at fast timescales but are damped out once the longest timescale is reached.

Conclusion

We have shown that three representative protein folding MSMs exhibit sloppy behavior with respect to their transition probability parameters. In general, the stationary and slow components of the eigenspectrum incurred only small deviations upon perturbation, whereas the high-frequency eigenvalues and eigenvectors were less robust under our framework. Especially near the lag time rate, eigenvalues and eigenvectors experienced deviations more than twice as large as those seen in slowly relaxing kinetic components.

With these conclusions about robustness in mind, we would like to discuss the new implications for MSMs that sloppiness holds.

Implications for models derived from simulation

Every MSM eigenvector analyzed in this study demonstrated sloppy characteristics. Although the degree of sloppiness varied from vector to vector, sensitivities of every observable spanned at least two orders of magnitude with reasonable uniformity.

We would like to reiterate that the robust behavior we report is not necessarily unique to models of protein folding. Topologically, protein folding MSMs are networks that contain many kinetically relevant but few thermodynamically relevant states. It is likely that the reported eigenvector robustness would be observed in any network sharing these characteristics. The important conclusion here, however, is that protein folding MSMs do exhibit this robustness to perturbation. This means that parts of MSM transition matrix eigenspectra (and observables that can be calculated from them) are not highly sensitive to uncertainties in many of the transition matrix elements themselves.

Although improvement in MSM accuracy is an ongoing task, we can thus be comforted that even moderate uncertainties in transition probabilities will have little impact on parts of the transition matrix eigenspectrum. Such confidence, however, comes with a few caveats. First, we have emphasized that the slowly relaxing aspects of mechanism are more robust to perturbation than the quickly relaxing ones. Observations that are contingent on high-frequency eigenvectors should be more closely scrutinized. Second, it is clear the number of parameters with large uncertainties still needs to be limited. Because transition probabilities are coupled together, too many successive errors in transition matrix elements could have drastic effects on the quantitative predictions of a MSM. In particular, if perturbations are large enough to substantially change the slow eigenvalues (i.e., the important timescales) of the model, a breakdown in mechanism will follow. Needless to say, methods for reducing transition probability uncertainties in MSMs remain intensely interesting subjects for investigation.

One area in which perturbation might change mechanistic properties resides in the choice of force field to be used in MD simulation. Shaw et. al. (29) have shown that folding mechanism can vary greatly depending on the force field used: in particular, whereas variants of the AMBER force field performed relatively consistently, discrepancies between variants of the CHARMM force field were large. Work on simulating the Fip35 WW domain has also raised questions regarding the mechanistic predictions made with the CHARMM force field (30).

Implications for interpreting biophysical experiments

The perturbation of model parameters has connections to concepts in protein folding biology that would be of general interest to an experimentalist. In particular, are experimental observations about protein folding robust? Although experiments, of course, are not concerned with simulation-specific perturbations like force field and space discretization errors, analogous perturbations in environmental conditions (temperature, pH, salt conditions, sequence mutations, presence of other proteins, etc.) as well as statistical uncertainty need to be considered in an experimental setting. The robustness seen in protein folding simulations predicts a similar robustness in the interpretation of protein folding experiments, with analogous caveats as discussed previously in the context of simulations. This is important for the comparison of simulation to experiment, comparison between experiments, and also for the interpretation of the experimental data, because less robust aspects of the system are most susceptible to small variations in experimental conditions.

Our analysis can shed light on which elements one would expect to be most robust. For instance, measurements of equilibrium properties (e.g., through thermal or chemical melting studies) are subject to perturbations in temperature and salt conditions. Nevertheless, because the properties of interest in these cases are stationary, an experimentalist should be relatively confident that such perturbations have little impact. In experiments that are time-resolved (from nanoseconds to milliseconds) and at the single molecule level, however, conclusions about kinetics and mechanism should be tempered with considerations of perturbative robustness. Elements of mechanism that occur on relatively slow timescales might be trusted with some surety, but, as with MSMs, conclusions based on high-frequency processes must be more closely scrutinized.

Why can one say that protein folding is mechanistically robust?

The title of this work is Protein Folding Is Mechanistically Robust. One might ask the question, given that some parts of the MSM eigenspectrum are not robust: is the title's statement justified?

Not all MSM observables are robust to perturbation. We should note again, however, that the nonrobust elements of MSM observables belong to fast parts of the transition matrix eigenspectrum. As indicated earlier, one's primary interest in studying folding protein mechanisms resides in understanding the important pathways that occur on the folding timescale. The slow eigenvectors contain this long timescale information and thus contain the most important mechanistic information. Because the slow eigenvectors changed very little upon perturbation, we conclude that protein folding mechanisms, in the context of MSMs, are robust.

Even though folding robustness is an important concept for experimentalists to consider, it has few applications to actually solving problems in biology. Rather, it is nonrobustness in systems that fundamentally gives rise to interesting behavior. The rare events that induce significant conformational changes (related, using a sloppiness framework, to the stiff parameters of a model) are the particular focus of many biologists. The question as to which parameters in a model are the stiffest, thus, may prove to be much more interesting than previously thought. Determining which parameters are stiff enough to change a low-frequency eigenvector in an MSM, for instance, could indicate which states are important in inducing a particular population shift. Therefore, changes in these parameters could simulate the binding of a ligand or substrate or some perturbation in the cellular environment. This analysis could have applications in conducting in-depth mechanistic simulations of concepts like allostery, calalysis, and inhibition, topics that truly define contemporary biology.