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Water mobility, denaturation and the glass transition in proteins
Biochimica et Biophysica Acta 1824 (2012) 785–791
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Water mobility, denaturation and the glass transition in proteins David Porter ⁎, Fritz Vollrath Dept. of Zoology, University of Oxford, Oxford, OX1 3PS, UK
a r t i c l e
i n f o
Article history: Received 18 January 2012 Received in revised form 12 March 2012 Accepted 15 March 2012 Available online 24 March 2012 Keywords: Protein hydration Denaturation Glass transition Lysozyme Myoglobin DFT
a b s t r a c t A quantitative mechanism is presented that links protein denaturation and the protein–water glass transition through an energy criterion for the onset of mobility of strong protein–water bonds. Differences in the zero point vibrational energy in the ordered and disordered bonded states allow direct prediction of the two transition temperatures. While the onset of water mobility induces the same change in heat capacity for both transitions, the order–disorder transition of denaturation also predicts the observed excess enthalpy gain. The kinetics of the water and protein components through the glass transition are predicted and compared with dielectric spectroscopy observations. The energetic approach provides a consistent mechanism for processes such as refolding and aggregation of proteins involved in protein maintenance and adaptability, as the conformational constraints of strong water–amide bonds are lost with increased molecular mobility. Moreover, we suggest that the ordered state of peptide–water bonds is induced at the point of protein synthesis and could play a key role in the function of proteins through the enhancement of electronic activity by ferroelectric domains in the protein hydration shell, which is lost upon denaturation. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Two main transition processes are associated with the structure, dynamics and activity of proteins: denaturation and the protein-water glass transition. While denaturation is usually associated with loss of activity through unfolding or aggregation of the protein at temperatures above about 330 to 340 K [1], conversely, the dynamics of hydrated proteins should increase above temperatures around 180 K that is attributed to the protein-water glass transition [2,3]. Since control of biological structure and activity is linked to both these processes, it is interesting that these two processes have not been related at either the functional or molecular level, even to the point of whether the two processes are mutually exclusive or one process might lead reversibly or irreversibly to the other after a given time-temperature history. Over a long period of time, there has been considerable interest in the detailed role of protein instability in important biological areas such as protein misfolding and maintenance, with more recent interest in adaption for medicine and evolutionary biology [4–6]. Given the general importance of protein denaturation as the most significant instability process, it is not surprising that an enormous body of experimental and theoretical literature on the subject is available, and reviews such as Robertson and Murphy give clear dissemination of key observed thermodynamic features [7]. Models for instability range from empirical thermodynamic [8,9] to detailed molecular dynamics simulations [10,11]. However, quantifying an underlying
⁎ Corresponding author. Tel.: + 44 1865271216. E-mail address: [email protected] (D. Porter). 1570-9639/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.bbapap.2012.03.007
electrochemical mechanism to test hypotheses about the onset of instability, the resultant torsional bond changes, and the consequent reformation of new stable and active protein structures seems to have received little attention. At the simplest qualitative level, a reversible unfolding of a native protein is initiated at a denaturation temperature, Ti, which is followed by a slower irreversible step (in many cases aggregation) to yield a final state that is unable to revert back to the native folded structure [12]. At the quantitative level, it is fair to say that at present the general view is that we cannot predict ab initio either the initiation conditions, such as Ti, or the thermodynamic parameters, such as enthalpy and entropy through the denaturing transition in terms of molecular structure. The same comments apply also to the peptide-water glass transition, Tg, as another instability that has received much recent attention [2,3]. The review by Doster gives an excellent overview of the current status of ideas on this specific form of Tg and links theories to a wide range of experimental observations such as the characteristic frequencies of the analytical techniques, but again does not give ab initio models to predict initiation conditions, thermodynamics, or kinetics for experimental observations and does not show how Tg might be linked to denaturation. Although simulation techniques such as molecular dynamics may reproduce some of the features of denaturation [10] and Tg [11], it must be remembered that such techniques are mainly based upon empirical forcefields that can be adjusted for just that purpose of reproduction, not ab initio prediction; particularly in complex systems containing water and highly polar proteins with complex hydrogen bonding [13]. Modelling large assemblies of molecules also gives simulation results that are essentially equivalent to experimental observations, and extraction
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of key atomic level events and their characteristic time scales that are the root cause of a particular effect is sometimes difficult. We suggested a relatively simple quantitative explanation of denaturation by calculating the transition energy from a potential energy well for specific peptide-water interactions that had been derived using density functional theory, dft [14]. The potentially reversible transition is located at the Born elastic instability point as the energy where the force between molecules is a maximum, or the stiffness between the interacting groups is zero, so that the water is free to move relative to the peptide [15,16]. At that point, the stronger peptidepeptide hydrogen bonding can form as peptides collide without water screening to form irreversible aggregates. This approach successfully estimates the denaturation temperature, but does not make any links either to structural disordering normally associated with denaturation, the detailed thermodynamics of the transition, and only makes reference to Tg processes as the bulk effect of water on observed thermomechanical processes, which was also developed by Fu for silk fibres using mean field modelling methods [17]. It is important not to confuse this bulk thermomechanical softening transition with the specific protein–water instability transition discussed in this work. The two effects must, inevitably, be linked due to the mobility of the tightly bound water, but here the specific peptidewater instability is taken to precede the bulk glass transition, and act as a rate limiting step for the bulk process through the onset of mobility of the water molecules. Here, we extend the basic denaturation model to the protein– water glass transition and make a direct quantitative link between the two, which includes the thermodynamics of the transitions and a first attempt at their kinetics as frequency-temperature relations. We first discuss the more general examples of transitions in polymers and water, where both the crystal melt and glass transition temperatures can be predicted in self-consistent models. 2. Mobility and transitions In order to quantify dynamic transitions in materials, we adopt a very simple description of a transition as the onset of mobility of molecules relative to each other [16,18]. This can occur when the stiffness (second differential of interaction energy with respect to bond length) of the intermolecular bonds tends to zero, and is known as the Born elastic instability condition [15,16]. Mathematical processing of the potential energy well for the interactions usually gives an energy at which this condition occurs, and this non-minimum energy value is attained by the combination of the zero point vibrational energy, Ho, and the energy of thermal vibrations of the molecular segments, HT, which can be quantified by Einstein and Debye theories [14,18]. It is important to note here that this energy condition for bond mobility is very different from bond strength as the energy to completely dissociate a bond. The hypothesis tested in this paper is that denaturation and the protein–water glass transition are two manifestations of the same transition condition of water molecules that are strongly bonded to the amide segments of a protein backbone chain by hydrogen bonds. The glass transition is defined as disordered, and denaturing is a change from an ordered to a disordered state. Therefore, the first problem here is to quantify order and disorder in the protein-water interaction energy condition for a transition to occur. Previous work has shown that order and disorder can be quantified for polymers and small molecules by the zero point vibrational energy term, Ho, and has been used to predict thermomechanical properties of crystal and amorphous forms [14,18]. The frequency of the molecular vibrations, νo, is calculated from the potential energy well or by using molecular modelling, and Ho values for the ordered and disordered states are defined by the first two quantised energy levels of hνo/2 and 3hνo/2 per vibration mode for ordered and disordered states respectively, where h is Planck's constant. Since the
disordered value of Ho is higher, then the thermal energy (temperature) to reach the transition condition must be lower. Also, for a given family of materials with characteristic forms of the potential energy well, the ordered and disordered transition temperatures usually have a consistent ratio of values, such as polymers where the crystal melt temperature is about 1.5 times the glass transition temperature [19]. Polymers are an elegant and well validated example of this approach to predicting transitions [18]. The energy of molecular interactions, E, is adequately described by the simple 6–12 Lennard-Jones potential energy well in terms of intermolecular distance, r, with a depth given by the cohesive energy, Eo. The transition condition, d 2E/dr2 = 0, occurs when the positive energy terms Ho + HT = 0.213Eo, such that the transition condition is less than a quarter of the total bonding energy, Eo. Using a version of the Debye model for heat capacity adapted for chain molecules such as polymers, simple and self-consistent analytical expressions can be derived for the crystal melt temperature, Tm, and the glass transition temperature, Tg, in terms of their theta temperature of skeletal mode vibrations, θ1, their cohesive energy, Eo, and the number of skeletal degress of freedom, N [18].
