Chemistry Review 8-23-2010

99:163 Chemistry Review, Shea

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99:163 Medical BiochemistryHandout for

Optional Chemistry Review SessionMonday, 8/23/2010 at 5-6 pm

Spivey Auditorium (Aud.2), BSB (Bowen)

Madeline A. Shea, Ph.D.Prof. of Biochemistry335-7885, 4-450 BSB

[email protected]

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1. Biomolecules2. Intermolecular Forces

A. van der WaalsB. Hydrogen BondsC. Electrostatics

3. WaterA. Universal Solvent & ReactantB. Self-InteractionsC. Mixtures and the Hydrophobic Effect

4. Resonance Structures5. Acids and Bases

A. Arrhenius and Brønstead-Lowry Acids and BasesB. Water: Acid or Base?C. The pH ScaleD. Acid-Base Equilibria

6. KineticsA. Basic OverviewB. Rate ConstantC. Half-Lives

Table of Contents

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7. Chemical Equilibrium7. The Concept of Equilibrium8. The Equilibrium Constant9. Le Châtelier's Principle

8. Thermochemistry7. The Nature of Energy8. Energetic Driving Forces: Entropy & Enthalpy9. Thermodynamics

9. Electrochemistry7. Oxidation-Reduction Reactions8. Balancing Redox Reactions

10. Binding7. Graphs - Overview of Arithmetic vs. Logarithmic8. Ligand Binding to 1 or 2 sites on a Macromolecule

11. Reference Materials7. Common Values of Constants in Chemistry8. Info on Amino Acids and Enzymes

Table of Contents, cont’d.

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http://www.whfreeman.com/stryerbiochem5/content.htmCompanion site to Stryer’s Biochemistry 5th Edition, highly interactive.

http://wine1.sb.fsu.edu/chm1045/chm1045.htmhttp://wine1.sb.fsu.edu/chm1046/chm1046.htmBy Dr. Michael Blaber from Florida State University.An excellent full tutorial for General Chemistry.

http://ull.chemistry.uakron.edu/biochem/By Dr. James Hardy from the University of Akron, Ohio.Lectures for an undergraduate Biochemistry class.GREAT for learning the properties of Amino Acids & Proteins!

Please note - Students are NOT expected to memorize the numbers/constants inthis handout. They are provided for reference and background only.

Annotated ReferencesFigures from -1. Stryer 5e - Berg,Tymoczko, Stryer, W. H. Freeman & Co.2. Nelson & Cox, Lehninger Principles of Biochemistry, W.H. Freeman & Co., 20053. Alberts et al., Molecular Biology of the Cell, Garland Science, 20034. Lodish et al., Molecular Cell Biology, W.H. Freeman & Co., 2004

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Examples of Common Molecules

Illustrations from Biochemistry textbook “Stryer 5e”.Oxygen=magenta, Carbon=gray, Nitrogen=cyan,Yellow=sulfur, White=hydrogen

http://www.whfreeman.com/stryerbiochem5/content.htm

Common functional groups


Biological molecules are Complex• Functional groups have different chemical properties (polar vs

nonpolar, reactivity, pKa, etc.)• Biological molecules are usually composed of multiple functional

groups• Properties of molecules are determined by the intrinsic properties

of their functional groups, the interactions between groups andinteractions with their environment.

Molecular Asymmetry or Chirality

• A carbon atom that cannot be superimposed on its mirror image is said to bechiral (requires four separate substituents)

• Each chiral center can have one oftwo conformations: R or S.

• Sterioisomers have the same chemicalstructure but different conformations

Molecular Asymmetry

biological activity andamino acids

• Sterioisomers have different threedimensional shapes and thus(usually) have different biologicalactivities

• Amino acids have asymmetriccenters

• Classification system for aminoacids is D (dexro-rotatory) and L(levorotory)

• Proteins contain L amino acids

• Intermolecular binding occurs though multiple (weak) interactions• Multiple interactions between molecules result in complexes.• Intermolecular complexes can range from transient to very stable.

