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Decoding How Proteins Fold

Submitted:

14 March 2026

Posted:

17 March 2026

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Abstract
One of the most puzzling and unsolved challenges in molecular biology is understanding how proteins fold. Despite having advanced predictive tools that can accurately estimate the native structures of proteins, we still lack a comprehensive model that explains how amino acid sequences dictate folding pathways and trajectories. This manuscript introduces a novel treatment for the issue by employing the “principle of least action.” This approach enables us to explore an intriguing question: how does a protein achieve its native state at a constant folding rate and within a biologically plausible time frame? A response to this inquiry will help us understand why proteins must fold along specific pathways and identify the boundary conditions that limit their availability. Additionally, the principle of least action—together with the effective trajectory conjecture—enables us to explain why different proteins could exhibit the same folding rate. Finally, it will enable us to provide an in-depth description of the genesis and solution of Levinthal's paradox. Our results are expected to pave the way for a more profound understanding of how proteins fold, shedding light on how the amino acid sequence and its surrounding environment encode the protein's folding pathways and, consequently, the protein's three-dimensional structure.
Keywords: 
protein folding
;  
effective trajectory
;  
folding pathways
;  
folding trajectories
;  
least action principle
;  
Levinthal paradox
;  
folding rate
;  
Anfinsen dogma
Subject: 
Biology and Life Sciences  -   Biophysics

Introduction

The folding pathways problem originates from Levinthal (1968), who posed a paradox (Levinthal, 1969) bearing his name: how can a protein attain its natural state within a biologically feasible timeframe while randomly exploring the whole conformational space? A proper answer to this query is of foremost importance since protein folds from milliseconds to seconds rather than (~1052) years—as foreseen by an exhaustive enumeration of all possible conformations for a 100-residue protein (Karplus, 2011). As the reader must be aware, several possible solutions to this apparent contradiction—also known as Levinthal’s paradox—exist in the literature (Zwanzig et al., 1992; Karplus, 1997; Finkelstein & Badretdinov, 1977; Dill & Chan, 1997; Bogatyreva & Finkelstein, 2001; Rooman et al., 2002; Garbuzynskiy et al., 2012; Ben-Naim, 2012; Finkelstein & Garbuzynskiy, 2013; Martinez, 2014; Ivankov & Finkelstein, 2020; Vila, 2023a). Although we will not revisit each possible solution here, it is worth noting that solving this paradox is relevant because it could enable us to determine clear answers to the following key questions: Why and how can proteins reach their native state in a biologically reasonable time? Regarding the ‘why,’ we demonstrated—from a thermodynamic viewpoint—that the range of ‘slowest’ folding times for two-state monomeric proteins (τ) spans from ~milliseconds to ~seconds (Vila, 2023a), which closely aligns with the observed data (Garbuzynskiy et al., 2012; Ivankov & Finkelstein, 2020). These time scales for protein folding kinetics arise from the fulfillment of the thermodynamic hypothesis (Anfinsen, 1973), also known as the Anfinsen dogma, which allowed us to demonstrate—through a statistical-thermodynamics analysis (Vila, 2019)—the existence of an upper limit to the marginal stability of globular proteins (ΔG) of approximately 7.4 kcal/mol (Vila, 2022), beyond which proteins either unfold or lose their functionality (Vila, 2021). This threshold applies to any fold class, sequence, or protein size, and it is significant because it arises from a quasi-equilibrium of forces that occurs at the lowest accessible protein's free-energy minimum (Martin & Vila, 2020). As to the ‘how’—which mainly focuses on clarifying the mechanisms by which proteins achieve their native states, including the pathways and trajectories of folding—it is a query that remains largely unresolved, despite the extensive literature available on the subject (Anfinsen & Scheraga, 1975; Creighton, 1985; Šali et al., 1994; Thirumalai, 1995; Baldwin, 1994; Baldwin, 1995; Wolynes et al., 1995; Lazaridis & Karplus, 1997; Dill & Chan, 1997; Pande et al., 1998; Dobson et al., 1998; Honig, 1999; Bakk et al., 2000; Englander et al., 2007; Karplus, 2007; Karplus, 2011; Englander & Mayne, 2014; Wolynes, 2015; Neupane et al., 2016; Lapidus, 2021; Kikuchi, 2022; Zhao et al., 2023; Fakhoury et al., 2023; Zhao et al., 2024; Chang & Perez, 2025). For a more profound understanding of the complexity of this problem, we should keep in mind that the protein folding process cannot be represented in a single dimension; rather, it results from interactions across multiple dimensions, creating complex multidimensional pathways and trajectories. This feature highlights the importance of understanding the interplay of complex forces at work, such as van der Waals interactions, hydrogen bonds, solvent effects, and electrostatic forces, all of which influence obtaining an accurate solution to the protein folding problem in a manner that demands solving it as an ‘analytical whole’ rather than a sum of parts (Vila, 2023b).
It is essential to acknowledge two key points prior to moving forward. Firstly, the terms ‘pathway’ and ‘trajectory’ represent distinct concepts. The distinction between pathways and trajectories is that pathways consist of discrete sequences of events—whether parallel or linear—to go from point A (terminus a quo) to point B (terminus ad quem). In contrast, trajectories encompass the entire duration of the process (Vila, 2025). Secondly, it is well-documented experimentally that proteins can reliably attain the folded state at the same rate in a proper environment (Harrington & Schellman, 1956; Gromiha, 2005; Huang et al., 2008; Finkelstein & Garbuzynskiy, 2013), thereby suggesting that all folding pathways—if multiple exist—may be characterized by an effective trajectory. To gain a better grasp of this conjecture, we will analyze it from the standpoint of the “least action principle,” a fundamental physical principle (Fee, 1941) and a powerful model for describing how systems behave under the constraints mentioned above (Feynman et al., 1963; Hanc & Taylor, 2004). This principle's significance encompasses all domains of physics, rendering it the nearest approximation to a theory of everything, as it pertains to classical mechanics (Landau & Lifshitz, 1982), relativity (Landau & Lifshitz, 1987), quantum mechanics (Feynman, 2005), thermodynamics (Kaila & Annila, 2008), and biophysics (Simmons & Weiner, 2013). This approach will allow us to further explore the more probable pathways that proteins may follow to achieve their folded or unfolded states within a biologically reasonable timeframe. This endeavor also aims to tackle the Levinthal paradox from a fresh perspective, distinguishing the current study from a previous resolution (Vila, 2023a). Furthermore, we seek to uncover the basic physical principles that dictate how proteins fold in response to alterations in their environment or sequence.

