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Determination of Barrier Heights and Prefactors from Protein Folding Rate Data

Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia, Canada

Correspondence: Address reprint requests to S. S. Plotkin, E-mail: steve{at}physics.ubc.ca.

	ABSTRACT

TOP ABSTRACT INTRODUCTION METHODOLOGY RESULTS CONCLUSIONS AND DISCUSSION ACKNOWLEDGEMENTS REFERENCES

 
We determine both barrier heights and prefactors for protein folding by applying constraints determined from experimental rate measurements to a Kramers theory for folding rate. The theoretical values are required to match the experimental values at two conditions of temperature and denaturant that induce the same stability. Several expressions for the prefactor in the Kramers rate equation are examined: a random energy approximation, a correlated energy approximation, and an approximation using a single Arrhenius activation energy. Barriers and prefactors are generally found to be large as a result of implementing this recipe, i.e., the folding landscape is cooperative and smooth. Interestingly, a prefactor with a single Arrhenius activation energy admits no formal solution.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODOLOGY RESULTS CONCLUSIONS AND DISCUSSION ACKNOWLEDGEMENTS REFERENCES

 
In contrast to many exothermic reactions in organic chemistry, the log protein folding rate displays a significant linear trend with the relative stability of the product and reactant (folded and unfolded states; Fersht, 1999). This indicates a late transition state in the language of Hammond's postulate, and the slope of the log rate versus stability line quantifies the degree of native structural information in the transition state.

Native stability may be modified by adjusting temperature T or denaturant concentration c. Many proteins show linearity over the majority of the branches of their Chevron plot, implying a linear dependence of folding and unfolding barriers on denaturant concentration c (Jackson and Fersht, 1991),

(1a)

(1b)

with m_F > 0 and m_U > 0.

Subtracting Eq. 1b from Eq. 1a, and defining G G_FU = G_U – G_F and m = m_F + m_U, we have that

(2)

For two-state folders, the kinetically determined m above equals, to good approximation, the thermodynamically determined m-value from relative stabilities.

Applying Kramers rate theory, the log forward folding rate is given by

(3)

Eliminating c from Eqs. 2 and 3 gives

(4)

where the left-hand side of Eq. 4 depends on both (T, c), but the function on the right-hand side depends on c only through the prefactor. Empirically it was observed by Scalley and Baker (1997) that for the proteins CspB and protein L, the data for various c collapse onto a single curve when the left-hand side is plotted versus 1/T. This indicates that the right-hand side is a function of temperature alone and so ln k_o(T, c) ln k_o(T). Denaturant concentration does not have a significant effect on the rate at which the system escapes from local traps (at least for those proteins studied). We make this assumption here as well.

Because the prefactor is independent of c, the change in log folding rate with denaturant is directly proportional to the change in barrier with denaturant,

(5)

which, together with Eq. 4, gives

(6)

(7)

This quantifies the assertion above that log folding rates depend linearly on the relative stability of the products. If we let m_U/m Q, Eq. 6 can be rewritten as

(8)

which is the commonly used linear free energy relation (Bryngelson et al., 1995).

Inspection of rate-stability isotherms for several different proteins—cytochrome C (cyt-C; Mines et al., 1996), protein L (Scalley and Baker, 1997), cspB (Schindler and Schmid, 1996), N-terminal protein L9 (Kuhlman et al., 1998), and S6 (Otzen and Oliveberg, 2004)—shows linearity over ranges up to 25 kJ x mol^–1 10 k_BT, indicating large and robust folding barriers, which are substantially larger than the folding barriers seen in many simulations (for example, see Fig. 1).

View larger version (13K): [in this window] [in a new window]   FIGURE 1  Logarithm of the rate versus (minus) native stability for horse Cytochrome C, at two temperatures. The plots are well fit by straight line functions that are used in the analysis of the text. Adapted from Mines et al. (1996).

1. The prefactor increases at higher temperature (since activated escape from traps is further facilitated, and solvent viscosity is reduced).
   
