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“Strange Kinetics” of Ubiquitin Folding: Interpretation in Terms of a Simple Kinetic Model

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“Strange Kinetics” of Ubiquitin Folding: Interpretation in Terms of a Simple Kinetic Model Computational structural and functional proteomics 243 Chapter # “STRANGE KINETICS” OF UBIQUITIN FOLDING: INTERPRETATION IN TERMS OF A SIMPLE KINETIC MODEL Chekmarev S.F.*1, Krivov S.V.2, Karplus M.2, 3* 1 Institute of Thermophysics, SB RAS, Novosibirsk, 630090, Russia; 2 Laboratoire de Chimie Biophysique, ISIS, Université Louis Pasteur, 67000, Strasbourg, France; 3 Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138, USA, e-mail: [email protected] * Corresponding authors: e-mail: [email protected] Key words: ubiquitin, folding, folding times, rate constants, kinetic model, experiment SUMMARY Motivation: Interpretation of protein folding experiments in terms of the transitions between characteristic states of the system may allow valuable insight into the folding mechanism. Results: The ubiquitin mutant Ub*G folding experiments of Sabelko et al. (1999), in which “strange kinetics” were observed, are interpreted in terms of a simple kinetic model. A minimal set of states consisting of a semi-compact globule, two off-pathway traps, and the native state are included. Both the low and high temperature experiments of Sabelko et al. are fitted by a system of kinetic equations determining the transitions between these states. It is possible that cold and heat denaturated states of Ub*G are the basis of the off-pathway traps. The fits of the kinetic model to the experimental results provides an estimate of the rate constants for the various reaction channels and show how their contributions vary with temperature. Introduction of an on-pathway intermediate instead of one of the off-pathway traps does not lead to agreement with the experiments. INTRODUCTION Recent progress in experimental studies of protein folding on millisecond and faster time scales has increased the interest in the existence and role of intermediates in the folding process (Ferguson, Fersht, 2003). Ubiquitin is a benchmark system for protein folding studies due to its structural and physical properties (Went et al., 2004). To provide a fluorescence probe for monitoring folding of ubiquitin, Khorasanizadeh et al. (1993) replaced the largely buried Phe45 by a Trp. In the recent temperature-jump experiments of Sabelko et al. (1999), which we analyze here, a double mutant (F45W, V26G) of human ubiquitin (Ub*G) was used; the V26G mutation was added to destabilize core contacts and a critical helix to make cold denaturation possible. Sabelko et al. found that the time-dependent population of unfolded states varied with temperature from a double- exponential distribution at T = 2 °C to a stretched-exponential one at T = 8 °C. The interest of the latter is that it occurs on warming rather than cooling, and the high temperature spectrum of characteristic times is (quasi-) continuous and spans a broad region, corresponding to what has been referred to as “strange kinetics” (Shlesinger et al., 1993). In protein folding such kinetics have been associated with downhill folding through an array of intermediates separated by relatively low free energy barriers (e.g., Skorobogatiy et al., 1998). Sabelko et al. (1999) adduced arguments in favor of a BGRS’2006 244 Part 2 downhill folding scenario at T = 8 °C but also indicated that the kinetics at this temperature is well fitted by a three-exponential distribution. Thus, the presence of long- living intermediates could not be ruled out. Stretched-exponential distributions are convenient for approximation of experimental data, because they require just two parameters, a time constant and a power of the stretching exponent. A shortcoming of this approach is that the relation between the energy landscape and the parameters is not direct; i.e. knowledge of the parameters gives little information about specific features of the landscape. Multi-exponential distributions require more fitting parameters. However, if these distributions are related to a specific model and represent solutions of a system of kinetic equations, valuable insight into the folding process can be obtained. Recently, we simulated the Monte Carlo folding kinetics of a 27-bead square lattice protein model and showed that the results could be described by a simple kinetic model with off-pathway intermediates (Chekmarev et al., 2005). In this work we use a similar model to interpret the data on Ub*G folding. KINETIC MODEL We consider three kinetic schemes of the folding process, which include a semi- compact globule, off- and on-pathway intermediates and the native state. The schemes differ in the number and type of the intermediates involved: scheme #1 includes one off- pathway trap, scheme #2 two separate off-pathway traps, and scheme #3 