f(phi)-value analysis

f-value analysis is a method to understand the effects of a mutation in a protein upon the free energy of the individual native state, denatured state and transition state.

Fersht, A. R., Matouschek, A. & Serrano, L. (1992). The folding of an enzyme I. theory of protein engineering analysis of stability and pathway of protein folding. J. Mol. Biol. 224, 771-782.

I. The free energy diagram for two-state protein folding

A simple two-state, reversible, protein folding process can be represented as:

N Û D

Where, N is the native (folded) state, and D is the denatured (unfolded) state.

The following diagram represents the free energy of the native and denatured forms of a protein under conditions where the native state is favored (e.g. 0M denaturant, physiological pH, room temperature, etc.):

From the above diagram we can conclude the following:

• The native state (N) has a lower free energy than the denatured state (D) (in fact, the native state appears to be the global energy minimum). The system will spontaneously adopt an equilibrium that favors the native state.
• The free energy difference between the N and D states (DG_D-N) is a measure of the stability of the protein

• In this case, DG_N®D is defined as the free energy change in going from state 1 (N) to state 2 (D).
• "Delta" values are always defined as (value of state2 - value of state1), thus DG_N®D = (DG_D - DG_N) or DG_{D - N}
• From the above diagram this value should be a positive value
• A positive value indicates a non-spontaneous process as written (i.e. going to the right). Thus, N Û D is non-spontaneous in the right-hand direction, but the reverse direction is spontaneous as written (thus, the above diagram reflects a situation where the equilibrium favors the native state of the protein)

II. Rates of folding and unfolding and the free energy diagram

The folding transition state, ‡ , (which is actually likely to represent an ensemble of structures) describes the energy barrier between the N and D states, and determines the rate at which the N state converts to D (unfolding process), and the rate at which the D state converts to N (folding process)

• The energy barrier for unfolding is proportional to the height between the N®‡ states
• The energy barrier for folding is proportional to the height between the D®‡ states
• The rate at which system can change states is inversely proportional to the height of the energy barrier (i.e. the larger the energy barrier, the fewer molecules in the sample that will have the necessary energy to overcome the barrier, and therefore, the slower the overall rate of change of state)

In the above diagram, the N®‡ energy barrier (DG_{‡ - N}) is larger than the D®‡ energy barrier (DG_{‡ - D}), thus:

• The rate of unfolding should be slower than the rate of folding
• This situation results in an equilibrium condition that will favor the N state (i.e. the above diagram is representative of a protein that is stably folded)

The rates of folding and unfolding are a function of the rate constants for folding (k_f) and unfolding (k_u), thus, we have an overall diagram that looks like this:

III. Equilibrium denaturation methods

Equilibrium denaturation experiments report the extent of denaturation as a function of added denaturant (isothermal equlibrium denaturation by guanidine or urea), or added heat (differential scanning calorimetry).

• Such experiments provide information on the equilibrium constant for denaturation, and therefore, DG_{D - N}:

DG° = -RTln(K_eq)

• These experiments assume the system is always at equilibrium (e.g. samples are allowed to come to equilibration prior to measurement), thus, they provide no information on the folding or unfolding kinetics (i.e. folding or unfolding rate constants)
• Thus, using these methods alone, we cannot say what the effects of mutations are upon the kinetic properties

Example I:

The mutant protein free energy diagram is shown in the green broken line, and the free energy of the native, denatured and transition states is indicated by an asterisk.

• The mutation has affected the transition state. In particular, it has stabilized the transition state and has had no effect upon either the native or denatured states
• In stabilizing (i.e. lowering the free energy of) the transition state, the mutation will result in an increase in both the rate of unfolding and folding
• However, since the free energy levels of the native and denatured states are unchanged, the overall DG_{D - N} value is unchanged.
• Thus, equilibrium denaturation methodologies would report no difference between the mutant and wild type proteins, but, kinetic experiments would clearly indicate that the mutation has altered the rates of folding and unfolding.

f-value analysis compares the free energy data from equilibrium denaturation methodologies to free energy values derived from kinetic studies, and this comparison allows a determination of how a mutation has affected the free energy of the native, denatured or transition states of the protein

IV. Probing the structure of the transition state

Example II:

A mutation (indicated by an asterisk *) that does not affect the denatured state, or the transition state, but destabilizes the native state:

In this example:

· The values of the folding rate constants, k_f and k*_f, for wild-type and mutant are observed to be equal, therefore, DG_{‡ - D} and DG*_{‡ - D} are identical in value and therefore the value of DDG_{‡ - D} = 0 (i.e. DG_{‡ - D} - DG*_{‡ - D} = 0)

Note: it may seem that when you make a mutation that the mutant should be considered the "new state" (i.e. state 2) in comparison to the wild type "previous state" (i.e. state 1). Thus, any delta values relating a mutant to wild type should be of the form: (mutant value - wild type value). However, there is no strict adherence to this frame of reference, and effects of mutations are commonly calculated by subtracting mutant values from the wild type. The key thing is that you explicitly state how you are calculating the values for the mutant in terms of the wild type protein when you report the relevant delta values.