2. The thermodynamic weight of the entropic component to the barrier (which includes contributions from the solvent) increases as well, which may decrease the barrier height.
   

	METHODOLOGY

TOP ABSTRACT INTRODUCTION METHODOLOGY RESULTS CONCLUSIONS AND DISCUSSION ACKNOWLEDGEMENTS REFERENCES

 
In what follows, we apply Kramers rate theory together with energy landscape ideas to extract barrier heights and prefactors from experimental rate data.

The temperature-dependence of the stability is given by the Gibbs-Helmholtz expression (Fersht, 1999; Jackson and Fersht, 1991) as

(9)

Then at equal stabilities G(T_o, c_o) = G(T, c),

(10)

For two-state folders, the heat-capacity ratio C_PU/C_P is approximately equal to m-value ratio m_U/m, giving the fractional solvent accessibility of the transition state. We assume this equality here as well, which gives for Eq. 10,

(11)

Inserting Eq. 11 into Gibbs-Helmholtz expressions for the barrier heights G_U at (T, c) and (T_o, c_o) gives the change in barrier height at fixed stability,

(12)

which is independent of c and depends only on the temperature difference between the two fixed-stability states (and thermodynamic parameters). This equation applies to points A and B in Fig. 1, for example.

For changes in temperature of a few degrees, the change in barrier height 'G_U is only a few percent of the total barrier height, when the rates versus temperature and denaturant are fit to a model to extract thermodynamic parameters, as in Kuhlman et al. (1998), Otzen and Oliveberg (2004), Scalley and Baker (1997), and Schindler and Schmid (1996). We used Eq. 12 for the change in barrier height when thermodynamic data were available. For the case of cyt-C we set 'G = 0.

The rates for pairs of states at the same stability G are given from Eq. 3 as

(13a)

(13b)

Random energy model for the temperature-dependent prefactor
At the mean field level for a landscape of uncorrelated states (random energy model or REM), the temperature-dependence of the prefactor in Eq. 3 is super-Arrhenius (Bryngelson and Wolynes, 1989; Onuchic et al., 1997). Moreover, the prefactor goes as the reciprocal of the viscous friction coefficient (Hanggi et al., 1990; Klimov and Thirumalai, 1997; Socci et al., 1996), so the log prefactors at (T_o, c_o) and (T, c) may be written as

(14a)

(14b)

To compare rate theories with experimental data we must introduce a fundamental timescale or rate constant k_oo, which is then modified by barriers representing the ruggedness of the energy landscape. Rates for short loop closure are 2 x 10⁷ s^–1 (Lapidus et al., 2000), comparable to helix formation rates of 10⁷ s^–1, and somewhat faster than rates of hairpin formation 10⁶ s^–1 (Eaton et al., 2000). Prefactors obtained from plots of experimental rate versus powers of chain length are of order µs (Li et al., 2004); however, these implicitly include any effects due to ruggedness. We take 10⁷ s^–1 as an estimate of the fastest local rate. Since 10–100 loops and/or secondary structural elements exist in a protein, we then take k_oo = 10⁹ s^–1. This estimate for k_oo may appear somewhat large; we will see later that smaller estimates for k_oo give smaller estimates for inferred folding barriers. We use the known temperature dependence of the viscosity in water (CRC, 2003). The quantity ² measures the ruggedness of the energy landscape. It may be eliminated from Eqs. 14a and 14b to give an equation relating the prefactors as

(15)

Equations 13a, 13b, and 15 constitute a system of three linear equations for three unknowns: G_U(T_o, c_o), ln k_o(T_o), and ln k_o(T), which can be solved analytically at any given stability, from linear fits to the log rate-stability data.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODOLOGY RESULTS CONCLUSIONS AND DISCUSSION ACKNOWLEDGEMENTS REFERENCES

 
The results of applying the method are shown in Fig. 2, for the data in Fig. 1, ranging from the stability of wild-type at 296 K (–74 kJ/mol) to zero stability at the transition midpoint. Barrier heights are plotted in units of kJ/mol; rates in prefactors are in units of s^–1.

