george ellis university of cape town with jean-phillipe ... · fundamental constants and biology...

Fundamental constants and biology

George Ellis

University of Cape Town

with Jean-Phillipe Uzan,Peter Atkins,David Sloan

Oxford Constants of Nature Meeting 2015

George Ellis Fundamental constants and biology

The broad contextFine tuning and constants

The issue of fine tuning

Life is only possible for some values of the fundamental constantsof physics (Brandon Carter, Bernard Carr and Martin Rees)

John Barrow and Frank Tipler, the Cosmological AnthropicPrinciple

Martin Rees, Our Cosmic Hanbitat, Just Six Numbers

Stephen Weinberg, the cosmological constant:Λ too large, no structures hence no life

Explanation: Multiverse?Bdernad Carr (Ed): Universe or Multiverse


Constants of physics

Fundamental constants of physics: parameters in the theory thatcannot be reduced to other parameters

Dependent on theoretical framework

Should be dimensionless to be physically meaningful

Example: not c or ~ or G

No agreement on number of constants, nor what they are

Comprehensive survey: Jean-Philippe Uzan“Varying Constants, Gravitation and Cosmology”Living Rev Relativity 14 (2011) 2 [arXiv:1009.5514]


Anthropic Limits: Tegmark

The effect of changing the dimensionalities of space and time:astro-ph/9702052

The effect of changing the CMB fluctuation amplitude Q ' 105:astro-ph/9709058

The effect of changing neutrino masses: astro-ph/0304536

The effect on changing the dark matter density, dark energydensity and CMB fluctuation amplitude: astro-ph/0511774

The effect of changing the masses of elementary particles:arXiv:0903.1024

Summary of effects: gr-qc/9704009. Annals of Physics 270, 1-51


Anthropic Limits: Tegmark’s diagramDimensions


Anthropic Limits: Barr and KahnQuark masses

S.M. Barr, A Khan (2007) “Anthropic tuning of the weak scaleand of mu/md in two-Higgs-doublet models”Phys.Rev.D76:045002 [arXiv:hep-ph/0703219]


The fine-structure constant

The fine-structure constant α is a fundamental physical constantcharacterizing the strength of the electromagnetic interactionbetween elementary charged particles. It is related to theelementary charge (the electromagnetic coupling constant) e,which characterizes the strength of the coupling of an elementarycharged particle with the electromagnetic field, by the formula

α =1

4πε0

e2

~c(1)

where

e is the elementary charge;~ = h/2π is the reduced Planck constant;c is the speed of light in vacuum;ε0 is the electric constant or permittivity of free space.

Being a dimensionless quantity, it has the same numerical value inall systems of units. The currently accepted value of is

7.2973525698× 10−3 ⇔ α−1 = 137.035 999 173. (2)


The Hoyle state

“Effects of the variation of fundamental constants on Pop IIIstellar evolution”S Ekstrom, A Coc, P Descouvemont, G Meynet, K A Olive, J-PUzan, E Vangioni (2009)A variation of the fundamental constants is expected to affect thethermonuclear rates important for stellar nucleosynthesis. Inparticular, because of the very small resonant energies of Be8 andC12, the triple α process is extremely sensitive to any suchvariations. We derive limits on the variation of the magnitude ofthe nuclear interaction and model dependent limits on thevariation of the fine structure constant based on the calculatedoxygen and carbon abundances resulting from helium burning. Therequirement that some C12 and O16 be present are the end of thehelium burning phase allows for permille limits on the change of thenuclear interaction and limits of order 10−5 on the fine structureconstant relevant at a cosmological redshift of z ' 15− 20.[arXiv:0911.2420]


Anthropic Limits: Tegmark’s diagramCoupling constants


Fine tuning and BiologyNot just physics

Cells,

Organic molecules:carbohydrates, lipids, proteins, nucleic acids (RNA, DNA)

functioning of living systems

What is possible is determined by possibility spaces

Evolutionary landscape: Waddington


The context: Possibility spacesEternal unchanging possibilities

Possibility spaces themselves exist as unchanging abstract(Platonic) spaces ΩP limiting all possible structures and motions ofphysical systems. They are the same at all places and times, Ourknowledge of them however is a representation of that space that ischanging with time. That is, we represent ΩP by some projection

P(t) : ΩP → EP

into a representation space EP where P(t)) depends on therepresentation we use, and changes with time. This does not meanthat physics itself, or the possibilities it allows, are changing: it isjust that our knowledge of it is changing with time. Ontology(what possibilities exist, as a matter of fact) is entailed by thenature of ΩP . Epistemology (what we know about it) isdetermined by the projection P(t). The representation space EP

will be represented via some coordinate system and set of unitswhich can be altered without changing the nature of the entitiesbeing represented.


