Biophysics II By A/Prof. Xiang Yang Liu Biophysics & Micro/nanostructures Lab Department of Physics, NUS

Outline Review of Energy, Enthalpy and Entropy and the correlation with Q and W Equilibrium and Equilibrium constant

Energy, Enthalpy and Entropy As dE = Q rev - PdV, At const V, dE = Q rev For single phase  E = Q rev =  Q rev =  mC v dT As dH = dE + PdV + Vdp = Q rev - PdV + PdV + Vdp = Q rev + Vdp At const P, dH = Q rev For single phase  H = Q rev =  Q rev =  mC p dT dS =  Q rev /T, single phase-  Q rev =  mC p (or C v )dT Isothermal reversible expansion (  E = 0,  H =  (PV),  S = W rev /T; for ideal gas,  S = nRln(V f /V i )) At constant T and P (such as melting and evaporation)  H = Q,  S=  H/T The Change of Enthalpy and Entropy in Different Processes

Chemical potential μ measures the availability of a particle species Goal: Understand how both concentration and internal energy of a molecular species enter its chemical availability. At equilibrium  A,  =  B,  matching role for macroscopic systems in equilibrium The corresponding standard thermodynamic quantities of H o, S o, G o G o = H o - TS o,  G o =  H o - T  S o A solution with multi-solutes- similar to that of mixtures of gases, but instead of partial pressure of 1atm, the concentrations for each solute are defined at 1M (or mole fraction = 1, etc. depending on the unit of concentration used.)

Ideal mixing + AB AB  What does it mean  AB -1/2 (  AA +  BB ) = 0  E = E fin – E ini = 4(1/2  AB ) – [2 (1/2  AA ) + 2 (1/2  BB )] = 4[  AB -1/2 (  AA +  BB )] = 0 No volume change (  V = 0)  H =  E +  (PV). At const P,  H =  E + P  (V) = 0

Ideal mixing In the mixing of a multi components solution,  E mix =  H mix = 0,  s mix-i = n i Rlnx i (i = 1, 2, …,),  S mix =   s mix-i = R  (n i lnx i )  G mix-i =  H mix + T  S mix = RT  (n i lnx i ) An ideal solution or ideal gases  G = G o +  G = G o + RT  (n i Rlnx i )  Chemical potential at const T, P,  i = [  G/  N i ] T,P, Nj, j  I   i =  i o + RTlnx i (  i o : Standard Chemical potential) The above expressions hold for a mixture of ideal gases, where x i = P i /P.

Equilibrium constant of two states N n N de K

Free energy… Denaturation of a protein or polypeptide- the reverse process of protein folding with some stabilizing effects. Heating proteins and adding surfactants/salts may lead to denaturation

Free energy… Native Denatured GG If a protein is denatured, it will be trapped in a local minimum and it will be difficult to get it back to the native state. How does it happen?

Free energy… Denaturation of a protein or polypeptide:   G den =  H den – T  S den   S = R ln (W den /W native ) Since W den /W native >> 1,  H > 0 (require E)  low T,  G > 0,  high T,  G < 0. >0. The breaking of the favorable interactions that hold the native conformation will surely require the input of energy, so  H >0.

Free energy… The ratio of molecules at equilibrium For the special case, n d /n n is the ratio of molecules at equilibrium,  G = 0

Free energy… For a denatured polypeptide, see how we can arrive at the conformation distribution at equilibrium g i ~ W i g n ~ W n ~ 1 g d ~ W d >> 1 Native Denatured EE nn nn

Free energy… Boltzmann distribution Actual distribution

Free energy… Putting quantities on a molar basis: At equilibrium,  G = 0, n d = n o d, n n = n o n K is the equilibrium constant  G o is the standard free energy

Chemical Reactions

Chemical Potential  i =  i o + RTln[i ] [i ]: concentration of ;  i o : chemical potential at P = 1 atm, T = 298K and [i ]  1.  i o depends on the unit of concentration selected, ie  If the unit of [i ] is mole fraction, x i,  i o : is the chemical potential at P = 1 atm, T = 298K and x i  1;  If the unit of [i ] is “molar” (moles per liter), M i,  i o : is the chemical potential at P = 1 atm, T = 298K and M i  1.  The same applied to other concentration units

Chemical Potential  i =  i o + RTln[i ]  i : describing the availability of particles just as T describes the availability of (internal) energy. The chemical potential is greater for molecules with more internal energy as they are more eager to dump that energy into the world as heat thereby increasing the world’s disorder). The chemical potential goes up when the concentration increases (more molecules available)

Chemical Potential A molecular species will be highly available for chemical reactions if its concentration is big or its internal energy is big.

