Water, H 2 O

1. Water, H2O Part III. Intermolecular forces

2. Condensed Water: Intermolecular Forces We need to understand what happens when water molecules get close to one another, but do not engage in direct covalent binding. Although here we concentrate on water, of course in the long run we have to understand these forces if we are to have a hope of understanding protein structure and action. There are basically 4 kinds of “long range” interactions that are of importance in biological systems: 1. van der Walls (dispersion forces) between neutral atoms, 2. dipole-dipole forces between polar molecules, 3. hydrogen bonding and finally 4. coulombic (charged ions). We’ll ignore the coulombic part for right. We will discuss dispersion forces and dipole-dipole coupling, then concentrate on hydrogen bonding if we manage to convince you that the H-bond is the dominate part of the attractive forces holding water together.

3. Dispersion Forces - Dispersion forces result in an attractive force felt between electrically neutral atoms. - Dispersion forces are strictly quantum mechanical in nature and exist between ALL atoms and molecules, independent of their charge state or the presence of a large dipole moment. - In the case of two metal plates separated in vacuum this force is known as the Casimir force. - The force that allows Argon to form the solid is due to dispersion forces. - The name “dispersion force” arises from the frequency dependence of the force, which has a maximum value in the UV to optical spectral range, where the “dispersion” (the derivative of the dielectric constant with frequency) is greatest. Intuitively, dispersion forces arise from the time-dependent dipole moments arising from the movement of an electron around a nucleus. This flickering dipole moment induces a dipole moment in a molecule nearby and thus a force exists between the two molecules.

4. This is clearly an enormously complex interaction, but we can very roughly estimate (in a Bohr-like mix of classical and quantum mechanics) the strength and frequency dependence of this force. Consider an electron orbiting a proton, as in hydrogen. Let the electron orbit at the Bohr radius Ro. The radius Ro of the orbit has an energy E(R) given by: equal to the energy of a photon of frequency f which can ionize this state. The orbiting electron has a (instantaneous) dipole moment  = eRo. This dipole moment will polarize a neutral atom which is a distance r away, giving rise to a net attractive force and potential energy of interaction w(r). To compute w(r), we need to know the electronic polarizability of a neutral atom. The polarizability of an atom is defined as the coefficient between the induced electric dipole p in response to an applied field E:

7. Dipol-Dipol Coupling The water is polar. Thus, it has an electric dipole moment, which one would guess would give rise to a stronger interaction than the induced dipoles governed by the dispersion forces. How can we estimate the strength of dipole-dipole coupling? We can (and will) do a simple dipole-dipole coupling calculation, but note well that for a molecule with a large dipole moment like water the effect of the electric field of the dipole moment on neighboring molecules leads to substantial changes in the polarization of the original molecule; in fact at some critical value the system can undergo a ferroelectric transitionwhich leads to effects totally not predicted by a simple isolated dipole-dipole calculation. If we have simple isolated dipoles the calculation is very simple. Let molecule 1 have electric dipole moment p and be a distance R from a second molecule of the same dipole moment. The electric field at a distance R from a dipole is:

9. Enhanced polarization So, we see that in the case of an isolated pair of dipoles the cohesive energy of water cannot be explained. The next step is to consider the effect of the dipole moments on each other for the case of many dipoles. The basic problem is to include the effect that the local field makes on the overall polarization. Our goal here is to find the net dielectric constant that we would expect water to have given the dipole moment of 1.83 D that we know from the previous section. If the dielectric constant comes out to be significantly higher than this, then we know that significant corrections need to be made. We need to calculate the local field F that the dipoles feel, since that is the electric field that aligns the dipoles. As we all know, if a dielectric is placed between the plates of a capacitor with charge density ±σon the plates there is a displacement field D which is determined by the free charge alone, and a field E within the dielectric which is less than D because of the induced polarization I within the dielectric. The relationship between D, E and I is:

10. Now, I is related to the dipole moment of the polar molecule,  since the internal field in the dielectric wants to align the dipole moment as we just saw. But, the field that actually aligns the dipole is not the macroscopic field E in the dielectric but instead the local field F that the dipole feels. It is confusing to consider yet another field, yet the crux of the problem is that the fields D, E and I are all macroscopic fields which are in effect an average over space and never deal with the microscopic and atomic nature of the real polar material.