Noncovalent Interactions mediate Complexes

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Intermolecular forces - usu. much weaker than covalent(intramolecular) bonds.

Only 16 kJ/mol of energy is required to overcome theintermolecular attraction between HCl molecules in the liquid state(i.e. the energy required to vaporize the sample). However, 431kJ/mol of energy (~25x more) is required to break the covalentbond between the H and Cl atoms in the HCl molecule

When a molecular substance changes states (gas to liquid, liquidto solid), the atoms within the molecule are unchanged.

The temperature at which a liquid boils reflects the kinetic energyneeded to overcome the attractive intermolecular forces (likewise,the temperature at which a solid melts reflects attractive forces).

The strength of the intermolecular forces determines the physicalproperties of the substance.

Intermolecular Forces

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Example of Weak Interactions

• For hydrogen, the attractions are so weak that the moleculeshave to be cooled to 21 K (-252°C) before the attractions areenough to condense the hydrogen as a liquid.

• For helium, intermolecular attractions are even weaker - themolecules won't stick together to form a liquid until thetemperature drops to 4 K (-269°C).


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http://www.chemguide.co.uk/atoms/bonding/vdw.htmlAttractions are electrical in nature. In a symmetrical molecule likehydrogen, however, there is no apparent any electrical distortion toproduce positive or negative parts. That is true on average.

The ellipse above represents a small symmetrical molecule (i.e. H2).The even shading shows that on average there is no distortion. But,the electrons are mobile, and at any one instant they might findthemselves towards one end, making that end partially negative. Theother end will be temporarily short of electrons and becomes partiallypositive. An instant later, the electrons may have moved to the otherend, reversing the polarity.

van der Waals Bonds

- ++ -

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If there is more than one, the polarity of all the molecules reverses,but there is still the + attracting -. As long as the molecules stayclose, the polarities will continue to fluctuate in synchronization sothat the attraction is always maintained.

This diagram shows how a whole lattice of molecules could be heldtogether in a solid using van der Waals dispersion forces. An instantlater, the distribution would change to show a quite differentarrangement of the electrons as they shifted in synchronization.

Polarity & Alignment of Dipoles

http://www.chemguide.co.uk/atoms/bonding/vdw.html

+ -+ -+ -

+ -+ -+ -

+ -+ -+ -

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Hydrogen atoms in a polar bond (i.e. H-F, H-O, H-N) canexperience an attractive force with a neighboringelectronegative molecule or ion which has an unsharedpair of electrons (usually an F, O or N atom on anothermolecule). Bonds between F, O, or N are very polar:

Because the hydrogen is stripped ofelectrons on one side, it is attracted toan electronegative atom in a nearbymolecule.

Hydrogen (H) Bonds

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Hydrogen Bonds: Biochem. Basting Stitches

• H-bonds are– low in energy (1-2 kcal/mol)– strongest when linear– the glue of local protein structure– possible in all biomolecules but esp. proteins and DNA


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Remember opposites attract& vice versa?

(q1•q2 )/r2

Seems like a good way tohold molecules together.

But most bodily fluidscontain ions competing forthe surfaces of biomoleculesand water binds too.

Ionic interactions are esp.important in “Active Sites” ofenzymes or ion channels orpoly- ion interactions (DNA,RNA) under conditions wherewater is excluded fromcontact surface.

Electrostatics: Coulomb’s Law

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Polar Nature of Water:separation of partial charges(wedge) gives rise to dipolemoment.

This is critical to its successas a solvent of ions andother polar molecules, suchas alcohols.

Water can solvate chargedor polar solutes, or serve asa bridge between + and -ions.

H+

O

H+δ+ δ+

2 δ-

105˚

-

+

Water: Universal Solvent & Reactant



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Anyone can walk on water at the righttemperature …

At low temp., water ,motion slows andoptimizes hydrogen bonds(number & linearity) - making arepeating array. This takes morespace (expansion), and lowers densitywhich allows ice to float.

Marine life survives at the bottom of alake while ice floats on top. Cells burstwhen frozen, which is why steaks get“freezer burn” and why organs can’t befrozen and stored.

