Physical Biochemistry Kwan Hee Lee, Ph.D. Handong Global University
Week 9
CHAPTER 4 Physical Equilibria
Basic Concepts Biological organisms are highly inhomogeneous. Different regions of each cell have different concentrations of molecules and different biological functions. Cover the equilibrium and transport of molecules among different regions of a system. Examples of transport: cell compartments separated by membranes, the inside and outside of a membrane, a liquid or solid in contact with a gas, and two immiscible liquids. Chemical potential will tell us whether two or more compartments are in equilibrium with one another or whether they are not in equilibrium.
If a species has a different chemical potential in two phases in contact with one another, it will move to the phase with the lower chemical potential. This will continue until the species reaches equilibrium, and its chemical potential becomes the same in all phases. It will allow us to distinguish and characterize active and passive transport in cells, the equilibrium concentrations of molecules separated by semipermeable membranes, and the equilibria of molecules between solids, liquids, and gases that determine fp, solubilities, bp, and osmotic pressure.
Applications: Membrane and Transport. Most biological membranes consist of a lipid bilayer that contains proteins and other molecules that serve as recognition sites, signal transmitters, or ports of entrance and exit. Membranes are so thin, having thickness of only one or two molecules, that they are often considered to be two-dimensional phases. The thermodynamic properties of membranes are then described in terms of surface properties, such as the surface chemical potential and the surface tension or pressure.
Membrane and Transport. Membranes not only separate the contents of a compartment from its surroundings but also permit the controlled transport of molecules and signals between the inside and outside. Differences between the inside and outside of a cell influence the exchange of metabolites and electrical signals, the flow of heat, and changes in shape. Temperature differences cause heat flow, pressure differences cause changes in shape, and electrochemical potential differences cause molecular transport and electrical signals.
Applications: Ligand Binding Noncovalent interactions that bind ligands like O2 to hemoglobin, substrates to enzymes, and complementary strands of DNA or RNA to one another underlie essential dynamic processes in living cells. Equilibrium dialysis provides a method of exploring the binding between macromolecules and small ligand molecules. In equilibrium dialysis a semipermeable membrane allows a ligand to reach equilibrium between two phases, one of which contains a macromolecule.
Ligand Binding The difference in concentrations of the ligand on opposite sides of the membrane depends on the interaction of ligand and macromo9lecule. This provides a very useful and easy method for studying equilibrium binding consatnts.
Applications: Colligative Properties The chemical potential must be the same for each component present in two or more phases at equilibrium with one another. The component can be the solvent or each of the solutes in a solution. The phase are solids, liquids, gases, or solution compartments separated by a semipermeable membrane. Any change in a property such as temperature, pressure, or activity in one phase that results in a change in chemical potential must be accompanied by an equal change in chemical potential in the other pahses, for the system to remain in equilibrium.
Colligative Properties The fundamental fact allows us to explain the colligative properties, such as the freezingpoint lowering, the boiling-point elevation, the vapor-pressure lowering, and the increase of osmotic pressure when a solute is dissolved in a solvent. Colligative properties are used to determine the concentrations and molecular weights of solutes in solution; they can be used to measure association and dissociation equilibrium constants of biopolymers.
Phase equilibria The transfer of a chemical from one phase to another is illustrated by the evaporation of liquid water into the vapor phase; the heat removed from our bodies by evaporation of sweat is essential to survival in hot climates. Transport of ions from inside a cell to outside is important to nerve conduction. Equilibrium is a dynamic process. When an organism dies, it approaches closer to equilibrium.
One-Component Systems If a system consists of two or more phases of the same substance in equilibrium at the same temperature and pressure, then the difference in chemical potential between the substance in the different phases is zero. If the system standard state is used for a substance in two different phases, its activity in the two phases will be the same at equilibrium.
Gibbs Phase Rule F = C-P+2, F )number of degree of freedom), C(components), P(number of phases) Most substances have three phases: solid, liquid, gas.