11. In fact, the calculation of F is an extremely difficult problem. We will first follow Debye’s simple calculation that • ignores correlations at a local level, and then • do a very simple mean field calculation that attempts to take local correlations into effect. • The idea here is to carve out a small sphere (small means small compared to the macroscopic dimensions of the capacitor but large compared to the atomic parts of the dielectric). We do this in the hopes that the dielectric has a small enough aligment that only large numbers of molecules summed together will give rise to an appreciable field. We can split the local field F up into 3 parts: • F1 is due to the actual charge on the plates, • F2 is due to the polarization of the charge on the dielectric facing the plates plus the net charge on the surface of the sphere, and • F3 is the local field due to the little sphere that we carved out.

12. We know that:

13. That is a negative number! This means that the large dipole moment of water results in a huge self interaction and that it should be by our analysis a ferroelectric. Looks like our attempt to rationally explain water is all wet, which shouldn’t be too surprising since the C-M relationship is a mean-field theory and which will break down for dense materials, especially water which is 55 molar and has a huge dipole moment! The C-M relation has a divergence built into it: since the term:

14. cannot exceed unity, there is a divergence in the dielectric constant for a very finite value of the dipole moment of the molecule, and in fact water is well past that limit, hence the amazing result that the predicted dipole moment of water from the known dielectric constant is a negative number. In fact, neither water nor ice are ferroelectrics. The divergences in C-M can be removed including the increase in the dipole moment of the molecule due to the internal field, and this has been done by Onsager. The key to Onsager’s work was the realization that the internal field at the site consists of two pieces: a modified external field G due to the dielectric medium, and a local field R due to the reaction of the polarizing medium back on the dipole in question.

16. The Ice Problem and Entropy We saw in our sp hybridization scheme that we had 4 bonding orbitals forming a tetrahedral symmetry around the oxygen atom. There is a partial charge water molecule model called the Bjerrum model which is based on the model of bond hybridization. Given the dipole moment of the water molecule, and the angle of 105o between orbitals, and the bond length of 1 Ǻ, one quickly finds that each arm contains a partial charge of 0.2e. Industries have arisen trying to calculate this number to ever higher precision, but for now this will suffice. Now, we let other water molecules form a tetrahedral environment around the water. A strange kind of lattice is formed. Any two of the four lobes contain positively charged hydrogen atoms, while the other two lobes contain negatively charged lobes of excess electron density. Note that there is an intrinsic amount of disorder contained in such a lattice. Charge neutrality requires that any oxygen atom can have at most 2 hydrogen atoms near it. The lattice that forms has a residual amount of entropy due to the possible ways of arranging the hydrogen atoms and still obey the so-called “ice rule”: For the four nearest neighbor hydrogens surrounding the oxygen atom, two are close to it and two are removed from it”. The residual entropy of ice is quite substantial: S/kB is 0.4 extrapolated to 0 K! Thus, in the case of ice it isn’t true that as T goes to 0 that the entropy S goes to 0.

17. We show some of the possible patterns that are possible in an ice lattice that satisfy the ice rules. Note that the hydrogen atoms are not at all free to individually move back and forth between the two equivalent positions that can be seem to exist between the oxygens, but rather the motions in the ice lattice are by necessity of a collective nature. In some respects, this lattice shows many aspects of frustration that play such a predominate role in spin-glass systems. In this model, there are no charged defects. In reality the ice lattice we have presented cannot represent the true lattice that is formed by water, since such a lattice has ferroelectric transitions which we know do not occur in a real ice or water system.