View larger version (15K): [in this window] [in a new window]   FIGURE 2  Barrier height GU and prefactors ko at two temperatures, as obtained from the REM approximation (see text), are plotted as a function of minus stability for cyt-C. The wild-type protein has a stability of G 74 kJ/mol. Numerical values are given in Table 1. Prefactor attempt rates are in s–1, and barrier heights are in kJ/mol. The short dashed line gives the barrier for a temperature-independent solvent viscosity. All logarithms are natural (base e).

The slope G_U/G 0.8 is also larger than its empirical value of m_U/m 0.4 (Mines et al., 1996), thus the barriers vanish at weaker stabilities than the wild-type protein. This indicates a breakdown in the validity of the theory at higher stabilities (larger G).

There are two parameters in the theory for which we have put in approximate values: the value of the attempt frequency k_oo = 10⁹ s^–1, and the value of 'G_U, which we have set to zero for cytochrome C in the absence of an empirically determined value. Increasing k_oo or decreasing 'G_U raises barriers, but does not change the slope G_U/G. The value of –G where the barrier vanishes linearly decreases as 'G_U is decreased below zero, with the barrier vanishing at the stability of the wild-type when 'G_U is –1.6 kJ/mol. This is not an unreasonable number compared to experimental numbers for other proteins (see below); however, it is somewhat disconcerting that barrier heights are such a strong function of the barrier change 'G_U. We will see later that this sensitivity is not present when a correlated landscape model is used for the prefactor.

Fig. 2 also shows that at least for the REM approximation it is important to account for changes in the viscosity of the solution with temperature, as the barrier substantially decreases when the viscosity is held constant versus temperature.

Equations 14a or 14b may now be solved for , giving a number 15 kJ/mol, that only weakly depends on stability G or barrier change 'G_U. Estimating the chain conformational entropy as 100 k_B (D'Aquino et al., 1996; Leach et al., 1966), we can give an estimate for the glass temperature T_G for this system,

(16)

which is also a fairly robust number as a function of stability or barrier change, as shown in Fig. 3. At the stability of wild-type cyt-C, T_G 150 K, giving T/T_G 2.0 at 296 K.

View larger version (10K): [in this window] [in a new window]   FIGURE 3  (A) The temperature TG that emerges from the REM analysis for cyt-C (see text and Eq. 16) varies only moderately with barrier height change at constant stability, 'GU (the value of which is not known for this protein). For this plot the stability is set to midway between zero and the stability of the wild-type (37 kJ/mol). (B) TG also changes little as native stability G is varied (for this plot 'GU = 0).

(17a)

(17b)

Here S is the chain entropy at the transition state, and and ß are parameters measuring the mismatch between entropy and energy giving the typical free energy barrier governing trap escape. The values for a bulk polymer 0.5, ß 1.8 are used below (Plotkin and Onuchic, 2002a,b; Wang et al., 1997). The temperature T_G was adjusted to the value that reproduced the experimentally determined slope of barriers versus stability, m_U/m. In Table 1 this number is compared to the value of T_G that emerges from the REM analysis. A mismatch of these two values may indicate a breakdown of the REM approximation for states in determining prefactors, i.e., a breakdown in the validity of Eqs. 14a and 14b. For cyt-C the value of T_G giving the correct slope is 1.2 kJ/mol, versus 1.0 kJ/mol from the REM analysis.

The entropy S may be eliminated from Eqs. 17a and 17b, giving an equation that relates the prefactors, which replaces Eq. 15,

(18)

Equations 13a, 13b, and 18 again define a system of three linear equations for three unknowns: G_U(T_o, c_o), ln k_o(T_o), and ln k_o(T), which may be solved analytically. The results are shown in Fig. 4.