Outlining the System Being Studied

The existence of metamorphic proteins (Murzin, 2008) indicates that the role of environmental factors cannot be overstated (Vila, 2020), as even slight changes in conditions can lead to dramatically different folding outcomes (Dishman & Volkman, 2018). This viewpoint emphasizes that the dynamic nature of proteins means their behavior is not solely determined by intrinsic properties—encoded in their amino acid sequence (Anfinsen, 1973)—but also heavily influenced by external stimuli. This complexity highlights the necessity of studying protein behavior under varied conditions to understand the principles of folding and stability fully. A single-point change in the amino acid sequence—such as due to mutations—only exacerbates this problem by giving rise, among other things, to epistatic effects (Vila, 2024a).
In summary, based on the considerations mentioned above, we will now focus on analyzing the folding of a single two-state protein in a fixed environmental setting. The analysis of two-state proteins (Jackson, 1998) is chosen for a specific reason: the native state and unfolded states are separated by an energetic barrier that exceeds the energy of thermal fluctuations (Akmal & Muñoz, 2004; Kuwajima, 2020). In other words, the folded and unfolded states are separated by a collection of high-energy native-like structures known as the Transition State Ensemble (TSE), which represents (as shown in Figure 1) the energetic barrier (ΔG) for the process (Matouschek et al., 1989; Itzhaki et al., 1995; Englander, 2000; Ding et al., 2002; Vendruscolo et al., 2003; Akmal & Muñoz, 2004; Shakhnovich, 2006; Englander & Mayne, 2014; Lange et al., 2008; Kuwajima, 2020; Stiller et al., 2022; Li & Gong, 2022). In this simple folding model, there are no stable intermediate states necessary to complete the process. Therefore, the treatment of protein folding and unfolding can be viewed as interchangeable processes (Lazaridis & Karplus, 1997; Jackson, 1998; Ivankov & Finkelstein, 2020)—as long as the Gibbs free-energy barrier (ΔG) remains the same for both cases. The folding and unfolding data from 108 proteins, of which 70 display two-state kinetics (Glyakina & Galzitskaya, 2020), support these claims, as the logarithm of the folding and unfolding rates demonstrates a strong correlation (R ~0.8). Henceforth, these terms shall be considered synonymous from this point forward.

