biophysics ii by a/prof. xiang yang liu biophysics & micro/nanostructures lab department of...
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Biophysics IIBiophysics II
ByByA/Prof. Xiang Yang LiuA/Prof. Xiang Yang LiuBiophysics & Biophysics &
Micro/nanostructures LabMicro/nanostructures LabDepartment of Physics, Department of Physics, NUSNUS
Outline
Review of Energy, Enthalpy and Entropy and the correlation with Q and W
Equilibrium and Equilibrium constant
Energy, Enthalpy and Entropy
As dE = Qrev- PdV, At const V, dE = Qrev
For single phase E = Qrev= Qrev=mCvdT
As dH = dE + PdV + Vdp = Qrev- PdV + PdV + Vdp = Qrev+ Vdp At const P, dH = Qrev
For single phase H = Qrev= Qrev=mCpdT
dS = Qrev/T, single phase- Qrev=mCp(or Cv)dT
Isothermal reversible expansion (E = 0, H = (PV), S = Wrev/T; for ideal gas, S = nRln(Vf/Vi))
At constant T and P (such as melting and evaporation) H = Q, S= H/T
The Change of Enthalpy and Entropy in Different Processes
2
2ln
)(,
,
2
1T
TmCdT
T
mCS
mCormCQT
QS
T
QS
pCConst
T
T
p
vprevrev
rev
p
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 Ho, So, Go
Go = Ho - TSo, Go = Ho - TSo
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
+
A B AB
What does it mean AB -1/2 (AA + BB) = 0
E = Efin – Eini = 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, Emix = Hmix = 0, smix-i = niRlnxi (i = 1, 2, …,), Smix = smix-i = R(nilnxi)
Gmix-i = Hmix + T Smix= RT(nilnxi)
An ideal solution or ideal gases G = Go + G = Go + RT(niRlnxi) Chemical potential at const T, P, i = [G/Ni]T,P, Nj, jI
i = io + RTlnxi (i
o: Standard Chemical potential)
The above expressions hold for a mixture of ideal gases, where xi = Pi/P.
Equilibrium constant of two states
Nn Nde
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 Denatured
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: Gden = Hden – TSden
S = R ln (Wden/Wnative)
Since Wden/Wnative >> 1, H > 0 (require E)
low T, G > 0, high T , G < 0.
The breaking of the favorable interactions that hold the native conformation will surely require the input of energy, so H >0. >0.
Free energy…
The ratio of molecules at equilibrium For the special case, nd/nn 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
gi ~ Wi
gn ~ Wn ~ 1
gd ~ Wd >> 1
Native
Denatured Denatured
E
n
n
Free energy…
kTWkTGwith
kTGkTGn
kTGkTWn
iioi
ioii
iiii
ln~
~~ln
~lnln
kTGkTGn
kTGkTGnonnn
oddd
~~ln
~~ln
Boltzmann distribution kTWkTgn iiiio
i expexp
kTGkTGn
n
kTGGkTGGn
n
o
n
d
on
odnd
n
d
~~ln
~~~~ln
Actual distribution
Free energy…
Putting quantities on a molar basis:
At equilibrium, G = 0, nd = nod, nn = no
n
RTGRTGn
n
GGRk
o
n
d
ln
~,
KkTGn
n oo
od
n
ln~
ln
K is the equilibrium constantGo is the standard free energy
Chemical Reactions
Chemical Potentiali = i
o+ RTln[i ] [i ]: concentration of ; i
o: chemical potential at P = 1 atm, T = 298K and [i ] 1.
io depends on the unit of concentration
selected, ie If the unit of [i ] is mole fraction, xi, i
o: is the chemical potential at P = 1 atm, T = 298K and xi 1;
If the unit of [i ] is “molar” (moles per liter), Mi, io: is
the chemical potential at P = 1 atm, T = 298K and Mi 1.
The same applied to other concentration units
Chemical Potential
i = io+ 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.
