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