fine tuning and the molecules of life · pdf filefine tuning and the molecules of life ......

Fine tuning and the molecules of life

George Ellis

Department of MathematicsUniversity of Cape Town

together withJean-Philippe Uzan, IAP, ParisDavid Sloan, Oxford University

June 22, 2017

Final Conference on the Physics of Fine TuningCrete, 18th – 23rd June 2017

Ellis (UCT) Fine tuning and the molecules of life June 22, 2017 1 / 50

Table of Contents

1 The basic idea

2 The Schrodinger equation

3 The Hydrogen Molecule

4 The Water Molecule

5 What about conformal invariance?

6 Organic molecules

7 The basic conundrum


Table of Contents

1 The basic idea





6 Organic molecules



The basic idea

Fine tuning relates the existence of life to the values of physical constants

Fundamental constants of physics:

parameters in the theory that cannot be reduced to other parameters

Dependent on theoretical framework

Should be dimensionless to be physically meaningful

Examples: α = e2/(~c), µ = me/mp. Not c or ~ or G .

No agreement on number of constants, or what they are

Comprehensive survey: Jean-Philippe Uzan“Varying Constants: Gravitation and Cosmology”Living Rev Relativity 14 (2011) 2 [arXiv:100.5514]


The basic idea

Fine tuning discussions conventionally involve only issues to do with

Existence of galaxies

Existence of atoms out of which organic molecules can be made:C, H, N, O, P, S (as a result of explosions of first generation stars)

Existence of 2nd generation stars with planets

Some of those planets have an atmosphere and water

Theme

While all these are necessary, they are not sufficient. They do not touchthe nature of life itself. That is what we touch on here.

NB: work in progress! Much to be done


The basic idea

One needs also to consider existence of essential molecules for life:

Water with suitable dipole

Nucleic acids: suitable DNA and RNA

Ssuitable proteins

One also needs lipids (fatty acids) and carbohydrates (sugars, starch,cellulose): but their structure is not so crucial.

Proteins are the key molecules doing the needed work in all sorts of ways:

catalysis: speeding up reactions by a huge amount (enzymes),

controlling gene expression (gene transcription factors, genecorrections),

controlling flow of ions into and out of axons (voltage gated ionchannels).

DNA is important only because it creates proteins at the right time andplace (because of gene transcription networks)


Biological context

The biomolecules must function in their microbiology contexts.

Specifically, they must enable the phenotype-genotype maps described byAndreas Wagner in his book Arrival of the Fittest:

namely, those for

Metabolic networks

Gene regulatory networks

Signal transduction networks

Proteins

These are eternal unchanging possibilities spaces for biology, whosecharacter is determined by the underlying physics.

Their nature (vast dimension, space-filling level surfaces) solves thetimescale problem for evolutionary biology.


Table of Contents

1 The basic idea





6 Organic molecules



The Schrodinger equationThe basic form

The Schrodinger equation

The link between physics and chemistry


The Schrodinger equationThe dimensionless form


The non-relativistic Schrodinger equationThe dimensionless form

The relevant case for chemistry is the electromagnetic force:V = e2/(4πεor)⇒ V /(mec

2) = α~/(mec)r .


The non-relativistic Schrodinger equationThe dimensionless form

Nb: α is the ‘fine structure constant’ for spectroscopy,but it is one of two main constants for atomic structure


Table of Contents

1 The basic idea





6 Organic molecules



The Hydrogen MoleculePeter Atkins and David Sloan

Hydrogen molecule: Simplest molecule to explore effects

Two protons and two electrons: radius of electron orbit compared withsize of nucleus

Figure: Units: Angstroms. But note that the metre depends on αEllis (UCT) Fine tuning and the molecules of life June 22, 2017 14 / 50


Explain why we choose 6% as an anthropic bound later!


Table of Contents

1 The basic idea





6 Organic molecules



The Water MoleculeLengths, angles, and dipole

Water as a significant biological molecule

Water molecule

Two protons one oxygen nucleus and 18 electrons


The dipole momentFrom Wikipedia

“Water is primarily a liquid under standard conditions, which is notpredicted from its relationship to other analogous hydrides of the oxygen

family in the periodic table, which are gases such as hydrogen sulfide. Theelements surrounding oxygen in the periodic table, nitrogen, fluorine,

phosphorus, sulfur and chlorine, all combine with hydrogen to producegases under standard conditions. The reason that water forms a liquid isthat oxygen is more electronegative than all of these elements with the

exception of fluorine. Oxygen attracts electrons much more strongly thanhydrogen, resulting in a net positive charge on the hydrogen atoms, and a

net negative charge on the oxygen atom.These atomic charges give each water molecule a net dipole moment.Electrical attraction between water molecules due to this dipole pulls

individual molecules closer together, making it more difficult to separatethe molecules and therefore raising the boiling point ....

Water is also a good solvent, due to its polarity..”


The Water MoleculeDetailed numerical calculations: Key paper

PHYSICAL REVIEW A 81, 042523 (2010)

“Chemistry as a function of the fine-structure constant and theelectron-proton mass ratio”

Rollin A. King, Ali Siddiqi, Wesley D. Allen, and Henry F. Schaefer III

“The dramatic advances of recent decades in electronic structure methods,numerical algorithms, and raw computing power permit the determination

of solutions very close to the ab initio limit for molecular systems ofreasonable size. “

- But we want more intuitive and transparent calculations based inperturbations methods


The Water MoleculeNon-relativistic equations


The Water Molecule


Relativistic effects

The dimensionless ratios that have consequences for chemistry areα = e2/(~c) and the electron-proton mass ratio µ = me/mp = β.