Biological Possibility spaces

Andreas Wagner: The Arrival of the Fittest: Platonic possibilityspaces for micro biology[Ard Louis talk]

Genotypes and phenotypes

Gene transcription regulatory circuits

Metabolic networks

Protein genotype networks

Controlled by biological molecule shape: 3-dimensional folding,lock and key mechanism, promoters and effects

These are what control biological possibilities

How do they relate to the fundamental constants?


Lock and key mechanism

The functioning depends on intricate 3-d folding.George Ellis Fundamental constants and biology

DNA structure

Very tight constraints in terms of bond lengths and angles(Crick and Watson: Watson, The Double Helix)


DNA structure

C R Calladine, H R Drew, B F Luisi, and A A TraversUnderstanding DNA: The Molecule and how it works (Elsevier)

The distance between adjacent sugars or phosphates in the DNAchain is 6 Angstroms. It must be between 5.5 and 6.5 Angstromsfor it to work (p.20).


The doable problemViability bites

We can’t do DNA!

Nor even the water molecule ..

Which is a key molecule for life, particularly because of its dipole[Ball, Life’s Matrix: A Biography of Water ; Ard Louis and SimonConway Morris]

We’ll have to settle for the length of the bond in a Hydrogenmolecule ...

and hope that DNA will work out similarly!

- its a reasonable first stepGeorge Ellis Fundamental constants and biology

The equation: The Schrodinger equationDimensionless formulation: Uzan

Let us start with the standard Schrodinger equation. It takes theform

i~∂tψ(r , t) = Hψ(r , t) (3)

where H is the Hamiltonian.Among the constants of the problem we choose (me , ~, c) toconstruct our standard units. The units of time, length, and energyare thus given by

`a =~

mec, ta =

~mec2

, Ea = mec2. (4)

For instance, the ionisation energy of hydrogen, given as

−EI =1

2mec

2α2 = 13.60580 eV,

in standard units, is

−Ei =1

2α2 (5)

in these atomic units.George Ellis Fundamental constants and biology

The equation: The Schrodinger equationDimensionless formulation: Uzan

The Schrodinger equation for a particle of mass m is then

i∂τψ(ρ, τ) = −(

1

2

me

m∆ +

V

mec2

)ψ(ρ, τ) (6)

in dimensionless form with potential V , where the space and time(dimensionless) coordinates are defined as

τ = t/ta, ρ =r

`a(7)

and ∆ is Laplacian associated with ρ.


The hydrogen atomHamiltonian: Uzan

At the lowest level, the Hamiltonian takes the form

H0 =P2

2me− e2

4πε0r(8)

so that the dimensionless Schrodinger equation takes the form

i∂τψ(ρ, τ) = −(

1

2

me

m∆ +

α

ρ

)ψ(ρ, τ) (9)

Thus at the lowest level, rescaling α→ Aα, ρ→ Aρ leaves secondterm invariant, but not the first (∇ scales as ρ−2). Hence thehydrogen atomic size will not just rescale with α.

Hypothesis: Changing α, chemistry will not work the same, withlarger molecules. It will change conformation and so function.


Peter Atkins: Molecular Quantum Mechanics

Three combinations of constants:The fine-structure constant:

α = µ0e2c/2h (10)

and two others for future convenience:

β = ~(2mec), γ = c~ (11)

Because µ0 = 1/ε0c2, the fine-structure constant is also

α = e2/2ε0ch = e2/4πε0c~ (12)

We shall need

e2/4πε0 = αc~ = αγ, ~2(2me) = β~c = βγ (13)

Note that the Bohr radius is

a0 = (4πε0~2)/(mee2) = ~2/(αc~me) = 2β/α (14)


The hydrogen atomPeter Atkins: Molecular Quantum Mechanics Oxford Univerity Press

The energies (MQM p 89; Z = 1) are

En = −(mee4)/(32π2ε20~2).1/n2 = −me/(2~2)(e2/(4ε0))2.1/n2

so

En = −γα2

4β.

1

n2(15)

The orbital of lowest energy (1s; n = 1, l = 0,ml = 0) is (MQM, p88 and Y from p78)

ψ1s(r , θ, φ) =1

π1/2

(1

a0

)3/2

e−ra0 =1

π1/2(α

2β)3/2e−2βrα

The most probable distance of the electron from the nucleus is theBohr radius, or 2β/α.


The hydrogen molecule ion (H+2 )

Peter Atkins: Molecular Quantum Mechanics: Oxford Univerity Press

But an atom is not a molecule. From MQM p265 in the LCAO(Linear Combination of Atomic Orbitals) approximation, for aninternuclear separation R in H+

2 , with S = R/a0 = αR/2β:

E+ − E1s =j0R− j ′ + k ′

1 + S, j0 =

e2

4ε0= αγ

E+ − E1s =αγ

R− j ′ + k ′

1 + S

j ′ =j0R

[1− (1 + s)e−2s

]=αγ

R

[1−

(1 +

1 + αR

2β

)]e−αR/2β

S =

[1 + s +

1

3s2]e−s =

[1 +

αR

2β+

1

3

(αR

2β

)2]e−αR/2β

k ′ =j0a0

[1 + s] e−s =2α2

β

[1 +

αR

2β

]e−αR/2β.


Depedance on αPeter Atkins, David Sloan

1 2 3 4 5

0.5

1.0

Graph of energies of system as a function of radius. Binding willcorrespond to a minimum. Various values of α are shown: the blueone is the physical value of α. Green and yellow are ±6% changein the binding radius.


Dependence on αDavid Sloan

Variation of energy with radius (on the x-axis marked 0 to 4) andchange in fine structure constant on the y-axis. For each value ofα, the minimum energy gives the binding radius.


Dependence on αDavid Sloan

2.6 2.7 2.8 2.9 3.0 3.1

-0.04

-0.02

0.00

0.02

0.04

Finding the minimum: Graph of dE/dr as a function of radius.Various values of α are shown: the blue one is the physical value ofα. Green is α reduced by 5.6%, and red is α increased by 6.4.%.Both give ±6% change in the binding radius.


Different anthropic limits

What are the effects of varying α on the possibility spacesdescribed by Wagner?

Which gives the tightest limits?

Limits from biochemistry

Limits from physics

Limits from astrophysics

Which is the tightest?

Do the physics ones include the biology ones, or vice versa?

It may be that physics gives the tighter limits.

Does physics preview the existence of life?


Fine tuning and constantsThe tightest limits: physics or biology?

It may be that the physics constraints may be tighter than thosefrom life (compare with Tegmark figure).


Spin effects

Does electron spin affect biology? Do we have to take finestructure interactions into account?

Electron Spin Interactions in Chemistry and Biology :Fundamentals, Methods, Reactions Mechanisms, MagneticPhenomena, Structure Investigation Likhtenshtein and Earle (2014)

The present status of the theory of electron spin effects infundamental processes such as spin exchange and dipole-dipoleinteractions, electron transfer, triplet-triplet energy transfer,andannihilation intersystem crossing is reviewed. These effects form abasis for the understanding of the molecular mechanisms essentialto chemical and biological reactions including photosynthesis andmagnetic field influence, and for the creation of advanced organicmagnets and catalysts, as well as the development of new methodsof studyingthe structural and molecular dynamics of biological andnon-biological objects.


Spin effects

Welcome to the Spin Chemistry Websitehttp://spinportal.chem.ox.ac.uk/

Broadly defined, Spin Chemistry deals with the effects of electronand nuclear spins in particular, and magnetic interactions ingeneral, on the rates and yields of chemical reactions. It ismanifested as spin polarization in EPR and NMR spectra and themagnetic field dependence of chemical processes. Applicationsinclude studies of the mechanisms and kinetics of free radical andbiradical reactions in solution, the energetics of photosyntheticelectron transfer reactions, and various magnetokinetic effects,including possible biological effects of extremely low frequency andradiofrequency electromagnetic fields, the mechanisms by whichanimals can sense the Earths magnetic field for orientation andnavigation, and the possibility of manipulating radical lifetimes soas to control the outcome of their reactions.