View larger version (16K): [in this window] [in a new window]   FIGURE 4  Barrier heights and prefactors as obtained from the correlated landscape analysis (see text), plotted as a function of minus native stability for h-cyt-C. Numerical values are given in Table 1. Prefactor attempt rates are in s–1, and barrier heights are in kJ/mol. The dotted line gives the barrier for a temperature-independent solvent viscosity. Note prefactors are approximately constant (as is physically reasonable) and solvent viscosity plays a minor role.

The REM value of T_G resulted from approximating a value of 100 k_B for the chain entropy S_o, so it is feasible that this estimate for the REM T_G could differ from the T_G that gives the correct m_U/m. The parameters and ß could, in principle, have been adjusted to best match the experimental slope; however, it can be shown that this results in the same solution of Eqs. 13a, 13b, and 18 as that determined by varying T_G.

In contrast to the REM approximation, the effects of the temperature dependence of viscosity were not significant here (Fig. 4). Nor were there any significant effects due to barrier height difference—as 'G_U changed from –2 kJ/mol to 0 kJ/mol, the barrier changed by <2%. The effects due to T_G are modest as well: over the range of T_G values in Fig. 3 B, the barrier height changed by <15%. Lastly, the prefactors of the correlated landscape model are nearly constant over the range of experimental stabilities (Fig. 4), consistent with empirical observations (see the comments below Eq. 4).

Equation 17a or 17b may now be solved for S as a check, giving S 40 k_B, or 40% of the unfolded chain entropy assumed in finding the REM T_G. Alternatively, we can estimate the unfolded entropy S_o from the value of S as S (1 – m_U/m)S_o, then Eq. 16 gives 14 kJ/mol. Since the variances of individual residues add to give ², ² N(1 – m_U/m)b², where b is a non-native energy scale per residue, here as 0.7 k_BT₃₀₀.

Fig. 5 shows that the inferred barriers and prefactors increase as the value of the bare reconfiguration rate k_oo increases. The prefactor ln k_o(T_o) closely follows the bare reconfiguration rate ln k_oo; i.e., they are approximately equal. The barriers at the transition midpoint and at the stability of the wild-type protein increase linearly, as 2k_BT_o ln k_oo.

View larger version (14K): [in this window] [in a new window]   FIGURE 5  Barrier heights and prefactors extracted from the recipe for the correlated energy landscape (see text) increase as the bare reconfiguration rate (appearing in Eqs. 17a and 17b) increases. The increase is linear. is the barrier at the transition midpoint, is the barrier at the stability of the wild-type protein, and ko(To) is the prefactor at temperature To in s–1.

For NTL9, the solution of the REM gave a T_G that monotonically decreased from a value of 0.4 at the stability of the wild-type protein, to zero at a stability of 11 kJ/mol. Similarly, the prefactor monotonically increases from 10⁸ s^–1 at the stability of the wild-type to 10¹⁰ s^–1 at zero stability. We note that these problems are not present if the stability difference 'G_U is set to zero, if the prefactor is two or more orders-of-magnitude slower, or if the temperature dependence of the viscosity is neglected. We take this sensitivity as a shortcoming of the procedure of rigorously demanding that the landscape theory fit to a limited subset of the experimental data. In this sense, a best (but not exact) fit to experimental rate surfaces as a function of both T and c as in Kuhlman et al. (1998), Otzen and Oliveberg (2004), Scalley and Baker (1997), and Schindler and Schmid (1996), is likely to give more accurate numbers. Likewise in the correlated model for NTL9, the prefactor increased from 10⁸ s^–1 at the stability of the wild-type to unphysical values at zero stability. A similar situation exists in the REM recipe for protein S6; however, it is resolved in the correlated landscape model for that protein.