Chemical Reactions Biomineralization & Demineralization Ca 2+ + CO 3 2- CaCO 3 ↓ H+H+ 5Ca (PO 4 ) 3- + OH - Ca 5 (PO 4 ) 3 OH (HAP)↓ H+H+

Chemical Reactions Chemical Potential  Chemical forces Chem. Potential difference Ca 2+, CO 3 2- CaCO 3 ↓ Chemical potential  Chem. Potential difference CaCO3 ↓ Ca2+, CO32- C > C eq Ca 2+ + CO 3 2- CaCO 3 ↓ H+H+

Chemical Reactions The reaction will stall when  i B =  i B or  = 0. Chemical equilibrium is the point where the chemical forces balance.

Chemical Reactions  G gives a universal criterion for the direction of a chemical reaction. Example Ca 2+ + CO 3 2- → CaCO 3 , after reaction, how much Ca 2+ and CO To find the condition for equilibrium, to find the Gibbs free energy change for between the final and initial states.  G =  (  ) fin -  (  ) ini  G = (  CaCO3 ) - (  Ca(2+) +  CO3(2-) ) At equilibrium, the concentrations of all species should fulfill the eqs. Let be the equilibrium constant, then pK eq = - log K eq

Chemical Reactions K eq :  temperature and Pressure dependent  It depends on the unit of concentration 2H 2 + O 2  2H 2 O (8.8)

Chemical Reactions General reactions v k : the stoichiometric coefficients  G-the net chemical force driving the reaction A reaction will run forward if  G 0. Standard free energy change:

Chemical Reactions

Dissociation: Ionic and partially ionic bonds dissociate reality in water Association constant K a and Dissociation constant K d.. When K a > 1 (logK a > 0, K d < 1, log logK d < 0), A + and B - tends to associate to AB. When K d > 1 (logK d > 0, K a 0), AB tends to dissociate to A + and B -. The large pK a, the easier the dissociation of the protein will be. A + + B - AB KaKa KdKd

Dissociation: Ionic and partially ionic bonds dissociate reality in water n i : the number of species i. AB A + + B - KdKd KaKa

Dissociation: Ionic and partially ionic bonds dissociate reality in water At equilibrium Let Standard Specific Gibbs Free change in association

Dissociation of water The dissociation of water: H 2 O  H + + OH - For pure water [H + ] = [OH - ] = M Dissociation Const: K d = [H + ][OH - ]/[H 2 O]. As [H 2 O] is constant at a given T, we have then K w = K d [H 2 O]. K w = [H + ][OH - ] = (10 -7 ) 2. Ion product of water at room temperature  pH = -log K w  pH = 7-neutral pH.  pH < 7, acidic (an acidic solution).  pH > 7, basic (a basic solution).

Dissociation: Ionic and partially ionic bonds dissociate reality in water A measure of the energy of single charges in a particular medium is its self-energy E s -the energy of a charge in the absence of its counter ion. r s : Stoke’s radius-the radius of charge distribution D: Dielectric const.

Dissociation: Ionic and partially ionic bonds dissociate reality in water The interior of a globule protein It is difficult to bury a charge in the interior of a globular protein due to the hydrophobic environment. To estimate the effect of the self energy of amino acids in solution as opposed to being buried in the interior of a protein Association constant K a: the large pK a, the easier the dissociation of protein will be. Globular protein Hydrophobic interior A + + B - AB

The charge on a protein varies with its environment ProtonatedDeprotonated Acidic side chain -COOH-COO - + H + Basic side chain -NH 3 + -NH 2 + H +

The charge on a protein varies with its environment Propobility of protonation

The measurement of the degree of disordering and the freedom The direction of change in thermodynamic system 2 nd law Review

Chapters in Textbook Chapter 8, in Biological Physics