At any temperature, bio-molecules arecompeting for water-water interactions,but water is 55 M. Outnumbers allelse.

Ice

Water Self-Interactions:Less Dense at Low Temperature

http://www.whfreeman.com/stryerbiochem5/content.htm20

What drives 2 molecules to bind and stay bound in an aqueousenvironment?

The hydrophobic effect is considered to be the major driving force forthe folding of globular proteins. It results in the burial of thehydrophobic residues in the core of the protein. It is furtherexemplified by the fact that oil and water do not mix.

Water is freed from surfaces that form intermolecular interface:Energetically favorable - increase in entropy of solution.

Mixtures & the Hydrophobic Effect

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Equivalent Lewis structures are called resonance structures orresonance forms.

Example: The Nitrate (NO3-) ion

These Lewis structures are equivalent except for the placement ofelectrons (i.e. the location of the double bond).

Resonance Structures

http://www.whfreeman.com/stryerbiochem5/content.htm 22

Arrhenius definitions:

Acids - substances that when dissolved in water release H+ ionsBases - substances that when dissolved in water release OH- ions

The definition of an Arrhenius acid and base emphasizes the H+ andOH- ions in water

Brønstead-Lowry definitions:

Acids - a substance that can transfer a proton to another substanceBase - a substance that can accept a proton from another substance

These definitions emphasize proton transfer, and can includesolvents other than water (aqueous solutions are not part of thedefinition, proton transfer is the key feature).

Acids and Bases

Water and pH

• Water is in equilibrium:H20 H+ + OH-

• Concentration of pure water is 55.5 M

• Keq= [H+] [OH-] / 55.5 M = 1.8 * 10-

16M

• Ion product of water (Kw) = 1 * 10-14 M2

= [H+] [OH-] so in neutral/pure water[H+] = [OH-] = 1 * 10-7 M

• pH = negative log of [H+]= - log [H+]

• Pure water [H+] = 1 * 10-7 M pH=7

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HCl(g) + H2O(l) -> H3O+(aq) + Cl-(aq)

In the reaction of HCl with H2O, HCl is a Brønstead-Lowry acid(donates a proton to H2O), and the H2O (in this particularreaction) is a Brønstead-Lowry base (accepts a proton from theHCl).

NH3(aq) + H2O(l) -> NH4+(aq) + OH-(aq)

In the above reaction of ammonia with water, the ammonia is aBrønstead-Lowry base, and the H2O is acting as a Brønstead-Lowry acid.

A molecule that can act as both a Brønstead-Lowry acid or aBrønstead-Lowry base (depending on the reaction in question)is termed amphoteric.

Water: Acid or Base? BOTH!


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The concentration of H+ ion in solution low and can vary over awide range (“many orders of magnitude” = many powers of ten).The H+ ion concentration in an aqueous solutions is representedby pH - negative logarithm (base 10) of [ H+ ion].

pH = -log [H+]: neutral (water) = 7 = - log [10-7]-pH = log [H+]

10^(-pH) = [H+]Other Scales:

pOH:pOH = -log [OH-]

-log[Kw] = pH + pOH = 14.0

Similar concept for acid and base dissociation: pKa & pKb:pKa = -log [Ka] pKb = -log [Kb ]

The pH Scale Acids and Bases

• Definition: acids release H+ and bases bind H+

• The equilibrium constant for an acid or base is called Ka and corresponds to the pH at themidpoint of the titration in which H+ are absorbed (or released).

• pKa = - log Ka

• Every acid and base has a characteristic pKa

Ka = [H+] [A-]

[HA]

pKa = - log Ka

Acids & Bases Buffer Changes in pH• Near its pKa, an acid (base)

releases (absorbs) H+

• Since pH depends on [H+],hydrogen ions associating with anacid or base will not affect pH

• Thus near its pKa, an acid orbase will limit pH change orbuffer the solution.

• Buffering works well within theblue box - 1 pH unit on eitherside of the pKa

Biomolecular Substituents help stabilize pH

• Each acid and base buffers ata different pH range (~1 pHunit on either side of its pKa)

• All living systems are buffered

• In serum, carbonate is theprimary buffer but othercomponents also contribute(phosphate, metabolitesamino acids, proteins, etc.)