In conventional, nonrelativistic quantum chemistry within theBorn-Oppenheimer approximation, it is assumed that both ratios are

negligibly small. The most important relativistic effects in chemistry canbe investigated by means of the Cowan-Griffin Hamiltonian in which H0 is

augmented with one-electron mass-velocity and Darwin terms:

H1 = α2

−1

8

∑i

∇4i +

π

2

∑I ,j

δ(ρIi )

(1)

The consequences of finite me/mp ratios on chemical systems can beprobed by means of the diagonal Born- Oppenheimer correction (DBOC)

EDBOC = −µ2

∑I

1

µI

⟨Ψe |∇2

I |Ψe

⟩(2)


The Water MoleculeDetailed numerical calculations

Figure: The water molecule bond length: Function of α/α0.Query: left hand limit??



Figure: The water molecule angle: Function of α/α0



Figure: The water molecule dipole moment: Function of α/α0. Changing α by afactor 6 causes dramatic changes to macroscopic properties of water: meltingpoint, temperature of maximum density, solvent properties, viscosity.



Figure: The water molecule bond lengthFunction of µ/µ0

Figure: The water molecule angleFunction of µ/µ0


Table of Contents

1 The basic idea





6 Organic molecules



What about conformal invariance?Maybe its just a change in size (Sloan) or of units (Uzan)

Simple quantum mechanical system which has only a single force lawacting, with a homogeneous potential term V (λr) = λnV (r), which has alinear dependence on some coupling constant β, e.g. V (r) = β|r1 − r2|n.

Use in the Schrodinger equation for a system of particles:

−~2

2m∇2ψ(r) = (V (r) + E )ψ (3)

If we change β we can find a second solution to our equations by defininga second wavefunction ξ(r) = ψ(λr) which will solve the Schrodingerequation with a different eigenvalue (energy level) but the same shape

(we’ve just rescaled |r| everywhere, not changed any angles).

Conformal change

Rescale β, lengths together: its just a larger molecule.Biological function is unchanged.


What about conformal invariance?Spin effects in biology (Uzan) will break this conformal symmetry


What about conformal invariance?Spin effects in biology


What about conformal invariance?Spin effects in biology will break this conformal symmetry


Table of Contents

1 The basic idea





6 Organic molecules



Organic MoleculesThe structure of DNA

The structure of DNA

Very tight constraints in order that it can function


Organic MoleculesDNA

Figure: Very tight constraints so it can function. Very small change can add up.Ellis (UCT) Fine tuning and the molecules of life June 22, 2017 37 / 50

Organic MoleculesProteins: Lock and key mechanism

Figure: Lock and Key mechanism: depends on intricate 3-d folding


Organic MoleculesVoltage gated ion channels

Figure: Voltage gated ion channels enable action potentials in neurons to progress


Organic MoleculesVoltage Gated Ion Channels

Figure: Voltage Gated Ion Channel(side view)

Figure: Voltage Gated Ion Channel(top view)


Organic MoleculesMessing with complex molecules

The important biomolecules are very complex and have very tightlycontrolled interactions involving molecular recognition at binding sites and

resultant conformational changes.

They involve very long chains of base pairs (DNA/RNA) and amino acids(proteins) so that small changes might accumulate disastrously over long

lengths of these chains. On the other had they might not.

We need to investigate effect of changes of constants on tertiary andquaternary protein structure (3-dimensional folding and assembly).

Simple molecules as indicators

Effects on hydrogen and water are an indication, but far from sufficient todecide the outcome decisively.

Studying the organic molecules? May be possible via repeating subunits


Table of Contents

1 The basic idea





6 Organic molecules



The basic conundrum

Which sets the tighter limits: physics/astrophysics or biology?


The basic conundrum


Physics as we know it allows molecules that include the molecules of life.


The basic conundrum

A multiverse provides a situation where it will still work out in somedomains, if the constants vary from domain to domain


The basic conundrum

What if there were a final theory of physics, uniquely implying thestandard model of particle physics?

If there was a unique fundamental theory it would be fine-tuned to expectlife: a deep conundrum. It would have the pre-image of life written into it.


ConclusionFine tuning for life relative to that for physics


It appears that the physics constraints are indeed tighter than theconstraints from life.

As regards α:

Stellar nucleosynthesis limits , holding everything else fixed, 10−3

Much tighter than the inferred from H2 applied to DNA: ±6%.

This may or may not be tighter than true limits from DNA andproteins. They themselves need to be investigated.

As regards µ:There are only combinations of limits. Limits on µ per se are unclear.

Why? Conundrum:

Physics that fulfils other anthropic requirements such as from stellarnucleosynthesis seems fine-tuned to expect life.


Conclusion

The consideration of fine tuning effects on chemistry and hencebiomolecules is a key part of the fine tuning project overall.

There is a lot to be done. It is a challenging project. This talk is just someindications of how to proceed.

Caution: As Walter Kohn points out in his Nobel prize acceptance lecture(1998), to solve a system of just 100 electrons would require minimizing a

function across 10150 dimensions.

”Traditional wavefunction methods ... are generally limited tomolecules with a small total number of chemically activeelectrons .... In general the many electron wave functionΨ(r1, ..., rN) for a system of N electrons is not a legitimatescientific concept when N ≥ N0, where N0 ' 103.”

Rather one needs density functional methods.


The water density-temperature relationFrom Wikipedia

Figure: The unusual density curve and lower density of ice than of water is vitalto life—if water was most dense at the freezing point, then in winter the verycold water at the surface of lakes and other water bodies would sink, the lakecould freeze from the bottom up, and all life in them would be killed


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