Vapor-liquid Equilibria and Clausis-Clapeyron Equation Δ vap V = V m, gas V m, liquid V m, gas V m, gas = RT/p ( T/ P) vap = T(RT/p)/ (V vap H m ) = RT 2 /p V vap H m dp vap /p vap = (Δ vap H m )/R (dt/t 2 ) ln p vap = (Δ vap H m )/RT + C = (Δ vap H m )/RT + (Δ vap S m )/R
Causis-Clapeyron equation For a pure liquid whose molar enthalpy of vaporization and normal boiling point are known, we can use this equation to obtain the equilibrium vapor pressure at other temperature.
Causis-Clapeyron equation Clausis-Clapeyron equation can be used to calculate a normal boiling point temperature from two measured vapor pressure at different temperatures. First the enthalpy of vaporization is calculated from this equation, then the temperature corresponding to a vapor pressure of 1 atm is calculated. An easy way to apply the Clausis-Clapeyron equation is to plot ln P versus 1/T. The slope is equal to ΔH vap /R. the vapor pressure at any temperature, or the boiling point at any pressure, can be read from the graph.
Causis-Clapeyron equation In using this equation, there are limitations: the assumption that ΔH vap is independent of temperature. Over a wide enough temperature range, ln P versus 1/T will be a curve; the slope at any point will give ΔH vap at that temperature. The assumption of gas ideality and the neglect of pressure dependence of the liquid activity are usually valid. We can increase the activity and therefore increase the vapor pressure of a liquid by applying an external pressure.
Solutions of Two or More Components For solutions containing two or more components present in two or more phases at equilibrium, μ A (phase 1) = μ A (phase 2) =.; μ B (phase 1) = μ B (phase 2) =. For equilibrium between liquid and vapor, there is equality of the chemical potentials of each component in two phases. If A is the solvent in a liquid solution, μ A (solution) = μ A (g, P A ), μ A (solution) = μ A (g, P A ) is for pure solvent.
Solutions of Two or More Components For liquid solvents below their boiling temperature, P A will be less than 1 atm. For an ideal gas, μ A (g, P A )- μ A (g, P A ) = RT ln P A /P A Any nonideality in the gas at pressures near 1 atm can be ignored, because it will be small compared to nonidealities in solution. Therefore,, μ A (solution)- μ A (pure solvent) = RT ln P A /P A
Solutions of Two or More Now relate vapor pressures to the activity of the solvent. μ A (g, P A )- μ A (g, P A ) = RT ln a A. Therefore, at equilibrium with gas, RT ln (P A /P A )=RT ln a A a A = P A /P A Components All of these must be measured at the same temperature. Vapor pressure measurement thus provide a simple method for measuring the activity of the solvent.
Henry s Law P B =k B X B : Henry s law, where P B is vapor pressure of solute B, X B = mole fraction of B in the solution, and k B = Henry s law constant for solute B in solvent A. Henry s law shows that gas solubility is directly proportional to the gas pressure. Gas solubility is a function of temperature and depends on the solvent. Oxygen is about 20% less soluble at 37 C than at 25 C.
Henry s Law Constants
Raoult s law In a compete description of the two-phase (gas, liquid) system containing two components (O 2, water), we note that the volatile solvent (water) will be present also in the vapor phase. At equilibrium, the vapor pressure of the solvent above the solution is given by Raoult s law: P A = X A P A For dilute solutions, X A is typically near 1.0.
Raoult s law It is reasonable that in this limit the solvent vapor pressure will approach that of pure solvent, P A, which is the proportionality constant for Raoult s law. By contrast, solute B in the same solution is surrounded almost entirely by A molecules, which is typically a very different environment from that in pure liquid B.
Raoult s law Vapor-pressure lowering is a practical method for determining the amount of a solute in solutions. It is a good way to determine the molecular weight of a protein in solution. For a system like air-saturated water to be in equilibrium, each component must have the same chemical potential in each phase. If the concentration of O 2 in the liquid-water phase is so low that the chemical potential of the dissolved O 2 is less than that of the O 2 in the gas phase, then O 2 will be transferred from the gas phase until equilibrium is reached.