CspB showed some difficulties that arose from its unusually late transition state (m_U/m 0.9) (Perl et al., 2002). The parameter T_G in the correlated model could not be adjusted to reproduce the high slope of barrier versus stability, without giving negative barriers. Again this may be an artifact of the exact fitting method mentioned above, i.e., more experimental data may also be needed to obtain more accurate numbers, or it may indicate that a simple mean field prefactor does not fully adequately describe the folding dynamics of this protein. In this case we took the temperature T_G = 1.81 kJ/mol that induced the barrier to vanish at the stability of the wild-type protein (while acknowledging that other fits give large barriers (Perl et al., 2002)). This has a steep barrier-stability curve, with slope m_U/m = 0.8 (as opposed to 0.9 observed empirically), very small barrier (7 kJ/mol at zero stability), and rugged landscape with very slow prefactor (10² s^–1). Such small barriers are consistent with estimates taken from simulations using C-models (Shea and Brooks, 2001).

The Arrhenius model generally admits no solution
A model often proposed for the prefactor assumes an Arrhenius temperature-dependence with single activation energy E_A, so that Eqs. 14a and 14b are replaced by

(19a)

(19b)

from which E_A may be eliminated, yielding

(20)

This equation, relating the prefactors together with Eqs. 13a and 13b, constitutes the new system of equations to be solved.

Eliminating G_U from Eqs. 13a and 13b gives another equation relating the prefactors:

(21)

Equations 21 and 20 both have ln k_o(T) on the left-hand side and (T_o/T) ln k_o(T_o) on the right. Subtracting them gives an equation that is independent of any variable to be solved for

(22)

which cannot be true in general, in particular because the left-hand side depends on c and the right-hand side does not.

A geometric analog may be helpful in understanding the situation. The solution to three equations in three variables is equivalent to finding the point where three planes intersect. Letting

Eqs. 13a, 13b, and 20 may be recast as

(23a)

(23b)

(23c)

where

Since A B in general, Eqs. 23a and 23b depict two parallel planes. Thus there is no point of intersection and the system of equations is ill-posed. For the special case of A = B there is a whole family of solutions consistent with the rate equations, but as mentioned above this scenario can only hold under very special circumstances.

	CONCLUSIONS AND DISCUSSION

TOP ABSTRACT INTRODUCTION METHODOLOGY RESULTS CONCLUSIONS AND DISCUSSION ACKNOWLEDGEMENTS REFERENCES

 
We have proposed here a method of testing energy landscape theory by mapping Kramers rate theory, with prefactors given from the statistics of energies of states, to experimental data on protein folding rates. We considered three models for the prefactor here: one where ruggedness is treated with a random energy approximation; one where correlations are taken into account; and an Arrhenius model with a single barrier governing reconfiguration times.

The numerical values of the barriers obtained from the above recipes should be taken with a grain of salt; however, it consistently emerged that folding barriers were large (except for CspB): the average barrier at the transition midpoint for the REM analysis is 19 k_BT, and the corresponding barrier in the correlated model is 18 k_BT. If CspB is omitted, the barriers are 21 k_BT and 22 k_BT, respectively. Wild-type S6, a protein known to fold very cooperatively (Lindberg et al., 2002), had the highest barriers.

With the exception of CspB, the prefactors in the correlated model tended to be quite high—approximately the bare reconfiguration rate for the whole protein (10⁹ s^–1). The folding barrier obtained from the recipe decreases as estimates for the bare reconfiguration rate decrease (Fig. 5). The prefactors from the REM recipe varied considerably.

All of the proteins analyzed here are considered two-state folders, so we would expect a Kramers theory to describe them. In lower temperature regimes the distribution of first passage times may be more relevant to study (Plotkin and Onuchic, 2002b; Zhou et al., 2003).

We found that in practice it was quite important to have accurate fits for the empirical rate-stability curves. For example, as temperature increased, the slope of the log rate versus stability curve had to remain approximately constant or tend to increase, to obtain reasonable solutions of the rate equations. Otherwise we found an unphysical situation where barriers did not increase as stability decreased. This sensitivity to the experimental data may favor a less stringent fit to the experimental constraints.