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1.01.8-2.02.2-2.42.22.8-3.42.9-3.33.0-3.53.74.0-4.55.65.8-6.46.0-6.56.1-6.4

6.46.5-7.56.5-7.07.07.3-7.57.6-8.08.39.210.511.012.013.014.0

battery acid (sulfuric acid)limeslemon juicevinegar (acetic acid)fruit jelliesapple juice, colastrawberriesorange juicetomatoesunpolluted rainpeascornbutter

cow’s milkhuman salivamaple syrupdistilled waterhuman bloodegg whitesbaking sodaboraxmilk of magnesialaundry ammonialime waterlye

Common pH Values

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• Strong acids have weak conjugate bases• Strong bases have weak conjugate acids

The ammonia (NH3)/ammonium ion (NH4+) conjugate acid/base pair

Ammonia can act as a base and accept a proton in water

NH3(aq) + H2O(l) NH4+(aq) + OH-(aq)

Ammonia can act as an acid anddonate a proton in water

NH4+(aq) + H2O(l) NH3(aq) + H3O+(aq)

The product of the acid-dissociation constant for an acid and the base-dissociation constant for itsconjugate base is the ion-product constant for water:

1x10-14 = Ka + Kb = Kw

Acid-Base Equilibria


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Common pKa Values

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Kinetics is the study of reaction rates. The rates of chemical reactions can berelatively fast, or slow, and can also be influenced by various factors, including:

Concentrations of reactants - for rxns. with mulitple solutes, usu. thehigher the concentration of reactants, the faster the reaction

Temperature - the higher the temperature, the faster the reaction

The presence of a catalyst - catalysts speed up reaction rates, however,the catalyst itself is neither created or destroyed in the process. They don’tcause reactions or change the final equilibrium distribution of reactants andproducts; they just speed them up.

The surface area of the reactants or catalyst - reactions that involve solidsoften proceed faster if the solid is a fine powder (higher surface area per unitmass). (We typically will try to start a fire using kindling, not a big log.)

Kinetics: How fast is it going?

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A k BAt the start of a reaction, A molecules make up 100% of the sample.A is converted into B over time. Thus, the number of A molecules inthe sample decreases and the number of B molecules increases.The reaction rate is a measure of how quickly A is consumed, orhow quickly B is produced.

Note: The reaction rate is not constant but changes with time.

Kinetic Rate Constant

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Need to normalize for reactions in which the stoichiometryis not 1:1 (in other words, non-isomerizations).

To correct the reaction rates for the stoichiometry,we divide the rate for each component by theircoefficient in the balanced equation.

General Rate of Reaction

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This equation can be used for the following:• The concentration of a reactant remainingat any time after the reaction has started2. The time required for a given fraction of asample to react3. The time required for a reactantconcentration to reach a certain level - as in"half-life" calculations

The half-life of a reaction, also known as t1/2,is the amount of time it takes for the soluteconcentration to drop to one-half of its initiallevel. In other words, the time when

[A]t = [A]0/2**Half-life is independent of concentration.”

Decay/Clearance Times - Half-Lives Likelihood of Interaction Measured byEquilibrium Constant (Keq)

Free energy of a reaction (ΔG) = Energy products - Energyreactants

ΔG°’ the standard free energy of a reaction under standardconditions (25°C, pH 7, 1 M initial reactants …); under nonstandardconditions use ΔG; ΔG = -RT ln Keq


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The condition where the concentrations of all reactants and productsno longer change with time is called equilibrium.

At equilibrium, the forward rate of the reaction, that producesproduct(s), and the backwards rate of the reaction, that producesreactant(s) are equal. One condition for reaching equilibrium is thatthere is no process by which the reactant(s) or product(s) areremoved from the system.

Forward Rate = Reverse Ratekf [A] = kr [B]

At equilibrium the rate at which products are formed from reactantsequals the rate at which products break down to form reactants.