Tm Tg
¼ 0:224θ1 þ
0:084 0:0513
Eo N
ð1Þ
Although this polymer model has been successfully applied to proteins and the effect of hydration water as a plasticiser on amorphous bulk proteins [17], the Ti and Tg transitions in this work involve specific and strong hydrogen-bonded peptide-water interactions that need to be considered as a special case. A relevant example is that of water-water interactions, which has been presented in detail in a previous paper and is outlined here to illustrate the principles of the model [14]. We used dft quantum methods (a summary of the method is given in the Appendix A) to calculate a transition energy condition for individual water–water hydrogen bonds of 4.4 kJ mol − 1 per H-bond from the potential energy well, which was also used to calculate a vibrational frequency and associated theta temperature of νo = 2.5 × 10 12 Hz and θ = 107 K: note that our value of the pairwise water-water hydrogen bonded energy was 20 kJ/mol, in agreement with recent published values also calculated using dft simulations [13], and the transition energy to bond energy ratio of 0.22 is coincidentally similar to that of 0.213 for Lennard–Jones interactions in polymers. The water–water bond was considered as a simple Einstein oscillator with two degrees of freedom per vibration, N = 2, with the two different quantised energy levels for Ho and the thermal energy, HT, is given in terms of the gas constant, R, by HT ¼
NRθ exp Tθ −1
ð2Þ
The two zero point energy values predict Tm = 260 K and Tg = 150 K for the crystal melting point and glass transition temperature of water, which are in good general agreement with observed values of 273 and 140 K, respectively, and illustrated in Fig. 1. Since the bonding mechanisms are very similar for water-water and water-peptide interactions and the transition temperatures are of the same general order, it seems appropriate to apply the same approach to the problem of denaturation and protein-water glass transitions. 3. Protein–water transition temperatures As mentioned in the introduction, we made a first limited suggestion for predicting Ti by formulating the hypothesis that denaturing is initiated by the onset of mobility of specific tightly-bound hydrogenbonded water molecules to the polar amide groups in the backbone chain of the protein macromolecules: this bonding is shown in the
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Temperature (K) Fig. 1. (a) Potential energy well for water-water interactions with energy levels marked, and (b) prediction of Ti and Tg using the energy levels for ordered and disordered states.
insets to Fig. 2, where the water molecule bridges and constrains the ϕ and ψ torsional angles [14]. Using density functional theory (dft) calculations of the energy of a water molecule as a function of distance from amide groups in the simplest polyglycine chain, we calculated an elastic instability point where the force between the molecules reaches a maximum, such that the stiffness of the intermolecular bond is zero and the water is free to move relative to the larger and less mobile protein. Fig. 2 shows the results of such simulations, with a mean instability energy of Ei = 5.7 kJ/mol water at a separation distance of about 0.4 Å. The full binding energy of the water-amide bond is predicted to be about 59 kJ/mol for beta-type conformation torsional angles in the protein chain, which appear to be an excellent
10
topologial match for the hydrogen bonded water bridge. The difficulties of modelling hydrogen bond interactions of this kind are discussed in detail by Wendler [13] with particular reference to cooperativity, also discussed by others for peptide interactions, including the bonding used here [20,21]. To convert this instability condition into an instability temperature, the water–amide interaction is considered as an Einstein oscillator with a reference theta temperature, θ = 150 K, which is calculated from the relation hνo ≡ kθ (where h and k are the Planck and Boltzmann constants, respectively) using a frequency of νo = 3.5 × 1012 Hz from the bond stiffness of 15 Nm− 1 and mass of the vibrating water molecule 18D as a simple harmonic oscillator at the minimum of the potential energy well of Fig. 2, and is discussed in detail in reference [14]. Fig. 3 shows energy as a function of temperature for the Einstein oscillator of Eq. (2) in two states, designated here as ordered and disordered, which have two different zero point energy values of NRθ/2 and 3NRθ/ 2 (again using hνo ≡ kθ to convert from hνo/2 and 3hνo/2, respectively) for the quantised oscillator respectively, where R is the molar gas constant and N = 2 for two vibrating hydrogen bonds per water-amide interaction. The upper ‘ordered’ state has a denaturation transition temperature Ti = 336 K and the new disordered state has a lower instability glass transition temperature, Tg = 180 K, with both temperatures agreeing well with experimental observation [1,3]. Importantly, both transitions are different states of the same instability condition, but with two clearly distinguished quantised zero point vibrational energy levels. This is the first direct quantitative link between Ti and Tg with a clear physical mechanism of the energetics of peptide–water bonding. It also tells us that the torsional bonds ϕ and ψ are liberated from the constraint of the water bonding bridge above these temperatures for the onset of chain dynamics and potential randomisation of the protein chain. However, this randomisation points to important differences between the two processes: denaturation may change from order to disordered states, whereas Tg remains disordered with no change. 4. Thermodynamics 4.1. Denaturation With these transition temperatures and a quantitative mechanism for denaturation, we can now predict the thermodynamic energetics of protein instability. At the transition, water changes from a bound oscillator to a freely mobile molecule. With this change of mobility, we would expect a heat capacity change of 6 new degrees of freedom per water molecule (three translational and three rotational). The associated change in heat capacity is ΔC = 6R ≈ 50 J/K/mol, and is associated
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Temperature (K) Fig. 2. Energy and force as a function of separation distance for a water–amide bond, with the elastic instability condition marked by a dashed line, predicted for the structures shown as an inserts using dft simulations.
Fig. 3. Predicted energy–temperature relations for the ordered and disordered states of an Einstein oscillator with θ = 150 K and zero point energies NRθ/2 and 3NRθ/2.
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more specifically with each water–amide segment interaction. This first approximation does not include any changes to the relatively immobile protein macromolecule, but we would expect a minor contribution as chain mobility increases. As a first suggestion, the change in heat capacity should be the same for both Ti and Tg. However, we have already noted that Ti may change from an intrinsically ordered to disordered state, without any structural constraints such as interchain crosslinking or strong hydrogen bonding between peptide side chains. If this order–disorder change occurs, there must also be a change in the zero point vibrational energy and kinetics, which effectively means an increase in enthalpy per peptide segment, which is observed as an excess enthalpy peak through the transition, ΔHi . To calculate ΔHi , we need to know both the change in zero point energy, ΔHo = NRθ, and the temperature interval over which the heat capacity changes, ΔTi, which is experimentally observed to be about 20 degrees, as shown in Fig. 4. ΔH i ≈ΔHo þ ΔC⋅ΔT i ≈3:5kJ=mol
ð3Þ
Thus, as the molecule becomes mobile, it gains an extra energy increment equivalent to ΔHi ≈ 3.5 kJ/mol per peptide segment through the transition region. The same procedure is applied to predict the change in entropy ΔS ≈ 10.4 J/K/mol through the denaturing transition. These parameters then allow calculation of the Gibbs free energy, which becomes a relatively small energy term. C S ¼ ∫ dT ¼ 10:4 J=K=mol peptide T
ð4Þ
These predictions agree quite well with the average observed values of ΔC = 58 ± 2 J/K/mol, ΔS = 8.8 ± 3 J/K/mol and ΔH = 2.92 ± 0.08 kJ/ mol per peptide segment normalised to 60°C [7], bearing in mind that proteins have many other structural features, such as polar side chains and potentially denatured or non-hydrated segments. Looking more specifically at a common benchmark protein, hen egg lysozyme has 123 peptide segments and Fig. 4 compares our predicted DSC scan for the change in thermodynamic properties through denaturation with experimental observation in the form of heat capacity as a function of temperature: observed characteristic values are ΔH = 427 kJ/mol, ΔC = 6.4 kJ/K/mol, and ΔS = 1.27 kJ/K/mol [7,22], relative to predicted values of ΔH = 430 kJ/mol, ΔC = 6.2 kJ/K/mol, and ΔS = 1.28 kJ/K/mol at Ti = 64 °C. These predictions embody both thermodynamic and kinetic features of the denaturing transition using parameters calculated directly from the potential energy well of Fig. 2. For the model energy
distribution in Fig. 4, a normal distribution with a standard deviation of 3.5 degrees fits the experimental data using the predicted thermal energy values; we suggested a predicted general form for this distribution previously, based upon activation probablity and kinetics [14].
4.2. Protein–water glass transition The thermodynamics of the protein–water glass transition from calorimetry experiments are outlined in the review by Doster, who points out that the change of heat capacity for the water component of hydrated myoglobin at the glass transition is the same as suggested here of 6R per mole of water for denaturation [3]. The glass transition distribution is broader than the well defined ordered transition shown in Fig. 4, as would be expected for a disordered process with a range of local bonding conformations and environments. Again, we can model this increase in heat capacity quite simply as a change in the water from constrained vibrations in a solid (3 degrees of freedom) to free mobility of the liquid molecules (an extra 6 degrees of freedom) through the glass transition. The total 9 degrees of freedom gives the heat capacity of liquid water as the experimentally observed value of 9R = 76 J/mol/K or 1 cal/g/K. The slightly higher observed value of about 1.1 cal/g/K at higher temperatures shown in the points in Fig. 5 for myoglobin proteins is probably due to the contribution of protein macromolecular degrees of freedom, which is observed as mobility in neutron scatting experiments, for example. The model for change in heat capacity, Cp, is shown as a line in Fig. 5 and the change in degrees of freedom is a cumulative Gaussian distribution with a mean Tg = 195 K and a standard deviation of 20 degrees for a best fit, where the peak width is of the same order as a secondary relaxation peak at about 200 K in a silk protein tested by dynamic mechanical thermal analysis [23]: we are not currently able to calculate the distribution width from first principles due to the complexity of the effects. This increase in heat capacity is superimposed upon the predicted heat capacity of the same Einstein oscillator model with three degrees of freedom as used for the ordered denaturation transition. There is no change of state in the disordered solid–liquid transition, so no excess enthalpy is expected that is seen in the denaturation transition in Fig. 4. It is interesting to note here that the thermal energy, HT, has two regions above and below Tg that could be very loosely described as being approximately linear in temperature, but offset by temperature To ≈ 150 K [24], with proportionality to T–To at higher temperatures. This has important consequences for the apparent form of the kinetics relations, as we will show below.
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o
Temperature ( C) Fig. 4. Comparison of predicted (solid lines) and observed (dashed lines [7,22]) profiles of heat capacity through the denaturing transition for hen egg lysozyme. Dashed baseline (− ⋅ − ⋅ −) is taken from the experimental data [7].
0
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Temperature (K) Fig. 5. Comparison of predicted (line) and observed (points [3]) change in heat capacity through the protein-water glass transition for myoglobin, together with the predicted enthalpy increase, HT, as a function of temperature.
D. Porter, F. Vollrath / Biochimica et Biophysica Acta 1824 (2012) 785–791
5. Kinetics The problem of differentiating between different transition mechanisms is further complicated by their kinetics. Observations on the temperature dependence of characteristic relaxation times with a range of different analytical techniques suggest numbers of different processes that can be associated with water, protein-water interactions, or the protein mobility. Khodadadi simply states that “it is not clear what is the main structural relaxation of the hydrated protein that freezes around Tg ≈ 180 K” [25]. The points shown in Fig. 6 for lysozyme using dielectric spectroscopy and neutron scattering give consistent profiles for what are reasonably argued to be two different protein–water glass transition processes [25]. Fig. 6 also includes available dielectric data for water as both amorphous water at high pressure [26] and over the whole temperature range from Cappaccioli [27] to support the suggestion that the hydration water is the main contribution to the dominant faster form of relaxation [3]. The classical Arrhenius function for time-temperature effects is essentially a reference rate for unconstrained molecular vibrations, νo, that is scaled by an activation probability in terms of thermal energy relative to an activation energy, ΔH, for that rate to be active. At the simplest level, thermal energy becomes RT per vibrational mode, but for the more complex thermodynamics of the protein-water glass transition, we suggest simply using the thermal energy HT as a function of temperature in a modified Arrhenius for a frequency, f, with a form that was used successfully to model thermal degradation in proteins [28] −ΔH f ¼ ν o exp HT
ð5Þ
Relaxation Time [s]
Taking νo = 3.5 × 10 12 Hz from the potential energy well calculations for peptide–water interactions, and HT directly from Fig. 5, the line (“increased mobility model”) plotted in Fig. 6 for relaxation time, 1/f, uses an activation energy of 57 kJ/mol (or 750 cal/g to be consistent with reported thermodynamic data), which corresponds to the binding energy of the water–amide interactions minus Ho, consisting of two hydrogen bonds. Thus, the increased thermal energy due to the onset of water mobility at Tg increases the relaxation rate above that of a simple Arrhenius in RT, and thus resolves the dilema of the main structural relaxation process as the hydrated protein unfreezes at Tg ≈ 180 K, mentioned above, while maintaining a relaxation time of about 1 s at the glass transition temperature of water at
1.E+02 1.E+01 1.E+00 1.E-01 1.E-02 1.E-03 1.E-04 1.E-05 1.E-06 1.E-07 1.E-08 1.E-09 1.E-10 1.E-11 1.E-12 100
Main dielectic [25] Neutron Specroscopy [25] water 1GPa [26] Water dielectric [27]
Increased mobility model Slower dielectric [25] Polymer model
150
200
250
300
Temperature [K] Fig. 6. Relaxation time as a fuction of temperature for the protein-water glass transition. Points are experimental observations as specified and the lines are predicted by the modified Arrhenius function of Eq. (5) for the water and hydrated protein responses.
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about 140 K. We see now that the proportionality of HT with (T–To) gives the relaxation function a Vogel–Fulcher form, rather than simple Arrhenius [24] and the frequency-temperature relations are smooth and continuous through the glass transition, which has previously led to claims of no dynamic transition [3], and resolves some important inconsistencies in the water-protein glass transition observations. However, the conclusion here is that the faster relaxation process is simply water that has its mobility constrained by bonding to peptide. Fig. 6 also shows the ‘slow dielectric’ response from the same experimental data set as the main dielectric points [25]. This tells us that there is a much slower set of dipoles that relax relative to the main process quantified above for water. Since the transition in question is the mobility of a protein-water interaction, it seems reasonable that a dipolar relaxation process associated with mobility of the hydrated protein should accompany that of the water, which is most likely to be that of the strong amide dipole involved in the water interaction. The two relaxation processes identified here are a specific example of the same effect seen in a wide range of solution mixtures, including aqueous systems, where the faster one is a local specific relaxation of hydration water, similar to the beta-relaxation in supercooled liquids and mixtures, and the slower one is related to structural cooperative motions of protein segments and hydration water together [29]. The objective in this work is simply to quantify the mechanism of the specific peptide-water interaction glass transition process, which then becomes a rate limiting step for water mobility for the bulk glass transition events of the hydrated protein. For simplicity, take Eq. (5) with interaction energy value of ΔH = 57 kJ/mol, which is similar to the value of 55 kJ/mol suggested from observations by Capaccioli for more general aqueous systems [29], but use the thermal energy, HT, for a protein polymer with N = 2 degrees of freedom per water interaction and a typical skeletal mode theta temperature of θ1 = 400 K for νo = 9 × 10 12 Hz in the expression [14,18] θ −1 6:7T H T ¼ NR T− 1 tan θ1 6:7
ð6Þ
The line ‘polymer model’ in Fig. 6 shows the simple model predictions relative to experimental observations, which are in reasonably good agreement and made easier by the temperature range not including the Tg effects on thermal energy. Thus, we can now argue that both water and hydrated protein components of the protein-water glass transition can be identified separately in a self-consistent framework. Unfortunately, analysis of kinetics by means of techniques such as dielectric spectroscopy in Fig. 6 is made difficult for the denaturation transition, since the most common way to report rate effects associated with denaturation is in the enzyme reaction rate, which is difficult to uncouple from other factors involved in chemical reactions. Since denaturation is essentially an irreversible process of unfolding or aggregation, we have not yet found a suitable way to validate any model predictions for relaxation time parameters. Since the model for water relaxation for Tg in Fig. 6 using the thermal energy function, HT, shown in Fig. 5 seems to work quite well, Fig. 7 shows predictions of the relaxation times for denaturation compared to Tg, with the inset showing the appropriate form of the thermal energy, HT, from the heat capacity in Fig. 4 and scaling the thermal energy for the polymer model in Eq. (6) by 0.75 below Ti to account very approximately for the decreased mobility due to the water bonding constraints [18]. However, this is clearly just a modelling exercise to illustrate a principle, since dielectric spectroscopy suggests that characteristic relaxation time actually increases as temperatures approach Ti due to the tendency to aggregation and unfolding, which according to the authors (Thomas et al) “does not conform to any known model” [30].
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Relaxation Time [s]
790
1.E+02 1.E+01 1.E+00 1.E-01 1.E-02 1.E-03 1.E-04 1.E-05 1.E-06 1.E-07 1.E-08 1.E-09 1.E-10 1.E-11 1.E-12 100
Protein
Water
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Temperature [K] Fig. 7. Model relaxation times for the denaturation process for water and protein contributions (solid lines) compared to those for Tg in Fig. 6 (dashed lines). Inset shows thermal energy through Ti, calculated from the heat capacity changes in Fig. 4.
6. Discussion: order and function Our model mechanism for denaturation tells us that dehydration and conformational randomisation are coupled. The onset of water mobility allows unfolding to proceed as the hydrogen bonded water, which constrains the bridged ϕ and ψ torsional angles to their values designated at their point of synthesis, becomes mobile and allows the protein chain to rotate freely or aggregate, potentially into mis-folded secondary structures. The relative torsional rigidity of native proteins in solution should manifest itself in terms of differences in their macromolecular dimensions and solution viscoelasticity as a function of concentration between native and denatured proteins. The self-consistent model prediction of the lower temperature ‘disordered’ glass transition then has other important consequences, since the onset of low temperature motion in proteins has been associated very loosely with biological activity, but without clear functional mechanisms [2,3]. Why then is order and disorder important for biological activity other than more obvious topological changes in secondary and tertiary structure? LeBard and Matyushov have presented a detailed argument that ordered ferroelectric domains in the protein's hydration shell increase the energetic efficiency of protein electron transfer by up to an order of magnitude [31]. Without this activity promoter, they argue that functions such as photosynthesis and catalytic activity would proceed at a prohibitively slow rate. The ferroelectric domains cannot exist without stable ordered peptide-water interactions, and so the conversion of ordered to disordered hydration through denaturing is automatically associated in our model with loss of protein electronic activity. Inspection of protein glass transition literature suggests to us that all the effects reported are associated with denatured experimental protein samples, which is consistent with our suggestion that any specific peptide-water interaction is either uniquely ordered or disordered and cannot exist in both states simultaneously. Since a denatured protein is already disordered, even below Tg, this immediately negates the earlier suggestions that protein functional activity is associated with the onset of molecular mobility above Tg, other than in a relatively minor way. It also tells us that loss of tertiary structure is not the only cause of loss of protein function, and that, even if teriary structure is maintained (by crosslinking, for example) then disorder kills function through loss of electronic activity. Critically for our closing hypothesis is the deduction that native proteins have functionally active and ordered peptide-water bonding, while denatured proteins have low functional activity due to their intrinsic disordered state. According to our model hypothesis, the ordered native state of a protein shown in the inset in Fig. 2 is formed only at the point of synthesis, and denaturing is essentially irreversible without the input of chemical energy. We must conclude that ‘re-naturing’ a protein to the ordered state is only possible by a combination of peptide bond
reformation by agents such as ATP in conjunction with conformational chaperones (or constraining crosslinks) to template the peptide chains [32]. The renaturing process allows maintenance of a specific protein morphology by the appropriate chaperone acting upon a denatured segment. Alternatively, there is the possibility of adaptation, where a denatured segment is templated by a different chaperone [5,6,33]. The resultant new morphology is then held in place by the water bridge unless it is intrinsically unstable with a low Ti. These changes can then be malign, such as cancerous or amyloid forming, or positive, which allows new adapted proteins to evolve. The energetic model presented in its simplest form in Fig. 2 for poly(glycine) will allow exploration of whether different combinations of peptide groups and torsional angles in the primary sequence have different denaturation stabilities to account for protein features such as dehydrons [4,5]. The model can, in principle, include sensitivity to external stresses, such as mechanical loading and, of particular interest, environmental aspects such as solution pH through their effect on the instability energy, Ei [14]. Moreover, increased effective amide–water bond energy due to polar or hydrogen bonding groups on the peptide side chains may give increased protein–water stability for proteins such as extremophiles to survive in extreme environments, such as elevated temperatures [34]. Similarly, increased stability may be imparted by larger hydrogen bonded ‘solvent’ molecules such as sugars [35], which can increase the dehydration stability of proteins, and may be included in the model by reducing the thermal energy of the solvent molecules due to the decreased mobility and minor adjustments to hydrogen bond energy. These aspects need to be explored in more detail with specialist quantum mechanics simulation methods. As part of this analysis, we need also to explore in much more detail the protein-water complex structures at an electronic structure level in the formation of a peptide from amino acids to see if there is a specific ‘ordered’ and ferroelectric hydrated intermediate transition state that is different from the conventional Pauling resonant peptide structure, which we would consider inert and inactive, and which would allow reformation of the ordered polar structure during renaturing without cleaving the peptide chain bonds. 7. Conclusions The water mobility model developed in this work provides a consistent quantitative link between protein denaturation and the proteinwater glass transition. The difference in zero point vibrational energy in the ordered and disordered states of the strong protein–water interactions allows prediction of the transition temperatures using the same energy criterion for the onset of water mobility. The thermodynamics of the two transitions are shown to be self-consistent, in that the same change in heat capacity is due to onset of water mobility. However, the order–disorder transition of denaturation induces an excess enthalpy gain. Components of kinetics of the protein-water glass transition are clarified as being due to the water and hydrated protein molecules independently, although the kinetics of the denaturation process are not resolved. The model provides a mechanism for unfolding and aggregation of proteins through the transitions, as the stabilising strong water-amide bonds become mobile. These stabilising ordered peptidewater bonds are suggested to be formed at the point of protein synthesis, and could play a key role in the function of proteins through the enhancement of electronic activity by ferroelectric domains in the protein hydration shell. At the simplest level, the model clarifies the links between different mechanisms involved in denaturation events that have remained illdefined since Kauzmann1 and provides a quantitative definition of active native proteins and inactive or dysfunctional states of a protein. Moreover, the general energetic approach to electrochemical contributions to protein stability, denaturation, refolding, activity, maintenance,
D. Porter, F. Vollrath / Biochimica et Biophysica Acta 1824 (2012) 785–791
and adaptability offer exciting new routes to understand the role of protein structure in a wide range of biological effects. Acknowledgements We thank Dr. Atsuko Sato, Dr. Chris Holland and Dr. Cedric Dicko for valuable discussions and insights and the European Research Council (SP2-GA-2008-233409) and AFOSR (F49620-03-1-0111) for financial support. Appendix A. Methods summary Simulations for the energy of intermolecular bonding in this work were performed with the Dmol 3 dft method, as reported previously [14,36]. The Perdew-Wang 1991, GGA (P91), nonlocal functional was used with a DNP basis set that includes polarisation functions and unrestricted spin polarisation. The medium numerical integration grid was used and thermal occupation of orbitals was applied to facilitate convergence. References [1] W. Kauzmann, Some factors in the interpretation of protein denaturation, Adv. Protein Chem. 14 (1959) 1–63. [2] A.L. Lee, J. Wand, Microscopic origins of entropy, heat capacity and the glass transitions in proteins, Nature 411 (2001) 501–504. [3] W. Doster, The protein solvent glass transition, Biochim. Biophys. Acta 1804 (2010) 3–14. [4] N. Pietrosemoli, A. Crespo, A. Fernandez, Dehydration propensity of order–disorder intermediate regions in soluble proteins, J. Proteome Res. 6 (2007) 3519–3526. [5] A. Fernandez, M. Lynch, Non-adaptive origins of interactome complexity, Nature 474 (2011) 502–505. [6] N. Tokouriki, D.S. Tawfik, Protein dynamism and evolvability, Science 324 (2009) 203–207. [7] A.D. Robertson, K.P. Murphy, Protein structure and the energetics of protein stability, Chem. Rev. 97 (1997) 1251–1276. [8] C. Schnarnagl, M. Reif, J. Friedrich, Staility of proteins: temperature, pressure and the role of solvents, Biochim. Biophys. Acta 1749 (2005) 187–213. [9] A. Cooper, Heat capacity of hydrogen-bonded networks, Biophys. Chem. 85 (2000) 25–39. [10] A. Caflish, M. Karplus, Molecular dynamics simulation of protein denaturation, PNAS 91 (1994) 1746–1750. [11] P. Kumar, Z. Yan, L. Xu, M.G. Mazza, S.V. Buldyrev, S.H. Chen, S. Sastry, H.E. Stanley, Glass transition in biomacromolecules and the liquid-liquid critical point of water, Phys. Rev. Lett. 97 (2006) 177802.
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[12] D. Milardi, C. La Rosa, D. Grasso, Extended theoretical analysis of irreversible protein thermal unfolding, Biophys. Chem. 52 (1994) 183–189. [13] K. Wendler, J. Thar, S. Zahn, B. Kirchner, Estimating the hydrogen bond energy, J. Phys. Chem. A 114 (2010) 9529–9536. [14] D. Porter, F. Vollrath, The role of kinetics of water and amide bonding in protein stability, Soft Matter 4 (2008) 328–336. [15] M. Born, Thermodynamics of crystals and melting, J. Chem. Phys. 7 (1939) 591–603. [16] J. Wang, J. Li, S. Yip, D. Wolf, S. Phillpot, Unifying two criteria of Born, Physica A 240 (1997) 396–403. [17] C. Fu, D. Porter, Z. Shao, moisture effects on antheraea pernyi silk's mechanical properties, Macromolecules 42 (2009) 7877–7880. [18] D. Porter, Group Interaction Modelling of Polymer Properties, Marcel Dekker, New York, 1995. [19] D.W. Van Krevelen, Properties of Polymers, third ed. Elsevier, Amsterdam, 1990. [20] D.A. Dixon, K.D. Dobbs, J.J. Valentini, Amide-water and amide-amide hydrogen bond strengths, J. Phys. Chem. 98 (1994) 13435–13439. [21] S.-Y. Sheu, D.-H. Yang, H.L. Selzle, E.W. Schlag, Energetics of hydrogen bonds in peptides, PNAS 100 (2003) 12683–12687. [22] Characterising protein stability by DSC, Life Sciences Application Note, document number 20211021306 February 2006, Calorimetry Sciences Corporation www. calorimetrysciences.com. [23] F. Vollrath, D. Porter, Spider silk as archetypal protein elastomer, Soft Matter 2 (2006) 377–385. [24] W. Doster, M. Settles, Protein-water displacement distributions, Biochim. Biophys. Acta 1749 (2005) 173–186. [25] S. Khodadadi, A. Malkovskiy, A. Kisliuk, A.P. Sokolov, A broad glass transition in hydrated proteins, Biochim. Biophys. Acta 1804 (2010) 15–19. [26] G.P. Johari, O. Andersson, Vibrational and relaxational properties of crystalline and amorphous ices, Thermochim. Acta 461 (2007) 14–43. [27] S. Capaccioli, K.L. Ngai, Resolving the controversy on the glass transition temperature in water, J. Chem. Phys. 135 (2011) 104504. [28] D. Porter, F. Vollrath, K. Tian, X. Chen, Z. Shao, A kinetic model for thermal degradation in polymers with specific application to proteins, Polymer 50 (2009) 1814–1818. [29] S. Capaccioli, K.L. Ngai, N. Shinyashiki, The Johari-Goldstein β-relaxation of water, J. Phys. Chem. B 111 (2007) 8197–8209. [30] G.E. Thomas, S. Bone, G. Drago, Determination of protein denaturation and glass transition temperatures using high-frequency time domain reflectometry, J. Phys. Chem. B 112 (2008) 15903–15906. [31] D.N. LeBard, D.V. Matyushov, Protein-water electrostatics and principles of bioenergetics, Phys. Chem. Chem. Phys. 12 (2010) 15335–15348. [32] D. Nath, U. Rawat, R. Anish, M. Rao, a-crystallin and ATP facilitate the in vitro renaturation of xylanase, Protein Sci. 11 (2002) 2727–2734. [33] S.L. Rutherford, Between genotype and phenotype: protein chaperones and evolvability, Nat. Rev. Genet. 4 (2003) 263–274. [34] M. Tehei, G. Zaccai, Adaptation to high temperatures through macromolecular dynamics by neutron scattering, FEBS J. 274 (2007) 4034–4043. [35] N. Teramoto, N.D. Sachinvala, M. Shibata, Trehalose and trehalose-based polymers for environmentally benign, biocompatible and bioactive materials, Molecules 13 (2008) 1773–1816. [36] Dmol3 is supplied by Accelrys. Information available on www.accelrys.com.
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Water mobility, denaturation and the glass transition in proteins David Porter ⁎, Fritz Vollrath Dept. of Zoology, University of Oxford, Oxford, OX1 3PS, UK
a r t i c l e
i n f o
Article history: Received 18 January 2012 Received in revised form 12 March 2012 Accepted 15 March 2012 Available online 24 March 2012 Keywords: Protein hydration Denaturation Glass transition Lysozyme Myoglobin DFT
a b s t r a c t A quantitative mechanism is presented that links protein denaturation and the protein–water glass transition through an energy criterion for the onset of mobility of strong protein–water bonds. Differences in the zero point vibrational energy in the ordered and disordered bonded states allow direct prediction of the two transition temperatures. While the onset of water mobility induces the same change in heat capacity for both transitions, the order–disorder transition of denaturation also predicts the observed excess enthalpy gain. The kinetics of the water and protein components through the glass transition are predicted and compared with dielectric spectroscopy observations. The energetic approach provides a consistent mechanism for processes such as refolding and aggregation of proteins involved in protein maintenance and adaptability, as the conformational constraints of strong water–amide bonds are lost with increased molecular mobility. Moreover, we suggest that the ordered state of peptide–water bonds is induced at the point of protein synthesis and could play a key role in the function of proteins through the enhancement of electronic activity by ferroelectric domains in the protein hydration shell, which is lost upon denaturation. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Two main transition processes are associated with the structure, dynamics and activity of proteins: denaturation and the protein-water glass transition. While denaturation is usually associated with loss of activity through unfolding or aggregation of the protein at temperatures above about 330 to 340 K [1], conversely, the dynamics of hydrated proteins should increase above temperatures around 180 K that is attributed to the protein-water glass transition [2,3]. Since control of biological structure and activity is linked to both these processes, it is interesting that these two processes have not been related at either the functional or molecular level, even to the point of whether the two processes are mutually exclusive or one process might lead reversibly or irreversibly to the other after a given time-temperature history. Over a long period of time, there has been considerable interest in the detailed role of protein instability in important biological areas such as protein misfolding and maintenance, with more recent interest in adaption for medicine and evolutionary biology [4–6]. Given the general importance of protein denaturation as the most significant instability process, it is not surprising that an enormous body of experimental and theoretical literature on the subject is available, and reviews such as Robertson and Murphy give clear dissemination of key observed thermodynamic features [7]. Models for instability range from empirical thermodynamic [8,9] to detailed molecular dynamics simulations [10,11]. However, quantifying an underlying
⁎ Corresponding author. Tel.: + 44 1865271216. E-mail address: [email protected] (D. Porter). 1570-9639/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.bbapap.2012.03.007
electrochemical mechanism to test hypotheses about the onset of instability, the resultant torsional bond changes, and the consequent reformation of new stable and active protein structures seems to have received little attention. At the simplest qualitative level, a reversible unfolding of a native protein is initiated at a denaturation temperature, Ti, which is followed by a slower irreversible step (in many cases aggregation) to yield a final state that is unable to revert back to the native folded structure [12]. At the quantitative level, it is fair to say that at present the general view is that we cannot predict ab initio either the initiation conditions, such as Ti, or the thermodynamic parameters, such as enthalpy and entropy through the denaturing transition in terms of molecular structure. The same comments apply also to the peptide-water glass transition, Tg, as another instability that has received much recent attention [2,3]. The review by Doster gives an excellent overview of the current status of ideas on this specific form of Tg and links theories to a wide range of experimental observations such as the characteristic frequencies of the analytical techniques, but again does not give ab initio models to predict initiation conditions, thermodynamics, or kinetics for experimental observations and does not show how Tg might be linked to denaturation. Although simulation techniques such as molecular dynamics may reproduce some of the features of denaturation [10] and Tg [11], it must be remembered that such techniques are mainly based upon empirical forcefields that can be adjusted for just that purpose of reproduction, not ab initio prediction; particularly in complex systems containing water and highly polar proteins with complex hydrogen bonding [13]. Modelling large assemblies of molecules also gives simulation results that are essentially equivalent to experimental observations, and extraction
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of key atomic level events and their characteristic time scales that are the root cause of a particular effect is sometimes difficult. We suggested a relatively simple quantitative explanation of denaturation by calculating the transition energy from a potential energy well for specific peptide-water interactions that had been derived using density functional theory, dft [14]. The potentially reversible transition is located at the Born elastic instability point as the energy where the force between molecules is a maximum, or the stiffness between the interacting groups is zero, so that the water is free to move relative to the peptide [15,16]. At that point, the stronger peptidepeptide hydrogen bonding can form as peptides collide without water screening to form irreversible aggregates. This approach successfully estimates the denaturation temperature, but does not make any links either to structural disordering normally associated with denaturation, the detailed thermodynamics of the transition, and only makes reference to Tg processes as the bulk effect of water on observed thermomechanical processes, which was also developed by Fu for silk fibres using mean field modelling methods [17]. It is important not to confuse this bulk thermomechanical softening transition with the specific protein–water instability transition discussed in this work. The two effects must, inevitably, be linked due to the mobility of the tightly bound water, but here the specific peptidewater instability is taken to precede the bulk glass transition, and act as a rate limiting step for the bulk process through the onset of mobility of the water molecules. Here, we extend the basic denaturation model to the protein– water glass transition and make a direct quantitative link between the two, which includes the thermodynamics of the transitions and a first attempt at their kinetics as frequency-temperature relations. We first discuss the more general examples of transitions in polymers and water, where both the crystal melt and glass transition temperatures can be predicted in self-consistent models. 2. Mobility and transitions In order to quantify dynamic transitions in materials, we adopt a very simple description of a transition as the onset of mobility of molecules relative to each other [16,18]. This can occur when the stiffness (second differential of interaction energy with respect to bond length) of the intermolecular bonds tends to zero, and is known as the Born elastic instability condition [15,16]. Mathematical processing of the potential energy well for the interactions usually gives an energy at which this condition occurs, and this non-minimum energy value is attained by the combination of the zero point vibrational energy, Ho, and the energy of thermal vibrations of the molecular segments, HT, which can be quantified by Einstein and Debye theories [14,18]. It is important to note here that this energy condition for bond mobility is very different from bond strength as the energy to completely dissociate a bond. The hypothesis tested in this paper is that denaturation and the protein–water glass transition are two manifestations of the same transition condition of water molecules that are strongly bonded to the amide segments of a protein backbone chain by hydrogen bonds. The glass transition is defined as disordered, and denaturing is a change from an ordered to a disordered state. Therefore, the first problem here is to quantify order and disorder in the protein-water interaction energy condition for a transition to occur. Previous work has shown that order and disorder can be quantified for polymers and small molecules by the zero point vibrational energy term, Ho, and has been used to predict thermomechanical properties of crystal and amorphous forms [14,18]. The frequency of the molecular vibrations, νo, is calculated from the potential energy well or by using molecular modelling, and Ho values for the ordered and disordered states are defined by the first two quantised energy levels of hνo/2 and 3hνo/2 per vibration mode for ordered and disordered states respectively, where h is Planck's constant. Since the
disordered value of Ho is higher, then the thermal energy (temperature) to reach the transition condition must be lower. Also, for a given family of materials with characteristic forms of the potential energy well, the ordered and disordered transition temperatures usually have a consistent ratio of values, such as polymers where the crystal melt temperature is about 1.5 times the glass transition temperature [19]. Polymers are an elegant and well validated example of this approach to predicting transitions [18]. The energy of molecular interactions, E, is adequately described by the simple 6–12 Lennard-Jones potential energy well in terms of intermolecular distance, r, with a depth given by the cohesive energy, Eo. The transition condition, d 2E/dr2 = 0, occurs when the positive energy terms Ho + HT = 0.213Eo, such that the transition condition is less than a quarter of the total bonding energy, Eo. Using a version of the Debye model for heat capacity adapted for chain molecules such as polymers, simple and self-consistent analytical expressions can be derived for the crystal melt temperature, Tm, and the glass transition temperature, Tg, in terms of their theta temperature of skeletal mode vibrations, θ1, their cohesive energy, Eo, and the number of skeletal degress of freedom, N [18].