The Least Action Principle and the Effective-Trajectory Conjecture

For a comprehensive and detailed exploration of the application of the principle of least action in classical mechanics, thermodynamics, relativity, and quantum mechanics, we recommend consulting the existing literature (Landau & Lifshitz, 1982; Landau & Lifshitz, 1987; Feynman, 2005; Kaila & Annila, 2008). We will skip over all those specific uses and instead provide, firstly, a layman's definition of the concept of least action and, later, a more concise definition of the principle from a physics-mathematics perspective.
The principle of least action in physics states that a system going from a starting to an ending point—such as the folding process of proteins—will naturally follow a path that minimizes the ‘action’ (a functional of the trajectory) among all feasible pathways connecting these two points. The implication here is that nature operates with optimal efficiency, rather than randomly searching among all feasible pathways.
Let us now focus on a brief description of the principle of least action within the framework of Lagrangian mechanics—an approach that enables us to frame the system using a function encompassing both kinetic and potential energy contributions (Feynman et al., 1963; Hanc & Taylor, 2004). Let us begin by characterizing the folding of a protein as a process that occurs in a high-dimensional dihedral-angle space, which could be represented by a smooth, continuous function: f   ( { φ , ψ , ω , χ } , t )—where { φ , ψ , ω , χ } denotes a set of internal dihedral angles of the protein at time t. This function determines how the protein conformation evolves along a given folding pathway under a given all-atom force field—which, in simple terms, refers to all the lines of force around and between atoms—such as the empirical conformational energy program for peptides (Arnautova et al., 2006). Additionally, we can define the time derivative of this function [ f ˙ { φ , ψ , χ } , t ], which identifies the folding velocity along a specified path that must be traversed within a well-defined range of folding rates (τ). Specifically, it must satisfy the following boundary constraint: ~milliseconds < τ < ~seconds (Vila, 2023a)—as per an analysis grounded in transition state theory (Ivankov & Finkelstein, 2020)—with the folding rate calculated as τ = τ0 exp (βΔG). Here, ΔG represents the protein's marginal stability (Vila, 2022), namely, the Gibbs free energy difference between the native state and the highest-energy native-like structures—of the transition state ensemble (TSE)—coexistent with it (see Figure 1); τ0 is a pre-exponential factor that indicates the folding speed limit of two-state proteins—also known as the barrier-less limit (Zana, 1975; McCammon, 1996; Mayor et al., 2000; Muñoz & Cerminara, 2016; Eaton, 2021)—and β = 1/RT, with R representing the gas constant and T being the absolute temperature. After providing some details of the folding process, the action (S), a functional of the trajectory, can be defined as the integral of the Lagrangian ( L ) along any folding pathway, namely as:
S = ∫ τ 0 τ L [ f φ , ψ , ω , χ , t ,   f ˙ φ , ψ , ω , χ , t ]   d t
The path integral spans the entire duration of the folding process, subject to two key constraints. First, the upper limit of the integration (t) must be less than a few seconds (Vila, 2023a). Second, the lower limit of the integration (t0) should fall between ten nanoseconds and ten microseconds, as this range represents the folding speed limit for two-state proteins (Zana, 1975; McCammon, 1996; Mayor et al., 2000; Muñoz & Cerminara, 2016; Eaton, 2021). It is important to highlight, on one hand, that the cap on the upper-bound limit of the integration (t)—applicable to any two-state protein, irrespective of its fold class, length, or sequence—has been demonstrated to stem from the validity of the thermodynamic hypothesis, also known as Anfinsen's dogma (Vila, 2023a). On the other hand, this constraint on the upper bound also provides explanations for why protein folding consistently occurs within a biologically reasonable time frame. In summary, the validity of the Anfinsen dogma delineates an integration limit that constrains viable folding pathways, as their trajectories are ruled by temporal constraints.
