Ugly SymmetryRevised Information Theory

and Its Application

Shu-Kun Lin

www.mdpi.org/lin/

A talk @ ACS Meeting

4 April 2001

Technical Intelligence

• Info searching, info analysis, decision making

• What is information?Relation with other concepts:similarity, diversity..., entropy, symmetry

• Quantitative assessmentInformation Theory (IT)

Matter–Energy–InformationConsider chemistry as an information science. ......bringing forward in chemistry the 3rd component of the

basic trilogy of the matter-energy-information. – Jean-Marie Lehn, 2000

• Information - Symmetry • Information - Entropy• Information - Stability

(Information Theory - Thermodynamics)(The Three Laws? The Three Laws)

However, the existing relationships are either confusing or wrong

Statistical mechanics Group theoryentropy symmetry

-Higher symmetry-higher order, lower entropy

Prigogine‘s Order out of ChaosGibbs paradox of mixing (ws - symmetry number)

Incorrect!It is incorrect to compare the stability and the symmetry of a dynamic system (a gas) and a static structure (crystal).

ln SS w

Information theory relationship of entropy S -information I

Regarding the term “entropy”,

“My greatest concern was what to call it. I thought of calling it ‘uncertainty’. When I discussed it with John von Neumann, he had a better idea: ‘You should call it entropy, for two reasons. In the first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, no one knows what entropy really is, so in a debate you will always have the advantage’ ”

-von Neumann’s private communication with Shannon We must revise IT

Information theory

Lewis' remark:"gain of entropy means loss of information, nothing more......"

- Lewis, G. N. (1930) Science 71, 569

Conversion?

Conservation?

S I

L S I

unit logarithms of probabilities

bit

nat ln logep p

2log p

Three laws of information theory?

The first law• The First Law

of Information TheoryThe logarithmic function L (or the sum of entropy and information) of an isolated system remains unchanged.

• The First Law of ThermodynamicsF-Helmholtz free energy (Helmholtz potential)

E E

E TS F

E K P

L S I e.g., a 1.44MB disk, L=1.44MB whether the disk is empty or occupied with a file of the size of 1.44MB. Let us use any available compression method to reduce the size of the original file to 0.40MB. Then, I=0.40MB and S=1.04MB and L=1.44MB .

The second law

• 2nd Lawof Information Theory

(constant L=S+I )Information I of an isolated system

decreases to a minimum at equilibrium.

Or: for an isolated system, entropy S increases to a maximum at

equilibrium.

• 2nd Law of Thermodynamics

E E(constant )E K P

EMinimum potential energy P

Maximum entropy S

universe = system + surroundings

• The function L of the universe is a constant.L=S+I

• The entropy S of the universe tends toward a maximum.

Information theory

• The energy E of the universe is a constant.

• The entropy S of the universe tends toward a maximum.

ClausiusThermodynamics

The third law

• The third Lawof Information TheoryFor a perfect crystal (at zero absolute thermodynamic temperature), the information is zero and the static entropy is at the maximum. Defines static entropy

• The third Law ThermodynamicsFor a perfect crystal (at zero absolute thermodynamic

temperature), the entropy is zero. Defines entropy

max

0

( ) static

static

S

S S

min

0

( )

S

S S

A more general form of the third law of information theory

“for a perfect symmetric static structure, the information is zero and the static entropy is the maximum”. - low temperature

- chemical reaction form static structure (a molecule):bond formation (connectivity) – staticsupramolecular structure - static

- self-organizationprotein folding - static

- self-assembling (many molecules)

Similarity

Symmetry nonsymmetryindistinguishability distinguishabilitythe same different

Similarity - Incomplete Symmetry

http://www.cindy.com/ Incomplete bilateral symmetry

Earth is not a perfect sphere

ln II wln SS wSymmetry number nonsymmetry number

Similarity (imperfect symmetry)

=1

=1

=1

ln

exp ln

ln ln leads to ( ln )

w

i ii

w

S i ii

w

i i Si

S p p

w p p

w p p w w L w

Entropy - information Symmetry - nonsymmetry

S I

Entropy-Similarity

The Similarity Principle

• The higher the similarity among the components is, the higher the value of entropy will be and the higher the stability will be.

Entropy

SimilarityS=0

Z=0 Z=1

Smax

The Similarity Principle

If all the other conditions remain constant, the higher the similarity among the componentsis, the higher value of entropy of mixing, assembling or chemical bond formation process will be.

