The ‘new view’ of proteins sees protein reactions as parallel processes occurring along funnelled energy landscapes. These landscapes are generally not smooth, but are superimposed by hills and valleys of different heights and widths leading to roughness on the energy surface. In the present paper, we describe the origins of protein energy landscape roughness, measurements of its scale and its implications.

Energy landscapes

An energy landscape maps out all possible phase space configurations of a system, usually as a plot of the free energy against the parameters that describe the system. For protein folding, parameters may include the end-to-end distance, the number of native contacts, an order parameter that describes the similarity of the structure to the native one and others. For protein binding, one may add the steric fitness of the interacting surfaces and the distribution of charges over them. The size and complexity of proteins renders their energy landscapes multidimensional. An example of a protein energy landscape is given in Figure 1(A). The landscape is limited to three dimensions by our ability to visualize objects. However, not only is it not possible to draw in more than three dimensions, but also it is not currently possible to construct experimentally the full hypersurface of a protein reaction. Nonetheless, we can extract certain global features of energy landscapes and, together with theory, use them to explain observed behaviours in protein reactions.

Funnel-shaped protein energy landscapes

Figure 1
Funnel-shaped protein energy landscapes

The energy landscape is a plot of the internal free energy (F) against reaction parameters (Φ1, Φ2). On the top, the free energy and entropy are high and the number of states (U1, U2…Un) is correspondingly large. The funnel shape of the landscape guides proteins to the low-energy native state (N) at the bottom of the funnel through multiple routes. (A) An idealized smooth energy landscape which would guide a protein reaction at the maximum attainable speed [27]. (B) A corrugated energy landscape which is traversed by multiple hills and valleys, brought about by energetic and topological frustration. The reaction must proceed through these barriers, which attenuate diffusion on the energy surface to a degree that scales exponentially with their height. Based on data by Dill and Chan [9].

Figure 1
Funnel-shaped protein energy landscapes

The energy landscape is a plot of the internal free energy (F) against reaction parameters (Φ1, Φ2). On the top, the free energy and entropy are high and the number of states (U1, U2…Un) is correspondingly large. The funnel shape of the landscape guides proteins to the low-energy native state (N) at the bottom of the funnel through multiple routes. (A) An idealized smooth energy landscape which would guide a protein reaction at the maximum attainable speed [27]. (B) A corrugated energy landscape which is traversed by multiple hills and valleys, brought about by energetic and topological frustration. The reaction must proceed through these barriers, which attenuate diffusion on the energy surface to a degree that scales exponentially with their height. Based on data by Dill and Chan [9].

For example, the modern picture of protein folding and binding, termed the ‘new view’ [1,2], is one of a funnel-shaped energy landscape (Figure 1). This shape is achieved by a concerted decrease of entropy and enthalpy such that the reduction in degrees of freedom (entropy) is continuously compensated for by a comparable gain in stabilizing interactions (enthalpy). The upper part of the funnel represents the unfolded state of a protein or the unbound state of a protein–ligand pair. Molecules in these states posses the highest conformational freedom, thus the number of possible states and, consequently, the number of points on the energy landscape is very large. This is reflected in the large area of the energy surface at high free energies. As the reaction proceeds, more and more constraints are applied, and the number of states available to the proteins decreases. The result is that the radius becomes smaller as the free energy decreases, resulting in the characteristic funnel.

Protein folding proceeds through random thermal fluctuations. Folding thus consists of stochastic migration through conformational states by Brownian motion of the polypeptide chain and can therefore be described as diffusion of the state along the energy surface. However, because of the funnel shape of the energy landscape, diffusion is biased towards the native state. The number of possible paths that connect the unfolded or unbound state to the native state is related to the largest radius in the funnel. As the reaction proceeds, and the radius of the energy surface decreases, the paths converge to the final state. Nevertheless, the number of pathways that can lead to the bottom of the funnel is immense. Unravelling these pathways requires acquisition of trajectories of individual molecules, which in turn entails the use of single molecule techniques.