Boiling Point and Freezing Point In liquids, molecules are in direct contact; in gases they are often far apart. So it is appropriate to approximate the gas as an ideal gas and to replace the activity of the gas by the pressure of the gas in atmosphere; a A (g) = P(atm) We can ignore the effect of pressure on the activity of the liquid and arbitrary take the activity as unity: a A (l) = 1. Temperature dependence of the equilibrium constant: ln P 2 /P 1 = -ΔH vap /R (1/T 2-1/T 1 ): Clausis- Clapeyron equation
Equilibrium between liquid and gas In figure on the right, both phases are at the same temperature and pressure, so the chemical potential of water is the same in both phases. If the temperature is raised, the water will establish a new equilibrium having a new vapor pressure and a new chemical potential that still will be identical for the two phases. The chemical potential do not depend on the amount of liquid or gaseous water, only on both phases being present and at equilibrium.
Boiling Point and Freezing Point Relationship between pressure and boiling point A (pure liquid) A(g) K = a A (g)/a A (l) Gases are always much more nearly ideal than liquids because the interactions of the molecules in the gas phase are always weaker than the interactions of the same molecules in the liquid phase.
Boiling Point and Freezing Point Effect of pressure on the freezing- and melting-point. Consider a solid and a liquid in equilibrium at temperature T and pressure P: G s = G l G s + d G s = G l + d G l d G s = d G l : if we change the pressure of a solid substance in equilibrium with its liquid, we must also change the temperature so as to keep the increment in free energy the same for the two phases.
Boiling Point and Freezing Point dg = -SdT + VdP -S s dt + V s dp = -S l dt + V l dp Rearrange: dt/dp = ΔV fus /ΔS fus ΔV fus = V l -V s, ΔS fus = S l -S s Since the process is reversible, ΔS fus = ΔH fus /T dt/dp = T ΔV fus / ΔH fus The effect of pressure on a melting point depends on the signs and magnitudes of two thermodynamic variables: ΔV fus and Δh fus The heat of fusion is always positive; heat is absorbed on melting. The volume of fusion can be positive or negative.
Equilibrium Dialysis If we had to rely only on normal saline solution flowing in our veins and arteries, the amounts of oxygen transferred from out lungs to muscle cells would be far too little to support life as we know it. The dissolved oxygen concentration is too low, even if we breathe pure O 2. Red blood cells solve this problem by packaging hemoglobin molecules, which are proteins containing a heme prosthetic group that binds O 2 and stores it until needed.
Equilibrium Dialysis Red blood cells are packed full of Hb and contained by a plasma membrane boundary. The plasma membrane is freely permeable to small, neutral molecules like O 2 but is impermeable to large protein molecules like Hb and to many charged ions. Red blood cells have many important functions in our bodies, carrying materials etc.
Equilibrium Dialysis Equilibrium dialysis is a laboratory method of studying the binding of a ligand (small molecules like O 2 ) by a macromolecules (protein, nucleic acid) quantitatively. Materials made from reformulated cellulose are typically used instead of a biological membrane. These polymers contain small pores that allow the free passage of small molecules, but prevent large molecules from passing.
Equilibrium Dialysis Tubes of cellulose can be filled with 100 ml or so of a macromolecular solution and tied off. The closed specimen solution in the dialysis bag can then be immersed in a beaker of solution containing the ligand. Immediately, the ligand starts to diffuse across the dialysis membrane, and this continues until equilibrium is reached. Stirring the contents speeds the approach to equilibrium. Once equilibrium is reached, all concentrations remain constant. Portions of the contents of the dialysis bag and of the outside medium may be removed for analysis.