The higher the similarity among the components is, the more spontaneous the mixing, the assembling or the chemical bond formation process will be, and the more stable the mixture, the assemblage or the chemical bond will be.

Lin, S.-K. (1997) Theorochem J. Mol. Struct. 398, 145-153

Gibbs Paradox of Entropy of Mixing

Entropy of mixing decreases discontinuously with similarity.(S)distinguishable =2R ln 2=11.53 JK–1

(S)indistinguishable =0

-distinguishable or nonsymmetry

-the same or not the same

-either symmetry or nonsymmetry

A BA

B

Entropy-Similarity

Curie-Rosen-symmetry principle

• The effects are more symmetric than the causes. (Curie)

• For an isolated system the degree of symmetry cannot decrease as the system evolves, but either remains constant or increases. (Rosen)

We have proved it (the 2nd law of IT and the similarity principle)A special case of the similarity principle

Symmetry definedSymmetry as a Greek word exactly means “same measure”

or measure of indistinguishability rotational symmetry

translational symmetrybilateral symmetry

permutation symmetry…

invariant transformations “microstates” of the same property

dynamic symmetry - static symmetry

SSw maxln

ISS max0Dynamic symmetry

Static symmetry

• Nonsymmetric state • Symmetric state

M

I N N

2

2ln (bits) > 0

M

I N

1

1 0ln

Symmetry - Stability

The Similarity Principle

If all the other conditions remain constant,the higher the similarity among the componentsis, the higher value of entropy of mixing,assembling or chemical bond formationprocess will be. The higher the similarity among the componentsis, the more spontaneous the mixing, theassembling or the chemical bond formationprocess will be, and the more stable the mixture,the assemblage or the chemical bond will be.

Symmetry - Information loss

The Similarity Principle

If all the other conditions remain constant,the higher the similarity among the componentsis, the higher value of entropy of mixing,assembling or chemical bond formationprocess will be. The higher the similarity among the componentsis, the more spontaneous the mixing, theassembling or the chemical bond formationprocess will be, and the more stable the mixture,the assemblage or the chemical bond will be.

Ugly Symmetry-Beautiful Diversity

Molecular Diversity Preservation International

www.mdpi.net

-to preserve permanently historically significant research chemical samples in the chemical museum in Basel, Switzerland

-to share and exchange any other rare chemical samples worldwide

-founded in 1996 in Basel, Switzerland

NN

MDPI‘s first sample

MoleculesA Journal of Synthetic Chemistry and

Natural Product Chemistry

ISSN 1420-3049

• Is an e-journal now in its 6th successful year of publication, available through free subscription at the Molecules website:

• http://www.mdpi.org/molecules/

Similarity Rule and Complementarity Rule

Similarity rulea component in a molecular

recognition process loves others of like properties.

hydrophobic interaction-stacking in DNA similarity in softness of the

well-known hard-soft-acid-base rules) predicts the affinity of individuals of similar properties

Complementarity rule predicts the affinity of individuals of certain contrast properties.

negative charge-positive charge, donor-acceptor, convex and concave

final structure is more "complete", more integral, more “solid” and more symmetric due to the property offset of the components

Similarity rulePhase separation

permutation symmetry

DDDDDDDDDDDDDDDDDDDDD

LLLLLLLLLLLLLLLLLLLLLLLL

Separation of two tataric acid enantiomers

- Louis Pasteur

Enzyme-Substrate Complementarity

DNA

Symmetry and beauty

Nature 1998, 396, 321-322

also

Nature 1994, 368, 239-42

Facial shape and judgements of female attractiveness.

Ugly Symmetry

A view of the earth

Ugly Symmetry

Stereochemical representation with false mirror symmetry: Both wedges have the thick ends at R3 and R4 placed identically away from the center

Shu-Kun Lin, Luc Patiny, Andrey Yerin, Janusz L. Wisniewski and Bernard Testa. Enantiomer 2000,

5, 571-583.

R4

R2R3

R1

However, none of the drugs (pharmaceuticals) discovered so far are very symmetric. Very few bioactive compounds are symmetric.

The highly symmetric paintings like this one found in many

famous modern art museums are the emperor's new clothes.

Conclusion

Information loss – entropy - Symmetry

(Information Theory-Thermodynamics)The Three Laws

Chemical processes: information, entropy and symmetry evolution can be considered

Higher symmetry – higher stability (good)

Higher symmetry – higher information loss (bad)

Beauty =stability+information