The origins of roughness

The picture described in Figure 1 is an idealized one. In the vast majority of cases, protein reactions involve competing interactions which cannot all be satisfied at once. When a protein folds, tightly bound water molecules need to be expelled, (stable) non-native contacts may be formed, energetically costly cavities may be generated due to imperfect fit of side-chains and locally favourable secondary structures may need to disassemble in order to form the final tertiary structure. Likewise, during pair formation in binding reactions, energetically favourable interactions with water molecules need to be broken and changes in conformation may be required, sometimes on a large scale. As discussed below, functionality adds additional constraints that are by no means less stringent and which are often contradictory to stability and kinetic foldability. In fact, it is highly unlikely that folding or binding of proteins will satisfy all of the constraints posed on the system optimally and thus the reaction is said to be frustrated. Frustration is a well-known attribute of complex systems such as glasses and social and economic networks. In terms of energy landscapes, frustration manifests itself as a series of hills and valleys of various heights and widths superimposed on the free-energy surface. A rough energy landscape is shown in Figure 1(B). The overall shape of the landscape is very similar to that shown in Figure 1(A): the landscape is funnelled towards the native state, and the overall gradient in free energy, which drives the process, is similar. However, the surface is rough or rugged. Diffusion along it must therefore proceed through multiple barriers, attenuating the speed of the reaction.

The idea that protein energy landscapes are rough is not new. The issue was addressed in theoretical treatments in the 1980s and subsequently [313]. There were also experimental clues for glassy behaviour of proteins, indicative of frustration, in the 1970s and early 1980s [14]. In 2000, Lapidus et al. [15] measured the end-to-end contact formation rate for a series of short flexible polypeptides. They found contact formation rate to be lower than expected for diffusion-limited processes, a discrepancy which they attributed to roughness estimated to be approx. 2 kBT.

Measuring roughness

Although intuitively clear, the effect of roughness on the speed at which protein reactions take place is not simple to quantify. Inspired by this problem, Zwanzig [16] calculated the attenuation of the diffusion coefficient brought about by roughness. In one dimension, he found the effective diffusion coefficient D* to be:

 
formula
(1)

where D is the diffusion coefficient in the absence of roughness, ε is the average roughness, and θ is a parameter that takes into account the distribution of barrier heights and widths. For a Gaussian distribution, θ was found to be equal to 2. Thus even a relatively small value of ε can lead to dramatic attenuation of the diffusion speed. The above relationship also points to a dependence of the effective diffusion coefficient on temperature. The higher the temperature at which a reaction takes place, the closer the effective diffusion coefficient is to that which is obtained in a smooth potential. This is because the higher the temperature is, the more likely a protein is to obtain enough thermal energy to overcome the barriers that traverse the energy surface, thus reducing their effect. Recently, the analysis above was extended to higher dimensions and yielded similar results [17].

Zwanzig's analysis was published 20 years ago [16], at a time when single-molecule experiments, which are vital to extract dynamic properties of protein reactions, were not available. Motivated by the now existing possibility of probing one molecule at a time, Hyeon and Thirumalai [18] have recently extended Zwanzig's theory in order to enable direct derivation of protein energy landscape roughness. The method they directed their theory towards was single-molecule force spectroscopy [19], in which individual protein molecules are unfolded by a mechanical force, f, applied by either optical tweezers or an AFM (atomic force microscope). They found that, when a protein is subjected to a pulling force that is applied at a constant rate, rf = , the most probable force for it to unfold at, f*, is given by:

 
formula
(2)

where Δx is the distance of the transition state from the native state, νD is a parameter which represents the escape rate at zero force and ΔF‡ is the overall energy barrier. Typically, the results derived from single-molecule pulling experiments are collimated into force spectra in which the most probable force for unfolding (or unbinding), f*, is plotted against the logarithm of loading rate, rf. However, it is not straightforward to extract roughness from such plots because some of the parameters on the right side of eqn (2) depend on f*. This difficulty can be overcome by comparing results obtained at different temperatures. As mentioned above, increasing the temperature enhances the ability of proteins to overcome free-energy barriers by increasing their kinetic energy. Thus the force at which unfolding occurs, at a given loading rate, is temperature-dependent. On the other hand, roughness is an inherent property of the energy landscape and does not depend on the temperature or the loading rate used. Therefore, by equating the right side of eqn (2) for different pairs of temperatures and loading rates that give the same value of f*, one can extract roughness (Figure 2).

