Born Lets now think about a somewhat different, but related problem that is associated with the name Born. Using a name (usually the name of the guy who first discovered or solved the problem) to describe a particular type of problem is a common in science. This nomenclature provides us with a shorthand code to talk about certain kinds of problems. If I say Born, you should from now on immediately associate this with the concept of electrostatic self-energy of a charged species. If I say Coulomb, you should think about the electrostatic work associated with moving charges relative to one another etc. But now to the topic of the class: The Born energy is the energy it takes to establish or maintain a charge on an object (i.e. the self-energy of that charge) The concept that a charge by its mere existence should possess energy seems pretty non-intuitive to many people, so here is a way to think about it. Lets start with a conducting, but neutral sphere. It will not cost us any energy to deposit a miniscule charge on its surface. But if we now add another very small charge of the same sign, the two charges will now repel each other. We have to expend work (as defined by Coulomb law) to bring these two charges together. As we add more and more charges we have to expend more and more work for each fractional charge we add to those that are already on the surface. This is, because we are working against a higher charge on the sphere. We can think of this as integration over some parameter λ that is 0 for the uncharged sphere and 1 for the fully charged sphere. ! "s(#) =#q4$%0Da&Gel= "s(#) ' q ' d#01(=#q4$%0Da' q ' d#01(=q24$%0Da#' d#01(=12'14$%0'q2Da This looks very similar to Coulomb doesn’t it? The only difference to Coulomb is the factor of 1/2. So we can immediately get to calculating numbers because we remember that we can pull together all the constants for the conversion of meter into angstrom etc. into one constant. So ! "GBorno=12#qint .2Da# 333kcal / mol Where a is the radius of the sphere we are charging. So how much energy does it cost to charge up a mole of 2 Å spheres with one charge on each of the spheres.! "GBorn=12#178.54 # 2# 333kcal / mol = 1kcal / mol How much in a membrane or a protein interior ! "GBorn=12#13# 2# 333kcal / mol = 27kcal / mol Wait, if we can calculate the energy of charging a sphere in a medium we can actually use a little thermodynamic cycle to calculate the energy for transferring an ion from one dielectric (e.g. water) into a medium with another dielectric (e.g. a membrane). Our energy for the transfer of an ion from a medium with dielectric constant D1 to a second medium with dielectric constant D2 is given by ! "Gtransfer( D1#> D2)0= #333$ q22aD1+333$ q22aD2=333$ q22a1/ D2#1/ D1( ) So transferring an ion with a single charge and a radius of 2Å from water to a protein interior (in this case lets say the protein has a dielectric of 3) requires the following free energy. ! "Gtransfer(D1#> D2)0=333$2 $ 21/ 3 #1/78.54( )= 26.7kcal /mol That’s why you hardly ever see a charge buried in a non-polar solvent. These numbers are amazingly good. In particular if you include the fact that the effective ion size is a bit larger than the VDW radius of the ion itself, you get almost perfect numbers. And because there is no entropy loss etc involved (we are not moving anything, We are not working against atmospheric pressure etc.) This is a true delta GUsing the concept of Born energy to understand the charge distribution in a protein active site. Now its time to put some of those equations to work and see if we can use them to help us understand how actual proteins work, if we can make sense of some experimentally measured data. Lets have a look together at the active site of my favorite protein, the photoreceptor protein PYP. PYP uses a para-hydroxycinnamic acid, which is buried in the protein’s interior, as a chromophore to sense blue light. In order for this to work properly the chromophore has to be negatively charged. The protonated chromophore absorbs light in the wrong part of the spectrum and undergoes the light-driven chromophore isomerization reaction, that ultimately leads to receptor activation, with minimal efficiency. Charged groups inside protein interiors are not as uncommon as the high Born energy of solvation might make us think. After all, charged groups are often required to carry out key functions in protein active sites. The proteins most certainly have to expend the energy to burry those charges during the folding process, but this is possible, as long as there are enough favorable interactions elsewhere in the structure that make up for the energetic penalty of burying a charge. So while burying a charge is not impossible, those charges that we find in the interior of a protein are there by random chance, but most likely serve a critical function that made it worthwhile for the protein to expend all that energy for burying that charge. Now lets look at the PYP active site. In it we find the chromophore located right next to a glutamic acid. And as a matter of fact the chromophore actually forms a hydrogen bond to the glutamic acid. What should puzzle you is that the chromophore is deprotonated, even though in water it has a much lower nominal pKa than the glutamic acid. The question you should be asking yourself is, “Why does that proton not simply jump over to the chromophore!” All that would be needed is a change in the hydrogen atom position of just 0.5 Å or so.A recent ultra-high resolution crystal structure (Getzoff, Gutwin & Genick, Nature Structural Biol. 2003) gave a hint. The structure shows that the chromophore adopts a charge state that is a hybrid between the phenolic and the quinonic form. In other words the charge is distributed evenly over the entire 12 atoms when it is on the chromophore. On the other hand the charge would be distributed over only three atoms (those of the carboxylic group) if it were localized on the glutamic acid. Think: chromophore = big sphere -> lower Born energy of carrying a charge Glutamate = little sphere -> higher Born energy of carrying a charge Could the difference in the Born energy quantitatively explain that when glu and the chromophore are buried inside the protein we find the charge on the chromophore, even though in solution the chromophore’s pKa is 4.5 pH units higher than that of the glutamic acid. To find out, lets