The task of solving equation (1) presents an enormous mathematical challenge that exceeds the current analysis's limits, as it requires solving—among other things—the Euler-Lagrange equations [ d d t d L d f ˙ −   d L d f = 0 ] that represent the solution to virtual variations on the functional S (Coopersmith, 2017); instead, we will focus on highlighting its main features, aiming to clarify the potential relationship between the effective-trajectory conjecture and the least action principle. To achieve this, it is important to note that the Hamilton principle (of stationary action) states that the variations (δ) that make the integral zero (δS = 0), by infinitesimal virtual variation on f, correspond to a stationary point of S—where the functional must be at a minimum or a saddle point, but not a maximum. All things considered, a pathway is considered a stationary point of the action (S) if—and only if—the Euler-Lagrange equations are satisfied (Coopersmith, 2017). Hence, protein folds by following a pathway that renders the action (S) stationary. However, how can we guarantee that the action has a unique path that makes it stationary? The latter represents an important question, as the number of folding pathways—specifically, whether there is one, a few, or a countless number—has been, and continues to be, a topic of debate (Baldwin, 1994; Baldwin, 1995; Lazaridis & Karplus, 1997; Dill & Chan, 1997; Bakk et al., 2000; Rooman et al., 2002; Englander & Mayne, 2014; Eaton & Wolynes, 2017; Englander & Mayne, 2017). Then, what if numerous pathways exist, each represented by trajectories that share the same folding rate (τ) as their sole common feature? This viewpoint associates the uniqueness of the solution to Eq. (1) with the time necessary for the protein to navigate the most efficient pathways—those that render the action (S) stationary—rather than suggesting the existence of a specific path (refer to Figure 2). This leads to the conjecture of an ‘effective trajectory,’ which encompasses all folding pathways characterized by a common single folding rate (τsingle), depending on the specific conditions described below. Firstly, the folding process must take place in a proper and controlled environment, which includes factors such as pH, temperature, and ionic strength, among others. Secondly, the single-folding rate must satisfy the following constraint: τsingle < ~seconds, as explained earlier.
The proposed effective-trajectory conjecture is based on a crucial observation: proteins unfold at a single-folding rate determined by DG (Vila, 2023a), which defines the protein's marginal stability while also serving as a threshold beyond which a protein may unfold or lose its functionality (refer to Figure 1). This scenario offers a simple and credible explanation for how multiple pathways may converge on the protein native state at the same rate (see Figure 2)—even in the presence of a degenerate Gibbs free energy minimum (see Figure 3), a possibility that cannot be ruled out (Vila et al., 2003; Martin et al., 2019). It is important to recognize that different proteins—whether they have varying sequences, differing numbers of residues, or both—can exhibit identical folding rates. Proteins with Protein Data Bank identifiers 1PBA, 1AYE, and 2VIK illustrate this phenomenon (Huang et al., 2008). At first glance, it may appear odd that such proteins can exhibit identical folding rates while possessing distinct folding pathways. However, upon recognizing two pivotal facts, this phenomenon becomes readily comprehensible. Firstly, the folding rates are limited—as previously mentioned—by the protein's marginal stability—regardless of fold class, sequence, or protein size (Vila, 2023a). This proposal is consistent with evidence suggesting that folding rates are insensitive to a protein's sequence details (Plaxco et al., 2000). Paths and trajectories are not synonymous, as previously noted. A path refers to a sequence of parallel or linear events linking the starting and ending points of a process, such as protein unfolding. On the other hand, a trajectory accounts for the time required for the protein to traverse that path, thereby defining the folding rate. The latter clarification is of paramount importance, as the thermodynamic hypothesis (1973), proposes that the amino acid sequence determines, in a proper environment, a protein's unique three-dimensional structure—a statement that remains valid even for metamorphic proteins (Vila, 2020)—regardless of how long it takes for the protein to fold or unfold. In other words, the amino acid sequence and its surrounding environment encode its folding pathways and consequently the protein's structure.
All in all, for more than half a century, chemists, biologists, mathematicians, and physicists have been unable to solve the protein folding problem analytically (Vila, 2023b). Consequently, as an alternative, we have employed a heuristic argument to both demonstrate the physical plausibility of the effective-trajectory conjecture and to illustrate the factors controlling the feasible folding pathways in a more intuitive manner, i.e., without relying on exhaustive mathematical rigor. The upcoming section will examine the validity range of the proposed solution, its relation to the Levinthal paradox, and its implications on critical issues in structural and evolutionary biology.