Reactions at Equilibrium

A B

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To determine the amount of each compound that will be present atequilibrium, you must know the equilibrium constant.To determine the equilibrium constant you need the following genericequation and formula:

na•A + nb•B ok nc•C + nd•D

Note: The same equilibrium condition for the concentrations ofreactants and products was reached from either direction (i.e. startingfrom either pure reactants, or pure product).

Equilibrium “Constant”

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If a system is in equilibrium, and this equilibrium is perturbed by achange in temperature, pressure or the concentration of a reactant orproduct, then the system will shift its equilibrium so as tocounteract the effect of this perturbation.

• If a substance (either reactant or product) is removed from asystem, the equilibrium will shift so as to produce more of thatcomponent (and once again achieve equilibrium).• If a substance (either reactant or product) is added to a system, theequilibrium will shift so as to consume more of that component (andonce again achieve equilibrium).• When heat is added to exothermic reactions at equilibrium,products will be consumed to produce reactants (shift to the LEFT).• When heat is added to endothermic reactions at equilibrium,reactants will be consumed to produce products (shift to the RIGHT).• When volume is decreased, then pressure is increased, and thereaction will shift to reduce the pressure.

Le Châtelier's Principle

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• Some reactions produce energy (exothermic).• Some reactions require energy (endothermic).

Thermodynamics – The study of energy and its transformations.Thermochemistry – The relationship between chemical reactions andenergy changes.

Energy is the capacity to do work or to transfer heat.Chemical energy is the potential energy stored in the arrangement ofelectrons and protons. Thermal energy reflects the kinetic energy of themolecules of a substance.

Systems and Surroundings• System – The flask or container the reactants are in.• Surroundings – Everything else.

** Usually the system is isolated from its surroundings such that there will bean exchange of energy, but not matter. **

Systems tend to attain as low an energy as possible. Systems with a highpotential energy are less stable and more likely to undergo change thanthose with a low potential energy.

The Nature of Energy

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Mixing tends to randomize the contents of two compartments -cell membranes allow for local gradients. But, keeping moleculesapart requires energy. More favorable (raises entropy) to mix.

Heat released upon mixing O2+H2 to make H2O. Enthalpy decreases.Compensates for reduced entropy (3 molecules are now 2).

Exothermic

Energetic Driving ForcesEntropy (∆S), Enthalpy (∆H)

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The First Law of Thermodynamics: Any energy lost by a systemmust be gained by the surroundings, and vice versa. Energy isneither created or destroyed by a process (it is transformed fromone form to another). Energy, E, can be associated with work, w(i.e. force•distance, displacing an object) or the transfer of heat, q.

∆E = q + w∆E = the change in the internal energy of a system

q = heat absorbed by the system from the surroundingsw = the work done on the system by the surroundings

The Second Law of Thermodynamics: In any process that movesfrom one equilibrium state to another, the entropy (∆S) of thesystem and environment together will either increase or remainunchanged. The total ∆S will not change during a reversibleprocess, but the entropy will increase in an irreversible process. ∆Sis constantly increasing (the universe is becoming moredisordered).

∆S universe = ∆S system + ∆S surroundings

Thermodynamics


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Oxidation - refers to the loss of electrons by a molecule, atom or ionReduction - refers to the gain of electrons by an molecule, atom or ion

“LEO says GER” and “OIL RIG” are common mnemonics

Oxidation number - the oxidation number of an atom is the charge thatresults when the electrons in a covalent bond are assigned to the moreelectronegative atom; it is the charge an atom would possess if thebonding were ionic

Chemical reactions in which the oxidation state of one or moresubstances changes are called oxidation-reduction reactions (orredox reactions)

Oxidation-Reduction Reactions

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Zn(s) + 2H+(aq) → Zn2+(aq) + H2(g)The oxidation numbers for the above elements and ions are:

Zn = 0, Zn2+ = +2 and H+ = +1, H2 = 0

Thus, the oxidation number of both the Zn(s) and H+ (aq) changeduring the course of the reaction, => this is a redox reaction.

In a redox reaction, both oxidation and reduction must occur.