Equilibrium Dialysis In a control where no macromolecule is present in the dialysis bag, the concentrations of all species will be the same inside and outside the bag. In a different control where a protein like albumin that does not bind O 2 is in the bag, then the O 2 concentration at equilibrium will be the same inside and outside, although the protein is inside and not outside. If a protein like myoglobin is in the bag, then the total concentration of O 2 inside will be significantly larger than the concentration of O 2 outside, where Mb is absent.
Equilibrium Dialysis The presence of the Mb inside serves to concentrate the O 2 by binding it. By a series of quantitative measurement, it is possible to determine both the binding equilibrium constant and the number of ligand binding sites per macromolecule. A convenient way of treating the data to extract these values is through the Scatchard equation.
Equilibrium Dialysis M + A M A K = [M A]/[M][A] c M = [M A] + [M] = total concentration of M (inside) c A (outside) = [A] c A (inside) = [A] + [M A] c A (bound) = c A (inside) c A (outside) = [M A] Assumption: the concentration of free A inside is the same as the total concentration of A outside. K = c A (bound)/[{c M -c A (bound)}c A (outside)]
The Scatchard Equation ν: average number of ligand molecules bound per macromolecule at equilibrium. ν = c A (bound)/c M K = ν/(1 - ν)c A (outside) = ν/(1- ν)[a] ν/[a] = K(1- ν) The equilibrium constant was written with the assumption that only one molecule of A was bound per macromolecule; it means that ν can vary only from 0 to 1.
The Scatchard Equation Many macromolecules have more than one binding site for a ligand. Then ν can vary from 0 to N, the number of binding sites on each macromolecule. If the sites are identical and independent, it is easy to generalize the equation. If there are N identical and independent binding sites per macromolecule, this means that the N sites have the same binding equilibrium constant K and that binding at one site does not change the binding at another site. ν/n/[a] = K (1- ν /N) ν/[a] = K(N- ν ): Scatchard equation
The Scatchard Equation The Scatchard equation is often used to study binding to a macromolecule. The binding can be measured by any suitable method, but equilibrium dialysis is often convenient. The value of [A] is the concentration on the solvent side of the dialysis membrane at equilibrium.
The Scatchard Equation The value of ν comes from the concentration of A on the macromolecule side at equilibrium. A plot of ν /[A] versus ν should give a straight line with slope of K, y-intercept of NK, and x-intercept of N. If a Scatchard plot does not give a straight line, it indicates that the binding sites are neither identical nor independent.
The Scatchard Equation The figure on the right is for the binding of O 2 to Mb. The fraction of binding sites occupied increases toward a maximum with increasing pressure of O 2. The pressure of O 2 at which the Mb sites are half-saturated, designated p50, is a measure of the strength of the binding. The fact that the straight lines are extrapolate reasonably close to ν =1 at ν/po2 = 0 is consistent with a picture that each Mb molecule has a single and necessarily independent binding site
The Scatchard Equation From the point where each line crosses the vertical corresponding to ν = 0.5, we see that the p 50 values increase with increasing temperature.
Binding of Mg-ATP The it fits the identical and independent-sites model. The slope of the plot gives an equilibrium constant K = 1.37 x 10 4 /M. The intercept gives N=4.2 Since the enzyme is known to have four identical subunits, the binding results are consistent with one Mg-ATP binding site per subunit.
Scatchard plot of the binding of Ethidium to DNA r/[a] = K(n-r), where r is the number of bound A per monomer unit of the macromolecule and n is the number of binding sites per monomer unit of the macromolecule. It represents the identical and independent sites model. The r intercept gives n = 0.23, or approximately one binding site per four nucleotides.
The result is consistent with the fact that ethidium binds to DNA by intercalation. Planar molecules slide between two adjacent base pairs in DNA so that the hydrophobic parts stacked on the base pairs: intercalation. At the lower temperature the line is steeper, indicating a greater binding constant. This temperature dependence of the binding constant yields a negative enthalpy change for the binding of the drug. The strong interaction between the drug and DNA is evidence that the target of the drug in vivo is the DNA.