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
Ca2+ + CO32- CaCO3 ↓
H+
5Ca2+ + 3 (PO4)3- + OH- Ca5(PO4)3OH (HAP)↓H+
Chemical Reactions Chemical Potential Chemical forces
Chem. Potential difference
0 cl
l
c
Ca2+, CO32-
CaCO3 ↓
Chemical potential
Chem. Potential difference
l
c
CaCO3 ↓
Ca2+, CO32-
C > Ceq
0 cl
Ca2+ + CO32- CaCO3 ↓
H+Ca2+ + CO3
2- CaCO3 ↓
Chemical Reactions
The reaction will stall when iB = 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 Ca2+ + CO32- → CaCO3 ,
after reaction, how much Ca2+ and
CO32-.
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.
23
23ln0
2233
COCa
CaCO
TkTk
G
B
o
Ca
o
CO
oCaCO
B
Let
TkK
B
o
Ca
o
CO
oCaCO
eq
2233exp
23
23ln
2233
COCa
CaCO
TkB
o
Ca
o
CO
oCaCO
be the equilibrium constant, then
23
23
COCa
CaCOKeq pKeq = - log Keq
Chemical Reactions
Keq: temperature and Pressure dependent It depends on the unit of concentration
TkK
B
o
Ca
o
CO
oCaCO
eq
2233exp
2H2 + O2 2H2O (8.8)
Chemical Reactions
General reactions
vk: the stoichiometric coefficients
G-the net chemical force driving the reaction A reaction will run forward if G < 0; or backward if G > 0. Standard free energy change:
mmkkkk XXXX ...... 1111
mmkkkkG ...... 1111
omm
okk
okk
ooG ...... 1111
Chemical Reactions
Dissociation: Ionic and partially ionic bonds dissociate reality in water
Association constant Ka and Dissociation constant Kd..
When Ka > 1 (logKa > 0, Kd < 1, log logKd < 0), A+ and B- tends to associate to AB.
When Kd > 1 (logKd > 0, Ka
< 1, logKa < 0 or p Ka > 0), AB tends to dissociate to A+ and B-.
The large pKa, the easier the dissociation of the protein will be.
A+ + B- AB
da KBA
ABK
1
]][[
Ka
Kd
log][log
/1loglog
ppKK
KK
aa
ad
Dissociation: Ionic and partially ionic bonds dissociate reality in water
ninjPTii
oii
n
G
ikT
,,
;ln
ni: the number of species i.
ad KAB
BAK
1
][
]][[
AB A+ + B-Kd
Ka
ABTkBTkATk
GGG
BoABB
o
BBo
A
ABBAinitialfinal
lnlnln
~~~
Dissociation: Ionic and partially ionic bonds dissociate reality in water
0
lnlnln
ABTkBTkATkG B
oABB
o
BBo
A
o
B
o
A
oAB
oaG ~
RT
G
Tk
G
AB
BA oa
B
oa
~
][ln
At equilibrium
Let
RT
GK
oa
d
ln
RT
GpKK
RT
GKK
oa
ad
oa
ad
303.2log
log303.2log303.2
Standard Specific Gibbs Free change in association
Dissociation of water
The dissociation of water: H2O H+ + OH-
For pure water [H+] = [OH-] = 10-7M Dissociation Const: Kd = [H+][OH-]/[H2O]. As [H2O] is constant at a given T, we have then Kw =
Kd[H2O]. Kw = [H+][OH-] = (10-7)2. Ion product of water at room
temperature pH = -log Kw 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 Es-the energy of a charge in the absence of its counter ion.
rs: Stoke’s radius-the radius of charge distribution D: Dielectric const.
ss DrqE 22
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 Ka:
the large pKa, the easier the dissociation of protein will be.
Globular protein
Hydrophobic interior
A+ + B- AB
]][[ BA
ABKa
The charge on a protein varies with its environment
Protonated Deprotonated
Acidic side chain -COOH -COO- + H+
Basic side chain -NH3+ -NH2 + H+
The charge on a protein varies with its environment Propobility of
protonation -COO-COOH-COOH- P -COO-COOH-COOH- P
COOH-HCOO-,
eqK
pKapHxwhere
KKP
x
pHaeqaeq
,
101
1
101
1
H1
1
,,
The measurement of the degree of disordering and the freedom
The direction of change in thermodynamic system 2nd law
Review
Chapters in Textbook
Chapter 8 , in Biological Physics
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