Spin effects

“Electron spin changes during general anesthesia in Drosophila”Turina, Skoulakisa, and HorsfieldProc Nat Aacd Sci 111 (2014) E3524-E3533

One hundred sixty years after its discovery, the molecularmechanism of general anesthesia remains a notable mystery. Avery wide range of agents ranging from the element xenon tosteroids can act as general anesthetics on all animals from protozoato man, suggesting that a basic cellular mechanism is involved. Inthis paper, we show that volatile general anesthetics cause largechanges in electron spin in Drosophila fruit flies and that the spinresponses are different in anesthesia-resistant mutants. We proposethat anesthetics perturb electron currents in cells and describeelectronic structure calculations on anesthetic-protein interactionsthat are consistent with this mechanism and account for hithertounexplained features of general anesthetic pharmacology.


Hydrogen atom: Fine structureUzan

The fine and hyperfine structures can be determined usingperturbation theory, that is by setting

H = H0 + W . (16)

The fine structure contains three contributions.

The spin-orbit interaction is described by

WS.O.

=α

2m2ec

2

~cr3

ge2L.S (17)

where ge is the electron gyromagnetic factor. It can be written indimensionless form as

WS.O.

mec2=

α

2ρ3ge2

L

~.S

~. (18)

The second correction arises from the (v/c)2-relativistic terms andis of the form

Wrel

= − P4

8m3ec

2. (19)

Its dimensionless form reads simply as

Wrel

mec2= −∆2

8. (20)


Hydrogen atom: Fine structureUzan

The third and last correction, known as the Darwin term, arisesfrom the fact that in the Dirac equation the interaction betweenthe electron and the Coulomb field is local. But, the non-relativistapproximation leads to a non-local equation for the electron spinorthat is sensitive to the field in a zone of order of the Comptonwavelength centered on r. It follows that

WD =π~2q2

m2ec

2δ(r). (21)

The average in an atomic state is of order〈W

D〉 = π~2q2/(2m2

ec2)|ψ(0)|2 ∼ mec

2α4 ∼ α2H0.Its dimensionless version is (by not forgetting that the Diracdistribution has a dimension)

WD

mec2= παδ(ρ). (22)


Hydrogen atom: Hyper-Fine structureUzan

Including fine and hyperfine structure, the study of the hydrogenatom can be written as

i∂τψ(ρ, τ) = −(

1

2

me

m∆ + V

)ψ(ρ, τ) (23)

with the dimensionless potential, no longer a function of (α/ρ),

V =α

ρ+ α

2ρ3ge2

L

~.S

~− ∆2

8+ παδ(ρ)

+αgp2

me

mp− 1

2ρ3L

~.I

~+

2π

3

ge2

I

~.S

~δ(ρ)

+1

4ρ3ge2

[3(S

~.n)(

I

~.n)− I

~.S

~

].

It is a function of 4 dimensionless constants: α, me/mp, ge and gp.


Fine tuning and constantsTo be done

Project:

Explore the effects of all the constants in the first and secondterms of this Hamiltonian on biological function:

Structure of DNA

Protein Folding

Wagner’s possibility spaces

See if they give tighter limits than the physics constraints, that is,the limits for atoms, nuclei, stars, and planets to exist


Fine tuning and constantsThe multiverse view

Fine tuning and the multiverse

c

c

c

c

c

Multiverse: Many variations of

fundamental physics realised

Fundamental physics Standard Model of

particle physics Organic molecules

Life


Fine tuning and constantsDreams of a Final Theory

c

c

c

c

c

Final Theory: Unique theory of

fundamental physics realised

Life

The problem with a Final Theory:

A Unique theory of fundamental physics would have the image of life written in


Conclusion

In summary,

The anthropic literature considers in detail the effect ofchanges in the fundamental constants of physics on physicalconstraints in order that life can exist

These constants, e.g. α, affect chemistry and hence affectbiological function - giving true anthropic constraints

Exploring this relation is wide open and very interesting

I thank Jean Philippe Uzan, Peter Atkins, Ard Louis, and DavidSloan for very helpful discussions on the topic.


george ellis university of cape town with jean-phillipe ... · fundamental constants and biology...

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