18. There must be considerable dynamics in this lattice, highly coupled as it must be, to explain the lack of dipole ordering transitions. There is a significant point to be made here, however. The ice model has built into it a signficant amount of entropy due to the empty orbitals which allow various bonding patterns to form. If defects are put into this lattice which disrupt the bonding patterns, they can actually decrease the entropy of the system since the bonds can no longer jump among a collection of degenerate states. Thus, the defects via a pure entropic effect can raise the net free energy of the system. This increase in free energy due to decrease in entropy is called the hydrophobic effect. It makes energetically unfavorable for the non-polar amino acids to expose themselves to bulk water, and thus contributes substantially to the free energy of various protein structures.

19. Hydrogen Bonds We are left with only coulomb interaction as the only force which can explain the strong coupling of water molecules to each other. It would appear that the positively charged hydrogen atom at any one site will be attracted to the negatively charged lone-pair orbital of the empty site facing it. This electrostatic attraction of the positively charged hydrogen for the negatively charged lone-pair orbital is loosely defined as a hydrogen bond. It isn’t a covalent bond but more of an electrostatic interaction. As we have presented it here, it is strictly electrostatic in nature. From the known crystal structure of ice it is easy to guess what the size of this attractive potential energy will be. We can guess that in ice the distance between lone pair orbital and the hydrogen atom is roughly 2 Ǻ, therefore the energy is trivially:

20. A typical H-bond

21. At this point we have been able to understand some of the consequences of • - the high density and • the large dipole moment • of the water molecule, and we have been able to do this by doing a strictly classicalelectrostatic calculations. However, the strictly electrostatic picture of water self-interactions is incomplete • and some form of covalent bonding (delocalization) of the hydrogen atom in condensed phases must occur.This semi-delocalized hydrogen state is the true hydrogen bond, and it is extremely important in biology (see e.g. the base-pairing in DNA).

22. You can do some simple calculations concerning what such bonds must look like. At the simplest levelthe hydrogen atom can be in either one of two (degenerate) sites. For example, in water it can either be1 Å away from an oxygen atom at site A or site B, and the two sites are separated by about 3 Å. Now,we have noted the hydrogen atoms are not free to make arbitrary transitions between these two equivalentsites, yet it is amusing to calculate the tunneling rate that you would expect in the no correlation limitfor hydrogen transfer. Note that hydrogen atoms will have the maximal tunneling rates between the twoequivalent sites, and so the disorder produced by tunneling will be maximized in the case of the hydrogenbonded system. Replacement of the hydrogen with deuterium will decrease the tunneling rate. A simple estimate of the tunneling rate can be made by appealing to the relatively simple double harmonic oscillator problem. The next figure shows the potential function we have in mind.

23. A double-welled parabolic potential surface.

24. TheSchrödinger equation is:

25. We can guess what this tunneling frequency might be. The infrared absorption spectrum of water has avery strong feature at about 3 μm which is due to the hydrogen vibrating against the oxygen atom.The infrared absorption spectrum of water reveals that the first excitedvibrational state of the hydrogen atom has a very large energy of about 0.3 eV, which we would guessmight be fairly close to the top of the energy barrier. Our tunneling calculation is actually pretty useless in this range, it would give an extremely high tunneling rate of the order of 1013 s-1. In fact, of course, the water is actually ionized at room temperature since the hydrogen “ion” is present at the 10-7 M concentration range. All these facts point to a picture of rapidly tunneling hydrogens in the icenetwork. There have been some measurements. A hole-burning technique was used to actually measure the tunneling rate of the hydrogens in a hydrogen bondedcrystal (benzoic acid) and it was found a tunneling rate of approximately 1010Hz, still very fast.