1. The compound that contributes the electrons is called thereducing agent. The reducing agent gives up electrons,causing another compound to be reduced (and is thereforeoxidized in the process)

2. The compound that accepts electrons is called the oxidizingagent. The oxidizing agent accepts electrons from anothercompound, causing the other compound to be oxidized (in theprocess, the oxidizing agent is reduced)

Redox Example: Zinc/Proton

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When considering redox reactions,gains and losses of electrons must be balanced.

Half-Reactions - It is common (some think easier) to consideroxidation and reduction as separate processes.

Example:Sn2+(aq) + 2Fe3+(aq) → Sn4+(aq) + 2Fe2+(aq)

This reaction can be considered as two separate processes:• The oxidation of Sn2+ (loss of electrons)

• The reduction of Fe3+ (gain of electrons)

Oxidation: Sn2+(aq) → Sn4+(aq) + 2e-

Reduction: 2Fe3+(aq) + 2e- → 2Fe2+(aq)

Balancing Redox Reactions

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MnO4- + C2O4

2- → Mn2+(aq) + CO2(g)

Begin by writing the oxidation and reduction half-reactions:Oxidation: C2O4

2- → CO2(g)Reduction: MnO4

- → Mn2+(aq)

Next, balance the atoms, ignoring Oxygen and Hydrogen atoms:C2O4

2- → 2CO2(g)MnO4

- → Mn2+(aq)

If the reaction is done under acidic conditions, H2O and H + willbe added. If basic conditions, H2O and OH- will be added tobalance the Oxygen and Hydrogen atoms:

C2O42- → 2CO2(g)

8H+ + MnO4- → Mn2+(aq) + 4H2O

**Note: Basic conditions are a bit more difficult to balance.

Balancing Reactions bythe Method of Half-Reactions

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Now that the atoms are balanced, we can balance charges withoutdisrupting our atoms by adding electrons:

C2O42- → 2CO2(g) + 2e-

5e- + 8H+ + MnO4- → Mn2+(aq) + 4H2O

Multiply the half-reactions by a factor to balance the electrons:5 [ C2O4

2- → 2CO2(g) + 2e- ]2 [ 5e- + 8H+ + MnO4

- → Mn2+(aq) + 4H2O ]

So now the half-reactions are:5C2O4

2- → 10CO2(g) + 10e-

10e- + 16H+ + 2MnO4- → 2Mn2+(aq) + 8H2O

The overall balanced reaction = 2 final half-reactions together:5C2O4

2- → 10CO2(g) + 10e-

10e- + 16H+ + 2MnO4- → 2Mn2+(aq) + 8H2O

5C2O42- + 16H+ + 2MnO4

- → 2Mn2+(aq) + 8H2O + 10CO2(g)

Balance charges after atoms …

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Ligand Binding to a Monomer

• Oxygen or Carbon Monoxide to Myoglobin

• A single ligation state (+ or - ligand).

His F8

His E7

Fe

Iron in a protoporphyrin IX planar ring.Fits in hydrophobic pocket & stabilized bynon-covalent interactions between His F8residue & Fe(II). Presence of hemestabilizes the tertiary structure of apo Mb!

Tight binding of oxygen to the hemerequires the pocket of the protein.Histidine (“His E7”) moves out of the way sooxygen can bind.

Heme Pocket in Globin Fold


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In a laboratory, we experimentally determine populated states* ofprotein or DNA to resolve equilibrium constants or freeenergies of binding (∆G’s) using methods such as those listedbelow.– fluorescence, UV/Vis, CD– filterbinding– dialysis– NMR & Footprinting (I-site)– gel mobility shift analysis (I-species)

Experimental Observables

* usually restricted to measuring sums of statesMust prove linear correlation between binding & signal

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Enumerate the macromolecular states

Molecular States: 1 Ligand Binding

Kassn= [MX]/([M]•[Xfree]*)∆G = –R•T•ln(Kassn) = Gibbs free energy

*activity of ligand, but ~ concentration

X

M# of X = j =0state = i = 1

MXj=1

s=i=2

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Fractional Saturation Depends onEnergetics of Binding Reaction