one off-pathway trap and one on-pathway intermediate, which connects the globule and the native state. Given the rate constants, the time-dependent populations of the states are determined by the systems of linear kinetic equations. Following the experimental conditions of Sabelko et al. (1999), we do not include the fully extended (denaturated) state of the system in consideration. Also, since the experimental population of unfolded states does not show a tendency to stabilization at long times, we infer that native state unfolding is negligible. Therefore, the folding time is equated to the first passage time. Solving the systems of kinetics equations yields the time-depending populations of the states as functions of the rate constants, among them, the population of the unfolded states, which was measured by Sabelko et al. (1999). Then the fit of the theoretical solution to experimental distribution can be used to estimate the rate constants. RESULTS AND DISCUSSION Testing the previously mentioned kinetic schemes against the experimental results of Sabelko et al. (1999), we have found that both the low and high temperature experiments (i.e. at T = 2 °C and T = 8 °C) are well described within the framework of scheme #2. However, at low temperature (T = 2 °C) only one trap is essential and scheme #2 reduces to scheme #1. Table 1 shows the corresponded values of the fitted rate constants, and Fig. 1 and 2 compare the theoretical solutions with the experimental results. Introduction of an on-pathway intermediate instead of one of the off-pathway traps (scheme #3) does not lead to agreement with the experiments. -1 Table 1. Ub*G folding: Rate constants rβα (μs ) and waiting times τβα = 1/ rβα (μs, in brackets) for the transitions from state α to β. Subscripts g, d1, d2 and f stand for the semi-compact globule, off-pathway traps 1 and 2, and the native state, respectively rd1,g ()τ d1,g rg,d1()τ g,d1 rd2,g()τ d2,g rg,d2()τ g,d2 rf,g()τ f,g T = 2°C 8.0×10-3 (125) 2.4×10-4 (4105) 4.1×10-9 (2.4×109) 1.7×10-2 (60) 3.7×10-2 (27) T = 8°C 2.0×10-3 (510) 5.6×10-4 (1799) 2.0×10-2 (51) 8.0×10-3 (125) 1.9×10-2 (52) BGRS’2006 Computational structural and functional proteomics 245 Figure 1. Populations of the unfolded states of Ub*G at T = 2 °C (panel a) and T = 8 °C (panel b). The triangles correspond to the experimental data of Sabelko et al. (1999), and the solid line to the theoretical solution with the rate constants from Table 1. With a knowledge of the forward and backward rate constants (Table 1), it is possible to calculate the free energies of the off-pathway traps with respect to the globule. Assuming detailed balance, r /r = exp ⎡⎤− F − FkT/ , where F is the free energy j, i i, j ⎣⎦( ji) B α of state α, and kB is the Boltzmann constant, we found Fd1 – Fg ≈ –1.9 and Fd2 – Fg ≈ 9.6 at T = 2 °C, and Fd1 – Fg ≈ –0.7 and Fd2 – Fg ≈ –0.5 at T = 8 °C, with the energy measured in kcal/mol. An additional assumption about the value of the prefactor Aj, i in the Arrhenius equation, rj, i = Aj, i exp ()−ΔFj,i/ kT B , made it possible to estimate tentative values of the effective free energy barriers between the states, ΔFj, i. For this, we assume -1 that the Aj, i are equal to the frequently used :speed limit: of 1μs , characteristic of small proteins (Kubelka et al., 2004). The results are shown in Table 2, and they are similar in magnitude to the free energy barriers obtained for a set of small proteins (Kubelka et al., 2004). Of course, the actual value of the barrier depends on the choice of the pre- exponential factor. The observed large variation of the effective free energies with temperature is indicative of the complexity of the system and the temperature dependence of the reduced energy landscape. In accord with the variation of the free energy surface with temperature, which was proposed for ubiquitin (Sabelko et al., 1999), it is possible that the cold and heat denaturated states of Ub*G are involved in the off-pathway traps. If so, when one passes from T = 2 °C, where cold denaturation dominates, to T = 8 °C, which corresponds to the optimum native stability (Sabelko et al., 1999), the role of the cold denaturated state in the folding process would decrease and that of the heat denaturated state would increase. Alternatively, a trap could be associated with a state having excess helix relative to the native state (M. Gruebele, personal communication). We hope the present analysis will stimulate further studies of ubiquitin, a model system for which a definitive folding mechanism is likely to be determined by additional experimental and theoretical analyses. Table 2. Ub*G folding: Effective free energy barriers between the characteristic states (kcal/mol). Subscripts g, d1, d2 and f are as in Table 1 ΔFd1g ΔFgd1 ΔFd2g ΔFgd2 ΔFfg T = 2 °C 2.6 4.6 11.8 2.2 1.8 T = 8 °C 3.5 4.2 2.2 2.7 2.2 BGRS’2006 246 Part 2 ACKNOWLEDGEMENTS We thank M.
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