· The unfolding rate constants are different between mutant and wild-type (faster for the mutant). Thus, DG_{‡ - N} values are different and the value of DDG_{‡ - N} (i.e. DG_{‡ - N} - DG*_{‡ - N}) is non-zero (positive in this case).

· The value of DDG_{‡ - N} can be determined from the wild type and mutant folding rate constants:

(Note: DDG_{‡ - N} is also referred to as DDG_unfolding or DDG_u)

· The DDG_{D - N} value for the mutant (i.e. the effect of the mutation upon stability) is determined experimentally using isothermal equilibrium denaturation data (at the same temperature as the kinetic studies, or using DSC data with DDG value determined by extrapolation of individual DG values to the temperature used for the kinetic experiments).

• Note that in the above case, the DDG_{D - N} value is equal to the value of DDG_{‡ - N}. In other words, it looks like if you make a mutation that affects the stability of the protein, and if this effect is characterized by changes upon the folding rate constants only, then the mutation has affected primarily the native state.

If then it means that the energetic changes between the wild type and mutant native states accounts for the entire energetic difference observed in the equilibrium stability study (and we conclude that the mutation has affected the native state exclusively)

• If a mutation affects the native state and transition states equally, then it is assumed that the mutation site is as folded in the transition state as it is in the native state (i.e. the mutation site adopts the native configuration in the transition state)
• If a mutation affects the denatured state and transition states equally, then it is assumed that the mutation site is as unfolded in the transition state as it is in the denatured state (i.e. the mutation site adopts the denatured configuration in the transition state)
• In the above example, the transition state and the denatured state are unaffected by the mutation - in other words, the mutation has affected these states equally - thus, we would conclude that the site of mutation is as unfolded in the transition state as it is in the denatured state

• In this case, the perturbation of the mutation upon the denatured state is equivalent to the perturbation of the transition state. Thus the site at this position is presumably unfolded in the transition state, or to put it another way – the site of mutation is as unfolded in the transition state as it is in the denatured state.

Example III:

In this example:

· The values of the unfolding rate constants, k_u and k*_u, for wild-type and mutant are observed to be equal, therefore, the values of DG_{‡ - N} and for DG*_{‡ - N} are identical and the value of DDG_{‡ - N} = 0

· The folding rate constants are different between mutant and wild-type (faster for the wild-type). Thus, DG_{‡ - D} values are different and the value of DDG_{‡ - D} is negative.

(Note: DDG_{‡ - D} is also referred to as DDG_folding or DDG_f)

• Note that in the above case, the DDG_{D - N} value is equal to the value of DDG_{‡ - D}. In other words, it looks like if you make a mutation that affects the stability of the protein, and if this effect is characterized by changes upon the unfolding rate constants only, then the mutation has affected exclusively the denatured state.

If then it means that the energetic changes between the wild type and mutant denatured states accounts for the entire energetic difference observed in the equilibrium stability study (and we conclude that the mutation has affected the denatured state exclusively)

• For this example, the perturbation of the mutation on the transition state is equivalent to the perturbation upon the native state. Therefore, the site of mutation is as folded in the transition state as it is in the native state.

V. Ambiguity…

Let’s revisit example II:

Let’s say we did not a priori know the exact effect of the mutation on N, D and the transition state, and only had data on the folding and unfolding rates. Here would be another possible interpretation that yields exactly the same kinetic result:

In this case, although the folding and unfolding rates would be the same as above, the interpretation is different. In this case, the mutation has not affected the N state, but has stabilized both the transition and D states to the same extent. Which interpretation is correct? In fact, we can shift the mutant free energy profile vertically and get the same kinetic effects. Thus, we are in a dilemma, what exactly is going on? This points to a limitation of the interpretation; using kinetic analysis, we can only make conclusions regarding the relative change in free energy of the N, D and transition states. If we have other data – in particular structural data, we may be able to make some conclusions as to whether the mutation has affected the N state of the mutant in comparison to the N state of the wild type; alternatively we might be able to rationalize some expected energetic effect upon the D state but not the N state (for example Ala to Pro mutations may not appreciably affect the structure or entropy of the N state, but would decrease the entropy of the D state – thus destabilizing the D state. Note that example III similarly has alternative, but kinetically identical, interpretations.

