Lecture 20

Chemical Potential

Reading:

Lecture 20, today: Chapter 10, sections A and B Lecture 21, Wednesday: Chapter 10: 10‐17 –end

MCB65 3/21/16 1 Pop Question 7 – Boltzmann Distribution

Two systems with lowest energy at 0 kBT. Energy levels separated by 3 kBT in system A, and 5 kBT in B. Calculate the partition coefficients for each system, then the probabilities of finding molecules in each energy level (use the energy levels from 0 to 15 kBT). System A: System B:

U = 15kBT p=0.000000291 U = 12kBTUp=0.00000584 = 15kBT p=0.00000030384 U = 9kBT p=0.000117 U = 10kBT p=0.00004509330 U = 6kBTUp=0.0024 = 5kBT p=0.006692438 U = 3kBTUp=0.0473 = 0kBT p=0.993245928 U = 0kBT p=0.9501

Simply apply the Boltzmann distribution: System A: System B: Q = e0+e‐3+e‐6+e‐9+e‐12+e‐15=1.0524 Q = e0+e‐5+e‐10+e‐15=1.0068 Relationship between Q and the overall distribution of particles? Smaller steps between levels → larger Q → molecules spread more. Less likely to find molecules at any given level

Why limit to <15 kBT? Because the contributions to the distribution (and the Q) of higher energy levels becomes MCB65 negligible –i.e. the higher energy levels are unpopulated for all practical purposes. 3/21/16 2 F0F1 ATP synthase performs an unfavorable reaction

~50 kJ/mol stored per ATP ~15,000 kJ/mol per second

Credits: John Walker MCB65 http://www.mrc-mbu.cam.ac.uk/research/atp-synthase 3/21/16 3 Today’s goals

• Explore the link between concentration and the free energy change

G     • Chemical potential i   Molar free energy G˚i ni  T ,P,n ji

• Equilibrium constant K: G o RT lnK

Q • Reaction quotient Q and mass action ratio Q/K: G  RT ln K

• Using G and K to look at biological systems: • ATP hydrolysis • Wednesday: acid‐base equilibria, protein folding MCB65 3/21/16 4 Chemical Potential

• The chemical potential (i) is defined as the rate of change of the free energy with respect to the number of molecules: G  G Units of energy i      G(for one molecule of i) Ni  Ni (e.g. J) T ,P,N ji

• Often in chemical reactions, we use moles (n): G     G Units of energy per mole i   i (e.g. J mol-1) ni  T ,P,n ji

• Molar free energy, G°i

MCB65 3/21/16 5 Chemical potential at play ‐ diffusion

[ ]in = [ ]out spontaneous dNout = -dNin in in out out ΔG <, >, or = 0? equilibrium: ΔG = ?

What happens to μin and μout at equilibrium? G  dG   dN   dN  0     in in out out i N  i T ,P,N ji in dNin  out dNin  0

dG   dN (in  out )dNin  0    in out MCB65 3/21/16 6 Figure from The Molecules of Life (© Garland Science 2008) Direction of spontaneous change and 

• System changing towards equilibrium: dG  (in  out )dNin  0

• For a spontaneous change, dNin and (in‐out) should have opposite signs

• If Nin decreases, dNin < 0

• (in‐out) > 0 and in > out • Molecules move spontaneously from regions of high chemical

potential to low chemical potential MCB65 3/21/16 7 Chemical potential and concentration

• In an ideal dilute solution, molecules do not influence each other and the enthalpy is independent of concentration.

• Assumption commonly used in biochemistry

• For an ideal dilute solution, we’ll show that the difference in chemical potential is related to the ratio of concentrations:

C2   2  1  kBT ln C1

• Where C1 and C2 are the concentrations of molecules MCB65 3/21/16 8 G1  H1  TS1 G2  H2  TS2

    G1 G2 A1    A 2    NA1  NA 2  T ,P,NB T ,P,NB H   S  H   S    1   T 1    2   T 2  NA1  NA1  NA 2  NA 2  T ,P,NB T ,P,NB T ,P,NB T ,P,NB

MCB65 3/21/16 9 Figure from The Molecules of Life (© Garland Science 2008) For an ideal solution,  depends on entropy

  2  1     H   S  H   S    2   T 2    1   T 1   NA 2  NA 2   NA1  NA1    T ,P,NB T ,P,NB   T ,P,NB T ,P,NB 

H  H  ideal solution  enthalpy changes are the same   2    1  NA 2  NA1  T ,P,NB T ,P,NB

    S2 S1   2  1 T   T  NA 2  NA1  T ,P,NB T ,P,NB

MCB65 3/21/16 10 Calculating the entropy:

