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Conceived and designed the experiments: SVK. Performed the experiments: SVK. Analyzed the data: SVK. Contributed reagents/materials/analysis tools: SVK. Wrote the paper: SVK.

The author has declared that no competing interests exist.

Protein folding dynamics is often described as diffusion on a free energy surface considered as a function of one or few reaction coordinates. However, a growing number of experiments and models show that, when projected onto a reaction coordinate, protein dynamics is sub-diffusive. This raises the question as to whether the conventionally used diffusive description of the dynamics is adequate. Here, we numerically construct the optimum reaction coordinate for a long equilibrium folding trajectory of a Go model of a

To understand dynamics of complex systems with many degrees of freedom, one often projects it onto one or several collective variables. Protein folding, the complex, concerted motion of a protein chain towards a unique three-dimensional structure, is one example of where such reduction of complexity is useful. It is usually assumed that the projected dynamics is diffusive. However, many experiments and simulations have shown that the projected dynamics is sub-diffusive, i.e., the mean square displacement grows slower than linear with time. It means that the dynamics has a memory; that the free energy surface together with diffusion coefficient do not properly define the dynamics; and that such projections cannot be used to accurately describe dynamics. Here, we show that if one carefully constructs the reaction coordinate by optimizing (maximizing) its free energy profile, one can use a simple (memory-less) diffusive description. Loosely speaking, when the complex dynamics is projected onto a simple coordinate, all the complexity of the original dynamics goes into the memory of the projected dynamics. If the dynamics is projected onto the (complex) optimum reaction coordinate, all the complexity of the original dynamics is in the reaction coordinate, and the projected dynamics is simple.

A free energy surface (FES) projected onto one or a small number of coordinates is often used to describe the equilibrium and kinetic properties of complex systems with a very large number (100 to 1,000 or more) of degrees of freedom. Studies of protein folding are an important case where this type of projected surface has been introduced and coordinates such as the number of native contacts and radius of gyration have been used

Employing the Mori-Zwanzig formalism

The equilibrium folding dynamics of the

Optimum one-dimensional reaction coordinates are constructed by numerically optimizing the mean first passage time to the native basin for a sufficiently broadly chosen functional form of a reaction coordinate (see

The first coordinate (

Pluses are for unoptimized

Black and red lines show the free energy profiles along the

The scaling exponent

The second coordinate is a linear combination of all interatom distances

The sampling intervals are

The sampling intervals are

The partition function of the cut based free energy profiles

We assume here that

However,

The two optimized reaction coordinates, while having very different functional forms, show very similar behavior (at the TS regions), e.g. the width and the height of the TS barrier is the same (

The analysis suggests that the higher is the free energy profile the closer is dynamics to diffusive. Evidently, the most optimal reaction coordinate is the one which has its free energy highest for every value of reaction coordinate. Consider invariant parametrization of reaction coordinate, namely the partition function of the configuration space from the initial value to the position x

To illustrate that the results presented are robust with respect to particular choice of the protein or the interaction potential, a protein with different secondary structure content (

Low free energy barrier

The analysis above just considers the dynamics around the transition state, i.e., at the top of the free energy barrier. The conclusion that the higher the free energy profile the closer the dynamics to diffusive is likely to be valid in general, e.g., for the barrier-less folding proteins. The quantitative analysis exploits the fact that at the very large sampling intervals, when the system flies ballistically over the barrier, the two free energy profiles for optimal and sub-optimal reaction coordinates are very similar, because the two coordinates distinguish equally well between the basins. It can be extended to the following general qualitative argument. The two sufficiently good reaction coordinates likely differ significantly only at relatively small spatial scales with the large scale description of the dynamics being very similar. As the sampling interval

The model of the protein employed in the analysis is relatively simple, thus allows for extensive simulation with large number of folding-unfolding events. More realistic simulation of protein folding would include explicit representation of solvent configuration degrees of freedom. The dynamics projected on the optimum reaction coordinate constructed by considering only protein degrees of freedom might be sub-diffusive because neglected solvent degrees of freedom could be important.

The analysis suggests that without specifying the reaction coordinate, the question why the dynamics is sub-diffusive is rather ill-posed. It is more appropriate to ask: is it possible, for a given trajectory, to construct the optimum reaction coordinate, so that the projected dynamics is diffusive?

In conclusion, we have shown that dynamics projected onto a reaction coordinate can be diffusive or sub-diffusive depending on the coordinate employed for the projection. If one has a flexibility in choosing the reaction coordinate, e.g. when describing protein folding, dynamics can be made diffusive (or close to it) by optimizing the reaction coordinate (making

The conventional way to construct the FEP, given the projection of a trajectory onto a reaction coordinate (the time-series of the value of the reaction coordinate)

A reaction coordinate (x) with a variable diffusion coefficient can be transformed to coordinate (y), called the natural coordinate

Other approaches have been suggested to characterize diffusive dynamics by computing the free energy profile together with the coordinate dependent diffusion coefficient

It is reasonable to assume that any “bad” choice of reaction coordinate, when different parts of the configuration space overlaps at projection onto this coordinate, will result in faster kinetics, i.e. in a smaller mean first passage time (mfpt). Clearly, the longest mfpt is obtained on the original FES or from a projection where no such overlapping occurs. Hence, we define the optimum reaction coordinate as the one that has the longest mfpt, which can be computed by Kramer's equation

The optimum reaction coordinates are constructed by numerically optimizing the mfpt functional for a sufficiently broadly chosen functional form of reaction coordinate. Starting with the initial set of parameters, which are sufficient to distinguish between the two free energy basins, the coordinate is iteratively improved by changing parameters and accepting the change if mfpt is increased. For the first reaction coordinate

For subdiffusion, the mean absolute displacement no longer scales as

Supporting information for “Is Protein Folding Sub-Diffusive?”.

(0.08 MB PDF)

I am grateful to Emanuele Paci for providing trajectories of the Go model simulations.