VI. f value analysis

The basis of f value analysis is to compare the overall free energy change for a mutation to the individual contributions of the folding and unfolding free energy change. The analysis usually is focused upon understanding whether a particular mutation site is folded or unfolded in the transition state (and in this way probes the "structure" of the transition state). This is valid regardless of the ambiguity discussed in section 5, because it is comparing the change in free energy of the transition state to either the N or D states, and is not concerned with absolute free energy values for these states.

• The DDG_{D - N} value is determined by denaturation equilibrium methods
• The value is determined from the unfolding kinetic constants of the mutant and wild type
• The value is determined from the folding kinetic constants of the mutant and wild type
• is called the "folding f value". If it equals 1.0 it means that the site of the mutation is native-like in the transition state (i.e. the site of mutation is as folded in the transition state as it is in the native state). A value of 0 means the opposite.
• is called the "unfolding f value". If it equals 1.0 it means that the site of the mutation is denatured in the transition state (i.e. the site of mutation is as unfolded in the transition state as it is in the denatured state. A value of 0 means the opposite.
• Fractional , or negative, values for f values are more difficult to interpret

Usually the choice of either f_f or f_u is based upon whether folding kinetic data or unfolding kinetic data can be more accurately determined.

VI. Cross-validation

DDG_{D - N} values can also be determined from the kinetic values:

• If the two-state model correctly describes the protein denaturation, then the above value should agree with the value from equilibrium denaturation studies

The above equation also suggests that If unfolding kinetic data for a mutant can be obtained, but folding kinetic data cannot, it can be predicted by comparing the equilibrium DG data and the known kinetic data:

VII. Derivations

The rate of folding is proportional to the free energy difference between the denatured state and the transition state:

Assumptions:

· k = 1.0

·

Boltzmann's constant, k_b = 1.380 x 10^-23 J K^-1

Temperature in K

Planck's constant, h = 6.626 x 10^-34 J sec

· n has units of (J K^-1)*(K)/(J sec) = sec^-1 (appropriate for a rate constant)

Calculation of DDG_‡-N from experimental rate constants of unfolding:

Assuming n and k are the same in both cases:

DDG_‡-D follows a similar derivation.

Chevron Plots

Denaturants such as urea and guanidine can unfold a protein. The addition of denaturants to a protein decreases the rate of folding and increase the rate of unfolding. Since the protein concentration is not altered, the observed rate change must be due to a change in the rate constant, k, as a function of denaturant:

The folding rate constant decreases exponentially as a function of added denaturant. Thus, the log (or, more commonly, natural log) of the rate constant as a function of denaturant is a linear function:

The value of ln(k_f) as a function of denaturant is a linear function:

y = m*x + b

ln(k_f) = m_f*[Denaturant] + ln(k_f0)

• The y intercept ln(k_f0) is the natural log of the value of the folding rate constant at 0M denaturant (i.e. assumed to be the physiologically-relevant rate constant)
• The slope is m_f, the so-called "folding m-value". This term describes the rate at which the folding rate constant decreases with added denaturant.

The expression for ln(k_f) can be converted to k_f by taking the exponential e of both sides, yielding:

k_f = k_f0*exp(m_f*[D])

The unfolding rate constant (referred to as k_u to distinguish it from the folding rate constant, k_f) increases exponentially as a function of denaturant concentration, and similarly, the natural log of ku as a function of denaturant concentration is a linear function:

And similar to k_f, the value of k_u is:

ln(k_u) = m_u*[Denaturant] + ln(k_u0)

k_u = k_u0*exp(m_u*[D])

Under conditions of low denaturant concentration, the native state is favored, and this is due to the relative magnitude of k_f compared to k_u. However, under high denaturant concentrations the denatured state is favored due to the relative magnitude of k_u compared to k_f. A global treatment of the relative effects of both rate constants is achieved by defining k_obs, the observed rate constant:

k_obs = k_f + k_u

k_obs = k_f0*exp(m_f*[D]) + k_u0*exp(m_u*[D])

ln(k_obs) = ln(k_f0*exp(m_f*[D]) + k_u0*exp(m_u*[D]))

ln(k_obs) is plotted as a function of denaturant concentration to produce a so-called "Chevron plot" of folding and unfolding kinetic data: