Lecture 2 and 3: Review of forces (ctd.) and elementary statistical mechanics. Contributions to protein stability Part I. Review of forces Covalent bonds Non-covalent Interactions: Van der Waals Interactions Electrostatics Hydrogen Bonds Hydrophobic Interactions Part II. Review of key concepts from Stat. Mech. Part III. Contributions to protein stability and binding
Non-Covalent Interactions: Hydrophobic Interactions Solvation/ Hydrophobic interactions The hydrophobic interaction is the pronounced attraction of non-polar solutes in water. Non-polar substances are poorly soluble in water (as is evident in mixing oil and water) and the free energies of transfer of non-polar substances to water are large and positive.
Origin of the Hydrophobic Effect (1) The origins of the hydrophobic effect are surprisingly still somewhat controversial. It is convenient to divide solvation processes into two steps: 1) the creation of a cavity in the liquid large enough to accommodate the solute, G cavity Bulk H20 (solvent) Solvent + cavity Because water is a strongly cohesive liquid, and because of its small size, the free energy of forming a cavity is higher than in other simple liquids (the probability of finding a reasonably large cavity is quite small). This is probably the main source of the anomalously low solubility of non-polar compounds in water (for polar and charged molecules, this cost is more than offset by the favorable electrostatic and hydrogen bonding interactions that can be formed; see the expression above for transferring an ion to a high dielectric solvent).
Origin of the Hydrophobic Effect (2) 2) the placement of the solute into the cavity. + G interactions w/ solvent Solvent + cavity Solute inside cavity The free energy changes associated with 2) are due to interactions between the solvent and the solute that we have already discussed--hydrogen bonding, van der Waals interactions, electrostatics (for non polar compounds, only van der Waals interactions are important). The van der Waals interactions between non-polar solutes and water are of the same order of magnitude as those between water molecules, but to retain hydrogen bonding in the vicinity of the non-polar solutes requires some ordering of water molecules. There is thus also a decrease in entropy associated with exposing non-polar compounds to water and a change in heat capacity which lead to anomalous temperature dependencies that characterize hydrophobic interactions. The first term is unfavorable for all solutes (it depends only in size), while the second is large and favorable for polar and charged solutes, but small for non polar solutes. The sum of these interactions is thus unfavorable only for non-polar solutes-hence the hydrophobic effect.
References The topics covered in this lecture are covered in more details in the following textbooks (available in Health Science Library and Biochemistry Library on reserve) Non-covalent interactions relevant to protein structure: Creighton, ch. 4 p139 The same, with an emphasis on Protein-protein interactions: Fersht, ch.11, p324 The three dimensional structure of proteins (α-helix, β-sheet) : Fersht, ch1 p1, Branden & Tooze ch. 1 The Generalized-Born Approximation for Macromolecules: Onufriev et al., Modification of the Generalized Born Model Suitable for Macromolecules, J. Phys. Chem. B. 2000, 104, 3712-3720
Some Simple but Key Results from Stat Mech (1) The energy of an isolated system is constant. (2) The entropy is proportional to the logarithm of the number of states of a system: S = k ln Ω (3) The entropy of an isolated system increases in any spontaneous process. (4) For a sub system in thermal equilibrium with a larger system (the outside world), the condition that the entropy of the combined system increases is equivalent to the condition that the free energy of the smaller system, G = E - TS + PV decreases.
The Boltzman distribution In most of biochemistry, the PV term is very small, and thus, to determine whether a reaction occurs spontaneously, we must consider the balance between the energy change and the entropy change: ΔG = ΔE - TΔS The probability of observing a particular state of a system with free energy G is: Prob exp [S tot /k] exp [-G/kT] This is a very important result as it relates free energy differences to differences in populations.
The Boltzman distribution (ctd.) Consider a protein with two different conformations, conf1 and conf2, that differ in free energy by some amount ΔG. From previous page, Prob(conf1) exp [-G(conf1)/ kt] Prob(conf2) exp [-G(conf2)/ kt] = exp [ -(G(conf1) - ΔG )/kt] The ratio between the populations (concentrations) of the two conformations at equilibrium is called the equilibrium constant (Keq). Keq = Prob(conf1)/Prob(conf2) = exp [-ΔG/kT] Taking the logarithm of both sides gives the familiar expression ΔG = -kt ln K eq which relates the free energy difference in a reaction to the log of the equilibrium constant.
The Boltzman distribution (ctd.) For example, if the free energy difference between the two conformations is 1kcal/mol, what is the ratio of the two populations (the equilibrium constant) at 300K? P(conf1)/P(conf2) = exp -(ΔG/kT) = exp -(1/0.6) =.19 It is useful to remember the free energy difference that corresponds to a ten fold difference in the populations of two states in equilibrium: ΔG = -kt ln K eq = -(0.6 ln 10) = -1.38 kcal/mol
Link with Protein Folding/Protein-protein Association Thus, to determine whether a protein will fold or whether two macromolecules will associate, one needs to determine the change in free energy in the process. ΔG = ΔE T ΔS At low temperature, the ΔE term dominates, but as the temperature is increased, the T ΔS term becomes increasingly important. We learned how to compute Δ E for processes involving changes in van der Waals interactions, hydrogen bonds, etc; but how to compute Δ S?
Entropic Change in Protein Folding From above, S = k ln (number of states), so we need to determine the change in the number of states during the process. This counting is simplest for amino acid side chains, which adopt a small number of discrete states called rotamers (each torsion angle has three possible values). Example: valine side chains have three possible rotamers (one torsion angle). How much entropy is lost in a change from a conformation in which the valine can adopt all three rotamers to a conformation in which only one rotamer is tolerated? ΔS = k ln 3 k ln 1 = 0.00198 ln 3 0 = 0.0022 kcal/mol K How much free energy does this correspond to? Δfree energy = - T ΔS = - 300 (0.0022) = -0.66 kcal/mol
Entropic Change in Protein Folding (cont d) The entropy changes in protein folding are estimated to be ~0.007 kcal/mol K per residue for the main chain, and ~0.003 kcal/mol K per residue for the side chains. For a 100 residue protein the total entropy change in folding is thus ~ 1kcal/mol K; at 300 degrees K (room temperature) this corresponds to ~300 kcal/mol. For a protein to fold, this large unfavorable contribution to the folding free energy must be compensated by the non-covalent interactions discussed last time, which are individually much weaker. (Current Opinion in Structural Biology, 7, 215-221)
Entropic Change in Macromolecular Association What is the entropic change caused by the association of two macromolecules? There are three components: 1) large decrease in translational entropy 2) decrease in rotational entropy 3) gain in entropy associated with intermolecular motions The net contribution of these three effects is estimated to be ~5-15kcal/mol at 300K.
The Boltzman distribution In most of biochemistry, the PV term is very small, and thus, to determine whether a reaction occurs spontaneously, we must consider the balance between the energy change and the entropy change: ΔG = ΔE - TΔS The probability of observing a particular state of a system with free energy G is: Prob exp [S tot /k] exp [-G/kT] This is a very important result as it relates free energy differences to differences in populations.
Last time, considered case where there were two alternative states. What happens if there are many alternative states? We know for each state: Prob exp [S tot /k] exp [-G/kT] The probabilities of all of the possible states must add up to one. This will be true if we write Prob(state) = exp [-G(state)/kT]/ Z where Z is the sum of the exp [G/kT] terms for each of the states Z is called the partition function
Concrete example Suppose a protein has three states: Unfolded state. 1000 different conformations, each with no interactions Intermediate state. 10 different conformations, each with two hydrogen bonds worth 1kcal/M Native state. 1 conformation, with twenty hydrogen bonds worth 1kcal/M What is the fraction of the protein which is folded as a function of temperature?
Putting everything together: Contributions to protein stability
General Considerations Since proteins are surrounded by water molecules, the energetics of protein folding and binding involve considerable tradeoffs between loss of protein-water interactions and gain of protein-protein interactions. For example, van der Waals interactions are gained between protein atoms and lost between protein and water, and similarly, hydrogen bonds formed between donors and acceptors within a protein chain come at the cost of breaking hydrogen bonds between these atoms and water. Because of these tradeoffs, the contribution of van der Waals interactions and hydrogen bonding to protein stability is relatively small. However, these interactions do have a very important negative influence on protein structures: van der Waals interactions and hydrogen bonds made with water in unfolded or unbound protein chains that are lost during folding or complex formation must be compensated by interactions within the protein or within the complex, or the free energy of folding/ complex formation will have large unfavorable contributions from the lost interactions with water. For this reason, protein structures rarely contain large cavities (which would involve a loss of van der waals interactions) or buried hydrogen bond donors or acceptors that do not make hydrogen bonds.
General Considerations (ctd.) In protein folding and binding reactions, the amount of non-polar surface exposed to water may change considerably, and thus the hydrophobic effect plays an important role in these processes. Unfolding, which involves exposure of hydrophobic side chains to water, may be viewed as equivalent to the transfer of these non-polar groups from a non-polar solvent to water. The requirements of retaining hydrogen bonding and attractive van der Waals interactions while minimizing the exposure of nonpolar atoms to water give rise to the hallmarks of globular protein structures: hydrophobic cores with few charged or polar atoms that are well packed to avoid loss of van der waals interactions, largely polar surfaces, and extensive alpha helix and beta sheet secondary structure which allow the polypeptide backbone to retain extensive hydrogen bonding while passing back and forth through the protein.
Entropy Loss and the Denatured State As noted in the previous lecture, the hydrophobic effect is probably the main driving force for protein folding: there is a large decrease in the amount of exposed hydrophobic surface area during folding. Opposing this large favorable contribution is an almost equally large unfavorable contribution from chain entropy loss. Recall that the free energy change Δ G = Δ H T Δ S and that the entropy change Δ S is proportional to the logarithm of the change in the number of accessible states. The native state has much lower entropy than the denatured state because the backbone is relatively fixed and most of the side chains adopt single conformations. There are also contributions from the change in entropy of the solvent when the non-polar groups of the protein become buried in the core of the protein. The net Δ S for folding is negative ( a reduction in the number of states), and thus at high temperature proteins unfold (ΔG for folding becomes positive). Denaturants denature proteins by reducing the strength of the hydrophobic effect. (see following table of transfer free energies of the amino acids from water to denaturants)
Experimental Data How well do these principles account for observed experimental data on protein stability and binding? Unfortunately, because the free energy of protein folding is the difference between two very large contributions: the large chain entropy loss upon folding and the large gain in hydrophobic interactions, it is not currently possible to predict overall protein stability even from high resolution crystal structures. Much more amenable to analysis are the changes in protein stability brought about by single amino acid changes. Studies of the effects of a large number of such sequence changes have led to the general conclusions listed on the next slide. It should be kept in mind, however, that while something may be inferred about effects of the mutations in the native state if a high resolution structure is available, there is considerable uncertainty about the effects of mutations on the denatured state. ΔG = G native - G denatured
Given these considerations, what do you expect would be the effect of mutations which 1) Change a non polar residue to a charged residue in the protein core 2) Change a non polar residue to a charged residue on the protein surface 3) Introduce a flexible residue in the protein core 4) Introduce a flexible residue on the protein surface 5) Change a non polar residue to a charged residue at a protein interface 6) Introduce a flexible residue at a protein interface 7) Increase the size of a residue in the protein core 8) Increase the size of a residue on the protein surface
Conclusions From Studies of Protein Stability (1) Sequence changes at buried sites almost always have much larger effects on stability than sequence changes at exposed sites. The small change at exposed sites is not surprising given that these residues are likely to have similar environments (ie, largely solvated) in both the denatured and native states. (See figure on λ repressor on next slide)
Conclusions From Studies of Protein Stability (ctd.) (2) Charged residues and to a lesser extent polar residues are disfavored at buried sites. This is expected given the large energetic cost of burying a charge. (3) Sequence changes which reduce the amount of hydrophobic burial are destabilizing. (see table on next slide)
Conclusions From Studies of Protein Stability (ctd.) Substitution G (kcal/mol) average Ile -> Val 1.3+/- 0.4 Ile -> Ala 3.8 +/- 0.7 Leu -> Ala 3.5 +/- 1.1 Val -> Ala 2.5 +/- 0.9 -CH2-1.2 +/- 0.9 Met -> Ala 3.0 +/- 0.9 Phe -> Ala 3.8 +/- 0.3
Conclusions From Studies of Protein Stability (ctd.) (4) Sequence changes which disrupt side chain packing in the interior or leave large cavities are unfavorable. Surface area of the cavity (5) Salt bridges between oppositely charged residues on the protein surface contribute relatively little to stability, probably because the more favorable electrostatic interactions are offset by the entropic cost of ordering the sidechains. Repulsive interactions between same charged residues on the surface can be quite destabilizing.
Conclusions From Studies of Protein Stability (ctd.) (6) Interactions between negatively charged residues at the N termini of alpha helices with the helix dipole formed by the lining up of all of the dipoles in the individual peptide bonds are stabilizing (this was first noted by one of the more senior lecturers in this course!) A method for predicting the effect of mutations on stability of proteins/protein interfaces is available on the web: http://robetta.bakerlab.org/ (Interface Alanine Scanning)
References Most of what has been covered in the lecture on Protein Stability can be found mainly in Creighton, ch4 especially sections 4.1, 4.3 and 4.4. (available on HSB and BIOCHEM libraries)