Express probabilities as a function of the activity offree ligand, [X]free

Y = fraction of occupied sites

Y = [MX][M ]+ [MX]

=filled sitesall sites

Y =Kassn • [M ][X ] free

[M ] + Kassn • [M ][X ] freeKassn = [MX]/{[M]• [X]free} 52

vacant: P1 = mols “M” / mols “Mtotal”

[M] /{[M] + [MX]}

saturated: P2 = mols “MX” /mols “Mtotal”

[MX] /{[M] + [MX]}

Biological Function depends onRelative Probabilities of States

P1 =[M]

[M]+ [MX]=

11 + Kassn[X ] free

P2 =[MX]

[M] + [MX]=

Kassn[X] free1 + Kassn[X ] free

Total = P1 + P2 = 1

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Langmuir Binding Isotherm

Y =Kassn• [X] free

1 + Kassn• [X] free

Y =K • X

1 + K • XY = hyperbola

Fractional SaturationIs experimental observable

in terms of 1 parameter, K

1 independent variable, X

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0

0.2

0.4

0.6

0.8

1

1 0-12 1 0-10 1 0- 8 1 0- 6 1 0- 4 1 0- 2 1 00

Frac

tion

al S

atur

atio

n

[X]free

Frac

tiona

l Sat

urat

ion

0

0.2

0.4

0.6

0.8

1

0 100 2 10- 5 4 10- 5 6 10- 5 8 10- 5 1 10- 4

Frac

tion

al S

atur

atio

n

[X]free

Frac

tiona

l Sat

urat

ion

Sample Binding Isotherms for Binary Reactions

Kassn = 106 M-1 Kdssn = 10-6 M = 1 µM*

[X] on linear scale [X] on log scale

* Why is this equal to [X] at the midpoint of titration?


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A graph represents the relationship between different variables.The shape and position describes that relationship. We rely heavily on“linear” and “log” plots of data (referring to X-axis).

Most scientific graphs show a mathematicalmodel fit to data. These may be called“line graphs” or “scatter plots”even when they are curved, to distinguishthem from bar graphs or pie charts.Curves on scientific graphs may notintersect any of the data points - they aregenerally drawn as best-fits, usingparameters that minimize the variancebetween the data and the equation of a model.

One of the most valuable uses for graphs is to "predict" data: Extrapolate:extending the graph, along the same slope, above or below measured dataInterpolate: predicting values between two measured data points from thevalue on the simulated curve.

Graphical Representations

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These are 2graphs of thesame data*, butthey look quitedifferent.

For data where Xor Y covers manyorders ofmagnitude, alogarithmic plotmay show arelationship moreclearly.

* y=100*2(x-1))

Arithmetic vs. Logarithmic Graphs

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The arithmetic chart makes the growth rate appear as it isincreasing rapidly. On an arithmetic graph, the vertical scale fromsay 10 to 100 would be plotted with an "equal distance" as 1,000to 10,000, bringing about the distortion in the later years. Equaldistance means that the vertical scale shows an equal distance foreach unit of change.

The same data plotted on log (logarithmic) scales show equaldistances for similar "percentage" moves. In the log plot, the y-axis has been converted to a logarithmic scale by re-computingthe data into either a base 10 logarithm or natural log. 10 to 100would be the same vertical "%" distance as 1,000 to 10,000.

Whenever you see a graph, be sure you can tell whether the axesare linear or logarithmic.

Log vs. Arithmetic Plots

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Chemical ReactionsDissociation, Association Constants & Rates

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Review Equilibrium: Sample Calculation

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2-site Binding Scheme

• Definitions of Intrinsic Affinity– Equal or Homogeneous k1 = k2

– Unequal or Heterogeneous k1 ≠ k2

• Definitions of Interactions– Independent

• k12 = kcooperativity= 1

– Cooperative• k12 ≠ 1


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Example: 2 Sites, 1 Ligand, 4 states