2

1

• We can use the probabilistic definition of entropy, with three states: N p  B (B molecules) 2 M N p  A1 (A molecules) 1 M M  (N  N ) p  A1 B (empty gridboxes) 0 M

MCB65 3/21/16 11 Figure from The Molecules of Life (© Garland Science 2008) 2

S1 MkB  pi ln pi i0 p does not depend on N  d/dN = 0   2 A1 A1 S1    MkB p0 ln p0  p1 ln p1  p2 ln p2 NA1  NA1 T ,P,NB

   M  (N A1  N B ) M  (N A1  NB ) N A1 NA1 MkB  ln  ln  N A1  M  M  M M 

MCB65 3/21/16 12 Applying simple rules for derivatives (chain rule, product rule, etc..) we get:     S1 NA1  M  (NA1  NB ) 1 T   kBTln  ln  NA1    M   M  T ,P,NB NB (solvent) >> NA1 (solute)     S2 NA 2  M  (NA 2  NB ) 2 T   kBTln  ln  NA1    M   M  T ,P,NB

 N  M  N   N  M  N         k Tln A 2  ln B  ln A1  ln B  2 1 B   M   M    M   M 

       N A 2 N A1 C2  kBTln  ln   kBT ln    M   M  C1 

C2 C1

MCB65 3/21/16 13 Molecular diffusion decreases chemical potential

C2   2  1  kBT ln C1

• If C2 > C1 then ln(C2/C1) > 0 and  is positive • The A molecules in Region 2 have a higher chemical potential • Makes sense – molecules will move spontaneously from Region 2 (high concentration) to Region 1 (low concentration) MCB65 3/21/16 14 Figure from The Molecules of Life (© Garland Science 2008) Chemical potentials

C2 • Switching to molar units:   RT ln C1

(multiply Boltzmann constant by Avogadro’s number  R = NAkB)

• Calculating the chemical potential relative to standard state:

o C o C     RT ln o    RT ln  C 1 • ***C/C° (and therefore C/1M) is unitless

MCB65 3/21/16 15 What is chemical potential?

• Chemical potential is proportional to the logarithm of concentration:

C2 • Comparing two solutions:   2  1  kBT ln C1 C C • Comparing to standard state:   o  RT ln  o  RT ln C o 1

• In mechanics, the direction of spontaneous change is always towards a reduction in potential energy • Similarly, in thermodynamics, the direction of spontaneous change is always towards a reduction in Gibbs free energy

• The partial molar Gibbs free energy (G˚i) of a type of molecule (i) is its “chemical potential” (˚ ) i MCB65 3/21/16 16 What are the concentrations at equilibrium?

MCB65 3/21/16 17 Defining a “reaction progress variable”, ξ • Reaction: aA  bB  cC  dD

• Change in free energy as the reaction progresses:

dG  A dnA  B dnB  C dnC  D dnD • These terms are NOT independent. To account for their coupling, we define the reaction progress variable (ξ or “xi”) which is a measure of how far the reaction has progressed

MCB65 3/21/16 18 Defining a “reaction progress variable”, ξ • E.g. 2A 1B  1C  2D

• 0 <  < 1



MCB65 3/21/16 19 Figure from The Molecules of Life (© Garland Science 2008) Defining a “reaction progress variable”, ξ • Reaction: aA  bB  cC  dD

• Change in free energy as the reaction progresses:

dG  A dnA  B dnB  C dnC  D dnD • These terms are NOT independent. To account for their coupling, we define the reaction progress variable (ξ) which is a measure of how far the reaction has progressed

nA  nA (0)  a dnA a(d) n  n (0)  b For a small dn b(d) B B step in the B reaction: nC  nC (0)  c dnC c(d)

nD  nD (0)  d dnD d(d) MCB65 3/21/16 20 Reaching equilibrium

• Substituting into the equation for dG: dni  (/)id

dG  A dnA  B dnB  C dnC  D dnD

A a(d)  Bb(d)  C c(d)  D d(d)

aA  bB  cC  dD d • At equilibrium, dG = 0 and d can be non‐zero

aA  bB  cC  dD  0

aA  bB  cC  dD

• Products of chemical potential and stoichiometric MCB65 coefficients are balanced 3/21/16 21 Equilibrium concentrations

a  c  [A] A c    o  RT ln b d A A 1 B D

o [B] B  B  RT ln 1 [A], etc, refer to the equilibrium concentrations [C]    o  RT ln C C 1 [A]/1M is dimensionless o [D] D  D  RT ln a  b  c  d  1 A B C D  [A]  [B] a o  aRT ln b o  bRT ln   A 1   B 1   [C]  [D] c o  cRT ln d o  dRT ln   C 1   D 1  

c d o o o o [C] [D] o cC  dD  aA  bB RT ln a b G [A] [B] MCB65  3/21/16 22 Defining the equilibrium constant