Avenues for Future Research

Knowledge of the single folding rate (τsingle) that characterizes the effective-trajectory solution of Eq. (1)—for a given environment—is crucial in practical applications. For example, it allows researchers to more precisely forecast changes in protein stability upon varying initial conditions (Vila, 2024b), such as those arising from single-point mutations (Vila, 2022). This analysis does not undermine the significance of understanding the pathway a protein could take, as this knowledge offers vital information regarding the key factors affecting protein evolvability (Vila, 2025). Unfortunately, as already mentioned, resolving this issue requires an analytical solution to Eq. (1), which—as previously indicated (Vila, 2023b)—presents a major unsolved challenge. This nuanced comprehension highlights the complexity of protein folding dynamics, suggesting that while multiple pathways might exist, all of them ultimately converge at a common destination: the native state, which must be attained at a given single folding rate (τsingle)—as determined by ΔGsingle (as shown in Figure 2). This convergence strengthens the idea that the folding process can be simplified—for practical applications—into an ‘effective trajectory.’ The latter aligns with the thorough analysis of the unfolding pathways of chymotrypsin inhibitor 2 conducted by Lazaridis and Karplus (1997). Their simulations indicate that a “… statistically preferred unfolding pathway…” is feasible, providing evidence that supports the proposed single effective-trajectory conjecture.
The whole analysis enables us, on the one hand, to bridge the effective-trajectory conjecture—originating from the validity of the least action principle—with the phenomenon of multiple pathways. This is a critical point, as it does not contradict the ‘new view’ of protein folding (Baldwin, 1994; Lazaridis & Karplus, 1997), which posits that there are multiple paths to the native state. On the other hand, it solves the Levinthal paradox (regarding the folding search timescale), as only those pathways exhibiting a common single folding rate (τsingle)—as provided by the least action principle—are allowed during the conformational search. This hypothesis dismantles the idea that the conformational search for the native state could be random. From this perspective, the paradox vanishes because each feasible protein folding pathway represents a solution.
We have thus far examined how proteins attain their native states regarding pathways and trajectories, but we have not addressed the mechanisms underlying that conformational search—such as the ‘funnel model’ (Dill & Chan, 1997) or the ‘foldon model’ (Bai et al., 1995)—as this falls outside the primary focus of our research. It is critical, however, to recognize these mechanisms' significance for comprehending a wide range of biological processes and illnesses. Thus, studying how these mechanisms affect protein pathways and trajectories may improve our understanding of protein folding and misfolding, which should be considered in future research.
Overall, future research in structural and evolutionary biology should prioritize solving Eq. (1). By solving this equation, researchers will gain crucial information that deepens our understanding of the main factors influencing protein folding and misfolding, as well as the feasible folding pathways. This knowledge will enable us to precisely and accurately determine how the amino acid sequence encodes its folding and, hence, which mechanism better represents the protein folding process.