In fact, reflection on the procedure raises a general issue on the rigorous application of experimental constraints to energy landscape theory. For example, if we were to add data at a third temperature T₁, two new equations would be introduced according to the recipe—one Kramers rate equation and one landscape equation for the prefactor, but only one new variable is introduced—the prefactor ln k_o(T₁). The system becomes overdetermined. Demanding equality rather than a best fit at several temperatures becomes too stringent a constraint on the theory, as long as the parameters in the theory (e.g., ² or E_A) are fixed. The more temperatures used, the more variables must be introduced into the theory, or the parameters must themselves become temperature-dependent. Nevertheless, the fact that the Arrhenius activation model fails in general to provide a solution for even two temperatures (two data points) should probably be seen as evidence against its strict applicability.

A perhaps more viable method would be to fit several temperatures with functional forms such as Eqs. 14a, 17a, or 19a to extract parameters such as ² and E_A. The difficulty in previous fits to data has been in the separation of E_A and the activation enthalpy H_U (Scalley and Baker, 1997). One can ask which temperature dependence (E_A/T or ²/T²) gives the best fit to the data, but there is not yet enough accurate data to distinguish between the two scenarios (Kuhlman et al., 1998; Scalley and Baker, 1997) by this method. However, the Arrhenius model becomes severely restricted by applying experimental constraints rigorously at two temperatures and denaturant concentrations, at the same stability. Because the activation energy in the prefactor can be absorbed into the enthalpic part of the barrier, and only the entropic part of the barrier is relevant in determining rate differences at fixed stability (by Eq. 12), the activation energy becomes irrelevant, and the difference in rates must then be due to quantities independent of denaturant concentration (entropic part of the barrier, temperature-dependent viscosity, etc.). All rate-stability curves for a given protein must be exactly parallel in the Arrhenius model—a situation not observed empirically.

Topological features of the native structure have been neglected in the rate theory. Including polymer physics into the theoretical model (Plotkin and Onuchic, 2000; Portman et al., 2001; Shoemaker et al., 1999) may also eliminate some of the sensitivity of the theoretically derived values in Table 1 on the experimental data.

Other methods have been used to estimate barrier heights. Adding a three-body contribution to a pairwise-interacting energy function to give best agreement with experimental -values, a barrier height for protein L of 16 kJ/mol was obtained (Ejtehadi et al., 2004). Other proteins such as FKBP and CI2 had larger barriers of 25 kJ/mol and 42 kJ/mol, respectively (Ejtehadi et al., 2004). The large barriers observed here also suggest that many-body interactions may be playing a significant role in the energy function. A variational theory for the free energy surface of -repressor gave a barrier of 12 kJ/mol (Portman et al., 2001). All-atom simulations of a three-helix bundle fragment of protein A in explicit water gave barrier heights 17 kJ/mol at the transition midpoint (Garcia and Onuchic, 2003). Applying Kramers theory with an experimentally determined estimate for the prefactor gave an estimate for the free energy barrier of 18 kJ/mol for the cold shock protein CspTm (Schuler et al., 2002). An analysis which took prefactors from experimental data, along with a thermodynamic analysis to extract enthalpic and entropic contributions to the barrier, gave typical barrier heights of 30 kJ/mol for the proteins analyzed (Akmal and Munoz, 2004). However, these last two methods found barrier heights under conditions of zero denaturant—the barrier heights at zero stability would likely be significantly higher. For example, the average (m_U/m)G for the proteins in Table 1 is 17 kJ/mol, to be added to the barrier height at conditions of zero denaturant.

Applying this method to a simulation model, where one knows the answers in advance, provides a good control for the study and is a topic for future work.

	ACKNOWLEDGEMENTS

TOP ABSTRACT INTRODUCTION METHODOLOGY RESULTS CONCLUSIONS AND DISCUSSION ACKNOWLEDGEMENTS REFERENCES

 
We thank José Onuchic, Reza Ejtehadi, Magnus Lindberg, and Matthias Huber for helpful discussions, and M. Oliveberg for sharing unpublished data.

S.S.P. acknowledges support from the Natural Sciences and Engineering Research Council and the Canada Research Chairs program.

Submitted on September 9, 2004; accepted for publication March 2, 2005.

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