Extracting roughness from force spectra

Figure 2
Extracting roughness from force spectra

Force spectra consisting of plots of the most probable force for unbinding, f*, as a function of the logarithm of the pulling rate, rf = , at three different temperatures. Data taken from [20]. Two points corresponding to the same value of f* (e.g. 90 pN, as marked by a horizontal line) are chosen, and the pulling rates and temperatures that match these points (indicated by vertical lines with values shown in parentheses) are used in eqn (2), yielding a set of equations which can be solved for the roughness ε.

Figure 2
Extracting roughness from force spectra

Force spectra consisting of plots of the most probable force for unbinding, f*, as a function of the logarithm of the pulling rate, rf = , at three different temperatures. Data taken from [20]. Two points corresponding to the same value of f* (e.g. 90 pN, as marked by a horizontal line) are chosen, and the pulling rates and temperatures that match these points (indicated by vertical lines with values shown in parentheses) are used in eqn (2), yielding a set of equations which can be solved for the roughness ε.

Following this scheme, we measured the energy landscape roughness of the unbinding, rather than unfolding reaction, of two proteins involved in the translocation of protein and RNA molecules through nuclear pore complexes [20]. One of the proteins was the prototypic nuclear import receptor, impβ (importin β). The second was the Ras-like GTPase Ran, which is implicated in many cellular processes including nucleocytoplasmic exchange during interphase, spindle assembly, chromosome congression and segregation during mitosis, as well as in post-mitotic nuclear envelope and pore reformation. The interaction between the two proteins involves complex dynamics and is characterized by slow kinetics, both on and off. This is because impβ, which interacts with multiple structurally distinct ligands, has a flexible structure that can accommodate grossly different conformations. This flexibility transcends to the complex itself, which was shown to alternate between two distinct conformational states whose relative population can be modulated by temperature and applied mechanical force [21], as well as by the Ran-binding protein, RanBP1 [22]. Given the above, it was predicted that the energy surface of the interaction between impβ and Ran is rugged.

To derive roughness, we obtained force spectra at three different temperatures: 7, 20 and 32°C (Figure 2). In addition, we modified eqn (2) to account for temperature-dependent variations in Δx owing to a Hammond effect [20]. νD was determined at each temperature by extrapolation of the force against loading rate curves to zero force, and Δx is obtained from the slope of the force spectra [23]. Using nine different pairs of temperatures and loading rates to determine roughness, we consistently obtained a value of approx. 6 kBT.

Three additional studies of protein energy landscape roughness were conducted, using the above described modified relationship. In the most recent one, the roughness underlying the well-studied streptavidin–biotin interaction was determined by Rico and Moy [24]. The force spectra obtained included two linear regimes corresponding to two main barriers: an inner barrier governing behaviour at high loading rates, and an outer barrier governing measurements at low loading rates. The values of ε obtained were ∼7.5 and ∼5.5 kBT along the outer and inner barriers respectively. The origin of roughness was attributed to competition of solvent water molecules with some of the hydrogen-bonds that stabilize the complex and to the flexibility of the ‘3–4’ loop of streptavidin, which may induce the formation of multiple conformational sub-states in the complex. The authors also proposed that roughness is a significant contributor to the unusually slow dissociation kinetics of the complex and may account for the discrepancies in the unbinding forces measured for this pair.

The first measurement of roughness of a folding reaction was performed on the fourth domain of the actin cross-linking protein filamin [25]. This Ig rod domain differs from the other (Ig) domains of the protein in that it unfolds at significantly lower forces and folds an order of magnitude faster. The authors observed a temperature-dependence of the force spectra which they assigned to a lowering of energy barriers with temperature brought about by a switch from hydrogen-bond-controlled interactions to those controlled by hydrophobic forces. However, the data could be alternatively explained as originating from a roughness of approx. ∼4 kBT in the free-energy surface.