Tm Tg
¼ 0:224θ1 þ
0:084 0:0513
Eo N
ð1Þ
Although this polymer model has been successfully applied to proteins and the effect of hydration water as a plasticiser on amorphous bulk proteins [17], the Ti and Tg transitions in this work involve specific and strong hydrogen-bonded peptide-water interactions that need to be considered as a special case. A relevant example is that of water-water interactions, which has been presented in detail in a previous paper and is outlined here to illustrate the principles of the model [14]. We used dft quantum methods (a summary of the method is given in the Appendix A) to calculate a transition energy condition for individual water–water hydrogen bonds of 4.4 kJ mol − 1 per H-bond from the potential energy well, which was also used to calculate a vibrational frequency and associated theta temperature of νo = 2.5 × 10 12 Hz and θ = 107 K: note that our value of the pairwise water-water hydrogen bonded energy was 20 kJ/mol, in agreement with recent published values also calculated using dft simulations [13], and the transition energy to bond energy ratio of 0.22 is coincidentally similar to that of 0.213 for Lennard–Jones interactions in polymers. The water–water bond was considered as a simple Einstein oscillator with two degrees of freedom per vibration, N = 2, with the two different quantised energy levels for Ho and the thermal energy, HT, is given in terms of the gas constant, R, by HT ¼
NRθ exp Tθ −1
ð2Þ
The two zero point energy values predict Tm = 260 K and Tg = 150 K for the crystal melting point and glass transition temperature of water, which are in good general agreement with observed values of 273 and 140 K, respectively, and illustrated in Fig. 1. Since the bonding mechanisms are very similar for water-water and water-peptide interactions and the transition temperatures are of the same general order, it seems appropriate to apply the same approach to the problem of denaturation and protein-water glass transitions. 3. Protein–water transition temperatures As mentioned in the introduction, we made a first limited suggestion for predicting Ti by formulating the hypothesis that denaturing is initiated by the onset of mobility of specific tightly-bound hydrogenbonded water molecules to the polar amide groups in the backbone chain of the protein macromolecules: this bonding is shown in the
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4 Order Ho
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a
0
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2 1
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Temperature (K) Fig. 1. (a) Potential energy well for water-water interactions with energy levels marked, and (b) prediction of Ti and Tg using the energy levels for ordered and disordered states.
insets to Fig. 2, where the water molecule bridges and constrains the ϕ and ψ torsional angles [14]. Using density functional theory (dft) calculations of the energy of a water molecule as a function of distance from amide groups in the simplest polyglycine chain, we calculated an elastic instability point where the force between the molecules reaches a maximum, such that the stiffness of the intermolecular bond is zero and the water is free to move relative to the larger and less mobile protein. Fig. 2 shows the results of such simulations, with a mean instability energy of Ei = 5.7 kJ/mol water at a separation distance of about 0.4 Å. The full binding energy of the water-amide bond is predicted to be about 59 kJ/mol for beta-type conformation torsional angles in the protein chain, which appear to be an excellent
10
topologial match for the hydrogen bonded water bridge. The difficulties of modelling hydrogen bond interactions of this kind are discussed in detail by Wendler [13] with particular reference to cooperativity, also discussed by others for peptide interactions, including the bonding used here [20,21]. To convert this instability condition into an instability temperature, the water–amide interaction is considered as an Einstein oscillator with a reference theta temperature, θ = 150 K, which is calculated from the relation hνo ≡ kθ (where h and k are the Planck and Boltzmann constants, respectively) using a frequency of νo = 3.5 × 1012 Hz from the bond stiffness of 15 Nm− 1 and mass of the vibrating water molecule 18D as a simple harmonic oscillator at the minimum of the potential energy well of Fig. 2, and is discussed in detail in reference [14]. Fig. 3 shows energy as a function of temperature for the Einstein oscillator of Eq. (2) in two states, designated here as ordered and disordered, which have two different zero point energy values of NRθ/2 and 3NRθ/ 2 (again using hνo ≡ kθ to convert from hνo/2 and 3hνo/2, respectively) for the quantised oscillator respectively, where R is the molar gas constant and N = 2 for two vibrating hydrogen bonds per water-amide interaction. The upper ‘ordered’ state has a denaturation transition temperature Ti = 336 K and the new disordered state has a lower instability glass transition temperature, Tg = 180 K, with both temperatures agreeing well with experimental observation [1,3]. Importantly, both transitions are different states of the same instability condition, but with two clearly distinguished quantised zero point vibrational energy levels. This is the first direct quantitative link between Ti and Tg with a clear physical mechanism of the energetics of peptide–water bonding. It also tells us that the torsional bonds ϕ and ψ are liberated from the constraint of the water bonding bridge above these temperatures for the onset of chain dynamics and potential randomisation of the protein chain. However, this randomisation points to important differences between the two processes: denaturation may change from order to disordered states, whereas Tg remains disordered with no change. 4. Thermodynamics 4.1. Denaturation With these transition temperatures and a quantitative mechanism for denaturation, we can now predict the thermodynamic energetics of protein instability. At the transition, water changes from a bound oscillator to a freely mobile molecule. With this change of mobility, we would expect a heat capacity change of 6 new degrees of freedom per water molecule (three translational and three rotational). The associated change in heat capacity is ΔC = 6R ≈ 50 J/K/mol, and is associated
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0 0
100
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Temperature (K) Fig. 2. Energy and force as a function of separation distance for a water–amide bond, with the elastic instability condition marked by a dashed line, predicted for the structures shown as an inserts using dft simulations.
Fig. 3. Predicted energy–temperature relations for the ordered and disordered states of an Einstein oscillator with θ = 150 K and zero point energies NRθ/2 and 3NRθ/2.
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more specifically with each water–amide segment interaction. This first approximation does not include any changes to the relatively immobile protein macromolecule, but we would expect a minor contribution as chain mobility increases. As a first suggestion, the change in heat capacity should be the same for both Ti and Tg. However, we have already noted that Ti may change from an intrinsically ordered to disordered state, without any structural constraints such as interchain crosslinking or strong hydrogen bonding between peptide side chains. If this order–disorder change occurs, there must also be a change in the zero point vibrational energy and kinetics, which effectively means an increase in enthalpy per peptide segment, which is observed as an excess enthalpy peak through the transition, ΔHi . To calculate ΔHi , we need to know both the change in zero point energy, ΔHo = NRθ, and the temperature interval over which the heat capacity changes, ΔTi, which is experimentally observed to be about 20 degrees, as shown in Fig. 4. ΔH i ≈ΔHo þ ΔC⋅ΔT i ≈3:5kJ=mol
ð3Þ
Thus, as the molecule becomes mobile, it gains an extra energy increment equivalent to ΔHi ≈ 3.5 kJ/mol per peptide segment through the transition region. The same procedure is applied to predict the change in entropy ΔS ≈ 10.4 J/K/mol through the denaturing transition. These parameters then allow calculation of the Gibbs free energy, which becomes a relatively small energy term. C S ¼ ∫ dT ¼ 10:4 J=K=mol peptide T
ð4Þ
These predictions agree quite well with the average observed values of ΔC = 58 ± 2 J/K/mol, ΔS = 8.8 ± 3 J/K/mol and ΔH = 2.92 ± 0.08 kJ/ mol per peptide segment normalised to 60°C [7], bearing in mind that proteins have many other structural features, such as polar side chains and potentially denatured or non-hydrated segments. Looking more specifically at a common benchmark protein, hen egg lysozyme has 123 peptide segments and Fig. 4 compares our predicted DSC scan for the change in thermodynamic properties through denaturation with experimental observation in the form of heat capacity as a function of temperature: observed characteristic values are ΔH = 427 kJ/mol, ΔC = 6.4 kJ/K/mol, and ΔS = 1.27 kJ/K/mol [7,22], relative to predicted values of ΔH = 430 kJ/mol, ΔC = 6.2 kJ/K/mol, and ΔS = 1.28 kJ/K/mol at Ti = 64 °C. These predictions embody both thermodynamic and kinetic features of the denaturing transition using parameters calculated directly from the potential energy well of Fig. 2. For the model energy
distribution in Fig. 4, a normal distribution with a standard deviation of 3.5 degrees fits the experimental data using the predicted thermal energy values; we suggested a predicted general form for this distribution previously, based upon activation probablity and kinetics [14].