26. This result has several significant consequences. First, since the proton can rapidly tunnel betweensites one expects that the protons should be highly delocalizedin ice. Now, I have to be rather carefulhere since (1) the ice rules allow substantial disorder even in the absence of tunneling (2) the ice rulesforce significant correlations in the tunneling process, so that the effect mass of the tunneling state mustbe substantially higher than the bare proton mass. However, it is true that X- ray diffraction of icereveals complete disorder in the protons and hence ice is termed a proton glasswith significant residualentropy at T=0 K as we mentioned. Second, since the hydrogen bond is quite “soft” and in fact is bestapproximated as a bistable double minimum we would expect that this bond is extremely non-linear, that is, very non-hookean in restoring force vs. displacement, particularly at large amplitudes of displacement. Infact, it is exactly this non-linearity in the displacement which Davidov has used in his theory of dynamicsoliton propagation in hydrogen bonded systems.

27. Percolation in understanding the phase transitions of the ice lattice, and later proteins. If you look at the ice model you might become worried about the stability of the structure, assuming thatit is the hydrogen bonds that hold the whole thing together. Only 2 of the 4 possible links can be filled atany one time. You might try to construct a toy of an object with 4 holes and only 2 links allowedper hole: is such a structure stable or not? Thus, we have a system which is held together in a rather fragile way and for which simple rotationsallow the scaffold to be cut out. The problem is to basically trace a path of connected hydrogen bonds fromone side of the bulk material to the other. This so-called percolation of bonds determines the rigidity ofthe object. The classic example is the so-called vandalized grid, where a disgruntled telephone employee cuts links at random in a 2dimensional resistor net. The resistance of such a net is surprisingly non-linear function of ρ, the fractionof uncut bonds, where we can have a “valency” Zc of resistors (or wires) per site. Again, surprisingly, in an infinite lattice if ρ is below some critical number ρc there is absolutely NO current flow in the net, andthe transition is quite accurately a second order phase transition. In fact, and we do not know how to provethis, for a 2-D lattice with Z=4 the critical percolation threshold is 0.5. Again, surprisingly, for D greaterthan 2 there are no analytical ways to find the critical threshold. Thenumerical values found by computers for several different types of lattices. Note that water with a Z=4diamond lattice in 3 D has a ρc (bond concentration) of 0.388, so ice seems safely solid.

28. The whole subject of bond percolation and the related issue of rigidity transitions is a fascinating fieldwhich links directly with many aspects of biology and networks. Polymers typically can undergo rigiditytransitions as the number of cross-link approaches a critical number per node. A useful web site whichdiscusses this subject can be found at http://www.pa.msu.edu/ people/jacobs/, the site of Prof. Jacobs inthe physics department of Michigan State University. There are two important points raised at this site: • there is NO algorithm known which can predict therigidity of a network in 3 dimensions, and • the rigidity is an inherently long ranged interaction. • The figure taken from http://www.pa.msu.edu/ people/jacobs/, gives a visualization of why 3 dimensions areso much more difficult than 2 dimensions.

29. Why 3 dimensions are so hard?

30. There are lots of amusing examples of percolation problems. • A great web site can be found at theBoston University Center for Polymer Studies, http://polymer.bu.edu / trunfio/ java/blaze/ blaze.html#applet. • They have a Java applet running there which (when it doesn’t crash your computer…) shows how forestfire speading can be viewed as a percolation problem with a rather sharp threshold. • Another fun thing arethe “happy and unhappy balls” which can be bought from Arbor Scientific (http://www.arborsci.com/).These are black polymer spheres that when squeezed seem to have identical elastic constants. Yet, if youdrop the balls you will find that one of the balls has almost no rebounding ability while the other is quiteresilient. This is an example of a systems where the dynamic behavior of the ball is quite different fromthe static behavior. I think it is due in the case of the balls to a phase transition in these polymer balls. Weprobably have a glass-rubber phase transition, which is the next step down from a gel. In other words, in agel you have a simple rigidity percolation which gives the solid a finite shear modulus, but one can quite • easily have rotational and translational freedom on a local scale which can make the object quite “soft”.If there is cross-coupling between the changes, as there always is and as we now know how to calculate,then yet another phase transition can occur which results in a glass state, which we mentioned.

