Modulation of a Protein Free-Energy Landscape by Circular Permutation

Circular permutations usually retain the native structure and function of a protein while inevitably perturbing its folding dynamics. By using simulations with a structure-based model and a rigorous methodology to determine free-energy surfaces from trajectories, we evaluate the effect of a circular permutation on the free-energy landscape of the protein T4 lysozyme. We observe changes which, although subtle, largely affect the cooperativity between the two subdomains. Such a change in cooperativity has been previously experimentally observed and recently also characterized using single molecule optical tweezers and the Crooks relation. The free-energy landscapes show that both the wild type and circular permutant have an on-pathway intermediate, previously experimentally characterized, in which one of the subdomains is completely formed. The landscapes, however, differ in the position of the rate-limiting step for folding, which occurs before the intermediate in the wild type and after in the circular permutant. This shift of transition state explains the observed change in the cooperativity. The underlying free-energy landscape thus provides a microscopic description of the folding dynamics and the connection between circular permutation and the loss of cooperativity experimentally observed.


Generation of folding trajectories
The reference structure for WT*T4L it the 1.25 Å resolution crystal structure 3DKE in the PDB.
The reference structure for CP13*T4L has been obtained by introducing a peptide bond between Both structure have been subjected to energy minimazation (2000 steepest descent) using a united atom model with implicit solvent [3].
Folding simulations of WT*T4L and CP13*T4L have been performed using a structure-based model that has one centre of interaction per amino acid in the C α position [1] implemented in CHARMM [2]. Interactions are attractive if they are present in the reference structure and repulsive otherwise. The magnitude and range of the interactions depend on the chemical properties of the residues, the separation between side-chains and the presence of hydrogen-bonds in the reference structure.  Construction of the optimal reaction coordinate Identication of the single coordinate that accurately describes complex folding process is challenging. In many cases, the standard progress variables (e.g. number of native contacts, radius of gyration, root mean square distance from the native structure) are not good reaction coordinates, because they do not preserve the barriers on the free-energy surface (FES) and thus may mask the inherent complexity of the latter [5].
A number of methods to construct good reaction coordinates have been suggested [612]. Here we employ a method which constructs the coordinate based on system dynamics by optimizing its cut-based free-energy prole (cFEP) [7,13].
Given a set of trajectories Y j (i∆t) recorded with time interval ∆t and a functional form dening the reaction coordinate P , the time series It is reasonable to assume that any bad projec- where a ij is either 1 or −1, r ij is the distance between atoms i and j and ∆ ij is the distance threshold, when contact between the atoms is considered to be formed; Θ is the Heaviside step function, whose value is zero for a negative argument and one for a positive argument. Numerical optimization was performed by iteratively picking a random pair of atoms ij, then scanning the whole parameter space for the pair (a ij = ±1 and ∆ ij = 0, 0.5, 1, . . . , 30) and nally selecting the one that gives the highest MFPT.

Equilibrium free energy prole
The equilibrium free-energy landscape is com- Clues on the nature of the transition state can also be gathered from the fraction of the native contacts present on average in the corresponding ensemble ( Figure S3). These can be considered as structural φ-values, and related to the experimentally measured φ-values [16]. For the wild type, φ-values is close to 0 (φ 0.2) conrm that the N-domain is almost completely unfolded. The C-domain (residue 1-12 and 60-164), on the contrary, is mainly folded , though not completely, with a φ-values closer to 1 (φ 0.8). The structural φ-values of the circular permutant are higher than for the wild-type, reecting the proximity of the transition state to the native state. In particular, it is true for the C-domain, which is almost folded.
Interestingly, the transition state of CP13*T4L is again compact (see the blue region in the Ndomain of T S I−N in Figure 2 and the high φ-values for residues around 20 in Figure S3). This implies that, in spite of dierences in the free-energy landscape, there is a common feature in the two variants: the closing of the protein determines the rate limiting step in T4 lysozyme and its circular permutant. frictions [17,18]. To check that the ndings reported here do not depend on the choice of friction coecient we repeated simulations with higher friction coecients (γ = 3; 5 and 10 ps −1 ) and analysed them using the optimal reaction coordinate found at γ=1 ps −1 ( Figure S4).