• We then define the equilibrium constant, K as:

o c d G RT lnK [C] [D] eq Keq  a b G o [A] [B]  RT Keq  e

Equilibrium constant provides a way to • Keq is measurable determine the concentrations, the extent of reaction, at equilibrium

• Keq is unitless • G° is in J/mol and RT is also in J/mol

MCB65 3/21/16 23 Extent of reactions at equilibrium • Hydrolysis of ATP:

ATP + H2O  ADP + Pi + energy

• G° = ‐28.7 kJ mol‐1 at pH 7.0 and 10 mM Mg2+

Extent of reaction [ADP][Pi ] K  [H2O]/[H2O]° ~ 1 for reaction with [ATP] a smaller G°?

G o  RT 28 / 2.478 5 K  e  e 10 ‐2 • [Pi] in cells is maintained at ~10 M, which means at equilibrium [ADP] 107 [ATP] MCB65 3/21/16 24 G –in a situation not at equilibrium

dG aA  bB  cC  dD d  0 dG a  b  c  d G d A B C D

Figure from The Molecules of Life MCB65 (© Garland Science 2008) 3/21/16 25 Direction of spontaneous change from observed concentrations?

MCB65 3/21/16 26 Reaction quotient, Q, describes observed conditions • Combining: o C G aA  bB  cC  dD     RT ln o  C • We obtain: Go Q  Reaction  quotient [C]c [D]d G  c o  d o  a o  b o  RT ln obs obs C D A B [A]a [B]b  obs obs G Go  RT lnQ Observed,  non-equilibrium

• Substituting: We get: Q Q G o RT lnK G  RT ln  2.3RT log  K K MCB65 3/21/16 27 Q/K is the mass action ratio

• The ratio Q/K is the mass action ratio 5.8 kJ/mol Q Q G o RT lnK G  RT ln  2.3RT log  K K

• The mass action ratio determines whether a reaction goes forward or backward:

Q 1 G < 0, reaction will go forward K Q 1 G > 0, reaction will go backward K

MCB65 3/21/16 28 Mass action ratio –Q/K

Q Q A  B G  RT ln  2.3RT log K K

mol-1

MCB65 Figure from The Molecules of Life (© Garland Science 2008) 3/21/16 29 G for ATP hydrolysis in cells

ATP  ADP + Pi + energy

K = [ADP][Pi] [ATP] [ADP] • G° = ‐28.7 kJ mol‐1 and at equilibrium: 107 [ATP]

• In cells, [ADP]/[ATP] = 10‐3

Q 103 G  RT ln ~ RT ln ~ RT ln1010 K 107 ~ 57 kJ mol1

MCB65 3/21/16 30 ATP synthesis is not spontaneous

ATP  ADP + Pi + energy

K = [ADP][Pi] [ATP] [ADP] • G° = ‐28.7 kJ mol‐1 and at equilibrium: 107 [ATP]

• How to drive ATP synthesis? • From the chemical potential concept

• Increasing the concentration of ADP and Pi could lead to ATP synthesis • Impractical for the cell

MCB65 3/21/16 31 An example of non‐expansion work

• Using a gradient of molecules across a membrane to synthesize ATP

Unfavorable reaction:

(ADP+Pi)•F (ATP) •F*

Favorable reaction:  B(high) B(low) (ADP+Pi)•F + B (ATP) •F*•B

MCB65 Figure from The Molecules of Life (© Garland Science 2008) 3/21/16 32 F0F1 ATP synthase uses proton chemical potential to synthesize ATP

Credits: John Walker MCB65 http://www.mrc-mbu.cam.ac.uk/research/atp-synthase 3/21/16 33 Oxidative phosphorylation in mitochondria

MCB65 3/21/16 34 Figures from The Molecules of Life (© Garland Science 2008) F0F1 ATP synthase couples ATP synthesis to a transmembrane proton gradient

+ + (ADP+Pi)•F + H  (ATP) •F*•H

K = [(ATP) •F*• H+] + [(ADP•Pi) •F][H ]

• U = q+ w, and q ~ 0 (under idealized conditions) • U = w • U is positive because ATP is produced • w is positive, corresponding to work done on the system by the surroundings • Chemical work as a consequence of transferring B molecules from high to low concentration, decreasing the free energy of B, and

storing this energy into synthesized ATP MCB65 3/21/16 35 Figure from The Molecules of Life (© Garland Science 2008) Some concepts to remember

• The chemical potential, , describes the rate of change of the free energy with respect to concentrations

• The equilibrium constant, Keq, provides a link between free energy and the concentrations at equilibrium

• The mass action ratio, Q/K, is related to the reaction free energy change, G, determining the driving force for the reaction

MCB65 3/21/16 36