• Reaction Scheme– M binds 2 mols of X– 2 sites • (0 or 1 ligand) = 4 states

• Enumerate 4 States & ∆G’s

s Species ∆Gs j1 M 0 (ref. state) 02 XM ∆G1 13 MX ∆G2 14 MX2 / XMX ∆G1+∆G2+∆G12 2

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Boltzmann Distribution* gives usExpression for Probability of State “s”

• Probability for each state depends on– affinity (equilibrium constant, free energy)– ligand activity (concentration)

Ps =exp(!"Gs / RT) • [X ] j

exp(!"Gs / RT )• [X ]j

s , j#

s ranges over all states of M and j counts # of ligands bound in each state

* consult your favorite Physical Chemistry textbook

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Fractional Saturation of Site 1

• States with Site 1 occupieds = 2 and 4 or species XM and XMX

• Molecules having Site 1 (i.e., ALL)s = 1, 2, 3, 4 or M, XM, MX and XMX

• Fraction of M with Site 1 occupied

Y1 =k1 • [X ]1 + k1k2k12 • [X ]2

(1 + (k1 + k2 )• [X ]1 + k1k2k12 • [X]

2

64

Fractional Saturation, cont’d.

• States with either site occupieds = 2, 3, 4 or species XM, MX and XMX

• mols of sitess = 1, 2, 3, 4; 2 x [ M], [XM], [MX],[ XMX]

• Average Degree of Occupancy

Yt =k1 • [X ]

1 + k2 • [X ]1 + 2k1k2k12 • [X ]

2

2 • (1 + (k1 + k2 ) • [X ]1 + k1k2k12 • [X]2 )

Yt =K1 • [X ]1 + 2K2 • [X ]2

2 • (1 + K1 • [X ]1 + K2 • [X ]

2 )where K1 = (k1 + k2 ) and K2 = k1k2k12

65

Unequal and Independent Sites

k1 = 108 M-1, k2 = 104 M-1 ; k12= kc= 1

0

0.2

0.4

0.6

0.8

1

1 0-11 1 0- 9 1 0- 7 1 0- 5 1 0- 3 1 0- 1

Frac

tiona

l Sa

tura

tion

[X]free

Fractional Saturation: Site 1, Average & Site 2

Site 1

Site 2

Average

Fra

ctio

nal S

atur

atio

n

66

Ligation Levels:1 molecule of X bound at either site

• States with 1 ligands = 2 and 3 or species XM and MX

• Moleculess = 1, 2, 3, 4 or M, XM, MX and XMX

• Fraction of M with 1 ligand (Theta)

!1 =k1 • [X ]1 + k2 • [X]1

1 + (k1 + k2 )• [X ]1 + k1k2k12 • [X]

2


67

Unequal and Independent Sitesk1 = 108 M-1 , k2 = 104 M-1 ; k12=kc=1

0

0.2

0.4

0.6

0.8

1

1 0-11 1 0- 9 1 0- 7 1 0- 5 1 0- 3 1 0- 1

Frac

tiona

l Sa

tura

tion

[X]free

f0 f1 f2

Ligation Species: 0, 1 or 2 ligands bound

Fra

ctio

nal S

atur

atio

n

68

Avogadro Constant ……………. N ….. 6.02214 * 1023 mol-1

Atomic Mass Constant …………mu ….1.66054 * 10-27 kgBoltzmann Constant ……………k ….. 1.38066 * 10-23 J/K

Stefan-Boltzmann Constant ….. σ …. 5.67051 * 10-8 W/m2*K4

Molar Gas Constant ……………R ….. 8.31451 J/mol*K

Faraday Constant ………………F ….. 96485.31 C/molMolar Volume of Ideal Gas …… Vm … 22.4141 L/mol(at STP: 0oC, 1 atm) 359.039 ft3/lb-molPlanck’s Constant ………………h ….. 6.626176 * 10-31 J*sElectron charge ………………... e ……1.60219 * 10-19 CElectron mass …………………..me … 9.10953 * 10-31 kgProton mass …………….………mp …. 1.672649 * 10-27 kgNeutron mass ………….………. mn ….1.67495 * 10-27 kg

Common Chemistry Constants

Documents

Chemistry Review 8-23-2010