31. The hydrophobic effect: entropy at work • Let’s finish our discussion about water. There is much evidence that liquid water is • a highly ordered liquidbut • which strangely has a large amount of internal disorder due to its’ hybridization scheme. • Whensomething is introduced into the water which cannot form hydrogen bonds, it forces the water to form acage around the molecule which has a lower entropy than the bulk liquid itself. The free energy changeis predominantly due to entropic rather than internal energy changes. This negative entropy change associatedwith ordering water is called the “hydrophobic effect”.

32. The hydrophobic effect can be seen in the fact that - many aliphatic moleculesare quite soluble inalcohols and other reasonably polar molecules but very sparingly soluble in water, as opposed to othersolids made of strongly covalently bonded molecules which are sparingly soluble in almost any solvent tostrong interatomic bonds. Further, - the solubility of aliphatic molecules in water takes on a characteristictemperature shape. If we consider the transfer of a benzene molecule (for example) from a neat solutionto a water environment the free energy change ΔG can be written as:

33. The enthalpy change ΔH of bringing benzene into a highly polar environment is actually likely to benegative: all of the dispersion effects we considered make it better for the benzene to be in water. However,the fact that benzene cannot form hydrogen bonds means that the entropy change upon entering waterΔSis negative: the entropy is smaller for benzene in water, and overwhelms the negative enthalpy. In fact,solubility plots indicate that benzene is more soluble at low temperatures than high temperatures (up toa point), as you would expect for free energy change with rather small negative ΔH and large negativeΔS. However, beyond a point the effect of increasing temperature is to break down the lattice of hydrogenbonds formed in water and the effective ΔS decreases with increasing temperature.

34. There is a simple demonstration of this effect. In the bagis a very concentrated solution of sodium acetate which is incredibly soluble in water (1 gramm dissolves in 0.8 ml of water!). This high solubility means that it likes to form hydrogen bonds with water.When it crystallizes, the water is forced to form a shell around the incipient crystal and 1) the resultingnegative entropy change is quite unfavorable, on the other hand there is 2) a latent heat of crystallization (enthalpy change) due to the packing of sodium and carboxylic groups which is quite favorable. This term is related to thevolume. Suppose that a microcrystal of radius R forms. Then, in general, the net free energy change is: If B∙T is considerably greater than A then crystals up to some radius Ro = 3A/2BT actually raise the free energy and do not form. That is, there is a minimum nucleation size below which the crystals areactually unstable. If the B term is big enough, the barrier can actually be so high as to allow the liquid todrastically supercool.

35. Hydrogen Bonding and Protein Stability What does all this have to do with proteins and nucleic acids? Clearly, everything since these macromoleculeare held together via hydrogen bonds and are dissolved in water. We can examine the role of thehydrogen bond on several levels. 1) The first point to make is that water molecules are strongly associated with the hydrogen bond donoramino acids which as we remarked are present in high concentration on the surface of the protein. Thereare of course internal hydrogen bonding amino acid residues which engage in intra-molecular hydrogenbonding to form things like the α helix and the β pleated sheets which are of great interest, but there isalso a rather random array of amino acids on the surface. 2) The second point is that if the water moleculesare not present on the surface of the protein then the amino acids will form a hydrogen bond network witheach other, and in fact undergo a rigidity transition. An example of such a rigidity analysis can also befound at Prof. Jacob’s web site, where he applies his 3-D pebble algorithm to determine rigidity. The next figure gives an example of such an analysis.

36. The crambin protein molecule decomposed into its rigid cluster units. Each rigid cluster is colored differently from its neighboring rigid clusters. A given bond will be colored half one color andhalf another color when it is shared between two distinct rigid clusters.