Conclusions

This work yields three principal conclusions. Firstly, we have been able to provide a reasonable answer to a critical question in evolutionary biology: how a protein always attains—through a single or multiple folding pathways—its native state within a biologically acceptable timeframe. We have used the ‘least action principle’ as a physical framework to rationalize this process, i.e., to disclose why proteins must fold by following specific pathways—those that make the ‘action’ stationary. All these specific pathways have in common a unique folding rate determined mainly by the protein's marginal stability (ΔG)—which ultimately is set by the equilibrium of native and native-like conformations populating the transition state ensemble. This indicates that if multiple pathways are present, they must ‘all’ be characterized by the same ΔG to ensure proteins fold at the most efficient rate, regardless of which path nature chooses to reach the native state. This proposal implies that proteins will achieve the native state within a biologically acceptable timeframe. The latter is assured by the validity of the Anfinsen dogma which determines a constraint for the slowest single-folding rate (τsingle < ~seconds). In general, the principle of least action should be recognized as a fundamental physical law that allows us to understand how proteins fold in accordance with rigorous physical-mathematical rules and specific boundary conditions—such as limiting the folding pathways to those that render the ‘action’ stationary and whose slowest folding rate is capped at a few seconds. Secondly, we concluded, after analyzing how proteins fold from the perspective of the principle of least action, that the Anfinsen dogma—“…that is, that the native conformation is determined by the totality of interatomic interactions and hence by the amino acid sequence, in a given environment (Anfinsen, 1973)”—should be rewritten as follows: “The amino acid sequence determines—in a proper environment—their folding pathways and, hence, their native conformation.” This rewriting of the Anfinsen dogma highlights the link that exists between the amino acid sequence and the native structure. The link specifically refers to the folding pathways, which are determined by all interatomic interactions, in accordance with the principle of least action (see Eq. 1). Additionally, it emphasizes that the amino acid sequence dictates the protein's native state rather than the latter being determined by the former. At first glance, the difference may seem subtle, but it is significant: the amino acid sequence alone is sufficient to determine—in a proper environment—the lowest accessible free-energy conformation in solution (also known as the protein native state). The analysis also shows us that all feasible folding trajectories share a common feature: the time required to traverse any feasible folding path—represented by the effective folding trajectory—must align with the protein-folding rate. This folding rate, by definition, may be the same for proteins with different sequences, lengths, or both, as it is primarily determined by the protein's marginal stability. Finally, our analysis, grounded in the principle of least action, indicates that neither the question of whether ‘there are pathways for protein folding’ (Levinthal, 1968) should have been viewed as a challenge nor the timescale for conformational search (Levinthal, 1969) as a paradox. It is important to note that the principle of least action, introduced by Maupertuis (Fee, 1941), predates Levinthal's query by more than 200 years. Therefore, had researchers considered this principle from the outset, it might have prompted them to propose—despite their limited understanding of the forces at play in the folding process at that time—that nature tends to pursue the most efficient pathways for folding rather than exploring them randomly, as suggested by some early attempts to tackle the protein folding problem. In conclusion, it is crucial to underscore the significance of the principle of least action for understanding both the genesis of the Levinthal paradox and its solution. At this point, it is worth noting that the real challenge is understanding the mechanisms that drive proteins to fold into their native states, and this question remains unanswered after more than 50 years of research in structural biology. The reason for these difficulties is that the central question of the protein folding problem remains unresolved: specifically, “how a sequence of amino acids—in a proper environment—encodes its folding pathways.” However, advancements in the state of the art of computational modeling, simulations, and experimental techniques are gradually shedding light on what these pathways could be, providing optimism for potential breakthroughs in our understanding of such essential biological processes.

Founding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Ethical approval

Not applicable.

Acknowledgments

The author acknowledges support from the Institute of Applied Mathematics San Luis (IMASL), the National University of San Luis (UNSL), and the National Research Council of Argentina (CONICET). I want to extend my gratitude to Pablo Garay for his assistance in preparing Figure 3.

Conflicts of Interest

The author declares no competing interest.