Janovjak, Knaus and Muller [26] derived the energy landscape roughness of single helices of the archaeal light-driven proton pump bacteriorhodopsin. Five of the seven transmembrane α-helices of the protein can be resolved in mechanical unfolding experiments. For these, roughness was found to be between 4 and 6 kBT. The high values obtained, which were unexpected for individual secondary structures, were suggested to reflect topological and curvature-associated frustration imposed by the surrounding lipid bilayers.

Thus the experimental studies conducted so far have revealed significant roughness in the energy landscapes that describe protein folding and binding. By ‘significant’, we mean that the values measured are a few times higher than kBT. Roughness must therefore have strong effects on protein folding and binding. How strong these effects may be can be realized by noting that the predicted speed limit for the folding of small globular proteins in a smooth energy funnel is ∼100/n μs, where n is the number of amino acids in the protein [27]. However, most natural proteins fold at least 2–3 orders of magnitude slower than expected from the above relationship, owing to energy landscape roughness.

Roughness and function

What is a good speed for a protein to fold at? One obvious consideration is that proteins need to be in an active folded state a significant portion of time, so folding must be correspondingly fast. In addition, proteins need to fold fast enough to avoid degradation which is promoted by the unfolded state. These points set a lower bound on acceptable folding speeds. This then raises the question of why roughness has, with its detrimental effect on reaction rates, been preserved by evolution. The answer to this question involves two issues: survival and function. It has been suggested that fast folders, which have smooth energy landscapes, are prone to aggregation and proteolysis since they also unfold very rapidly [28,29]. On the other hand, we note that too rough an energy landscape extends the folding/unfolding processes themselves due to the attenuation of diffusion along the surface, subjecting the protein to similar risks. We thus suggest that, in order for these two conflicting requirements to be simultaneously satisfied, an optimum energy landscape roughness must exist for each protein.

One should bear in mind that proteins are made to perform a task and not to fold as fast as possible (for an excellent discussion of this point please see the review by Gruebele [29]), and many times functional groups contribute to roughness. For example, polar or charged residues are often placed inside the hydrophobic core of proteins, where they serve in ligand exchange or catalysis. This, of course, is energetically unfavoured and thus leads to frustration and roughness. The same holds true for buried water molecules and prosthetic groups. Another example is loops. Long flexible loops come at an expensive entropic cost because they are able to accommodate a large number of conformations. Yet their flexibility makes them very useful for binding, in which they are frequently found to play a pivotal role. There have been attempts to engineer frustrating motifs out of proteins, decreasing roughness and speeding up folding, but this has turned out to be at the expense of function [2931]. Functional constraints may thus not be easy to remove, and the reactions must proceed at a pace which is slowed down relative to that expected from smooth funnels.

Function is not limited to the ability to perform a single task optimally. In a dynamic biological environment, function also requires diversity and adaptability on many time scales. In other complex systems, and most commonly in economic systems, it has been known for some time that frustration leads to rich dynamics. A well known example of this is given in the minority game theory where the inability to simultaneously satisfy all constraints leads to fluctuations in the state of a system around some value [32]. For proteins, frustration which corrugates energy landscapes transcends into the ability to continuously explore many states that may be quite distinct from each other in structure, but nonetheless have similar free energy. This ability, in turn, is crucial for many functions such as movement, binding and catalysis, all of which involve transitions between states or sub-states. The ability to accommodate multiple states also increases the repertoire of molecules with which a given protein can interact and provides robustness against deleterious mutations. The tendency to fluctuate continuously between semi-stable states can also enhance adaptability both on short time scales, ensuring a fast response to variations in cellular demands, and, on much longer time scales, facilitating the acquisition of new traits during evolution.

Thus, although smooth energy landscapes enable fast folding, evolution seems to generally prefer proteins with rough energy surfaces to optimize folding and unfolding speeds and to allow for function and adaptability, as well as for their own self-propagation.

2nd International Meeting on Molecular Perspectives on Protein–Protein Interactions: Independent meeting held at Hotel Croatia, Dubrovnic, Croatia, 27 June–1 July 2008. Organized and Edited by Colin Kleanthous (York, U.K.), Jacob Piehler (Frankfurt, Germany) and Gideon Schreiber (Weizmann Institute, Rehovot, Israel).

Abbreviations

     
  • impβ

    importin β

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