© 1999 Macmillan Magazines Ltd ................................................................. Interpreting the folding kinetics of helical proteins Yaoqi Zhou* & Martin Karplus*² * Department of Chemistry & Chemical Biology, Harvard University, Cambridge, Massachusetts 02138, USA ² Laboratoire de Chimie Biophysique, ISIS, Universite  Louis Pasteur, 67000 Strasbourg, France .............................................................................................................................................. The detailed mechanism of protein folding is one of the major problems in structural biology 1,2 . Its solution is of practical as well as fundamental interest because of its possible role in utilizing the many sequences becoming available from genomic analysis 3 . Although the Levinthal paradox 4 (namely, that a polypeptide chain can ®nd its unique native state in spite of the astronomical number of con®gurations in the denatured state) has been resolved 4±7 , the reasons for the differences in the folding beha- viour of individual proteins remain to be elucidated. Here a C a - based three-helix-bundle-like protein model with a realistic ther- modynamic phase diagram is used to calculate several hundred folding trajectories. By varying a single parameter, the difference between the strength of native and non-native contacts, folding is changed from a diffusion±collision mechanism 8 to one that involves simultaneous collapse and partial secondary-structure formation, followed by reorganization to the native structure. Non-obligatory intermediates are important in the former, whereas there is an obligatory on-pathway intermediate in the latter. Our results provide a basis for understanding the range of folding behaviour that is observed in helical proteins. All-atom models with explicit solvent have provided information on high-temperature unfolding 9,10 and the free-energy landscapes of several proteins 11,12 . However, corresponding simulations of the folding kinetics for ensembles of trajectories are beyond what can be done with available computers 13 , as folding times are millise- conds or longer for most proteins. To overcome such computational limitations, we use discrete molecular dynamics to study the folding of the small (46-residue) three-helix-bundle fragment of protein A (ref. 14; and Fig. 1a). The thermodynamic behaviour of the model (Fig. 1b) is in accord with simulations of crambin using an all-atom representation and the temperature-dependent X-ray diffraction results for ribonuclease A (ref. 15), as well as data for other proteins 16,17 . An important element of the model 18 is that the phase diagram can be varied by the choice of a single parameter, the bias gap (see Methods), which determines the relative stability of native and non-native contacts. For example, equilibrium collapse to a disorganized globule occurs for a small bias gap and to an organized (molten) globule for a large bias gap, as shown in Fig. 1b. To analyse the folding kinetics, 100 trajectories, each 100 ms to 1 ms in length, were calculated for a series of values of the bias gap parameter, starting from the fully denatured state under conditions where the native state is stable (see Methods). The folding times range from 60 ns to more than 0.1 ms for the highly optimized model (large bias, gap, g  1:3) and from 2 ms to more than 1 ms for a weakly optimized model (g  0:3). The range of folding behaviour is illustrated in Fig. 2. The kinetics of two trajectories for a small value of the bias parameter (g  0:3), both are very different, although identical model parameters are used and both reach the native state. The ®rst (Fig. 2a) has early helix formation (that is, about 70% of the helical contacts exist at 10 ns), whereas other progress variables (the fraction of native interhelical contacts and the inverse native fraction of the volume, for example) are still very far from the native state. The resulting near-helical peptide chain then undergoes a slow diffusion±colli- sion-like search 8 of the relative helix orientations until the native state is reached at 19 ms. In this folding scheme, there can be letters to nature 400 NATURE | VOL 401 | 23 SEPTEMBER 1999 | www.nature.com 0.4 0.6 0.8 1.0 1.2 Bias gap 0.0 0.4 0.8 1.2 1.6 Reduced temperature Disordered Coil globule Molten globule Surface–molten solid (native) Solid Collapse transition a b Figure 1 Structure and thermodynamics of the model three-helix bundle protein. a, Global minimum structure of the model obtained by annealing the NMR structure of the three-helix-bundle fragment (residues 10±55) of Staphylococcus aureus protein A (ref. 14); helix I, II and III are shown in green, blue and red, respectively. b, The phase diagram for models with different bias gap values. All temperatures are in units of the energy parameter e. Thermal transitions and collapse transition are indicated by open circles and open squares, respectively. The ®lled circles indicate a ®rst-order-like two-state transition. There are four signi®cant transitions: a collapse transition; a disordered to ordered (molten) globule transition; a globule to native-state transition; and the transition from the active native state to a frozen inactive state 15,22 .