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Figure 1. The Gibbs free-energy profile for the unfolding of a two-state protein is presented in an oversimplified format. The native state is illustrated by a green-ribbon diagram of a protein, while the highest point of the free-energy profile represents the Transition States Ensemble—a high-energy set of structures that coexist in fast dynamic equilibrium with the native state (McCammon et al., 1977; Vendruscolo et al., 2003; Lange et al., 2008). The Gibbs free-energy difference between these two states is represented by ΔG, indicating the protein's marginal stability (Martin & Vila, 2020)—a point beyond which the protein unfolds or becomes nonfunctional. The unfolding rate (τ) for a two-state protein—derived from first-principles calculations (Vila, 2023a)—is inset in the figure. For additional details, please consult the main text.
Figure 1. The Gibbs free-energy profile for the unfolding of a two-state protein is presented in an oversimplified format. The native state is illustrated by a green-ribbon diagram of a protein, while the highest point of the free-energy profile represents the Transition States Ensemble—a high-energy set of structures that coexist in fast dynamic equilibrium with the native state (McCammon et al., 1977; Vendruscolo et al., 2003; Lange et al., 2008). The Gibbs free-energy difference between these two states is represented by ΔG, indicating the protein's marginal stability (Martin & Vila, 2020)—a point beyond which the protein unfolds or becomes nonfunctional. The unfolding rate (τ) for a two-state protein—derived from first-principles calculations (Vila, 2023a)—is inset in the figure. For additional details, please consult the main text.
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Figure 2. Panel (a) illustrates some pathways, among the many possible unfolding routes, in a cartoon format. Thick (black and magenta) lines depict these pathways, which begin at the native state and conclude at the unfolded states. The speed at which each pathway is traversed—during the unfolding process— determines the unfolding rate (τ) for the corresponding trajectory, which is given by τx = τ0 exp (βΔGx), with x = 1 to 10. The figure also shows—in panel (a)—an arbitrary distribution of the Gibbs free-energy differences (ΔGx) associated with each pathway. Three of the ten trajectories, namely those that align with a given Gibbs free energy (ΔGsingle) value—defined by the folding rate resulting from a pathway that complied with the principle of least action—are highlighted in magenta (see main text for further details). The chosen arbitrary distribution for ΔG's in panel (a) mirrors the group of higher-energy-native-like structures populating the transition states ensemble (TSE)—as shown in panel (b).
Figure 2. Panel (a) illustrates some pathways, among the many possible unfolding routes, in a cartoon format. Thick (black and magenta) lines depict these pathways, which begin at the native state and conclude at the unfolded states. The speed at which each pathway is traversed—during the unfolding process— determines the unfolding rate (τ) for the corresponding trajectory, which is given by τx = τ0 exp (βΔGx), with x = 1 to 10. The figure also shows—in panel (a)—an arbitrary distribution of the Gibbs free-energy differences (ΔGx) associated with each pathway. Three of the ten trajectories, namely those that align with a given Gibbs free energy (ΔGsingle) value—defined by the folding rate resulting from a pathway that complied with the principle of least action—are highlighted in magenta (see main text for further details). The chosen arbitrary distribution for ΔG's in panel (a) mirrors the group of higher-energy-native-like structures populating the transition states ensemble (TSE)—as shown in panel (b).
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Figure 3. Panel (a) illustrates the Gibbs free-energy landscape for a two-state protein in a cartoon format. The lowest accessible free-energy minimum associated with the protein's native state is denoted as N. Panel (b) illustrates the scenario in which the lowest accessible free energy minimum is degenerate—a possibility that cannot be dismissed (Vila et al., 2003; Martin et al., 2019). In both panels, the green-filled dots highlight an arbitrarily selected starting point, and the dashed yellow lines delineate potential pathways ending at the native state (N).
Figure 3. Panel (a) illustrates the Gibbs free-energy landscape for a two-state protein in a cartoon format. The lowest accessible free-energy minimum associated with the protein's native state is denoted as N. Panel (b) illustrates the scenario in which the lowest accessible free energy minimum is degenerate—a possibility that cannot be dismissed (Vila et al., 2003; Martin et al., 2019). In both panels, the green-filled dots highlight an arbitrarily selected starting point, and the dashed yellow lines delineate potential pathways ending at the native state (N).
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