4.2. Protein–water glass transition The thermodynamics of the protein–water glass transition from calorimetry experiments are outlined in the review by Doster, who points out that the change of heat capacity for the water component of hydrated myoglobin at the glass transition is the same as suggested here of 6R per mole of water for denaturation [3]. The glass transition distribution is broader than the well defined ordered transition shown in Fig. 4, as would be expected for a disordered process with a range of local bonding conformations and environments. Again, we can model this increase in heat capacity quite simply as a change in the water from constrained vibrations in a solid (3 degrees of freedom) to free mobility of the liquid molecules (an extra 6 degrees of freedom) through the glass transition. The total 9 degrees of freedom gives the heat capacity of liquid water as the experimentally observed value of 9R = 76 J/mol/K or 1 cal/g/K. The slightly higher observed value of about 1.1 cal/g/K at higher temperatures shown in the points in Fig. 5 for myoglobin proteins is probably due to the contribution of protein macromolecular degrees of freedom, which is observed as mobility in neutron scatting experiments, for example. The model for change in heat capacity, Cp, is shown as a line in Fig. 5 and the change in degrees of freedom is a cumulative Gaussian distribution with a mean Tg = 195 K and a standard deviation of 20 degrees for a best fit, where the peak width is of the same order as a secondary relaxation peak at about 200 K in a silk protein tested by dynamic mechanical thermal analysis [23]: we are not currently able to calculate the distribution width from first principles due to the complexity of the effects. This increase in heat capacity is superimposed upon the predicted heat capacity of the same Einstein oscillator model with three degrees of freedom as used for the ordered denaturation transition. There is no change of state in the disordered solid–liquid transition, so no excess enthalpy is expected that is seen in the denaturation transition in Fig. 4. It is interesting to note here that the thermal energy, HT, has two regions above and below Tg that could be very loosely described as being approximately linear in temperature, but offset by temperature To ≈ 150 K [24], with proportionality to T–To at higher temperatures. This has important consequences for the apparent form of the kinetics relations, as we will show below.
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Temperature ( C) Fig. 4. Comparison of predicted (solid lines) and observed (dashed lines [7,22]) profiles of heat capacity through the denaturing transition for hen egg lysozyme. Dashed baseline (− ⋅ − ⋅ −) is taken from the experimental data [7].
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Temperature (K) Fig. 5. Comparison of predicted (line) and observed (points [3]) change in heat capacity through the protein-water glass transition for myoglobin, together with the predicted enthalpy increase, HT, as a function of temperature.
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5. Kinetics The problem of differentiating between different transition mechanisms is further complicated by their kinetics. Observations on the temperature dependence of characteristic relaxation times with a range of different analytical techniques suggest numbers of different processes that can be associated with water, protein-water interactions, or the protein mobility. Khodadadi simply states that “it is not clear what is the main structural relaxation of the hydrated protein that freezes around Tg ≈ 180 K” [25]. The points shown in Fig. 6 for lysozyme using dielectric spectroscopy and neutron scattering give consistent profiles for what are reasonably argued to be two different protein–water glass transition processes [25]. Fig. 6 also includes available dielectric data for water as both amorphous water at high pressure [26] and over the whole temperature range from Cappaccioli [27] to support the suggestion that the hydration water is the main contribution to the dominant faster form of relaxation [3]. The classical Arrhenius function for time-temperature effects is essentially a reference rate for unconstrained molecular vibrations, νo, that is scaled by an activation probability in terms of thermal energy relative to an activation energy, ΔH, for that rate to be active. At the simplest level, thermal energy becomes RT per vibrational mode, but for the more complex thermodynamics of the protein-water glass transition, we suggest simply using the thermal energy HT as a function of temperature in a modified Arrhenius for a frequency, f, with a form that was used successfully to model thermal degradation in proteins [28] −ΔH f ¼ ν o exp HT
ð5Þ
Relaxation Time [s]
Taking νo = 3.5 × 10 12 Hz from the potential energy well calculations for peptide–water interactions, and HT directly from Fig. 5, the line (“increased mobility model”) plotted in Fig. 6 for relaxation time, 1/f, uses an activation energy of 57 kJ/mol (or 750 cal/g to be consistent with reported thermodynamic data), which corresponds to the binding energy of the water–amide interactions minus Ho, consisting of two hydrogen bonds. Thus, the increased thermal energy due to the onset of water mobility at Tg increases the relaxation rate above that of a simple Arrhenius in RT, and thus resolves the dilema of the main structural relaxation process as the hydrated protein unfreezes at Tg ≈ 180 K, mentioned above, while maintaining a relaxation time of about 1 s at the glass transition temperature of water at
1.E+02 1.E+01 1.E+00 1.E-01 1.E-02 1.E-03 1.E-04 1.E-05 1.E-06 1.E-07 1.E-08 1.E-09 1.E-10 1.E-11 1.E-12 100
Main dielectic [25] Neutron Specroscopy [25] water 1GPa [26] Water dielectric [27]
Increased mobility model Slower dielectric [25] Polymer model
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Temperature [K] Fig. 6. Relaxation time as a fuction of temperature for the protein-water glass transition. Points are experimental observations as specified and the lines are predicted by the modified Arrhenius function of Eq. (5) for the water and hydrated protein responses.
789
about 140 K. We see now that the proportionality of HT with (T–To) gives the relaxation function a Vogel–Fulcher form, rather than simple Arrhenius [24] and the frequency-temperature relations are smooth and continuous through the glass transition, which has previously led to claims of no dynamic transition [3], and resolves some important inconsistencies in the water-protein glass transition observations. However, the conclusion here is that the faster relaxation process is simply water that has its mobility constrained by bonding to peptide. Fig. 6 also shows the ‘slow dielectric’ response from the same experimental data set as the main dielectric points [25]. This tells us that there is a much slower set of dipoles that relax relative to the main process quantified above for water. Since the transition in question is the mobility of a protein-water interaction, it seems reasonable that a dipolar relaxation process associated with mobility of the hydrated protein should accompany that of the water, which is most likely to be that of the strong amide dipole involved in the water interaction. The two relaxation processes identified here are a specific example of the same effect seen in a wide range of solution mixtures, including aqueous systems, where the faster one is a local specific relaxation of hydration water, similar to the beta-relaxation in supercooled liquids and mixtures, and the slower one is related to structural cooperative motions of protein segments and hydration water together [29]. The objective in this work is simply to quantify the mechanism of the specific peptide-water interaction glass transition process, which then becomes a rate limiting step for water mobility for the bulk glass transition events of the hydrated protein. For simplicity, take Eq. (5) with interaction energy value of ΔH = 57 kJ/mol, which is similar to the value of 55 kJ/mol suggested from observations by Capaccioli for more general aqueous systems [29], but use the thermal energy, HT, for a protein polymer with N = 2 degrees of freedom per water interaction and a typical skeletal mode theta temperature of θ1 = 400 K for νo = 9 × 10 12 Hz in the expression [14,18] θ −1 6:7T H T ¼ NR T− 1 tan θ1 6:7
ð6Þ
The line ‘polymer model’ in Fig. 6 shows the simple model predictions relative to experimental observations, which are in reasonably good agreement and made easier by the temperature range not including the Tg effects on thermal energy. Thus, we can now argue that both water and hydrated protein components of the protein-water glass transition can be identified separately in a self-consistent framework. Unfortunately, analysis of kinetics by means of techniques such as dielectric spectroscopy in Fig. 6 is made difficult for the denaturation transition, since the most common way to report rate effects associated with denaturation is in the enzyme reaction rate, which is difficult to uncouple from other factors involved in chemical reactions. Since denaturation is essentially an irreversible process of unfolding or aggregation, we have not yet found a suitable way to validate any model predictions for relaxation time parameters. Since the model for water relaxation for Tg in Fig. 6 using the thermal energy function, HT, shown in Fig. 5 seems to work quite well, Fig. 7 shows predictions of the relaxation times for denaturation compared to Tg, with the inset showing the appropriate form of the thermal energy, HT, from the heat capacity in Fig. 4 and scaling the thermal energy for the polymer model in Eq. (6) by 0.75 below Ti to account very approximately for the decreased mobility due to the water bonding constraints [18]. However, this is clearly just a modelling exercise to illustrate a principle, since dielectric spectroscopy suggests that characteristic relaxation time actually increases as temperatures approach Ti due to the tendency to aggregation and unfolding, which according to the authors (Thomas et al) “does not conform to any known model” [30].
