The exclusive effects of chaperonin on the behavior of proteins with 52 knot

The folding of proteins with a complex knot is still an unresolved question. Based on representative members of Ubiquitin C-terminal Hydrolases (UCHs) that contain the 52 knot in the native state, we explain how UCHs are able to unfold and refold in vitro reversibly within the structure-based model. In particular, we identify two, topologically different folding/unfolding pathways and corroborate our results with experiment, recreating the chevron plot. We show that confinement effect of chaperonin or weak crowding greatly facilitates folding, simultaneously slowing down the unfolding process of UCHs, compared with bulk conditions. Finally, we analyze the existence of knots in the denaturated state of UCHs. The results of the work show that the crowded environment of the cell should have a positive effect on the kinetics of complex knotted proteins, especially when proteins with deeper knots are found in this family.


I. STRUCTURAL AND SEQUENTIAL COMPARISON OF UCHS
A. Detailed analysis of investigated proteins: 3IRT, 2LEN, 4I6N, 4I6N-m  The secondary structure of 4I6N has a eleven amino acids long gap. The missing sequence of amino acids 142 to 152 belongs to the mobile part of 4I6N, the so called cross over loop 2 . The missing amino acids were added using Modeller. The best conformation, based on the Discrete Optimized Protein Energy (DOPE) calculations, was used in the further studies as a representative of 4I6N, and as a template for 4I6N-m.  FIG. E. Dependence of mean potential energy of native state (green curve), structure with only C-terminus structured (blue) and only N-terminus structured (red). For high temperatures (unfolding) the protein stays in the native state only in the begging of simulation, therefore only the most native-like structures are included in the mean, resulting in decrease of green curve for higher temperatures.  The averaged and smoothed dependencies Q(t) for each given temperature and each condition (bulk/confinement) were used to fit the equation:

III. INFLUENCE OF TEMPERATURE ON KNOTTING
With all A i and k i positive and y 0 being the vertical shift (always close to 0). The number n ∈ {2, 3, 4} of exponential functions fitted was dependent on the fitting errors obtained. We fitted the maximal number of exponential functions, for which the errors were maximally of order of magnitude of obtained parameters. The fitting was done using QtiPlot with unscalled Levenberg-Marquardt algorithm and tolerance 10 −5 . Obtained values are stored in the Tab. E and F. ln

B. Determination of folding temperature
The most complete trace of Chevron plot were fitted with the equation where first part describes folding, second unfolding. As the limbs of Chevron plot in the Fig. 5 main text are both curved, we included the second order effect (dependence of x 2 ). The argument x represents the "concentration of denaturant", which in our case is the value − /k b T . The fitting resulted in parameters given in Tab. G.

Conditions
m  The minimum of fitted curve was found to determine the (kinetic) equilibrium temperature which was used in main text (approximatelly 114 for bulk and 120 for confinement). H. The characteristic times for the folding and unfolding processes of 3IRT in different temperatures. τ knot denotes the mean knotting time, τ Q is the mean time at when the system firstly hits the most probable value of Q in the folded state, and τ tot = Q|τ knot denotes the total folding time of 3IRT. Times for unknotting and unfolding are denoted as τ unknot and τ U Q ), respectively. The system is studied in both bulk and confinement, which is represented by 'bulk' and 'chap' respectively in this Table. Folding

C. Folding and unfolding times
Unfolding Note, that this temperature differs slightly from the temperature determined from kinetics. The free energy profile shows broad maximum, which is a convolution of different folding pathways and plasticity of intermediate states.

X. CONTACT BREAKING PROBABILITY
The contact breaking rate was calculated by comparing the set of contacts between consecutive frames. For each contact (e.g. beetween residues 1 and 5) we identified the frames in which this particular contact was broken, i.e. the situations in which in one frame the contact existed, but it disapeared in the next frames. This number was then divided by the total number of frames in which a given contact existed. This fraction has a meaning of the conditional probability of contact breaking, given that the contact existed. Next, we calculated the mean probability for all contacts as a function of temperature. The same analysis has been done in case of any pairs of beads separated by at least 4 residues, which during simulations were closer than 6Å, but do not form a native contact (in total 23197 such pairs). Mean breaking rate (conditional probability) of contacts as a function of temperature. A: native contacts, B: pairs of beads not forming native contacts. The breaking probability for native contact for temperatures larger than 110 is always higher for bulk, then for confinement, therefore confinement stabilizes native contacts. On the other hand, the separation proability for pairs of beads nt forming native contact is higher for confinement meaning, that confinement destabilizes structures with spatial proximity of beads not forming native interactions.