^{1,a)}, Margarida M. Telo da Gama

^{2,b)}and Patrícia F. N. Faísca

^{3,c)}

### Abstract

We perform extensive Monte Carlo simulations of a lattice model and the Gō potential [N. Gɵ and H. Taketomi, Proc. Natl. Acad. Sci. U.S.A.75, 559563 (1978)] to investigate the existence of folding pathways at the level of contact cluster formation for two native structures with markedly different geometries. Our analysis of folding pathways revealed a common underlying folding mechanism, based on nucleation phenomena, for both protein models. However, folding to the more complex geometry (i.e., that with more nonlocal contacts) is driven by a folding nucleus whose geometric traits more closely resemble those of the native fold. For this geometry folding is clearly a more cooperative process.

Two of the authors (R.D.M.T. and P.F.N.F.) thank Fundação para a Ciência e Tecnologia (FCT) for financial support through Grant Nos. SFRH/BPD/27328/2006 and SFRH/BPD/21492/2005, respectively. This work was also supported by FCT through Project Nos. POCI/FIS/55592/2005 and POCTI/ISFL/2/618.

I. INTRODUCTION

II. MODEL AND METHODS

A. Gɵ model and simulation details

B. Native geometries

C. Probability to fold,

III. EXPLORING THE HIDDEN “ARCHITECTURE” WITHIN A LATTICE PROTEIN

A. Target conformations

B. Interresidue contact correlation analysis reveals distinct protein sections

C. Section’s geometric traits

IV. FOLDING PATHWAYS

V. SECTION FORMATION AS A FUNCTION OF THE FOLDING PROBABILITY

VI. FROM MACRO- TO MICROSTRUCTURAL FORMATION: EVIDENCE FOR NUCLEATION PHENOMENA

VII. COOPERATIVITY AT THE LEVEL OF MACROSTRUCTURAL FORMATION

VIII. CONCLUSIONS

### Key Topics

- Geologic folds
- 63.0
- Proteins
- 49.0
- Protein folding
- 29.0
- Folding pathways
- 20.0
- Bond formation
- 14.0

## Figures

Three dimensional representations of geometry 1 (top, left) and geometry 2 (bottom, left), and their respective contact maps (right). In the contact maps each circle represents a native contact. Nonlocal LR contacts are shown in white.

Three dimensional representations of geometry 1 (top, left) and geometry 2 (bottom, left), and their respective contact maps (right). In the contact maps each circle represents a native contact. Nonlocal LR contacts are shown in white.

Probability distribution for the fraction of native contacts, , for geometry 1 (left) and geometry 2 (right) as a function of . A conformation is considered unfolded when .

Probability distribution for the fraction of native contacts, , for geometry 1 (left) and geometry 2 (right) as a function of . A conformation is considered unfolded when .

(Color online) Time to fold as a function of the reaction coordinate . Long lived trapped states are observed in geometry 1 (left) at very high , but are absent in geometry 2 (right). To measure of each conformation we considered only folding events in which the protein folded before unfolding. is the mean time-to-fold averaged over these folding events. The horizontal black lines indicate the cutoff times below which a conformation is committed to fold. For geometries 1 and 2 there are, respectively, 4724 and 4162 conformers with .

(Color online) Time to fold as a function of the reaction coordinate . Long lived trapped states are observed in geometry 1 (left) at very high , but are absent in geometry 2 (right). To measure of each conformation we considered only folding events in which the protein folded before unfolding. is the mean time-to-fold averaged over these folding events. The horizontal black lines indicate the cutoff times below which a conformation is committed to fold. For geometries 1 and 2 there are, respectively, 4724 and 4162 conformers with .

Density plots of the probability (left column) and fraction of conformations (right column), where contact is present and is not for geometry 1 (top) and geometry 2 (bottom). Native contacts are ordered according to their relative values of (the order is the same for the and plots). The groups of contacts forming sections A, B, and C are identified. Contacts that were not assigned to any section (“free” contacts) are identified by the letter F. The range of lies between 0 (black) and 1 (white), while varies between 0 (black) and 0.54 (white) in geometry 1 and between 0 (black) and 0.64 (white) in geometry 2.

Density plots of the probability (left column) and fraction of conformations (right column), where contact is present and is not for geometry 1 (top) and geometry 2 (bottom). Native contacts are ordered according to their relative values of (the order is the same for the and plots). The groups of contacts forming sections A, B, and C are identified. Contacts that were not assigned to any section (“free” contacts) are identified by the letter F. The range of lies between 0 (black) and 1 (white), while varies between 0 (black) and 0.54 (white) in geometry 1 and between 0 (black) and 0.64 (white) in geometry 2.

(Color) Protein sections identified for geometry 1 (top row) and geometry 2 (bottom row). Native contacts forming sections A, B, and C are, respectively, colored red, blue, and green, in the three dimensional representations (left) and contact maps (right). Note that the protein sections identified as groups of correlated native bonds are grouped together in the protein’s three dimensional native structure.

(Color) Protein sections identified for geometry 1 (top row) and geometry 2 (bottom row). Native contacts forming sections A, B, and C are, respectively, colored red, blue, and green, in the three dimensional representations (left) and contact maps (right). Note that the protein sections identified as groups of correlated native bonds are grouped together in the protein’s three dimensional native structure.

(Color) Density plots of the probability for having a certain as a function of for the sections A, B, and C in geometry 1 (top) and geometry 2 (bottom).

(Color) Density plots of the probability for having a certain as a function of for the sections A, B, and C in geometry 1 (top) and geometry 2 (bottom).

(Color online) Average fraction of native bonds in each protein section, , as a function of in geometry 1 (a) and geometry 2 (b). Also shown is the dependence of the protein’s average fraction of native bonds on the reaction coordinate, , for both geometries. Note that when folding is near completion at high , there is a sharp increase in the fraction of native contacts for geometry 1.

(Color online) Average fraction of native bonds in each protein section, , as a function of in geometry 1 (a) and geometry 2 (b). Also shown is the dependence of the protein’s average fraction of native bonds on the reaction coordinate, , for both geometries. Note that when folding is near completion at high , there is a sharp increase in the fraction of native contacts for geometry 1.

## Tables

Absolute contact order (ACO), fraction of long-range (LR) contacts, optimal folding temperature , and folding time for geometries 1 and 2.

Absolute contact order (ACO), fraction of long-range (LR) contacts, optimal folding temperature , and folding time for geometries 1 and 2.

Number of native bonds forming each protein section, absolute contact order (ACO), and fraction of long-range (LR) contacts of each protein section.

Number of native bonds forming each protein section, absolute contact order (ACO), and fraction of long-range (LR) contacts of each protein section.

Folding pathways at the macrostructural level of section formation (showing the first, second, and third sections to fold) and their relative probabilities of occurrence. The probabilities do not add to one, since there are some events in which two sections fold simultaneously. The average time elapsing between the formation of the first section and the formation of the last section in each pathway is given in units of .

Folding pathways at the macrostructural level of section formation (showing the first, second, and third sections to fold) and their relative probabilities of occurrence. The probabilities do not add to one, since there are some events in which two sections fold simultaneously. The average time elapsing between the formation of the first section and the formation of the last section in each pathway is given in units of .

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