D. Porter, F. Vollrath / Biochimica et Biophysica Acta 1824 (2012) 785–791
Relaxation Time [s]
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Temperature [K] Fig. 7. Model relaxation times for the denaturation process for water and protein contributions (solid lines) compared to those for Tg in Fig. 6 (dashed lines). Inset shows thermal energy through Ti, calculated from the heat capacity changes in Fig. 4.
6. Discussion: order and function Our model mechanism for denaturation tells us that dehydration and conformational randomisation are coupled. The onset of water mobility allows unfolding to proceed as the hydrogen bonded water, which constrains the bridged ϕ and ψ torsional angles to their values designated at their point of synthesis, becomes mobile and allows the protein chain to rotate freely or aggregate, potentially into mis-folded secondary structures. The relative torsional rigidity of native proteins in solution should manifest itself in terms of differences in their macromolecular dimensions and solution viscoelasticity as a function of concentration between native and denatured proteins. The self-consistent model prediction of the lower temperature ‘disordered’ glass transition then has other important consequences, since the onset of low temperature motion in proteins has been associated very loosely with biological activity, but without clear functional mechanisms [2,3]. Why then is order and disorder important for biological activity other than more obvious topological changes in secondary and tertiary structure? LeBard and Matyushov have presented a detailed argument that ordered ferroelectric domains in the protein's hydration shell increase the energetic efficiency of protein electron transfer by up to an order of magnitude [31]. Without this activity promoter, they argue that functions such as photosynthesis and catalytic activity would proceed at a prohibitively slow rate. The ferroelectric domains cannot exist without stable ordered peptide-water interactions, and so the conversion of ordered to disordered hydration through denaturing is automatically associated in our model with loss of protein electronic activity. Inspection of protein glass transition literature suggests to us that all the effects reported are associated with denatured experimental protein samples, which is consistent with our suggestion that any specific peptide-water interaction is either uniquely ordered or disordered and cannot exist in both states simultaneously. Since a denatured protein is already disordered, even below Tg, this immediately negates the earlier suggestions that protein functional activity is associated with the onset of molecular mobility above Tg, other than in a relatively minor way. It also tells us that loss of tertiary structure is not the only cause of loss of protein function, and that, even if teriary structure is maintained (by crosslinking, for example) then disorder kills function through loss of electronic activity. Critically for our closing hypothesis is the deduction that native proteins have functionally active and ordered peptide-water bonding, while denatured proteins have low functional activity due to their intrinsic disordered state. According to our model hypothesis, the ordered native state of a protein shown in the inset in Fig. 2 is formed only at the point of synthesis, and denaturing is essentially irreversible without the input of chemical energy. We must conclude that ‘re-naturing’ a protein to the ordered state is only possible by a combination of peptide bond
reformation by agents such as ATP in conjunction with conformational chaperones (or constraining crosslinks) to template the peptide chains [32]. The renaturing process allows maintenance of a specific protein morphology by the appropriate chaperone acting upon a denatured segment. Alternatively, there is the possibility of adaptation, where a denatured segment is templated by a different chaperone [5,6,33]. The resultant new morphology is then held in place by the water bridge unless it is intrinsically unstable with a low Ti. These changes can then be malign, such as cancerous or amyloid forming, or positive, which allows new adapted proteins to evolve. The energetic model presented in its simplest form in Fig. 2 for poly(glycine) will allow exploration of whether different combinations of peptide groups and torsional angles in the primary sequence have different denaturation stabilities to account for protein features such as dehydrons [4,5]. The model can, in principle, include sensitivity to external stresses, such as mechanical loading and, of particular interest, environmental aspects such as solution pH through their effect on the instability energy, Ei [14]. Moreover, increased effective amide–water bond energy due to polar or hydrogen bonding groups on the peptide side chains may give increased protein–water stability for proteins such as extremophiles to survive in extreme environments, such as elevated temperatures [34]. Similarly, increased stability may be imparted by larger hydrogen bonded ‘solvent’ molecules such as sugars [35], which can increase the dehydration stability of proteins, and may be included in the model by reducing the thermal energy of the solvent molecules due to the decreased mobility and minor adjustments to hydrogen bond energy. These aspects need to be explored in more detail with specialist quantum mechanics simulation methods. As part of this analysis, we need also to explore in much more detail the protein-water complex structures at an electronic structure level in the formation of a peptide from amino acids to see if there is a specific ‘ordered’ and ferroelectric hydrated intermediate transition state that is different from the conventional Pauling resonant peptide structure, which we would consider inert and inactive, and which would allow reformation of the ordered polar structure during renaturing without cleaving the peptide chain bonds. 7. Conclusions The water mobility model developed in this work provides a consistent quantitative link between protein denaturation and the proteinwater glass transition. The difference in zero point vibrational energy in the ordered and disordered states of the strong protein–water interactions allows prediction of the transition temperatures using the same energy criterion for the onset of water mobility. The thermodynamics of the two transitions are shown to be self-consistent, in that the same change in heat capacity is due to onset of water mobility. However, the order–disorder transition of denaturation induces an excess enthalpy gain. Components of kinetics of the protein-water glass transition are clarified as being due to the water and hydrated protein molecules independently, although the kinetics of the denaturation process are not resolved. The model provides a mechanism for unfolding and aggregation of proteins through the transitions, as the stabilising strong water-amide bonds become mobile. These stabilising ordered peptidewater bonds are suggested to be formed at the point of protein synthesis, and could play a key role in the function of proteins through the enhancement of electronic activity by ferroelectric domains in the protein hydration shell. At the simplest level, the model clarifies the links between different mechanisms involved in denaturation events that have remained illdefined since Kauzmann1 and provides a quantitative definition of active native proteins and inactive or dysfunctional states of a protein. Moreover, the general energetic approach to electrochemical contributions to protein stability, denaturation, refolding, activity, maintenance,
D. Porter, F. Vollrath / Biochimica et Biophysica Acta 1824 (2012) 785–791
and adaptability offer exciting new routes to understand the role of protein structure in a wide range of biological effects. Acknowledgements We thank Dr. Atsuko Sato, Dr. Chris Holland and Dr. Cedric Dicko for valuable discussions and insights and the European Research Council (SP2-GA-2008-233409) and AFOSR (F49620-03-1-0111) for financial support. Appendix A. Methods summary Simulations for the energy of intermolecular bonding in this work were performed with the Dmol 3 dft method, as reported previously [14,36]. The Perdew-Wang 1991, GGA (P91), nonlocal functional was used with a DNP basis set that includes polarisation functions and unrestricted spin polarisation. The medium numerical integration grid was used and thermal occupation of orbitals was applied to facilitate convergence. References [1] W. Kauzmann, Some factors in the interpretation of protein denaturation, Adv. Protein Chem. 14 (1959) 1–63. [2] A.L. Lee, J. Wand, Microscopic origins of entropy, heat capacity and the glass transitions in proteins, Nature 411 (2001) 501–504. [3] W. Doster, The protein solvent glass transition, Biochim. Biophys. Acta 1804 (2010) 3–14. [4] N. Pietrosemoli, A. Crespo, A. Fernandez, Dehydration propensity of order–disorder intermediate regions in soluble proteins, J. Proteome Res. 6 (2007) 3519–3526. [5] A. Fernandez, M. Lynch, Non-adaptive origins of interactome complexity, Nature 474 (2011) 502–505. [6] N. Tokouriki, D.S. Tawfik, Protein dynamism and evolvability, Science 324 (2009) 203–207. [7] A.D. Robertson, K.P. Murphy, Protein structure and the energetics of protein stability, Chem. Rev. 97 (1997) 1251–1276. [8] C. Schnarnagl, M. Reif, J. Friedrich, Staility of proteins: temperature, pressure and the role of solvents, Biochim. Biophys. Acta 1749 (2005) 187–213. [9] A. Cooper, Heat capacity of hydrogen-bonded networks, Biophys. Chem. 85 (2000) 25–39. [10] A. Caflish, M. Karplus, Molecular dynamics simulation of protein denaturation, PNAS 91 (1994) 1746–1750. [11] P. Kumar, Z. Yan, L. Xu, M.G. Mazza, S.V. Buldyrev, S.H. Chen, S. Sastry, H.E. Stanley, Glass transition in biomacromolecules and the liquid-liquid critical point of water, Phys. Rev. Lett. 97 (2006) 177802.
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